A Heteroscedasticity–Covariance‐Structured Multivariate Probabilistic Seismic Demand Model With Closed‐Form Fragility and Analytical Loss‐Moment Propagation

ABSTRACT Traditional probabilistic seismic demand models typically describe one structural response at a time with constant conditional dispersion, limiting their ability to represent the joint evolution of drift, floor acceleration, and residual drift. This paper develops a heteroscedasticity–covariance‐structured multivariate demand model with intensity‐dependent mean, marginal dispersion, and positive‐definite correlation. The fitted joint distribution yields closed‐form fragility for affine log‐demand limit states and analytical first and second moments of an illustrative conditional loss index. Full‐dimensional synthetic studies verify multidemand fragility and loss propagation, while reduced‐order studies clarify the mechanisms associated with dispersion and dependence simplifications. Nonlinear time‐history analyses of an equivalent three‐story steel moment frame under scaled far‐field ground motions evaluate the model hierarchy, marginal fragility calibration, and covariance‐sensitive loss dispersion. The full model achieves the lowest held‐out joint negative log‐likelihood among scalar, multivariate Gaussian, residual‐covariance, and copula benchmarks. Grouped cross‐validation at record and earthquake‐event levels supports the bounded saturating variance layer in the full non‐collapse database. The shared rank‐one correlation structure also outperforms pair‐specific and rank‐two alternatives, while dynamic‐versus‐constant correlation gains remain modest at the available event count. Fragility and conditional mean‐loss‐index results remain stable across the tested variance forms, with greater sensitivity concentrated in loss‐index variance. Censoring‐aware diagnostics contextualize the high‐intensity survivor‐dispersion behavior. Dedicated residual‐drift exceedance, collapse‐mixture, and direct‐simulation modules provide routes to repairability, collapse‐sensitive, and upper‐tail decision quantities. The resulting formulation provides an estimable and analytically tractable demand‐to‐decision chain for multivariate seismic assessment.