PSI - Issue 64

Antonino Maria Marra et al. / Procedia Structural Integrity 64 (2024) 2117–2124 Marra et al./ Structural Integrity Procedia 00 (2019) 000 – 000

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5. Model updating A finite-element model (FEM) of the deck was realized, through the software CSi Bridge© (Fig. 6). The model was implemented based on the information from the design drawings of the structure. The FE-modeling was first used for the preliminary estimation of the natural frequencies and of the modal shapes of the deck, to determine efficiently the position for the installation of the accelerometric sensors and to support the identification of the structural dynamic features from the ambient vibration measurements. The elastic moduli used for the reinforced concrete slab was 20,000 MPa. The updating procedure starts with the definition of the prior distribution of the elastic stiffness ( ) of the horizontal springs added to the roller supports located at one end of the deck ( 0 ( ) ). A uniform distribution was adopted by considering a range of from 0 (roller support) to 10 7 N/mm. The upper extreme of the stiffness was fixed once the natural frequency provided by the FEM was very close to the one obtained by using the pinned support. Then, the measurement errors of the estimated first natural period provided by the experimental measurements must be defined. Thus, the function ( ̅| , ) , corresponding to the measurement errors, was defined by employing a Gaussian distribution centered at the measured first natural period = 0.196 s, from the first natural frequency identified from full-scale measurements ( ,1 = 5.1 Hz), with a standard deviation of 0.003 s (Fig. 7). Finally, the modeling uncertainties, expressed by the term ( | ) , were accounted for with a Gaussian distribution centered on the prediction of the structural analysis software concerning the first natural period, and characterized by a standard deviation of 0.0035 s. The estimation of this term has required the implementation of an ad hoc MATLAB algorithm to automatically run CSiBridge ©. The FE code has been run one thousand times by changing the elastic modulus to obtain the predicted natural period. Once the behavior of the first natural period with the elastic stiffness was defined, a resampling of the function was conducted to obtain a step Δ = 100 N/mm. Fig. 8 shows the results of the BMu for the elastic stiffness of the roller supports based on the first natural frequency measured. The median value of the updated distribution function is 21000 N/mm. The posterior distribution curve ( ̅ ( ) ) exhibits a concentration of the probability in the proximity of the lowest values of , meaning that the horizontal constraints at one end of the deck are closer to a roller than a pinned support. The value is also reported in Fig. 8 (vertical dashed line), that is the value corresponding to a deterministic calibration, without any uncertainties of measurements and modeling. It should be noted that the dashed curve, which represents the uniform prior distribution, lies very close to the horizontal axis, as its value is two orders of magnitude smaller than the peak of the posterior distribution.

Fig. 6. FEM of the curved bridge deck.