Coupled Motion in the Dynamic Analysis of a Three Block Structure

A mathematical model is presented for the in-plane dynamic analysis of a structure composed of three rigid blocks, simply supported on each other, in presence of dry friction. In order to investigate the dynamics excited by a given ground motion, a Lagrangian formulation is proposed, with displacements, velocities and reactive forces restricted in bounded fields, due to the incompenetrability of the elements. Special attention is devoted to the modelling of the impacts, and to the role of Coulomb friction during the smooth dynamics. Different failure modes are obtained, characterized by a different participation factor of the degrees of freedom. Regions of coupling between rocking and sliding-displacements can be identified, as a function of the features of the excitation and of the friction coefficients. Survival domains decreasing with time can be obtained. They seem to indicate that, in presence of an earthquake, excessive relative displacements in the contact surfaces between the elements would be the most probable cause of failure of such a structure.