Issue 51

R. Landolfo et alii, Frattura ed Integrità Strutturale, 51 (2020) 517-533; DOI: 10.3221/IGF-ESIS.51.39 518 description of the masonry structure and its non-linear behaviour. A number of approaches for the prediction of settlement- induced damages were proposed in the literature that consider masonry either as an assembly of discrete blocks [1-9] or as a continuum medium [10-19]. The comparison between discrete and continuous models to study masonry behaviour is also a widespread topic [20-22]. Aiming at comparing the outcomes of the two modelling strategies, in this study, a discrete Rigid Block Limit Analysis (RBLA) and a continuous Finite Element Analysis (FEA) are used. The adopted RBLA model considers the structure as a collection of rigid blocks interacting through no tension contact surfaces, under the assumption of infinite compressive strength. This model is part of the concave contact formulation, where the interaction takes place in the contact points located at the block vertexes. Due to the assumption of infinite compressive strength, the failure condition is reached through joint opening and sliding among the blocks at the contact interfaces. The adopted non-linear three-dimensional FEA model represents the masonry wall as an elasto-perfectly plastic homogenized Love-Kirchhoff plate [23, 24]. The model, formulated in the framework of multi-surface plasticity, is implemented in a FE code for the path-following analysis, by means of a minimization algorithm directly derived from the Haar-Karman principle. The output of such a formulation is represented by capacity curves derived from specific load and constrain conditions. In a previous paper [23] the authors carried out a comparison for masonry walls subjected to lateral forces, with the purpose of simulating the seismic behavior. In the present paper, the comparison is extended to the case of settlements, using appropriate loading and boundary conditions. More precisely, a masonry façade arranged in a running bond pattern and subjected to uniform settlement is investigated. Since the aim of the study is more on the prediction of the behaviour of masonry up to failure, the simplest simulation of the ground settlement is considered, in which part of the support moves downwards up to the attainment of failure in the structure. This simulation was originally proposed in [25] and supported by simple experiments to provide an overview of the damage pattern in existing masonry buildings. Clearly, the failure pattern strongly depends on the position and size of the moving support. The influence of the following parameters is investigated: block size and shape, joint frictional properties, width of the settled area. The influence of the openings and of additional live loads on the structural behavior under settlement is also examined. The results from the two numerical models are compared in terms of failure pattern induced by the settlement and corresponding reduction of the ground support reaction that is requested to activate the motion. The paper is organized as follows: the rigid block model and the formulation governing the settlement-induced collapse prediction are presented in the following section. Then the formulation of the homogenized anisotropic constitutive model adopted for FEA is briefly outlined. Finally, a case-study is presented, consisting in the façade of an historic masonry building and the results obtained from RBLA and FEA are illustrated and compared. M ODELLING OF SETTLEMENT he ground movement induced settlement is numerically simulated, according to [25], by applying a vertical downward displacement to the ground support and detecting the displacement of the structure, considering a cohesionless frictional interface at the base. The imposed displacement induces a progressive reduction of the vertical reaction at the base of the masonry structure, up to a limit value corresponding to the self-weight of the portion of the wall involved in the settlement, according to the attained failure configuration. The RBLA returns the expected failure pattern together with the value of the base reaction in terms of load factor (or collapse multiplier), expressing the reaction which activates the failure mechanism. The path-following FEA provides the capacity curve in term of vertical base reaction s f versus the displacement imposed to the movable block and the strain pattern at each step of the analysis up to failure. T HE RIGID BLOCK MODEL FOR LIMIT ANALYSIS he first formulation adopted deals with a rigid block model for limit analysis of historic masonry structures modelled as assemblages of polyhedral elements. The formulation assumes a frictional behaviour along blocks interfaces and considers no-tension and infinite compressive strengths at contact interfaces. The model is formulated for 3D assemblages and is herein used to investigate the in-plane behaviour of wall panels. The interaction between blocks i and i +1 is simulated by means of a concave contact formulation, where the internal forces are associated to the contact points k located at the corners of the interfaces j (Fig. 1). The internal forces are collected in a T T