Issue 49

M. L. Puppio et alii, Frattura ed Integrità Strutturale, 49 (2019) 725-738; DOI: 10.3221/IGF-ESIS.49.65 730 in a smeared sense. This leads to a composite model with anisotropic properties. This method is adequate when the dimensions of the finite element are large enough in comparison to the unit/block dimensions [24]. The Multi-Directional Fixed Crack (MDFC) model with damage and plasticity constitutive laws provided by the software DIANA FEA [25] are used. With this model, the cracking process is not represented by an actual detachment between elements of the same mesh, but it is obtained by smearing the damage on the finite element interested by the strain concentration, by degradation of mechanical properties of the elements involved. The MDFC model is based on the decomposition of the total strain into an elastic strain and a crack strain (Litton 1975), a criterion for crack initiation depending on a tension cut-off and a limit threshold angle between two consecutive cracks. The conventional cracking parameters are the reduction factor of the Young’s modulus µ and the reduction factor on the shear modulus β [26]. It is clear that such a modelling assumes for masonry an intermediate trend between the brittle and the plastic behaviour. This softening features has been defined by Lourenço [27] as a parabolic trend, but can be numerically described in several ways, the simplest of which is clearly that of a linear tension softening. In the case of the Volterra’s walls, after a preliminary analysis with the linear softening branch, the Moelands and Reinhard model [28] is used (exponential trend), because it is better suited to the plastic and damage model proposed in [29]. As for the tensile behaviour, the most relevant in brittle material such as masonry, the post-peak response can be simulated by two parameters: the cracking energy G f and the fracture width h . The stresses are determined as a function of the strain in a coordinate system oriented in the cracking direction; the crack orientation was assumed to be fixed whereas for the shear behaviour it was taken into account considering it as damage based. A crucial aspect considered in the modelling is the soil-structure interaction. As shown in (Fig. 3 - b), the walls lay on densified fine sand with calcarenite layers, whereas the backfill derives from anthropic modifications and it is therefore highly non-homogeneous. Thus, for it, laboratory tests are carried out obtaining the soil volume density (γ), its friction angle ( Φ ), the transversal contraction coefficient (ν), the cohesion (c) and the edometric module (M0), where the test has inhibited transversal dilation. For modelling purposes, the normal elastic modulus (E), with free transversal dilatation, is necessary. This can be obtained directly from the edometric modulus from the expression [30]: (1 )(1 2 ) 1        Ed E E (1) Soil is modelled with a surface interface that is managed by setting specific material parameters related to Mohr-Coulomb law with zero dilatancy. Soil Parameters Base Soil Backfill Soil E [kN/m 2 ] 23400 16240 υ 0.35 0.35 Density [T/m 3 ] 2 1.9 Porosity 0.4 0.4 Cohesion [kN/m 2 ] 8 8 Friction Angle φ [deg] 12 12 Initial Stress K 0 0.398 0.412 Table 1 : Soil mechanical parameters. [31] Adopted mechanical parameters As for the linear material properties a Young’s modulus of E = 1.2E+06 [kN/m 2 ], a Poisson’s Ratio of ν= 0.2 and a mass density of γ= 2.1 [T/m 3 ] are selected. For the masonry post-peak behaviour, it is basically impossible to define actual values of fracture energy, but some indications (although related to brick masonry) are given in [6]. In the parametric analyses, the ratio between the external walls strength and that of the inner core was kept constant and primarily equal to 0.5, secondly to 0.25. The following initial values are considered: tensile strength, f t = 0.35 MPa, tensile fracture energy, G f = 0.02; compressive strength, f c = 1.95 MPa. Different values of fracture energy are assumed, from 0.00075 kN/m to 0.03 kN/m;