Issue 51

A. Chiozzi et alii, Frattura ed Integrità Strutturale, 51 (2020) 9-23; DOI: 10.3221/IGF-ESIS.51.02 17 Figure 6: (a) τ b -slip constitutive law; (b) σ-τ s- τ t failure surfaces for masonry-FRP interfaces. In this case, flow rule (7) is specialized to:                                                        1 1 1 [ ] M F PL M F PL M F PL N M F m m m k s N k M F k t m m m k n N M F m m m A u u B u C u (15) Eqn. (15) represents further equality constraints to the LP problem, in which    M F m is the m -th plastic multiplier associated to the m -th linearizing plane. The Italian design code for FRP reinforcement suggest specific  -  s -  t failure surfaces for FRP-masonry interfaces, as depicted in Fig. 6(b), in which b f is the interface shear strength and mt f describes masonry tensile strength. For each point of each FRP-masonry interface  M F PL N unknown plastic multipliers are introduced. Therefore, the total number of unknown plastic multipliers for FRP-masonry interfaces is equal to    M F M F M F PL P I N N N . On each FRP-masonry interface i , associated to the surface i S , the internal dissipation rate is computed in the local reference system as:       int, j M F j S P dS σ u (16) Moreover, the non-negativity of each plastic multipliers must be enforced by means of the additional constraint:    0. m (17) Finally, we must impose a normality condition, requiring that the power dissipated by a unitary live load is equal to one, i.e.:   1 1 P (18) Therefore, the LP problem associated to the proposed upper-bound formulation reads: