PSI - Issue 11

Michela Monaco et al. / Procedia Structural Integrity 11 (2018) 388–393 Author name / Structural Integrity Procedia 00 (2018) 000–000

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et al., 2018). In this last case the contribution of in-plane shear resistance of the masonry walls is a determinant factor for the stability of the whole structure (Sacco et al., 2018). The most diffused procedure for the analysis of unreinforced masonry buildings are in fact based on a comparison between the displacement capacity of the structure and the displacement demand corresponding to a design earthquake, by means of a force-displacement curve (Calderoni et al., 2016). This essentially require suitable models able to describe the inelastic behaviour of the material in order to give information about the displacement capacity of the structure (Calderini et al., 2009). Different strategies have been proposed in literature to analyze this feature in case of out of plane mechanisms can be excluded (Gesualdo et al. 2017). A diffused approach, particularly suitable for the analysis of standard masonry buildings, in which regular arrangement of openings can be recognized, takes into account the structure as an ‘equivalent frame’, made up by a set of masonry panels. The behaviour of piers, i.e. the vertical panels resisting to both dead and seismic loads is distinguished by that of spandrels, placed above the openings and coupling piers in the case of seismic loads (Calderoni et al., 2011). The failure modes of masonry piers under in-plane loads (shear walls) depends on masonry quality (mechanical characteristics and workmanship), the boundary conditions, the applied loads and the aspect ratio. Different combinations of these features can lead to different failure modes (Betti et al., 2015). This approach is perhaps more suitable for engineering practice than the analysis of the structure by means of finite element procedure, for which suitable constitutive models (Monaco et al., 2014; Gesualdo and Monaco, 2015) and significant computational effort are needed (Gesualdo and Monaco, 2010). For this reason this paper is focused on the determination of stress and displacement state in a masonry panel subjected to in-plane loads, to model the behaviour of a masonry pier. The proposed approach is based on a minimum energy procedure, implemented in a Mathematica© code (Wolfram, 2003). 2. Description of the model A key prerequisite of a constitutive model is the capacity of describing the real behaviour of a structure with a consistent tool (Gesualdo et al., 2010). Sophisticated models are readily available in the literature, but their use in practice is limited, because in general complex identifications of the material parameters are needed (Guadagnuolo and Monaco, 2009). Moreover, as they are implemented in highly specialized numerical codes, they can only be applied by numerical analysis specialists. In what follows, the minimum energy principle is applied to the one dimensional model of Euler-Bernoulli beam to model the in-plane behaviour of a masonry panel. Reference is made to Figure 1.a, where a masonry panel loaded with a force R is represented. The force vector R is the resultant of a vertical triangular distribution with resultant Nn and an horizontal force T . It forms an angle ϕ with the vertical axis of the masonry panel. The vertical triangular load is distributed on a length A, while B and H are the panel dimensions. a b

Fig. 1. (a) Masonry panel with in-plane load; (b) Shape of the reacting structure.