PSI - Issue 78

Pooria Mesbahi et al. / Procedia Structural Integrity 78 (2026) 1839–1846

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Table 1: Comparison between ground truth values and inferences by DE-MC and DRAM DE-MC algorithm DRAM algorithm

Parameter True value Mean

CV [%] Rel. error [%]

Mean

CV [%] Rel. error [%]

E c [Pa] E s [Pa] f c 0 [Pa] f y [Pa]

3.00E + 10 2.89E + 10 4.79 2.10E + 11 2.02E + 11 6.48 2.60E + 07 2.54E + 07 3.41 4.30E + 08 4.19E + 08 3.89

3.74 3.97 2.27 2.54 3.29 0.29

2.78E + 10 0.30 2.09E + 11 0.79 2.43E + 07 0.53 4.80E + 08 0.89 2.40E + 04 0.58 4.45E-02 0.41

7.33 0.37 6.48 0.12 1.00 11.63

γ [N / m 3 ] 2.40E + 04 2.32E + 04 4.72

ξ [-]

4.50E-02 4.51E-02 1.44

case. While the BN updates beliefs dynamically across nodes via propagation and back-propagation whenever new evidence is introduced.

3. Numerical case study

In this study, two reinforced concrete (RC) bridge models (details in Fig. 3), B1 and B2, are made using fiber-based elements in OpenSees to demonstrate the functionality of the proposed framework . The only di ff erence between the bridges is the height of the piers (H): 6.706 m for B1 and 7.5 m for B2. B1 is assumed to be equipped with an accelerometer at the deck level, whereas B2 has no sensors. A seismic scenario corresponding to the Norcia earthquake (EVENT ID: EMSC-20161030 0000029, MAGNI TUDE W: 6.6, STREAM: HGE, Noicia, Italy downloaded from https: // ran.protezionecivile.it / IT / quakelive.php) is applied at the site of both bridges, so the seismic input is known for both bridges. First, a sensitivity analysis was performed to identify the model parameters that most significantly influence their dynamic behavior among candi date parameters from this analysis six model parameters were selected (initial elastic modulus of the concrete ( E c ), elastic modulus of the reinforcement ( E s ), concrete compressive strength ( f c 0 ), yield strength of the reinforcement ( f y ), weight per unit volume of concrete ( γ ), and damping ratio ( ξ )). Then, some values (presented in the table 1) were assumed to be the ground truth for the model parameters and the deck-level acceleration response of B1, contaminated with 0.5% g root-mean-square (RMS) noise, was used as synthetic measurements. Bayesian FEMU was then performed for B1. The results obtained using MCMC with Delayed Rejection Adaptive Metropolis (DRAM) algorithm (8,000 iterations with 2,000 burn-in) exhibited sampling stagnation (Fig. 4a), moti vating the use of the DE-MC algorithm, which better explores the parameter space. DE-MC was implemented with 20 chains, 300 burn-in, and 1,500 iterations, (Fig. 4b). Although the chains still exhibit a high rejection ratio, leading to flat segments in some parts, the results demonstrate improved uncertainty quantification when compared through histograms (not shown here for brevity). Convergence stability could be further enhanced by updating the standard de viation of the noise, tuning the hyperparameters, and introducing an adaptive penalty for cases in which the generated random samples cause the model to fail before reaching the full duration of the measurements. As shown in Table 1, the DE-MC algorithm consistently yields lower relative errors (computing based on mean values) compared to MCMC for most parameters, highlighting its superior accuracy in model updating. The improve ment is particularly significant for E c , f c 0 , ξ and f y , where DEMC reduces the error drastically compared to MCMC. However, it is worth noting that MCMC achieves slightly better accuracy for E s and γ , indicating its relative strength in estimating these specific parameters. Finally, the updated model parameters of B1 are presented in Fig. 5, where uniform priors are compared against the obtained posterior distributions (Gaussian distribution was fitted to the his togram). The posterior distributions not only show significantly reduced uncertainty but also have mean values closer to the assumed true values for all parameters.