Sparse Bayesian PC-Kriging modeling with multi-criteria active learning for structural reliability: Application to shield tunnel structures

The conventional PC-Kriging surrogate model in structural reliability analysis often encounters challenges such as limited sample utilization and the trade-off between convergence rate and accuracy. This study proposes an enhanced PC-Kriging model integrated with sparse Bayesian learning and an active learning strategy, which incorporates a dynamic sampling method that balances Expected Improvement (EI), Entropy, and Sobol indices, promotes diversity through batch dispersed sampling, and updates samples using spectral projection. These advancements establish a reliability analysis framework enabling adaptive limit-state refinement and multi-criterion joint convergence. Validation via benchmark cases demonstrates superior accuracy and convergence robustness compared to mainstream single-criterion approaches. The method is further applied to assess the structural reliability of shield tunnels under combined compression-bending conditions. Key findings include: (1) The framework achieves efficient reliability evaluation within fewer iterations; (2) Preserving continuity between bending moment and axial force variables is critical, as disrupting their correlation introduces significant deviations in failure probability estimates; (3) Distinguishing eccentric compression states in segmental linings exhibits pronounced impacts on both limit-state functions and failure probabilities compared to other variables. This work extends the applicability of multi-criterion active learning in engineering reliability analysis and provides a systematic methodology for shield tunnel safety assessment.