Stochastic block rocking on nonlinear flexible foundations: Lyapunov-based statistical linearization with an enhanced contact law

Rocking of rigid blocks on flexible foundations is intrinsically nonlinear and non-smooth, driven by unilateral contact, uplift, and impacts. This work presents a semi-analytical framework for such systems under modulated, filtered white-noise base excitation. The base–foundation interaction is modeled with a generalized contact law that combines a linear Winkler-type restoring component, a single-parameter tunable nonlinear stiffness, and a Hunt–Crossley-type deformation- and rate-dependent damping mechanism. This structure preserves the parsimony of classical models while enabling amplitude-dependent hardening/softening, uplift-induced asymmetry, and state-dependent stiffness and dissipation beyond linear idealizations. The originally coupled, piecewise nonlinear rocking relations are reduced to a single governing equation via a quasi-static condensation scheme, with smooth analytical surrogates for the restoring moment and damping across no-uplift (inner) and uplift (outer) regimes. Building on this reduction, a compact statistical linearization approach embedded in Lyapunov moment equations yields time-varying equivalent stiffness and damping together with second-order response statistics, thereby avoiding repeated time integration of highly nonlinear equations. These equivalent elements are physically interpretable, providing direct insight into the system’s evolving dynamic character. Comparisons with Monte Carlo simulation show close agreement for the variances of rotation and rotational velocity across ranges of excitation intensity and foundation parameters, while delivering orders-of-magnitude computational savings for ensemble statistics. The framework offers a concise, performance-oriented tool for analysis and parametric screening of rocking systems under realistic, time-varying excitation and complex soil–structure interaction, and it provides a tractable basis for data-driven identification and resilience-oriented applications.