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Damage identification of cracked reinforced concrete beams through frequency shift


The safety evaluation of reinforced concrete (RC) bridges is of the outmost importance, both for the early warning of critical states below a given safety margin and owing to plan maintenance cycles of the infrastructural network. Structural health monitoring based on dynamic testing has become widespread in the last 20 years, leading to very effective operational algorithms able to extract valuable structural features from the recorded signals. However, although in principle it is possible to identify position and severity of the damage by using a finite element model, still some identification issues are unresolved due to the non-linear nature of the oscillations of a cracked beam. In fact, the available experimental data show, for a given damage pattern, a significant underestimation of the natural frequencies given by cracked beam numerical models. This paper presents an approximate solution for the problem of a vibrating damaged RC beam with opening–closing (breathing) cracks. The solution is based on the static equivalence of the kinetic energy and allows incorporating most of the features of a beam loaded above the cracking limit and oscillating under the self-weight with breathing cracks. The comparison with a wide data set collected in the literature points out the predictive capability of the developed analytical formulas. An independent test confirms the theoretical results.

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  1. 1.

    Wikipedia, List of bridge failures. Accessed May 2014

  2. 2.

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

    Article  Google Scholar 

  3. 3.

    Salawu OS (1997) Detection of structural damage through changes in frequency: a review. Eng Struct 19(9):718–723.

    Article  Google Scholar 

  4. 4.

    Doebling SW, Farrar CR, Prime MB, Shewitz DW (1996) Damage identification and health monitoring of structural and mechanical systems from changes in their vibration characteristics: a literature review. LA-13070-MS, UC 900. Los Alamos National Laboratories, New Mexico

  5. 5.

    Yao YY, Shue-Ting TE, Glisic B (2014) Crack detection and characterization techniques—an overview. Struct Control Health Monit 21(12):1387–1413.

    Article  Google Scholar 

  6. 6.

    Magalhães F, Cunha A, Caetano E (2012) Vibration based structural health monitoring of an arch bridge: from automated OMA to damage detection. Mech Syst Signal Process 28:212–228.

    Article  Google Scholar 

  7. 7.

    Lianga Y, Lic D, Songd G, Fenga Q (2018) Frequency Co-integration-based damage detection for bridges under the influence of environmental temperature variation. Measurement 125:163–175.

    Article  Google Scholar 

  8. 8.

    Askegaard V, Langsœ HE (1986) Correlation between changes in dynamic properties and remaining carrying capacity. Mater Struct 19(1):11–20

    Article  Google Scholar 

  9. 9.

    Van Den Abeele K, De Visscher J (2000) Damage assessment in reinforced concrete using spectral and temporal nonlinear vibration techniques. Cem Concr Res 30(9):1453–1464.

    Article  Google Scholar 

  10. 10.

    Maeck J, Wahab MA, Peeters B, De Roeck G, De Visscher J, De Wilde WP, Ndambi J-M, Vantomme J (2000) Damage identification in reinforced concrete structures by dynamic stiffness determination. Eng Struct 22(10):1339–1349.

    Article  Google Scholar 

  11. 11.

    Neild SA, Williams MS, McFadden PD (2003) Nonlinear vibration characteristics of damaged concrete beams. ASCE J Struct Eng 129(2):260–268.

    Article  Google Scholar 

  12. 12.

    Tan CM (2003) Nonlinear vibrations of cracked reinforced concrete beams. Ph.D. thesis, University of Nottingham.

  13. 13.

    Koh SJA, Maalej M, Quek ST (2004) Damage quantification of flexurally loaded RC slab using frequency response data. Struct Health Monit 3(4):293–311.

    Article  Google Scholar 

  14. 14.

    Massenzio M, Jacquelin E, Ovigne PA (2005) Natural frequency evaluation of a cracked RC beam with or without composite strengthening for a damage assessment. Mater Struct 38(10):865–873.

    Article  Google Scholar 

  15. 15.

