Simple SQP approach for out-of-plane loaded homogenized brickwork panels accounting for softening

A simple homogenized model for the non linear analysis of masonry walls out-of-plane loaded is presented. In the model, the panels are assumed to behave as Kirchhoff–Love plates. A rectangular running bond elementary cell (RVE) is subdivided into several layers along the thickness and, for each layer, a discretization where bricks are meshed with plane-stress three-noded triangular elements and joints are reduced to interfaces is assumed. Non linearity is concentrated on brick–brick and joint interfaces, which exhibit a frictional behavior with limited tensile and compressive strength with softening. Finally, macroscopic curvature bending moment diagrams are obtained integrating along the thickness in-plane micro-stresses of each layer. Homogenized masonry flexural response is then implemented at a structural level in a FE non linear code based on a discretization with three-noded elements and elasto-damaging interfaces. Three different models of increasing accuracy are presented. The first (EPP) relies in assuming an elastic-perfectly plastic behavior for the interfaces. The incremental problem is solved at a structural level through a well known quadratic-programming approach. The second (ED) accounts in an approximate way for the softening behavior and consists in a preliminary homogenized limit analysis of the structure, which allows to identify the failure mechanism and in the subsequent FE non linear analysis of the whole structure assuming that all the non linearity is concentrated on the yield line defining the failure mechanism. The last (EPD) is a sequential quadratic programming approach. Here, deteriorating bending moment curvature curves obtained through homogenization are approximated through a linear piecewise constant discontinuous function. At each load step, all interfaces are assumed to behave as an elastic-perfectly plastic material and the discretized non linear problem is solved by means of the quadratic programming algorithm used for the EPP model. The two step model proposed is validated both a cell level and at a structural level comparing results provided with both experimental data and existing macroscopic numerical approaches available in the literature.