PSI - Issue 64

Carmelo Gentile et al. / Procedia Structural Integrity 64 (2024) 677–684 Author name / Structural Integrity Procedia 00 (2019) 000–000

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Table 1. Statistical description of the modal parameters identified from 16/10/2018 to 15/10/2019. Mode Id. f ave [Hz] σ f [Hz] f min [Hz] f max [Hz] MAC ave MAC min Id. Rate C1 1.383 0.011 1.359 1.423 0.997 0.980 99.8 % C2 1.677 0.020 1.605 1.739 0.995 0.979 99.3 % C3 1.993 0.015 1.956 2.059 0.993 0.960 99.3 % C4 2.530 0.021 2.477 2.632 0.984 0.950 97.9 % C5 2.668 0.021 2.609 2.757 0.981 0.941 96.4 % C6 2.778 0.034 2.683 2.923 0.978 0.939 88.9 % C7 3.170 0.051 3.066 3.383 0.969 0.911 71.4 % C8 4.177 0.038 4.094 4.350 0.950 0.902 36.9 %

Fig. 5. Automatically identified natural frequencies of the Milan Cathedral versus time (from 16/10/2018 to 31/10/2023).

Figure 5 reveals a remarkable variation in natural frequencies, likely influenced by fluctuating outdoor and indoor temperatures. Specifically, during colder seasons, frequencies for all modes generally increase as temperature decreases (indicating a negative correlation). Conversely, in warmer seasons, frequencies typically increase with rising temperatures, except for mode C2. This peculiar behaviour suggests a correlation between frequency and temperature, possibly due to the opposing effects of thermal expansion on metallic ties and masonry: (a) In colder seasons, the extensometers installed in the tie-rods indicate a generalized strain increase with decreased temperature, so that the corresponding increased forces in the tension bars conceivably exert a stiffening action on the overall structure; (b) during the hot season, the thermal expansion of the masonry blocks has a dominant effect on most frequencies, causing an inversion in behaviour (and frequencies increasing with increased temperature). 4. Structural condition assessment As previously stated, a classic SHM strategy based on natural frequencies is currently applied: the adopted methodology involves the subsequent application of linear PCA (Jollife, 2002) to minimize/remove the EOV effects and multivariate Hotelling’s control chart (Montgomery, 2013) to the frequency residuals. It is further noticed that: (a) to define the PCA-based regression, the first year was assumed as training period; (b) to maximize the occurrence of damage-detection indicators, the PCA analysis is carried out adopting the natural frequencies of the first 6 modes, whose identification rate is higher than 88% (Table 1); (c) 3 principal scores have been retained in the PCA model. After the definition of the PCA-based regression model, it was used for newly collected frequency estimates. Figure 6 shows the cleansed frequencies during the entire monitoring period herein considered: the effectiveness of PCA in removing the fluctuations is highlighted by comparing Figs. 5 and 6.