Issue 47

E. Grande et alii, Frattura ed Integrità Strutturale, 47 (2019) 321-333; DOI: 10.3221/IGF-ESIS.47.24 323 characterized by the derivation of the explicit solution of a system of differential equations obtained by considering the equilibrium of an infinitesimal portion of the reinforcement and the mortar layers composing the strengthening systems. In order to model the slip between the reinforcement and the upper and lower mortar layers, two approaches are considered. The first approach (denoted in the following approach 1), considers a nonlinear behavior of the lower reinforcement/mortar interface only, by considering a shear stress-slip constitutive law characterized by a linear brittle behavior with a residual strength in the post-peak phase. On the other hand, the approach 2 assumes a nonlinear behavior for both the lower and the upper reinforcement/mortar interface, still considering a shear stress-slip constitutive law characterized by a linear fragile behavior with a residual strength in the post-peak phase. Moreover, in the latter approach, a calibration of the shear strength of the upper interface is proposed in order to implicitly account for the effect of the damage of the mortar on the contribution of this component of the strengthening system. In addition to these approaches, in the second part of the paper is presented the analytical solution in case of a shear stress-slip law characterized by a linear softening behavior in the post-peak phase. In particular, a step function approximating the law together with the procedure carried out from the approach 2 are used in order to derive the analytical solution. The proposed approaches are validated in the paper by considering experimental results derived from the literature. Moreover, the results are also compared with the ones obtained by the model recently proposed by [9,16], where, differently from the proposed approaches, the damage of the upper mortar was explicitly introduced in the model by assuming a nonlinear behavior in terms of normal stress-strain for the upper mortar layer. Although this assumption allows to account for the phenomena generally observed, it leads to a computational effort significantly greater than the one characterized the two approaches proposed in this paper. A CCOUNTED MODEL AND APPROACHES he model here considered for the study of the bond behavior of FRCM systems externally applied on masonry or concrete supports is based on the work in [9, 16]. Indeed, considering the scheme shown in Fig. 1, the main components characterizing the model are: a cohesive support, a lower mortar layer, a lower interface, the strengthening, an upper interface and an upper mortar layer. Introducing a reference axis x in the direction of the reinforcement system and fixing the origin in correspondence of the unloaded section, the equilibrium of forces characterizing an infinitesimal portion of the reinforcement and the upper mortar layer (see Fig. 1) leads to the following system of differential equations governing the problem of the bond behavior:       0 0 p e e i i p p p e e e e c p c p d b t s s b dx d b t s b dx                     (1) where p  and e c  are the normal stresses in the reinforcement and in the upper mortar, respectively; p t and e c t are the thicknesses of the reinforcement and the upper mortar, respectively; i  and e  are the shear stresses at lower and upper interfaces, respectively, both depending on the corresponding slips i s and e s ; p b is the width of the reinforcement. Introducing the following hypotheses: - the support and the lower mortar layer are assumed to be rigid; - the (lower and upper) mortar/reinforcement interfaces are modeled as zero-thickness elements with only shear deformability; - the upper mortar layer and the reinforcement are assumed deformable only axially. it is possible to write the displacements of both the reinforcement and the upper mortar layer (namely p u and e c u , respectively) as functions of the slip of the lower and upper interfaces: i p e i e c u s u s s    (2) T