Surrogate models for seismic response analysis of flexible rocking structures

Seismic design tools are based on surrogate models of the designed structure and the responses of those surrogates to earthquake ground motions. To design symmetric flexible rocking structures, a surrogate model that includes rocking and flexure is needed. In this paper, we derive the equation of motion of a flexible multi-degree-of-freedom (MDOF) structure rocking on its base in modal coordinates. Then, we introduce a set of two-degree-of-freedom (2DOF) surrogate models that accounts only for the first elastic vibration mode of the multimass structure and its rotation about the base pivot points. We investigate the surrogates' ability to represent the dynamics of an elastic MDOF structure that uplifts and rocks and the interaction between rocking and flexure. Therein, we detail the simplifications for the equations of motion of the 2DOF surrogate models and the adopted rocking impact model, and develop and check the sliding initiation condition. We show that the simplified 2DOF surrogate model responses compare well to experimental results. Then we assess the 2DOF surrogate model accuracy in representing the earthquake response of the MDOF model using Cloud Analysis and the coefficient of determination $R^2$ approach. We find that simplified 2DOF surrogate of the MDOF model is quite accurate ($R^2=0.99$) in estimating its maximum relative top displacement and acceptably accurate ($R^2=0.90$) in estimating its maximum base rotation based on a thousand randomly generated flexible rocking structure earthquake response analyses. Lastly, we discuss using the simplified 2DOF surrogate model of symmetric flexible rocking structures in preliminary seismic design, and give examples featuring a continuous elastic hollow semi-conical chimney and an inelastic flexible MDOF structure, both with a base that may uplift.