PSI - Issue 66

Umberto De Maio et al. / Procedia Structural Integrity 66 (2024) 459–470 Author name / Structural Integrity Procedia 00 (2025) 000–000

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Fig. 4. Natural frequency degradation trend for the proposed model (a) and the bilinear frequency (Chati et al., 1997) (b).

In order to reproduce the dynamic experimental behavior of the cracked beam, and to overcome the problems caused by the non-linearities, in this work, it has been proposed a numerical formula for natural frequencies as a linear combination of the frequency at the loading phase , L i f (that is the crack open configuration) and the frequency at the unloading phase , UL i f (that is the crack closed or partially closed configuration), for different load levels and for each mode shape:

f a f  

b f

(14)

 

, UL i

, L i

i

The parameter 0.08334 b  have been determined for the first mode shape, using as objective function the Mean Squared Error (MSE), between the optimized frequencies and the numerical experimental frequencies, and the ordinary least squared (OLS) is used as optimization function to minimize the MSE. The results in terms of normalized frequency, with respect to the undamaged natural vibration frequency, are shown in Fig. 5, considering four mode shapes (1, 4, 6, and 7). The common dynamic damage indicators, available in the technical literature, have been employed to assess their capability in damage detection of strengthened structures and to obtain further information about the location and the extent of the damage. Two different indicators have been adopted, the first one is the Modal Curvature (MC), presented in Fig. 6, for the four mode shapes considered. The MC results show that in the early load levels, the constant bending zone is most affected by the slope changes of the modal curvature. Subsequently, significant peaks in the curvatures are detected in the shear zones of the beam. 0.861661 a  and