PSI - Issue 44

Riccardo Martini et al. / Procedia Structural Integrity 44 (2023) 657–664 Riccardo Martini et al. / Structural Integrity Procedia 00 (2022) 000–000

662

6

0.25%, in agreement with the experimental values. The non structural loads (i.e. kerbs and road pavement) are assigned as distributed loads on the girders. The seismic isolators are modelled as vertical restraints and the horizontal deformability of the devices are included by using elastic springs, properly calibrated. Trucks are represented by masses placed on the deck slab and connected through vertical links, characterized by a stiffness of 2050 kN/m which is evaluated once the mass and the frequency of the purely vertical vibration mode of the truck are determined. In order to reproduce the DT1 (unloaded bridge), the modal analysis is firstly performed on the FE model with no trucks, and the frequency of the first bending vibration mode (Fig. 4(b)), which is equal to 2.38 Hz, is obtained. In this case (DT1), the experimental and analytical results show a good match and suggest a good calibration of the FE model. Before simulating the DT2, it may be useful to preliminary assess the importance of the BTI phenomena adopting the tool provided in Fig. 1. The � ratio is calculated considering the unloaded bridge frequency (2.35 Hz), obtained from the DT1, and the vertical truck frequency, which is assumed to be 1.08 Hz, according to data available in the literature (Gara et al. (2020)). For the case at hand, the � = 2.18 . In DT2, trucks with a total mass of 440 t are located over the first lateral span, which has a total weight of 487 t; thus, the mass ratio � is equal to 1.1. Once the � and � ratios are defined, by entering the graph in Fig. 1 with the � = 2.18 , and by interpolating the � curves for the value of 1.1, the expected frequencies are obtained in a non-dimensional form; in detail, ( 2 ⁄ ) = 0.41 and ( 2 ⁄ ) = 1.1 . The two expected frequencies of the loaded bridge are thus estimated by multiplying the first frequency of the unloaded bridge with the two circular frequency ratios; the first expected frequency is 0.99 Hz and the second one is 2.58 Hz. The recorded accelerations for the loaded configuration (DT2) are shown in Fig. 5(a) in terms of first Singular Value (first SV), obtained by performing the Singular Value Decomposition (SVD) of the Power Spectral Densities (PSDs). In order to compare the experimental and analytical results obtained from DT2, two different FE models are adopted, one implementing the compliance of the trucks and the other one assuming the trucks to be rigid systems. In both models, steady state analysis are performed by applying a linear combination of harmonic vertical displacements and rotations at the base restraints with the aim of detecting the fundamental frequencies of the system by applying a white noise. The approach is consistent with the assumption made in performing AVT, for which the ambient noise is assumed to be a signal with a flat spectrum. The steady state analysis are performed in the frequency range 0-4.5 Hz and the absolute values of the FRFs of the vertical accelerations at A1 accelerometer position, as shown in Fig. 3(b), are outlined in Fig. 5(b); the latter are normalized with respect to the highest values obtained in the analysis. As reported in Fig. 5(a), the first SV (green line) shows a large number of peaks and the identification of the frequency of the first bending mode is not an easy task. In order to support the identification process, the response of the FE model characterized by flexible trucks (black line in Fig. 5(b)) is analyzed. The first two peaks of the FRF (highlighted with blue dots) represent the first bending mode of the deck, which is split in two distinguished peaks due to the BTI phenomena: one is referred to the bending mode, characterized by a in counterphase vibration of the masses (bridge and trucks), and the other by a in phase vibration. As the results show, the 2-DoF system graph presented above in Fig. 1 provides a worthwhile approach to assess the relevance of the BTI phenomena with respect to the more advanced FE model, with a maximum error of about 5 % on the expected frequencies. In this case, the first SV peaks related to the first bending modes of the deck match quite well with the numerical model ones (maximum error 2 %) and with the 2-DoF system prediction (maximum error 3 %). On the other hand, if the BTI phenomena are neglected, the FE model response with rigid trucks (dashed black line) is far from reality and may lead to misinterpretations of bridge dynamic. It is worth noting that the first SV shows also other peaks, which are not investigated in this work because referred to higher modes that require a large number of sensors during the AVTs in order to avoid the spatial aliasing issues.

(a)

(b)

1st Flexural Mode: 2.38 Hz

Fig. 4. (a) bridge FEM view; (b) first flexural shape mode.