Issue 47

E. Grande et alii, Frattura ed Integrità Strutturale, 47 (2019) 321-333; DOI: 10.3221/IGF-ESIS.47.24 332 Then, considering the equations at the basis of the problem and the boundary conditions, both presented in the previous section, the analytical solution has been derived and presented in Fig. 8 in terms of applied load vs. the slip. In the same plot it is also reported the curve previously obtained by applying the approach 2 and then considering a linear-brittle behavior also for the lower interface. By examining the curves of the plot it emerges a similar value in terms of peak load and post-peak behavior. This outcome is due to the fact that the law assumed for the approach 2 and the one with softening are characterized by the same value of fracture energy:  f =0.54 N/mm. On the other hand, the greater initial stiffness and the smoother post-peak branch of the case with softening are strictly related to the shape of the shear stress- slip laws. In Fig. 9, it is reported for the case with b p =60 mm the distributions along the bond length of shear stresses at the interfaces and normal stresses at the upper mortar layer for the load step corresponding to the attainment of the slip s1 at the lower interface. The plots underline the length of the four zones (a 1 =10mm; a 2 =28.5mm; a 3 =54mm; a 4 =25mm) characterizing the behavior of the lower interface. Moreover, it also emerges that: the maximum value of normal stresses in the upper mortar layer remains equal to the tensile strength because of the assumption of a reduced bond strength for the upper mortar; after the zone a 1 , since shear stresses in the upper interface are equal to zero, also the normal stresses at the upper mortar layer are zero. a) b) c) Figure 9 : Results in terms of distribution along the bond length of a) slips, b) shear stresses, c) normal stresses at the step corresponding to the attainment of the slip value si=sf=1.2 mm at the lower interface: case with softening. C ONCLUSIVE REMARKS n this paper a one-dimensional simplified model for studying the bond behavior of FRCM strengthening systems externally applied to masonry structures is proposed. The model is based on the study of an infinitesimal portion of the strengthening system composed by the reinforcement and the mortar layers, computing the explicit solution of a system of equilibrium differential equations. Interfaces are introduced between the reinforcement and the upper and lower mortar layers to model the possible slip phenomena. A nonlinear shear-stress slip laws, characterized by a brittle fracture with a residual strength in the post-peak stage, is adopted for the interfaces to derive the two approaches presented in the first part of the paper. These approaches differ only for the behavior of the upper interface: in the first one the upper interface is characterized by a linear behavior, while, in the second one by the proposed nonlinear response. Both the approaches have been applied to two case studies available in the literature. From the results, approach 2 is able to better describe the experimental results, both in terms of peak load and post peak behavior, with respect to approach 1. In approach 2 the value of the peak shear stress in the upper interface is calibrated in order to take into account the damage occurring in the upper layer of mortar. The results are also compared with the ones obtained by the model proposed by [9,16]. From a computational point of view, the presented model results simpler than the one proposed in [9,16], as it doesn’t model directly the damage mechanism in the upper layer of mortar but it is able to take it into account by suitable setting the peak shear stress in the constitutive law of the upper interface. In the second part of the paper it has been presented the analytical solution in the case of softening behavior of the lower interface. In particular, a step function has been introduced in order to approximate the softening branch. Also in this case the approach has been applied to the same case studies of literature. From the obtained results it is emerged a response similar to the one obtained in the approach 2 due to the important role played by the fracture energy, parameter common to the two models. I