A Semi-Analytical Approach for the Contact Dynamics of Ancient Columns Modeled as Rigid Blocks

The problem of conservation and protection of architectural heritage against dynamic inputs such as the case of earthquakes has represented for years an important task in the field of structural engineering with the aim of describing and understanding the behaviour of monumental structures. This work deals with monumental structures made of large and heavy stone blocks, simply supported to each other, a typical technology used to build the Greek temples. The plane dynamics of a rigid rectangular block simply supported on a horizontally moving ground, with friction and unilateral constraints at ground contact, is modeled by means of a new semi-analytical approach for non-smooth contact dynamics recently developed by one of the authors [1]. The dynamics - formulated as a contact problem governed by Signorini's and Coulomb's laws - is converted into an equivalent problem of static balance by making use of specific projective techniques in the configurations space. The procedure is then applied to a numerical model, so that a code can be implemented where the dynamics of the block is fully modeled according to the various possible mechanisms considering both phases of smooth and non-smooth contact dynamics, moreover normal and tangential contact reactions during smooth dynamics, and corresponding impulses during non-smooth, are estimated at each step. Either extended or single point contacts are then possible with consequent sliding, rocking or sliding-rocking modes, so as the possibility of impacts and also of free dynamics when contact is lost. Analyses are conduced and then compared to corresponding cases previously treated in literature, in order to validate the proposed numerical method for the rocking mode. Moreover, dynamic evolution of the block is represented graphically in order to observe different configurations at each instant of time. The goal is to understand the dynamic behaviour of the block under several input intensities in order to prevent excessive sliding or overturning and to localize with sufficient precision the position of the block after the earthquake.