This paper addresses the seismic retrofitting of masonry churches with timber roofs by designing a ductile roof diaphragm with a new energy-based methodology. The proposed approach relies on nonlinear dynamic analyses conducted on an equivalent structural model. In this model, masonry nonlinearity is represented by rotational plastic hinges at the base of the equivalent wall elements. Roof system nonlinearity is modeled by shear plastic hinges simulating the energy dissipation of steel connections. In the equivalent model, the earthquake is implemented using a set of spectrum-compatible accelerograms. The dynamic response of the aforementioned plastic hinges is analyzed in terms of equivalent damping during the seismic events by extracting the relevant hysteresis cycles. This allows for the evaluation of both dissipated and strain energy. The estimation of the equivalent damping ratio provided by the roof diaphragm is based on multiple design configurations. After identifying the maximum achievable damping ratio, the study suggests ways to determine the corresponding roof stiffness, which defines the optimal retrofit configuration. This configuration is then implemented in a three-dimensional model that includes nonlinear properties for both masonry and connection elements, allowing a validation of the seismic response obtained from the initial equivalent model with a more complex and detailed model. Finally, a seismic response comparison is conducted between the optimized dissipated energy configuration, based on damping ratio evaluation, and an overstrength design variant determined considering the elastic behavior of the roof connections.
