PSI - Issue 84

Domenico Liberatore et al. / Procedia Structural Integrity 84 (2026) 1151–1158

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The comparison of the ten measurements highlights recurrent frequency peaks at approximately 3.67 Hz, 7.59 Hz and 12.41 Hz in the longitudinal direction (instrumental North-South, Fig. 3a); 3.75 Hz, 6.40 Hz and 10.47 Hz in the transverse direction (instrumental East-West, Fig. 3b); 6.42 Hz, 9.50 Hz, 12.73 Hz and 19.98 Hz in the vertical direction (Fig. 3c). As shown in Fig. 3, the most significant modes occur predominantly in the transverse direction, identified as the structurally weaker direction. Longitudinal modes are less pronounced and appear only at higher frequencies, indicating that the transverse behavior governs the dynamic response of the bridge. Vertical spectra (Fig. 3c) show less pronounced peaks, suggesting a mixed-mode behavior rather than purely vertical. Free-field measurements on Tiber Island exhibit low-frequency values without significant resonance peaks. The Frequency Domain Decomposition (FDD) technique is then applied, obtaining good agreement with FFT based results in both frequency values and vibration directions. The modal frequencies and the directions of the modal shapes are listed in the third and last columns of Table 2, respectively. 4. Numerical model and modal validation The Finite Element numerical model of the bridge is developed using the software FEAP (Finite Element Analysis Program), developed at the University of Berkeley (Taylor 2017). A 3D 8-node hexahedral solid element, with three translational degrees of freedom at each node, is adopted to build the FE mesh. The FE model (Fig. 4) considers structural symmetries and material variations from the laser scanner survey. In particular, the upstream transverse section of the bridge is extruded along the depth of the structure, distinguishing between the external and the opus caementicium of the internal core. Regarding the central pier, the downstream portion, embedded in the soil, is excluded, and only the exposed masonry is modeled. The horizontal surfaces at the base are fully restrained to simulate the possible presence of huge foundations realized at the Roman age, based on data related to similar historical buildings of the area. However, considering the presence of alluvial soil on both Tiber Island and the bankside, elastic springs are adopted for the connection between the spandrels and the ground, considering a stiffness value of 10 5 kN/m 3 and 6 10 4 kN/m 3 for the direction orthogonal and parallel to the surface, respectively.

Fig. 4. Numerical FE model of the bridge (a); downstream view (b); upstream view (c).

4.1. Constitutive law with damage Masonry is modeled as a continuous homogenous material, following a macromechanical approach. The following constitutive law accounting for material damage, firstly developed in Addessi & Sacco (2016) and used also in following studies, such as Addessi et al. (2022), is adopted: = (1 − ) (1) where and denote the 6-component stress and strain vectors, respectively; the damage variable D ranges between 0 and 1 for undamaged and fully damaged states, respectively, following the continuum damage mechanics and describing the softening branch of the curve. Damage simulates the formation and spread of microcracks due to tensile states. To this end, its evolution is driven by an equivalent strain measure based on the principal strains and defined as: =√〈∑ 〈 + 0 〉 +2 3 =1 − ∑ ∑ 1− 3 =1 2 〈 〉 − 〈 〉 − 3 =1 〉 + − 0 (2)