The current confidence in the ability to provide buildings with adequate resistance to horizontal actions does not extend back to historic and existing masonry structures. Furthermore, it has been shown that the high vulnerability of historical centers to horizontal actions is mostly due to the absence of adequate connections between the various parts, especially when wooden beams are present both in the floors and in the roof [
]. This characteristic leads to overturning collapse of the perimeter walls under seismic horizontal acceleration and combined in- and out-of-plane failures. Even if limit analysis is not sufficient for a full structural analysis under seismic loads, it can be profitably used in association with more sophisticated techniques (as for instance non-linear FE with damaging procedures) in order to obtain a simple and quick estimation of collapse loads and failure mechanisms. Up to now, simplified limit analysis methods are at disposal to the practitioners both for safety analyses and design of strengthening [
]. Nevertheless, in some cases these methods are based on several simplifications, one of which is an a-priori assumption of the collapse mechanics combined with the separation of in- and out-of-plane effects.
In this paper, the micro-mechanical model presented by the authors in [
] and [
] for the limit analysis of respectively in- and out-of-plane loaded masonry walls is utilized in presence of coupled membrane and flexural effects. In the model, the elementary cell is subdivided along its thickness in 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 3D structures. Several examples of technical relevance are discussed in detail and comparisons with standard FE codes are provided.