PSI - Issue 84

Laura Ierimonti et al. / Procedia Structural Integrity 84 (2026) 959–966

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Rigid links connected the pier-cap centroids to their top surfaces, where six-degree-of-freedom spring elements reproduced the sliding bearings between deck and piers; an equivalent configuration was adopted for the abutment bearings. In FEM B , the deck was modelled using shell elements with variable thickness and geometry to accurately reproduce the real bridge configuration. The piers were represented following the same modelling approach adopted in FEM A . The models were subsequently calibrated against reference modal properties extracted with the CoV-SSI algorithm implemented in the MOVA/MOSS software (García-Macías et al., 2020). The results of the calibration are summarized in Table 1. More details can be found in (Tomassini et al., 2025). 4.2. Surrogate model A surrogate model inspired by the modular feed-forward neural network (FNN) architecture proposed in Tomassini et al. , 2025 was developed to build the SM of the Volumni Bridge. The input to the FNN consisted of the non dimensional parameters , defined as functions of the stiffness multipliers of the deck with = 1,..,20, while the outputs were the corresponding normalized frequencies and mode shapes (more details can be found in Tomassini et al., 2025). The internal architecture comprises a shared block of fully connected layers for feature extraction, followed by two specialized modules: one dedicated to predicting the modal frequencies through GELU-activated layers, and a second composed of different branches devoted to mode-shape prediction using hyperbolic-tangent activations. According to this, the deck of the FEM A was discretized into 20 control regions (each span was subdivided into four equal segments, each one quarter of length of the span), and each region was associated with an independent stiffness multiplier varying within the range [0.8,1.05]. A Latin Hypercube Sampling (LHS) procedure was used for generating 1024 combinations of these parameters; for each configuration, the calibrated FEM A was used to extract the corresponding natural frequencies and mode shapes, generating the dataset used to train the surrogate model. Training was performed for 1000 epochs using a custom loss function that jointly accounts for frequency errors and MAC-based mode-shape similarity, with 15% of the samples reserved for internal validation. An independent validation set of 512 FEM A simulations confirmed the high accuracy of the surrogate model, with frequency predictions achieving 2 >0.99 and mode shapes reaching MAC values consistently above 0.98. These results demonstrate the suitability of the proposed FNN-based surrogate for efficient, stiffness-oriented damage assessment. W ith reference to Fig. 2, the spans are numbered C1 to C5 along the Florence direction, and the damage -sensitive parameters to be updated are denoted as , ( =1,…,19 -Florence direction). The selected surrogate sub-model competing candidates are defined as follows: (i) SM 1 , = θ 1 , . , θ 4 ; (ii) SM 2 , = θ 5 , . , θ 8 ; (iii) SM 3 , = θ 9 , . , θ 12 ; (iv) SM 4 , = θ 13 , . , θ 16 ; (iv) SM 5 , = θ 17 , . , θ 20 ; (v) SM 6 , = θ 1 , . , θ 20 . 5. Results and discussion Bayesian model class selection is performed to compare the six surrogate models defined in Section 4.2, with differing parameterizations of the elastic moduli. Mode correspondence between undamaged and damaged models is established using the Modal Assurance Criterion (MAC). A full MAC matrix is computed, and modal pairing is performed by maximizing MAC values, ensuring correct mode tracking. Damage is simulated by reducing the parameters θ 9 ,…, θ 12 by 20% in FEM B , corresponding to an equivalent reduction in the stiffness of the box girder. Figure 3 shows the evolution of the BIC across the iterative updating process. During the initial undamaged iterations (iterations 0-4), all surrogate models exhibit comparable BIC values, indicating limited discriminatory power due to insufficient information content in the data. Among all candidates, SM 3 consistently yields the lowest BIC values, indicating the strongest balance between goodness-of-fit and model complexity. The remaining low-dimensional models (SM 1 toSM 5 ) perform competitively but are systematically inferior to SM 3 . In contrast, SM 6 (the 20-parameter model), despite the potential flexibility of the parametrization, exhibits substantially higher BIC values throughout the iterations. This behavior reflects the strong penalization of unnecessary complexity when the additional parameters are not sufficiently supported by the data.