Mode shape curvature of multiple cracked beam and its use for crack identification in beam-like structures

Nguyen Tien Khiem


The problem of using the modal curvature for crack detection is discussed in this paper based on an exact expression of mode shape and its curvature. Using the obtained herein exact expression for the mode shape and its curvature, it is demonstrated that the mode shape curvature is really more sensitive to crack than mode shape itself. Nevertheless, crack-induced change in the approximate curvature calculated from the exact mode shape by the central finite difference technique (Laplacian) is much greater in comparison with both the mode shape and curvature. It is produced by the fact, shown in this study, that miscalculation of the approximate curvature is straightforwardly dependent upon crack magnitude and resolution step of the finite difference approximation. Therefore, it can be confidently recommended to use the approximate curvature for multiple crack detection in beam by properly choosing the approximation mesh. The theoretical development has been illustrated by numerical results.


multiple-cracked beams; crack detection; mode shape curvature; Laplacian approximation

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S.W. Doebling, C. R. Farrar, M. B. Prime, and D.W. Shevitz. Damage identification and health monitoring of structural and mechanical systems from changes in their vibration characteristics: a literature review. Technical report, No. LA-13070-MS, Los Alamos National Lab., NM (United States), (1996).

S. W. Doebling, C. R. Farrar, and M. B. Prime. A summary review of vibration-based damage identification methods. Shock and Vibration Digest, 30, (2), (1998), pp. 91–105.

E. P. Carden and P. Fanning. Vibration based condition monitoring: a review. Structural Health Monitoring, 3, (4), (2004), pp. 355–377.

W. Fan and P. Qiao. Vibration-based damage identification methods: a review and comparative study. Structural Health Monitoring, 10, (1), (2011), pp. 83–111.

O. S. Salawu. Detection of structural damage through changes in frequency: a review. Engineering Structures, 19, (9), (1997), pp. 718–723.

P. F. Rizos, N. Aspragathos, and A. D. Dimarogonas. Identification of crack location and magnitude in a cantilever beam from the vibration modes. Journal of Sound and Vibration, 138, (3), (1990), pp. 381–388.

J.-T. Kim, Y.-S. Ryu, H.-M. Cho, and N. Stubbs. Damage identification in beam-type structures: frequency-based method vs mode-shape-based method. Engineering Structures, 25, (1), (2003), pp. 57–67.

K. R. P. Babu, B. R. Kumar, K. L. Narayana, and K. M. Rao. Multiple crack detection in beams from the differences in curvature mode shapes. ARPN Journal of Engineering and Applied Sciences, 10, (4), (2015).

E. Sazonov and P. Klinkhachorn. Optimal spatial sampling interval for damage detection by curvature or strain energy mode shapes. Journal of Sound and Vibration, 285, (4-5), (2005), pp. 783–801.

M. Cao, M. Radziénski, W. Xu, and W. Ostachowicz. Identification of multiple damage in beams based on robust curvature mode shapes. Mechanical Systems and Signal Processing, 46, (2), (2014), pp. 468–480.

D. Dessi and G. Camerlengo. Damage identification techniques via modal curvature analysis: overview and comparison. Mechanical Systems and Signal Processing, 52, (2015), pp. 181–205.

J. Ciambella and F. Vestroni. The use of modal curvatures for damage localization in beam-type structures. Journal of Sound and Vibration, 340, (2015), pp. 126–137.

G. Raju and L. Ramesh. Crack detection in structural beams by using curvature mode shapes. IJIRST–International Journal for Innovative Research in Science & Technology, 3, (2), (2016), pp. 282–289.

A. C. Altunısık, F. Y. Okur, S. Karaca, and V. Kahya. Vibration-based damage detection in beam structures with multiple cracks: modal curvature vs. modal flexibility methods. Nondestructive Testing and Evaluation, 34, (1), (2019), pp. 33–53.

A. K. Pandey, M. Biswas, and M. M. Samman. Damage detection from changes in curvature mode shapes. Journal of Sound and Vibration, 145, (2), (1991), pp. 321–332.

M. M. A. Wahab and G. De Roeck. Damage detection in bridges using modal curvatures: application to a real damage scenario. Journal of Sound and Vibration, 226, (2), (1999), pp. 217–235.

C. P. Ratcliffe. Damage detection using a modified Laplacian operator on mode shape data. Journal of Sound and Vibration, 204, (3), (1997), pp. 505–517.

C. P. Ratcliffe. A frequency and curvature based experimental method for locating damage in structures. Journal of Vibration and Acoustics, 122, (3), (2000), pp. 324–329.

M. Cao and P. Qiao. Novel Laplacian scheme and multiresolution modal curvatures for structural damage identification. Mechanical Systems and Signal Processing, 23, (4), (2009), pp. 1223–1242.

M. Chandrashekhar and R. Ganguli. Structural damage detection using modal curvature and fuzzy logic. Structural Health Monitoring, 8, (4), (2009), pp. 267–282.

M. M. R. Taha, A. Noureldin, J. L. Lucero, and T. J. Baca. Wavelet transform for structural health monitoring: a compendium of uses and features. Structural Health Monitoring, 5, (3), (2006), pp. 267–295.

T. V. Lien and N. T. Duc. Crack identification in multiple cracked beams made of functionally graded material by using stationary wavelet transform of mode shapes. Vietnam Journal of Mechanics, 41, (2), (2019), pp. 105–126.

N. G. Jaiswal and D.W. Pande. Sensitizing the mode shapes of beam towards damage detection using curvature and wavelet transform. Int. J. Sci. Technol. Res., 4, (4), (2015), pp. 266–272.

N. T. Khiem and H. T. Tran. A procedure for multiple crack identification in beam-like structures from natural vibration mode. Journal of Vibration and Control, 20, (9), (2014), pp. 1417–1427.

T. G. Chondros, A. D. Dimarogonas, and J. Yao. A continuous cracked beam vibration theory. Journal of Sound and Vibration, 215, (1), (1998), pp. 17–34.

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