PSI - Issue 84

Gianluca Bruno et al. / Procedia Structural Integrity 84 (2026) 240–247

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Two separate experimental campaigns were carried out on the bridge, aimed at characterizing both material properties and dynamic behavior. Of interest is the dynamic monitoring of the bridge, which was performed through 18 single-axis accelerometers positioned along the longitudinal length of the deck, two dual-axis accelerometers installed on the two frame piers, and one triaxial accelerometer positioned on one of the abutments. The OMA was performed on experimental data, and results in terms of frequencies are reported in Table 1. The results show that the presence of the continuous slab modifies the ideal static pattern of simply supported beams, introducing a degree of continuity between adjacent spans. In engineering terms, this means effective constraints between spans (modelled here using parameters C1 and C2), which significantly influence the frequencies and distribution of the observed modal deformations.

Table 1. Values of the first three vibration modes for the case study structure. Vibration mode Frequency [Hz] 1° mode 1 = 4.7429 2° mode 2 = 5.7443 3° mode 3 = 6.6051

To apply the proposed algorithm, a high-fidelity finite element model of the bridge was developed using SAP2000 software (Computer and Structures 2023). The deck was modelled using a truss model, i.e., the main longitudinal beams and transverse beams were represented by frame elements, while the continuous slab was modelled as a plate element. The frame piers were modelled with frame elements constrained at the top of the deck and fixed at the base, while the abutments were idealised as rigid supports. The constraint scheme reproduces the presence of expansion joints at the ends of the structure and the possible continuity between the bays. In particular, the side spans were connected to the abutments by supports allowing rotation. The continuity or discontinuity between adjacent bays is controlled by two discrete parameters, 1 and 2 , which represent the constraint between the first and second bays and between the second and third bays, respectively, using the equal-degree-of-freedoms command. To complete the calibration process, the elastic modules of the concrete for three spans were considered as uncertain, and then the vector of uncertain parameters was defined as = [ 1 , 2 , 3 , 1 , 2 ] . The engineering ranges of the uncertain parameters were defined, considering for 1 , 2 and 3 ranging from 29000 to 37000 MPa, while constraints 1 ans 2 assumed two values, i.e., 0 (disconnected) and 1 (fully connected). Modal analyses were performed for different combinations of within the ranges, in order to construct the dataset necessary for generating the ROM. After, the ROM of the bridge was defined. In particular, the snapshot matrix was created, to which SVD was applied to extract a reduced orthonormal basis. By selecting the first =10 left singular vectors, corresponding to the most energetic modes, the projection matrix Φ was constructed, which was used to project the mass and stiffness matrices of the FOM into the reduced space. The order reduction leads to a model with 10 degrees of freedom (DOFs) from the initial 270 ones. The final definition of the matrices was carried out by employing the first-order expansion. The ROM was then processed in the DRL environment, in which the agent updated the parameters = [ 1 , 2 , 3 , 1 , 2 ] and then, the matrices and with the related frequencies. The DRL agent framework was implemented in Python (Python Software Foundation 2001), configuring a complete interaction with the created ROM. The action space was defined in a mixed manner. For the elastic modules of the three spans, 1 , 2 and 3 , a continuous action space was adopted, in which the agent can increase or decrease the parameter values within the defined ranges. For parameters 1 and 2 , a discrete action space was adopted, according to a Boolean criterion. The state of the environment includes, at each timestep , the natural frequencies simulated by the ROM ( ) = [ 1 ( ) , 2 ( ) , 3 ( ) , 4 ( ) , 5 ( ) , 6 ( ) , 7 ( ) , 8 ( ) , 9 ( ) , 1 0 ( ) ] for the current parameter configuration and the current value of ( ) = [ 1( ) , 2( ) , 3( ) , 1( ) , 2( ) ] . The reward function was defined as ( ) = − + √∆ 3 + (8) where is a constant equal to 1 and imposed with a negative sign to incentive the agent to reach the solution quickly, is a penalty given to the agent when performs actions exceeding the extreme ranges of uncertain parameters ( is equal to -1 to discourage the agent in pursuing physically implausible solutions), ∆σ= |(σ (t) −σ (t−1) )| , i.e., is the