A Multi-Pier-Macro MPM method for the progressive failure analysis of full scale walls in two way bending

The aim of the paper is to extend and put at disposal to common users a reliable numerical procedure called MPM (Multi-Pier Macro) approach, already presented in-plane, for walls subjected to two way bending. This approach is developed starting from a Multi-Pier MP approach proposed by the authors in the past. The MP approach for the out-of-plane analysis of masonry can be used for practical purposes, but requires a refined discretization with relatively huge computation effort and a specific training for the user. The MPM approach overcomes some well known limitations of the MP one, using a discretization that is totally independent from the actual unit cell dimension considered and totally unrelated to the specific texture of the masonry wall considered. The MPM method relies on a very unrefined discretization of a wall out-of-plane loaded with parallelepiped cells with large edges made by vertical and horizontal beams plus braces, assuming for each element equivalent geometric and mechanical properties that guarantee the elastic, peak and post peak rough equivalence between the original masonry wall and the mono-dimensional discretization proposed. Its main feature is to allow the analysis of large and complex masonry structures, shortening the calculation time and being accessible to practitioners without a specific background on masonry and who own a commercial software equipped only with non linear 1D elements. The procedure is benchmarked on four walls out-of-plane loaded in two way bending, in presence also of perforations for three of the four walls; a good predictivity in the non-linear range is observed when compared with much more sophisticated numerical approaches.