Nonlinear dynamics of rocking rigid blocks under stochastic base excitation: a stochastic-linearization approach

This paper introduces a semi-analytical framework for the nonlinear dynamics of rigid blocks rocking on nonlinear flexible foundations under stochastic base excitation. Unlike classical Housner and Winkler models that assume null or linear base-foundation interaction, the proposed approach incorporates uplift and velocity-dependent dissipation via a Hunt-Crossley-type law, augmented by a tunable cubic restoring component. The foundation is modeled as a distributed bed of nonlinear springs and dampers, combining Winkler-type linear stiffness with added nonlinearity to enable hardening or softening beyond the baseline response. Backbone amplitude dependence arises from contact geometry and is further shaped by the cubic stiffness term. The originally coupled, piecewise nonlinear rocking relations are condensed into a single governing equation using a quasi-static condensation scheme, with smooth analytical surrogates for restoring moment and damping across no-uplift and uplift regimes. A stochastic linearization scheme embedded in Lyapunov moment equations yields time-varying equivalent stiffness and damping, enabling second-order response statistics without repeated integration of highly nonlinear equations. Comparisons with Monte Carlo simulations show close agreement in rotation and rotational velocity variances across excitation intensities and foundation parameters, while achieving significant computational savings. The framework offers an efficient tool for parametric screening of rocking systems under complex soil-structure interaction.