    Baghiee N, Esfahani MR, Moslem K (2009) Studies on damage and FRP strengthening of reinforced concrete beams by vibration monitoring. Eng Struct 31(4):875–893.

    Article  Google Scholar 

  16. 16.

    Musiał M (2012) Static and dynamic stiffness of reinforced concrete beams. Arch Civ Mech Eng 12(2):186–191.

    Article  Google Scholar 

  17. 17.

    Hamad WI, Owen JS, Hussein MFM (2014) Modelling the degradation of vibration characteristics of reinforced concrete beams due to flexural damage. Struct Control Health Monit 22(6):939–967.

    Article  Google Scholar 

  18. 18.

    Belluzzi O (1960) Scienza delle Costruzioni, vol I–IV. Zanichelli, Bologna. ISBN 978-8-808-01256-2

    MATH  Google Scholar 

  19. 19.

    Benedetti A, Nichols JM, Tomor A (2016) Influence of environmental degradation on dynamic properties of masonry bridges. In: 16th IB2MAC conference, Padua, Italy

  20. 20.

    Newtson M, Johnson GP, Enomoto BT (2006) Fundamental frequency testing of reinforced concrete beams. J Perform Constr Facil 20(2):196–200.

    Article  Google Scholar 

  21. 21.

    Castel A, Gilbert RI, Ranzi G (2014) Instantaneous stiffness of cracked reinforced concrete including steel-concrete interface damage and long-term effects. J Struct Eng 140(6):1–9.

    Article  Google Scholar 

  22. 22.

    Xu T, Castel A (2016) Modeling the dynamic stiffness of cracked reinforced concrete beams under low-amplitude vibration loads. J Sound Vib 368:135–147.

    Article  Google Scholar 

  23. 23.

    Chondros TG, Dimarogonas AD, Yao J (2001) vibration of a beam with a breathing crack. J Sound Vib 239(1):57–67.

    Article  Google Scholar 

  24. 24.

    Bendat JS (1998) Nonlinear systems techniques and applications. Wiley, New York. ISBN 978-0-471-16576-7

    MATH  Google Scholar 

  25. 25.

    Bayissa WL, Haritos N (2005) Experimental investigation into vibration characteristics of a cracked RC T-beam. In: 4th Australasian congress on applied mechanics, Melbourne, Australia, pp 263–269

  26. 26.

    Kim H, Melhem H (2003) Fourier and wavelet analyses for fatigue assessment of concrete beams. Exp Mech 43(2):131–140.

    Article  Google Scholar 

  27. 27.

    Gao Y, Du Y, Jiang Y, Zhao W (2017) Research on cracking of reinforced concrete beam and its influence on natural frequency by expanded distinct element method. J Aerosp Eng 30(2):B4016015–B4016017.

    Article  Google Scholar 

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This research has received the financial support from the ERA-NET Infravation 2014 research program through the SHAPE project (Predicting Strength Changes in Bridges from Frequency Data—Safety, Hazard, and Poly-harmonic Evaluation) under Grant 31109806.004. The support of the Infravation program is gratefully acknowledged.

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Correspondence to Andrea Benedetti.

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The authors declare that they have no conflict of interest.

Ethical standards

The Project SHAPE and the research program Infravation completely comply with the ethical standards required by the European Community for research granting. This study does not contain any research done with humans or animals.

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Informed consent was obtained from all participants included in the study.



Tables 7, 8, and 9 contain the non dimensional data of the points and curves shown in Figs. 3 and 9.

Table 7 Non dimensional load–frequency curves of the references [8, 12, 14]
Table 8 Non dimensional load–frequency curves of the references [11, 15, 17]
Table 9 Non dimensional load–frequency curves of the references [9, 10, 13, 16]

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Benedetti, A., Pignagnoli, G. & Tarozzi, M. Damage identification of cracked reinforced concrete beams through frequency shift. Mater Struct 51, 147 (2018).

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  • RC beam
  • Breathing crack
  • Dynamic test
  • Damage detection
  • Frequency shift