A Fast Upper Bound FE Limit Analysis-Based Approach for the Design of Tie Rods in Historical Masonry Structures

The cultural heritage of many southern European countries is mainly constituted by historical masonry structures. Due to their intrinsic vulnerability to earthquakes and the high seismic hazard of such countries, retrofitting and strengthening interventions are useful to improve their seismic load carrying capacity. One of the most common and effective strengthening interventions is the utilization of tie rods. Such typology of retrofitting is allowed also for listed buildings, because of its reversibility. Through a fast upper bound FE limit analysis-based approach developed in the software MATLAB by the authors, it is possible to assess the seismic capacity and evaluate the best position for tie rods, to maximize their effectiveness. Any kind of historical masonry structure may be discretized with infinitely resistant hexahedron elements with quadrilateral interfaces, where all dissipation occurs. The structure can be subjected to different input loads, both in terms of horizontal force distribution (e.g. uniform or inverse linear in height) and seismic input angle. The collapse load is obtained in presence of scarcely resistant or no-tension material modelling for masonry. The procedure provides the failure mechanism, interfaces plastic multipliers and collapse multiplier directly from the primal problem, while the dual one gives information on the internal forces acting on interfaces. The possible presence of tie-rods is classically modelled using rigid-plastic truss elements. The main advantage of the approach proposed is that, with a small computational effort, it is possible to find the best configuration of the tie rods by a trial-and-error procedure, analyzing the response of the whole structure in a realistic way. A case study, namely a masonry church located in Italy, is considered to show the potential capabilities of the numerical ansatz proposed.