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

7.2. Bond-slip bridging of fibers on concrete cracks7.2 Bond-slip bridging of fibers on concrete cracksFracture opening processes in concrete activate bridging effects induced by fibers. Theaxial (tensile) stresses in fibers are balanced by bond developing on their lateral surfaceembedded in concrete matrix. Thus, a simple equilibrium equation can be writtendσ f [x]d x= − 4τ a[x]d f(7.16)where σ f is the axial tensile stress developed in fibers and considered in Eq. (7.2), τ a thelocal bond stress between fiber and matrix, and d f the fiber diameter. This approach isstrictly true in the case of considering synthetic fibers, while can be extended for steelones when the length l emb results in the condition that ∣ ∣ σf ,max∣ ∣ ≤ σy,s , where σ f ,maxand σ y,s represent the maximum axial stress and the steel yielding, respectively.A simplified bilinear bond-slip law is proposed to model the fiber-to-concrete debondingprocess as follows⎧⎪⎨ −k E s[x]s[x] ≤ s eτ a [x] = −τ y,a + k S (s[x] − s e ) s e < s[x] ≤ s u(7.17)⎪⎩0 s[x] > s uwhere s[x] defines the debonded displacement between the fiber and concrete (at thepoint of the abscissa x). The positive constants k E and k S represent the elastic andsoftening slopes of such bond-slip relationships, respectively; τ y,a is the shear bondstrength, while s e and s u are the elastic and ultimate slips, respectively.The complete derivation of this numerical model, and its validation against bond-slipexperimental tests, has been proposed in a previous work published by the authors,see Caggiano et al. [2012c], Caggiano and Martinelli [2012] and completely reported inChapter 4.7.3 Dowel action of fibers crossing the concrete cracksThe dowel mechanism, resulting in a shear transfer action across cracks, representsan important component on the overall interaction between steel fibers and concretematrix. A simple numerical sub-model for the dowel action is based on definingboth stiffness and strength of the generic fiber embedded in the concrete matrix andsubjected to a transverse force/displacement at the fracture level. On the one hand,147