PSI - Issue 78

Elisa Tomassini et al. / Procedia Structural Integrity 78 (2026) 1831–1838

1835

Author name / Structural Integrity Procedia 00 (2025) 000–000 5 Table 1: Frequencies f and dampings ratios ξ of the identified modes of the Marmore Bridge (December 12, 2024, 3:00 a.m).

Mode

Mode

Mode

f [Hz]

ξ [%]

f [Hz]

ξ [%]

f [Hz]

ξ [%]

1 2 3 4 5 6 7 8

0.54 0.96 1.14 1.41 1.53 1.68 2.10 2.33

3.11 1.40 0.39 3.11 0.42 0.82 0.53 0.93

9

2.46 2.76 2.94 3.13 3.69 3.90 4.27 4.58

0.75 1.65 0.57 0.59 1.16 0.80 1.21 0.60

17 18 19 20 21 22 23

4.67 5.13 5.66 6.04 6.32 7.12 7.48

1.92 0.94 0.40 0.41 1.25 0.99 1.32

10 11 12 13 14 15 16

stability of identified poles across successive model orders, a composite criterion was employed, combining the ab solute relative di ff erence in natural frequencies with the MAC error between the mode shapes. A threshold of 5% was applied to the sum of these two metrics, ensuring the selection of physically meaningful and numerically sta ble modes for subsequent analysis. The stabilization diagram in Fig. 2(a) reveals 23 stable modes, whose shapes are illustrated in Fig. 2(b). Corresponding modal parameters, summarized in Table 1, exhibit high MPC and low MPD values, confirming well-defined, low-complexity modes suitable for reference. The fundamental mode corresponds to the first transverse bending of the deck, while the second mode displays a coupled response between second-order transverse deck bending and first-order vertical arch bending. The third mode predominantly captures vertical bending of the secondary arch, with minor transverse contribution. Modes within the 1.4–2.7 Hz range are primarily vertical, while higher modes involve complex higher-order bending and torsional behaviors. Bending modes reflect localized responses of the deck acting as continuous spans, whereas torsional modes similarly exhibit localized distortions. Modes 9 and 10, localized near the abutments, likely stem from boundary e ff ects. No significant bending–torsion coupling was observed. Acceleration data recorded on the Marmore Bridge between June 30 th , 2024, and February 1 st , 2025, were analyzed to assess the e ff ectiveness of the SHM methodology for long-term monitoring of this structurally complex system. The natural frequencies identified in the reference analysis were continuously tracked over the seven-month period. As shown in Fig. 4(a), a clear seasonal trend emerges, with frequencies increasing during colder months and decreasing in warmer periods, indicating a negative correlation with temperature. Fig. 3 illustrates the temperature dependence of the identified natural frequencies. The relationships are notably nonlinear. Modes 1 and 2, primarily associated with transverse deformations, show dispersed trends, implying weak or complex temperature dependencies. In contrast, Modes 7, 8, 9, and 15 exhibit clearer bilinear or parabolic patterns. Nonlinearities in frequency–temperature behav ior—particularly below 0 ◦ C—are well documented and often attributed to ice formation, which locally increases sti ff ness (see e.g.Peeters [2001], Entezami et al. [2024]). In this case, however, inflection points occur near 20 ◦ C, indicating di ff erent underlying mechanisms. The structural complexity of the bridge—featuring a long-span steel deck supported by arches and piers, with numerous secondary elements—leads to intricate dynamic behavior. Frequency reductions at higher temperatures may result from the temperature-sensitive viscoelastic behavior of asphalt layers. Moreover, thermal expansion and contraction of steel elements can significantly influence internal stresses and boundary conditions, especially in long spans. These factors, coupled with variable live loads, contribute to the observed nonlinear response. To capture such nonlinear trends, a Gaussian Mixture Model (see Bishop [2006]) was applied to segment the data into two clusters, within which localized MLR models were used to describe frequency–temperature correlations. Finally, Hotelling’s T 2 control charts were constructed using this GMM-based model, focusing on a 24-day assessment window from February 1 st to 25 th . Two Upper Control Limits (UCLs), corresponding to 90% and 99% confidence thresholds, were defined using training data. A grouping size of 4 was adopted. Although the training period was limited to seven months—short of the standard one-year duration recommended to account for seasonal variability—the control charts yielded stable results, highlighting the robustness of the approach and its applicability in early-stage SHM scenarios.