A Method of Cells-Type Approach to predict the Macroscopic Strength of Masonry Walls in Two-Way Bending

Following an approach recently presented by the authors to estimate the in-plane homogenized failure surfaces of masonry walls with joints of finite thickness, here an extension is proposed to predict the macroscopic strength properties of walls subjected to out-of-plane loads. Similarly to the so-called Method of Cells for fiber-reinforced composites, a typical representative volume is subdivided into a few sub-cells, and a strain-rate periodic, piecewise differentiable velocity field, depending on a limited number of degrees of freedom, is defined. Upper bounds to the macroscopic strength domain of the wall in the space of the macroscopic bending and twisting moments are obtained by applying the kinematic theorem of limit analysis within the framework of homogenization theory for periodic media. By means of standard linear mathematical programming, several points of the approximated macroscopic failure surface are determined, each one representing an upper bound to the ultimate load bearing capacity of the wall under given moment combinations. The approximated macroscopic failure surfaces are in good agreement with those previously obtained, at a much higher computational cost, by alternative numerical approaches.