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.

翻译：不确定性量化与可靠性更新对于保障工程系统的安全与性能至关重要。本研究开发了一种层次贝叶斯建模框架，以利用数据实现不确定性量化与可靠性更新。通过利用HBM的概率结构，该方法为将模型不确定性和参数变异性纳入可靠性评估提供了稳健的解决方案。该框架在线性数学模型和动态结构模型中得到了应用。对于线性模型，推导了超参数与可靠性的解析解，为不确定性量化与可靠性评估提供了高效且精确的手段。在动态结构模型中，直接使用从HBM获得的超参数后验分布来更新可靠性。该方法依赖更新后的后验分布，以在可靠性预测中反映系统不确定性与动态行为的影响。所提出的方法通过处理静态与动态场景下的多源不确定性，展现出相较于传统贝叶斯推断的显著优势。本工作凸显了HBM框架的通用性与计算效率，确立了其作为结构健康监测及其他工程应用中不确定性量化与可靠性更新的有力工具。