Issue 22

S. Bennati et alii, Frattura ed Integrità Strutturale, 22 (2012) 39-55 ; DOI: 10.3221/IGF-ESIS.22.06 51 a) b) c) d) Figure 11 : Bending moment M ( z ): a) stage 1: 0 M M  ; b) stage 2: 0.9  u M M ; c) stage 3: 1.01  u M M ; d) stage 4: 1.01 u M M  . The shear force in the beam, T , is instead null everywhere. This follows from obvious reasons of equilibrium, considering moreover that by virtue of the hypotheses adopted, the FRP reinforcement strip is only able to bear axial stresses. Fig. 12 shows a plot of the axial force, N , on the reinforcement strip as a function of the abscissa, z , for three values of the applied couple, M , corresponding to the first three stages of behaviour described in the previous section. Note that with growing M and the consequent progressive detachment of the reinforcement, the length of the active part of the reinforcement is progressively reduced (falling to zero at the limit for  M ). Fig. 13 shows the length of the damaged region, c , as a function of the length of the region lacking reinforcement, a . During stage 2, c increases from 0 to u c , while the applied couple goes from 0 M to u M . During stage 3, the length c decreases from u c to 0 , while a increases from its initial value, 0 a , up to the half-length of the beam, l ( complete detachment of the reinforcement). Fig. 14 sums up the mechanical response of the FRP-strengthened beam as predicted by the proposed mechanical model. In particular, it shows a plot of the applied couple, M , as a function of the transverse displacement in the mid-span section,  . The different stages of behaviour are clearly distinguishable. In stage 1 ( 0 0   M M ), the strengthened beam exhibits a linearly elastic response, while in stage 2 ( 0   u M M M ), the system’s response takes on a weakly non-linear quality, due to the fact that a part of the interface has entered the field of softening response (though the non-linearity of such response is not clearly evidenced in the graph of the considered numerical example, it is however evident in the equations describing the model). In stage 3 (  u M M ), the response becomes strongly non-linear, due to the fact that increasing portions of the interface enter the debonding range, and the reinforcement accordingly detaches from the beam. In particular, at the beginning of stage 3, an abrupt increase in the mid-span transverse displacement can be observed to occur with almost constant applied load. Subsequently, the behaviour of the system tends asymptotically (  M ) to that of a beam lacking reinforcement, represented in the figure by the dashed blue line (stage 4).