A homogenization limit analysis model based on a plate and shell upper bound FE formulation is presented. In the model, the elementary cell is subdivided along its thickness into several layers. For each layer, fully equilibrated stress fields are assumed, adopting polynomial expressions for the stress tensor components in a finite number of sub-domains. The continuity of the stress vector on the interfaces between adjacent sub-domains and suitable anti-periodicity conditions on the boundary surface are further imposed. In this way, linearized homogenized surfaces in six dimensions (polytopes) for masonry in- and out-of-plane loaded are obtained. Such surfaces are then implemented in a FE limit analysis code for the analysis at collapse of entire three-dimensional structures and meaningful examples of technical relevance are discussed in detail.

The micro-mechanical model presented competes favourably with more traditional approaches, such as for instance full three-dimensional heterogeneous techniques and "at hand" calculations based on the assumption of zero resistance of masonry in tension. In fact, full three-dimensional analyses performed on entire buildings by means of the homogenization model presented, require a reduced computational cost (less than 150 seconds for a single optimization reported in Section 4), do not require an a-priori evaluation of the collapse mechanisms and can take into account important features of masonry at failure. Furthermore, limit analysis is able to give important information at failure, such as failure mechanisms, collapse loads, stress distribution, plastic dissipation zones, etc. Therefore, the model presented results in a valuable tool for practitioners involved in advanced analyses of full three-dimensional masonry structures subjected to seismic actions.

The limited computational effort of the optimization problems obtained in this framework allows one to tackle interesting engineering problems, such as for instance the evaluation of collapse loads stochastic distribution of masonry structures when mortar mechanical properties (i.e. input parameters) are assumed as random variables.

The combination of homogenization, limit analysis and response surface approximation allows one to obtain reliable predictions of failure loads distribution when metamodels are built starting from few points sampled randomly and making use of both traditional Monte Carlo approaches and Latin Hypercube sampling. In this way, a reliable estimation of output collapse loads distribution can be obtained avoiding performance of expensive Monte Carlo simulations with several points.

Return to the contents page
Return to the book description