tesi A. Caggiano.pdf - EleA@UniSA - Università degli Studi di Salerno

3.4. One-dimensional bond-slip model for fibersFigure 3.11: Pull-out of a single fiber.⎧⎪⎨ −k E s[x]s[x] ≤ s e• Bond constitutive law: τ a [x] = −τ y,a + k S (s[x] − s e ) s e < s[x] ≤ s u ,⎪⎩0 s[x] > s uwhere, assuming a bilinear τ a − s law, k E and −k S represents the slope of theelastic and softening branches of the bond-slip relationship, respectively, whereasτ y,a is the maximum shear stress. Thus, s e = τ y,ak Ethe ultimate slips, respectively.and s u represent the elastic andAs schematically reported in Table 3.2 and based on the approaches proposed forstudying FRP laminates under pull-out by Yuan et al. [2004] for long anchorages andCaggiano et al. [2012c] for both short and long ones, different states of the bond responsecan be defined. The fiber-to-concrete interface is in elastic bond state (E) ifs[x] ≤ s e , in softening state (S) when s e < s[x] ≤ s u , or the bond is crushed if s[x] > s u .A combination of these three stress states can occur throughout the bonding lengthduring the pull-out process of the single fiber (see Table 3.2).Fully elastic behavior of fibers is assumed. This is strictly true in the case of syntheticfibers, while can be accepted for steel ones when the length l emb results in the conditionof P i ,max ≤ σ y,s A f , where σ y,s is the yielding stress and A f the area of the fibertransverse section.For the sake of simplicity, the description of the complete analytical pull-out model isreported in Chapter 4, which is completely dedicated to modeling the bond behaviorof fiber-to-concrete joints under pull-out actions.59