Reliability-Calibrated Surrogate Models for the Nonlinear Seismic Assessment of Existing Reinforced Concrete Frames
Keywords:
structural reliability, surrogate models, Gaussian process regression, polynomial chaos expansion, existing reinforced concrete buildings, seismic fragility, active learningAbstract
Surrogate models are routinely substituted for nonlinear finite element models in the probabilistic seismic assessment of existing reinforced concrete buildings, and are accepted on the basis of global goodness-of-fit statistics such as the leave-one-out coefficient of determination. This paper shows that such statistics do not control the accuracy of the resulting reliability estimate, and derives the relation that does. A first-order expansion gives the error in the reliability index as the conditional mean surrogate error evaluated at the limit-state threshold, scaled by the ratio of the response density at that threshold to the standard normal density at the reliability index. The global error variance enters only at second order. The relation is tested on a four-storey, three-bay non-seismically detailed frame modelled with fibre beam-column elements, assessed at four drift-based limit states with reference reliability indices of -0.49, 0.71, 1.34 and 1.99, and with eleven random variables covering seismic intensity, materials, loads and model error. Across quadratic response surfaces, sparse polynomial chaos expansions, multilayer perceptrons, Gaussian process models and polynomial-chaos Kriging, the relation predicts the observed reliability-index error with a regression slope of 1.04 and a coefficient of determination of 0.93, whereas conventional fit statistics leave that error undetermined by up to two orders of magnitude. Variance decomposition attributes about 94% of the response variance to seismic intensity and demand model error, and less than 1% to all material variables combined. Three practical consequences are demonstrated: logarithmic transformation of the drift response changes the reliability-index error by an order of magnitude; misclassification-directed enrichment reaches an error below 10⁻³ within about 100 finite element evaluations; and the epistemic uncertainty introduced by the surrogate can be propagated into bounds on the failure probability that contain the reference solution at every training budget.
References
Surrogate models are routinely substituted for nonlinear finite element models in the probabilistic seismic assessment of existing reinforced concrete buildings, and are accepted on the basis of global goodness-of-fit statistics such as the leave-one-out coefficient of determination. This paper shows that such statistics do not control the accuracy of the resulting reliability estimate, and derives the relation that does. A first-order expansion gives the error in the reliability index as the conditional mean surrogate error evaluated at the limit-state threshold, scaled by the ratio of the response density at that threshold to the standard normal density at the reliability index. The global error variance enters only at second order. The relation is tested on a four-storey, three-bay non-seismically detailed frame modelled with fibre beam-column elements, assessed at four drift-based limit states with reference reliability indices of -0.49, 0.71, 1.34 and 1.99, and with eleven random variables covering seismic intensity, materials, loads and model error. Across quadratic response surfaces, sparse polynomial chaos expansions, multilayer perceptrons, Gaussian process models and polynomial-chaos Kriging, the relation predicts the observed reliability-index error with a regression slope of 1.04 and a coefficient of determination of 0.93, whereas conventional fit statistics leave that error undetermined by up to two orders of magnitude. Variance decomposition attributes about 94% of the response variance to seismic intensity and demand model error, and less than 1% to all material variables combined. Three practical consequences are demonstrated: logarithmic transformation of the drift response changes the reliability-index error by an order of magnitude; misclassification-directed enrichment reaches an error below 10⁻³ within about 100 finite element evaluations; and the epistemic uncertainty introduced by the surrogate can be propagated into bounds on the failure probability that contain the reference solution at every training budget
