Optimal FRP strengthening for transversally loaded masonry walls by means of a combined homogenization-topology optimization approach

A novel approach combining homogenization and topology optimization (TO) for the optimal placement of FRPs on masonry panels in two-way bending is presented. In the model, the heterogeneous assemblage of bricks and mortar is replaced by an equivalent homogeneous material exhibiting an orthotropic behaviour both in the elastic range and at failure. The macroscopic strength properties are defined by means of a simplified equilibrated limit analysis approach based on homogenization theory. The Representative Element of Volume (RVE) is subdivided along the thickness into several layers: each layer is discretized by means of a few Constant Stress Triangles (CST). Interfaces exhibiting a frictional behaviour with limited tensile strength represent either mortar joints or internal brick surfaces. The out-of-plane homogenized strength domain in the space of homogenized bending moments and torsion is thus obtained by means of the iterated solution of a linear programming problem involving a few optimization variables. Eventually, an approximation of the actual strength domain by means of two interpolating polynomial surfaces, to be used at a structural level, is obtained. At structural level, the amount of reinforcement required to fulfil the macroscopic strength criterion is minimized through a TO approach, with prescribed inequality constraints on the masonry internal actions. The structural element to be reinforced is discretized through Kirchhoff-Love plate finite elements. The stiffness matrix of each element consists of a contribution due to masonry, and a term related to the fiber-reinforcement. The optimal layout is iteratively sought using standard methods of mathematical programming. The capabilities of the proposed procedure are illustrated through applications on a series of windowed panels in two-way bending. The effectiveness of the numerically obtained optimal layouts is assessed through comparisons with traditional reinforcement layouts suggested by design codes.