The paper deals with the behavior of restrained rocking blocks under seismic actions. Structural or non-structural masonry or r.c. elements, such as building façades or pre-cast panels subjected to out-of-plane modes, may be assimilated to rocking blocks restrained by horizontal springs. Horizontal restraints can represent flexible floors or steel anchorages or any anti-seismic device designed to impede overturning probability. Their effect could improve, in most cases, the dynamic response of blocks in terms of reduction of rotation amplitude. Nevertheless, this effectiveness could vanish or, surprisingly, affect the response in negative way, resulting in overturning when low values of stiffness or one-sided motion in particular conditions are assumed. Two cases of horizontal restraints are analyzed: (i) concentrated restraint as single spring and (ii) smeared restraint as spring bed with constant or linearly variable stiffness. The single stabilizing or destabilizing terms of the formulation are here analyzed and commented, providing practical evaluations to obtain enhancement of response in static and dynamic perspective. A numerical example of a masonry façade with non-linear boundary conditions has been provided highlighting how the choice of stiffness values affects the oscillatory motion and rebound effects. Finally, unit stiffness for masonry/concrete walls and retrofitting techniques, such as steel tie-rods, has been calculated

This paper provides a contribution to the rocking analysis of masonry walls by making a comparison with the kinematic analysis suggested by the Italian code. It is shown that the latter approach is generally over-conservative and therefore potentially inappropriate for historic buildings, where rehabilitation can be expensive and can affect their cultural value. The equation of motion given by the Housner formulation, corresponding to the movement of a rigid block, is here modified to account for different boundary conditions at different heights of the wall. These boundary conditions or horizontal restraints can represent vaults, transverse walls, or retrofitting devices such as steel tie-rods. A systemic analysis of walls having different dimensions and slenderness is performed, and the results from the Italian code and rocking analysis are compared. Finally, the improvement in the response offered by retrofitting devices is discussed in terms of reduction of amplitude ratio

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:The assessment of the rocking and overturning response of rigid blocks to earthquakes is a complex task, due to its high sensitivity to the input motion, variations in geometry and dissipation issues. This paper presents a literature review dealing with classical and advanced approaches on rocking motion with particular reference to masonry walls characterized by a monolithic behavior. Firstly, the pioneering work of Housner based on the concept of the inverted pendulum is discussed in terms of the most significant parameters, i.e., the size and slenderness of the blocks, the coefficient of restitution and ground motion properties. Free and restrained rocking blocks are considered. Then, static force-based approaches and performance-based techniques, mostly based on limit analysis theory, are presented to highlight the importance of investigating the evolution of the rocking mechanisms by means of pushover curves characterized by negative stiffness. From a dynamic perspective, a review of probabilistic approaches is also presented, evaluating the cumulative probability of exceedance of any response level by considering different earthquake time histories. Some recent simplified approaches based on the critical rocking response and the worst-case scenario are illustrated, as well.

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