On the dynamics of rigid-block structures|bapplications to SDOF masonry collapse mechanisms

Abstract

This dissertation proposes a novel formulation for the rocking motion (RM) of rigid-blocks when no sliding mechanisms are present. Both theoretical and numerical analyses are included. The results obtained from a new formulation for single block dynamics, which have shown to be extremely powerful, led to the generalization for many degrees of freedom systems presented herein. That generalization was possible through a transformation of the Euclidean configuration space to a complex Riemannian manifold endowed with an Hermitian metric tensor. The present contribution applies also techniques derived from chaos theory to the RM problem for the understanding and quantification of the dynamic stability. The probability of overturn of rocking blocks under random loading is both experimentally and numerically investigated. Through an intensive use of the Poincaré surface of section technique and bifurcation analysis, the underlying structure of the phase space is highlighted. Delay coordinated, recurrence plots and recurrence quantification analysis are used to find the recurrences and patterns present in the dynamics at different time scales. An estimator for the quantification of chaos in the problem of rocking blocks under earthquake loading is proposed. Stochastic analysis is performed through an ensemble of input earthquake samples. The ensemble limits and other stochastic properties of the relevant physical magnitudes are analyzed. A set of experiments at a seismic table on four blue granite stones were conducted to validate the theoretical analyses. The response is investigated under different input actions and block geometries. Several applications to Earthquake Engineering and structural safety assessment are therefore derived. In particular, numerical thresholds for the probability of collapse of slender structures are found by means of different criteria. A model for the probability of collapse based on the stationary solution of the associated Fokker-Planck equation is proposed.