Rocking motion of two- and tree-dimensional freestanding rigid bodies: Formulation and parametrical analysis

In the past, many studies have been devoted to the analysis of the motion of rigid blocks or groups of rigid blocks. Due to the non-smoothness and to the complexity of the motion , the earliest models were subjected to many simplifying hypothesis that led, among the others, to the general bi-dimensional well-known formulation of slide-rocking rigid block presented in [1]. In latest years, the complexity of the models produced has been widely increased and different phenomena associated to the motion of rigid bodies discovered, such as the existence of the so called “survival regions”, deep investigated [2]. Despite this variety of bi-dimensional problems, only few works are related to three-dimensional formulations and are mainly dedicated to circular shaped tanks and the interaction with their inner liquids. The same authors of the present work extensively investigated the behaviour of the rigid block with a two-dimensional model, both in the case of free-standing rigid block and in the case of isolated rigid body. The influence of constraints on the base displacements and on the slip motion has been analysed, alternatively, the possibility for the rigid block to be partially out of the oscillating base has been considered. The effects of the slenderness, of the friction coefficient and of others geometrical parameter has been investigated, leading to the construction of different maps of behaviour similar to those obtained in [3]. For an impulsive horizontal acceleration applied to the ground exact and approximated (for damped systems) results have been obtained in closed form. Here a general formulation for a three-dimensional square-based rigid block is presented. The rigid body is assumed able only to rock, exact nonlinear equations of motion describing the rocking motion are obtained, starting, ending and impact conditions of the motion are found. Several analyses have been carried on, in order to highlight the influence of different geometrical parameters in the motion evolution. Some results have been compared to those obtained with the two-dimensional model presented in [3]. They proved interesting and will be used as base for further works with more complex models of three-dimensional rigid blocks. References [1] HW. Shenton and NP. Jones, Base excitation of rigid bodies. I: Formulation, Journal of Engineering Mechanics, 117(10), 2286-306, 1991. [2] M.D. Purvance, A. Anooshehpoor, and J.N. Brune, Freestanding block overturning fragilities: Numerical simulation and experimental validation, Earthquake Engineering & Structural Dynamics, 37, 791-808, 2008. [3] A. Di Egidio and A. Contento, Base Isolation of Sliding-Rocking Non-Symmetyric Rigid Blocks Subjected to Impulsive and Seismic Excitations. Engineering Structures 31, 2723-2734, 2009.