Issue 22

S. Bennati et alii, Frattura ed Integrità Strutturale, 22 (2012) 39-55 ; DOI: 10.3221/IGF-ESIS.22.06 48 the relative displacement at the interface and the corresponding tangential stress are always null. The system thus behaves as a beam lacking any reinforcement (Fig. 7). The solution to the problem in this case is given by Eqs. (22)–(25). By imposing the boundary conditions, ( ) 0, ( ) (0) 0, (0) 0        d d d d v l M l M T (44) we deduce the integration constants for this stage: 1 2 3 4 0, , , 0 2      M Ml C C C C EJ EJ (45) Figure 7 : Stage 4) Entirely debonded interface. N UMERICAL APPLICATION y way of example, let us consider a concrete beam of length 6 m  L with rectangular cross section of dimensions 30 cm  B and 40 cm  H . The width and thickness of the FRP reinforcement are respectively 25 cm  f B and 0.5 cm  f t . A segment of length 0 50 cm  a is initially devoid of any reinforcement. The other relevant lengths are / 2 300 cm   l L and 0 0 250 cm    b l a . The Young’s modulus of the beam and FRP strip are respectively 30000 MPa  E and 256000 MPa  f E , and the constants defining the interface constitutive law are 48 3 N/m  k , 0 4.2 MPa   , and 0.53 mm   u w [7]. From these values, we also deduce 0 0 / 0.0875 mm     w k and 0 0 / ( ) 9.49 2 N/mm       s u k w w . Using Eq. (28), we determine the value of the applied couple at the elastic limit, 0 144.73 kN m  M , corresponding to the end of stage 1. Eq. (35) furnishes the value of the characteristic length of the damaged region, 51.3 cm  c c . For the problem in question, the condition 0  c b c is thus clearly verified. Numerical solution of Eq. (34) gives the length of the damaged region at the end of stage 2, 37.6 cm  u c ( Fig. 8). From this latter value, through Eq. (32), we calculate the value of the applied couple at the end of stage 2, 356.21 kN m  u M . In practice, the same results could have been obtained using the approximate expressions given in Eqs. (29), (36), and (37). In order to show the trends of the computed quantities (displacements, internal forces and interfacial stresses) along the beam, we define a new abscissa, z , measuring the distance of the generic cross section from the left-hand support of the beam. The abscissa z , different from the abscissa s used to deduce the analytical solution, turns out to be more effective to present the obtained results. Fig. 9 shows a plot of the beam transverse displacement, v , as a function of z , for four values of the applied couple, M , corresponding to the four stages of behaviour described in the previous section. B