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This paper investigates the out-of-plane behavior of masonry walls interacting with roofs. Often, collapses of masonry portions supporting roofs may occur due to the roof thrust, which generates a destabilizing effect over motion. Nevertheless, the roof weight can produce a positive stabilizing effect for rotation amplitudes smaller than the critical value. The dynamics of a rocking masonry block interacting with roofs is discussed, by properly modifying the Housner equation of motion of the free-standing single degree-of-freedom block. The dependence of the restoring moment on the rotation angle is investigated and the minimum horizontal stiffness is calculated so that the same ultimate displacement as the system without roof thrust is obtained. Two case studies are presented as applicative examples of the proposed method: an unreinforced masonry structure tested on shaking table and a spandrel beam subjected to roof thrust that survived the Emilia Romagna earthquake. Inertia moments and radius vectors of different failure mechanisms are also provided to solve the equation of motion for different block shapes. Finally, a parametric analysis of a trapezoidal rocking block has been carried out by changing its geometrical shape. This analysis shows that the influence of the shape is relevant for the calculation of the failure load, although is not possible to determine an a priori most critical shape. [ABSTRACT FROM AUTHOR]

This paper deals with the need of extending results of deterministic rocking analyses to stochastic analyses on restrained masonry façades in one-sided motion. The purpose is to define the level of improvement achieved with any anti-seismic device of a given stiffness and strength, in terms of reduction of probability of exceedance of a certain limit state. The most efficient intensity measures (IMs) are identified for three masonry façades of churches in free and restrained conditions. A reliability analysis is carried out by considering over 70 earthquakes, of which 50 recorded during the recent 2016–2017 Central Italy Earthquake. Four limit states are taken into account: rocking initiation, limited rocking, moderate rocking and near-collapse condition. The yielding limit state is considered for the analysis with anti-seismic devices. Univariate and bivariate fragility curves (FCs) are compared in free and restrained configurations, to discuss the reduction of probability of exceedance depending on 15 intensity measures. The results show that the best IMs are velocity-based parameters, in particular the Fajfar Index and Peak Ground Velocity, together with Peak Ground Acceleration. In one-sided motion without restraints, the higher the compression stiffness of the sidewalls, the more unstable the wall is in probabilistic terms. Practical curves show, for each IM, the reduction of probability of exceedance obtained thanks to assumed horizontal restraints. These help to understand, in a stochastic perspective, to what extent the anti-seismic device can be beneficial or detrimental (in case of amplifications of motion) for given earthquake intensities. The comparison of univariate and bivariate FCs confirms the superiority of bivariate FCs. Indeed, often the univariate curves sensitively underestimate the probability of exceedance, especially for low-medium intensity earthquakes, and are not able to offer any information regarding the influence of other IMs. [ABSTRACT FROM AUTHOR]