Issue 49

M. L. Puppio et alii, Frattura ed Integrità Strutturale, 49 (2019) 725-738; DOI: 10.3221/IGF-ESIS.49.65 729 inclination angle (for values greater than 30°). As for the geometry, the change in the front face slope does not affect so much the thickness required for the stability of the wall. By contrast, the change in the rear face slope influences the minimum value of safe thickness. If the rear face is inclined towards the backfill, sensitive improvements in stability are achieved and a wall can be thinner even with a small inclination. As recalled by Terzaghi and Coulomb and other Authors [16,17], cohesion play a relevant role in the stability, being that related to the shear failure. Indeed, the cohesion can be sufficient to guarantee the stability and can be found in in-situ decomposed materials and in a smaller amount in un-satured backfills, as a result of soil suction. This is usually the reason why some ancient masonry retaining walls stand with a relatively reduced width. However, cohesion could be strongly reduced by the soil saturation and therefore it is not recommended to rely on it if the backfill soil is liable to be saturated. The amount of friction at the back of a retaining wall depends on the downward displacement of the soil with the respect to the wall. Under ordinary circumstances, the friction angle existing between the wall and the retained soil (δ) varies between 2/3 Φ and Φ [18,19]. M ETHODOLOGY FOR THE ESTIMATION OF WALL MECHANICAL PARAMETERS FROM NUMERICAL MODELS methodology in the assumption of constitutive laws and load scenarios is necessary to assess the mechanical parameters that played a relevant role in the collapse of the walls. Since the inner core of the wall is composed by rubble masonry (Fig. 3), it is complicated to identify realistic values of mechanical properties so that a homogenization process is needed to simplify the modelling, mediating the available notions interpolated by reports, pictures and historical documentation. The material of the wall is then considered as split into two homogenized materials, one for the external walls and another one for the inner core (Fig. 3a). The adopted process is that of trying to understand the parameters that most influence the masonry behavior from the results obtained from a physical model. Non-linear static analysis is carried out by evaluating the flood effects on the wall stability. By evaluating the stress state found in relevant points of the structure and carrying out several parametric analyses, as a first starting point, the field of variable quantities are basically reduced to two main parameters: maximum tensile strength f t and maximum fracture energy G f , which are crucial in the behaviour of historic masonry modelling [6,20]. Both these quantities are clearly linked to the masonry tensile strength behavior. It is indeed usually the tensile stress to determine the breakage of a brittle material such as masonry, even more in the presence of a rubble masonry, characterized by low values of tensile strength. The first parameter ( f t ) characterizes the tensile yielding stress, to which a very reduced yield strain is associated, whereas the second one ( G f ,) describes the plastic softening behaviour. The fracture energy is a function of the area subtended by the softening envelope of the tensile constitutive law. A linear softening branch is considered, in which the ratio G f /h between fracture energy and fracture width ( h ) represents the area under the constitutive law curve [6], and it is therefore closely related to the ultimate strain of the material. Once this point is attained, the material can be considered as broken and it is no longer able to withstand any additional stress. Hence, it is evident that, when these two quantities change, the retaining wall strength to the imposed loads varies and, as it will be shown later, also the collapse mechanism will be modified. The main parameter considered is the ratio of the external wall tensile strength to the inner core tensile strength, initially taken equal to 2. This ratio is assumed by considering similar studies carried out on multi-leaves walls [21] and based on the report on the material drafted for the purposes of reconstruction [22,23], which highlights how the inner core presents few traces of mortar and binders. For this reason, the internal material behavior could be still more reduced, considering it as a permeable material and modelling it with the same mechanical features of a good gravelly soil. Hence, an analysis with a strength ratio equal to 4 is also performed. In the conclusion, the results of the analysis are presented and discussed, highlighting the contributions of the above-mentioned assumptions on the failure modes. N UMERICAL MODEL OF THE AS - BUILT WALL Assumptions on modelling techniques and smeared crack model mong the possible techniques available for modelling masonry, micro-modelling, simplified micro-modelling and macro-modelling, the latter is chosen for the case under examination. Indeed, the Volterra’s walls show a complex texture, the units are irregular and masonry is difficult to be mechanically characterized, as discussed in the previous section. The objective of the numerical analysis is to describe the effect of external loads (flood event) on joints and units A A