Numerical Evaluation of the Equivalent Damping Ratio Due to Dissipative Roof Structure in the Retrofit of Historical Churches

This paper is focused on the numerical evaluation of the equivalent damping ratio (EDR) given by a dissipative wood-based roof diaphragm in the seismic retrofitting of single-nave historical churches. In the design phase, the EDR could be a key parameter to select the optimal roof structure configuration, thereby obtaining the maximum energy dissipation. In this way, the roof structure works as a damper to facilitate a box behavior of the structure during the seismic response. The EDR measures the energy dissipated by the nonlinear behavior of the roof’s steel connections and masonry walls during seismic events. In a preliminary retrofitting design phase, an initial implementation of the geometries of the walls and the chosen geometry for the roof is carried out by adopting an equivalent frame model (FEM) with inelastic rotational hinges for the nonlinear properties of the masonry walls and inelastic shear hinges for the nonlinear behavior of the roof’s steel connections. Since, for historical churches, the transversal response under seismic events is the worst situation for the single-nave configuration, the earthquake is applied as transversal accelerograms. In this way, the damped rocking of the perimeter walls due to the dissipative roof diaphragm can be described in terms of a hysteretic variable. By varying the value of the hysteretic variable, possible configurations of the roof diaphragm are tested in the design phase, considering the different shear deformation values of the inelastic hinges of the roof. Under these hypotheses, the EDR is evaluated by performing nonlinear Time History analyses based on the cyclic behavior of the inelastic hinges of the roof, the strain energy contribution due to the walls, and the lateral displacements of the structure. The EDR values obtained with the Time History method are compared with those obtained by applying the Capacity Spectrum Method by performing nonlinear static analyses, either for the coefficient method of FEMA 356 or the equivalent linearization technique of ATC-40. The EDR evaluations are performed by considering the following different hysteretic behaviors of the roof’s steel connections: the skeleton curves with stiffness degradation and the trilinear model with strength and stiffness degradation. Finally, the variation in the EDR values as a function of the hysteretic variable is presented as well so to evaluate if the maximum EDR value corresponds to the optimal value of the hysteretic variable able to reduce the lateral displacements and to contain the shear forces acting on the roof and the façade under a safety limit.