PSI - Issue 41

Andrea Pranno et al. / Procedia Structural Integrity 41 (2022) 618–630 Author name / Structural Integrity Procedia 00 (2019) 000–000

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highest damage level for the 7 th modes, but considering the experimental uncertainties given by the high-order natural vibration mode, is still an acceptable deviation. The curve trends highlighted that the investigated natural vibration modes show a decreasing trend with three different slopes: the slope that occurs between the percentage failure load 0% and 15% is slight, the slope that occurs between 15% and 50% is considerable, and the slope that occurs between 50% and 95% remains similar to the first.

Fig. 4. Normalized natural vibration frequencies of the modes 1, 4, 6, and 7 compared with the envelopes of the experimental data reported in (Hamad et al., 2015).

In order to evaluate the correlation between the modal shapes in the undamaged configuration L1' with those in damaged ones L2'-L10', the MAC coefficients were evaluated and reported in in Tab. 4. The coefficients were determined by using formula (11) applied to mode shapes normalized with respect to the mass matrix and evaluating the displacements relating to mode 1, 4, 6, 7 and 8 along the top of the beam at 51 points equally spaced to each other 60 mm. It worth noting that, as the level of damage increases, the correlation between the mode in the undamaged configuration and that relating to the damaged configuration decreases, as a consequence of the fact that the values of the diagonal coefficients of the MAC matrix (MAC ii ), deviate more from the unit value as the damage level increases. These deviations are not particularly relevant, in fact the relative maximum percentage deviation measured is equal to 2.11% corresponding to the level of damage L10 ' for the 8 th mode. In Fig. 5, we then investigated the Curvature Damage Factor (CDF) normalized with respect to the CDF which is related to the highest damage level (L10'). Such indicator was determined by summing the absolute variations of the Modal Curvatures (evaluated in fifty points placed on the whole length of the top beam and equally spaced) between the undamaged configuration L1' and the other investigated damage levels L2' - L10' over all the investigated mode shapes. It will be noted that, at the undamaged configuration (L1'), the CDF is equal to zero, but it oscillates for the subsequent damage levels, indicating the presence of damage localization. The obtained curves show several peaks in the center of the steel-reinforced concrete beam whose intensity is related to the damage level and therefore to the fracture length. Furthermore, CDF indicator was shown to be heavily affected by the local effect caused by the loads and stiff plates. In fact, we can see noise in the oscillatory trend of the CDF in these beam zones.