Issue 51

M.G. Masciotta et alii, Frattura ed Integrità Strutturale, 51 (2020) 423-441; DOI: 10.3221/IGF-ESIS.51.31

Either estimator allows to identify seven vibration modes in the frequency range 37–147 Hz with damping ratios below 2% for RS and six vibration modes in the frequency range 30–135 Hz with damping ratios below 5% for RSW. For the sake of brevity, only the modal features extracted by the SSI-UPCX are presented herein. A first analysis of the global modal parameters relevant to the two undamaged scenarios shows that, by increasing the overall weight of the arch of about 50%, frequency values decrease of around 10%, while damping values increase of circa 48%. This result is somehow expected assuming a classical Rayleigh damping formulation for the structure [29]. In what concerns the mode shapes, similar configurations are observed for the in-plane vibration response of the arch, with MAC values higher than 0.90 for modes 1, 3 and 4. On the other hand, due to the variation of in-plane and out-of-plane modal components induced by the change in loading conditions, the degree of similarity between corresponding torsional modes decreases, reading MAC values lower than 0.71 for modes 2, 5 and 7, whereas the symmetric mode in Z-direction at 125.06 Hz cannot be identified as a stable mode. The modal feature extraction process described above is then applied to the output signals collected after each displacement stage. As the support slides, cracks start to propagate, and the arch stiffness degrades with progressive damage. Despite the low displacement rate, this leads to significant changes in the arch eigenfrequencies, recording downshifts of about 36%, 15% and 22% for modes 1, 2 and 4, respectively. The frequency results for the six consecutive scenarios (RSW to DS5) are presented in Tab. 3 and Fig. 5. It is worth noting that considerable frequency drops are already observed during DS1, scenario in which the first crack c 1 appears in the left region of the keystone causing a sudden stiffness reduction in the structure that is reflected by a significant change in the eigenfrequencies of modes 1, 4 and 5. With regard to the modal vectors, the direct comparison between undamaged (RSW) and final damaged (DS5) scenarios reveals remarkable local changes in the high-frequency modes of the arch, but very little changes as far as the main vibration mode is concerned. This is clearly evident by visually inspecting the mode shapes juxtaposed in Fig. 6, whose MAC values are provided below each pair. The larger the differences in modal displacements due to local damage, the higher the MAC sensitivity and the lower its value. Such a result is a direct consequence of the vulnerability that vibration modes featuring a higher number of inflection points exhibit to local structural damage.

Frequencies [Hz]

Scenario

Mode 1

Mode 2

Mode 3

Mode 4

Mode 5 120.62 113.16 111.44 110.60 110.98 110.86 ‒ 8.09

Mode 6

Mode 7 134.02 136.40 130.91 128.84 129.60 129.00 ‒ 3.75

RSW DS1 DS2 DS3 DS4 DS5

30.06 26.31 23.40 21.44 20.20 19.16

50.95 50.28 48.86 45.14 44.05 43.19

59.44 59.04 58.14 58.27 57.77 57.54 ‒ 3.20

95.23 80.47 75.75 74.97 74.39 74.09

‒ ‒

121.50 118.71 116.68 115.78 ‒ 4.71

‒ 36.26

‒ 15.23

‒ 22.20

Δ f [%]

Table 3 : Eigenfrequencies variation over progressive damage scenarios.

Mode 1

Mode 2

Mode 3

Mode 4

Mode 5

Mode 7

1.00

0.75

0.50

RSW DS1

DS2

DS3

DS4

DS5

Figure 5: Relative frequency decay with increasing support settlements.

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