PSI - Issue 78

Francesco Mariani et al. / Procedia Structural Integrity 78 (2026) 875–882

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Fig. 3. Test results for the girder damage parameters: a) elastic modulus of the girder segment above pier n.1; b) elastic modulus of the girder segment above pier n.3; c) elastic modulus of the girder segment above pier n.5; d) elastic modulus of the girder segment above pier n.7

5. Surrogate model: Bayesian Neural Network training and testing

Following dataset generation, the o ffl ine phase proceeds with the training of the BNN. To reduce the dimensionality of the input space and improve the e ffi ciency of the learning process, each type of sensor data is pre-processed as follows: • Natural frequencies: the first five natural frequencies are extracted through modal analysis of the acceleration measurements. • Mode shapes: the raw accelerometer data, obtained from 62 sensors measuring responses in two directions (vertical and longitudinal) across five vibration modes, yield 620 features. This high-dimensional input could bias the model towards mode shape features over frequency-based ones. To address this, Modal Assurance Criterion (MAC) values are computed with respect to the undamaged condition for each of the five modes, e ff ectively compressing the mode shape data into five representative features. • Rotations: the dataset includes 62 rotation measurements from inclinometers under self-weight. Given the strong correlations among multiple sensors under varying structural states, Principal Component Analysis (PCA) is applied to reduce redundancy. The analysis shows that 21 principal components are su ffi cient to capture nearly 100% of the variance in the inclinometer data. After this pre-processing step, the BNN is configured to map the sensor inputs to the Young’s moduli of individual concrete segments. The optimal network architecture consists of a single hidden layer with 32 neurons and employs the hyperbolic tangent (Tanh) activation function. The Markov Chain Monte Carlo (MCMC) sampling method is used to estimate the posterior distributions of the network weights, generating 200 samples per weight. A train-test split is applied during training while varying the number of training samples. The analysis indicates that approximately 2,000 samples are su ffi cient to achieve reliable predictive performance. The remaining 500 samples are used to evaluate the generalization capability of the trained model. As illustrated in Figures 3 and 4, the test results exhibit strong agreement between predicted and actual values, with data points aligning closely along the bisector. In particular, the results also highlight the increased uncertainty in the predictions related to the material properties of the piers, reflecting the greater complexity in capture variations in the sti ff ness of these elements by measuring e ff ects on the superstructure.

6. Results: damage detection

After training, the BNN surrogate model was evaluated on a new dataset representing a range of realistic damage scenarios. For each case, the BNN generated 200 predicted values by sampling from the posterior weight distributions, resulting in a probabilistic output for each structural segment. To assess the model’s e ff ectiveness in identifying both