An energy-based methodology to estimate the ultimate condition of complex continuous masonry structures

Performance-based assessment procedures for existing structures based on pushover-like nonlinear static analyses require the definition of progressive damage states on the pushover curve, which may be associated with proper limit states for safety verification purposes. In Codes, the "ultimate condition" of the structure, i.e. a damaged state in which the structure is conventionally supposed to be no longer able to carry vertical and horizontal loads, is typically identified when the base shear shows a decay higher than a given threshold (usually 15-20%). In the case of masonry structures, this issue is not always straightforward depending on the complexity of the structure and the failure mode activated. The challenge especially arises in case of monumental buildings, such as complex continuous structures (e.g. towers) and buildings composed of many macro-elements (e.g. churches), dominated by flexural failure modes (e.g. overturning mechanisms). Indeed, in these cases, pushover curves may not show a significant softening for large horizontal displacements (e.g. masonry crushing is not activated). Accordingly, the application of the aforementioned approach results impossible. In this paper, an energy-based methodology to estimate the ultimate condition of masonry structures is introduced. Although the procedure is potentially applicable to any typology of masonry structure, it is tentatively applied here to continuous tower-like structures. A preliminary calibration phase to set the energy ratio thresholds is addressed by considering various simplified benchmarks subjected to several load scenarios. To this aim, the model is subdivided into several portions, and the ratio between the dissipated energy and the total energy is tracked in each portion of the structure along with the pushover analysis. The ultimate condition of the structure is then found as soon as such ratio reaches a pre-defined threshold in a portion, calibrated in such a way to highlight where in the structure the damage is mainly localized. The methodology is then tested on a real case study application, i.e., the bell tower of the San Francesco da Paola church in Rome, modelled through a suitable nonlinear continuum modelling approach. Results prove that the methodology is general, effective, and easy to use.