Issue 22

S. Bennati et alii, Frattura ed Integrità Strutturale, 22 (2012) 39-55 ; DOI: 10.3221/IGF-ESIS.22.06 47 Figure 6 : Stage 3) Elastic–Damaged–Deboned interface. The unknown lengths a and c can both be determined by requiring that the value of the relative displacement at the reinforcement strip’s extremity be that corresponding to incipient debonding of the interface and that the relative displacement in the transition section between the damaged and the elastic interface portions be equal to the value corresponding to the elastic limit, or in other terms, that 0 (0) ( ) and       u w w w c w (39) By substituting the expressions for the relative displacements into Eqs. (39), with the expressions for the integration constants listed in the Appendix, after some simplifications, we obtain the following equation set: 2 0 0 2 2 0 0 sin cos tanh ( )] ( ) tanh 0 tanh ( ) [ sin tanh ( ) cos ] tanh 0 [                                   c c l a c b l a c c l a c c b M M M M (40) By solving the first of Eqs. (40) for tanh ( )    l a c and substituting into the second equation, we get a quadratic equation for M : 2 2 2 2 2 0 0 0 0 2 2 1 (1 ) sin ] tanh (1 ) tanh 0 [ sin                 c M M M b M b c (41) Eq. (41) has two possible solutions, one of which is however to be discarded, as it leads to physically unacceptable values of length a . Consistently, we find 0 0 1 tanh sin      b c M M (42) and 1 arctanh( tan )        a l c c (43) Since, for physical reasons, length a can only increase, we may consider its derivative with respect to c . Thus, it can be seen that as long as a is increasing, during stage 3 of behaviour c must necessarily decrease from the value u c to 0 . At the same time, a grows from the value 0 a to l , that is, until complete detachment of the reinforcement. Actually, such situation is arrived at only asymptotically. In fact, from Eqs. (42) and (43) it can be seen that as c falls towards 0 , length a tends to l and the applied couple M increases boundlessly. Stage 4) Entirely debonded interface In stage 4 of behaviour, which is however reached only asymptotically for  M , the interface is entirely debonded and the reinforcement is completely detached from the beam except for the mid-span section where, for reasons of symmetry,