Quantifying uncertainty and updating reliability are essential for ensuring the safety and performance of engineering systems. This study develops a hierarchical Bayesian modeling (HBM) framework to quantify uncertainty and update reliability using data. By leveraging the probabilistic structure of HBM, the approach provides a robust solution for integrating model uncertainties and parameter variability into reliability assessments. The framework is applied to a linear mathematical model and a dynamical structural model. For the linear model, analytical solutions are derived for the hyper parameters and reliability, offering an efficient and precise means of uncertainty quantification and reliability evaluation. In the dynamical structural model, the posterior distributions of hyper parameters obtained from the HBM are used directly to update the reliability. This approach relies on the updated posteriors to reflect the influence of system uncertainties and dynamic behavior in the reliability predictions. The proposed approach demonstrates significant advantages over traditional Bayesian inference by addressing multi-source uncertainty in both static and dynamic contexts. This work highlights the versatility and computational efficiency of the HBM framework, establishing it as a powerful tool for uncertainty quantification and reliability updating in structural health monitoring and other engineering applications.

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