Keywords: no-tension structures, reinforcement, topology optimization, limit analysis, optimal.

Policies and procedures are required for designing and the employment of additional material on the surface of walls with reference to the strengthening of masonry panels through the application of strips of fibre reinforced polymer (FRP). It is often observed that the strips are glued in a somewhat messy way, practically covering the wall in all directions. Striped glass, carbon or other materials can be used to reinforce the masonry and, apart from some economic and technological characteristics, their mechanical function is essentially independent of the type of fibre used. The aim is to give the masonry (as well as the reinforced concrete) a tensile strength that can be very useful for dramatically improving its performance, especially under seismic loading conditions.

It is generally accepted that the best model for analysing the behaviour of walls is the so-called no-tension material (NT), which behaves very well in compression, but is unable to resist tensile forces. Actually, the walls are designed to balance loads through the activation of purely compressive stresses. So the overall strength of the structure may improve with a little traction capacity of the material. The analysis of reinforced masonry structures follows a similar logic approach as for reinforced concrete: masonry absorbs the compressive stress, and the reinforcement resists tensile stresses, if present.

With this perspective, the no-tension material approach is very effective and can be implemented by applying a design procedure for the reinforcement. On the other hand, from the literature, one realises that the methods such as computational variable-topology shape in structural optimization can be adapted to find the optimal distribution of reinforcement on a given surface. Topological optimization has been flourishing over recent decades, and a wide range of tools for the management and solution of such problems was proposed, leading to a widespread use of the methodology in the industry.

Methods using a fixed reference domain for the material distribution into the computational variable-topology shape design of continuous structures seem to fit very well with the aim, and in conjunction with a finite element model of the structure, produce a partition wall surface where the problem becomes the setting of the reinforcement. So that the geometric representation of the structure becomes similar to a grey-scale rendering.

A first approach to the problem with reference to the application of reinforcement on no-tension panels was developed by the authors with reference to concrete beams reinforced by steel rods. With reference to the masonry, the goal is to find the "minimum" distribution of the material that complies with all the conditions of equilibrium and compatibility. It is good to note that the reinforcement usually has higher quality characteristics compared to those of the basic material (e.g. masonry walls reinforced by woven C-FRP), for which the conditions for resistance are largely verified and they should not be included in the optimisation process. Typically, the problem can be addressed by controlling the amount of reinforcement with the aim of minimum of fractures, or controlling the fractures with the aim at minimum of reinforcement. An intermediate approach is to define an "index of performance" of the reinforcement, which mixes both fractures and reinforcement in an objective function, thus reducing the number of constraints in the minimisation process.

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