Issue 22

S. Bennati et alii, Frattura ed Integrità Strutturale, 22 (2012) 39-55 ; DOI: 10.3221/IGF-ESIS.22.06 46 0 0 0 [ cos coth ( ) sin ]tanh          M M c b c c b (32) Vice versa, by (numerically) solving Eq. (32), we can determine the length c as a function of M . It can thus be seen that c grows with M throughout the entire stage 2. The value of the applied couple, u M , at the end of this stage of behaviour is determined by imposing that the (maximum) relative displacement at the extremity of the FRP strip be equal to the value corresponding to interface debonding: (0)    u w w (33) By rendering Eq. (33) explicit and simplifying, we obtain 0 tanh ( ) tan       u u b c c (34) Eq. (34) can be solved numerically to determine the maximum value of c , indicated by u c , corresponding to the end of stage 2. To this purpose, noting that since the first member of Eq. (34) is always positive, the second must be likewise, hence the following condition must hold, 2     u c c c (35) where c c is a characteristic length. The system’s response is different in the three cases corresponding to a reinforcement length, 0 b , respectively, less than, equal to, or greater than c c . An analogous result has been found by Cottone and Giambanco [7] for the problem of a stretched strip. For our problem, however, considering current parameter values, it is likely that 0  c b c , so that we will focus on this case only in what follows. Furthermore, the hyperbolic tangent in Eq. (34) takes on values very near unity. Therefore, instead of solving Eq. (34) numerically, it is also possible to use the approximate expression 1 arctan     u c (36) By substituting the calculated value of u c into Eq. (32), we finally obtain the value of the couple, u M , applied at the end of stage 2. In particular, from the approximate value given by Eq. (36), we get 2 0 0 0 2 0 1 1          u u s w k M M M M k w (37) Stage 3) Elastic–Damaged–Debonded interface As the intensity of the applied couple M increases beyond the value u M , the FRP-strengthened beam enters stage 3 of behaviour. The interface portions near the reinforcement extremities become debonded and the reinforcement progressively detaches from the beam. In this stage, the beam can once again be divided into three parts: the total unreinforced portion (made up of the part initially lacking reinforcement and the part whose reinforcement has detached), of length a ( to be determined), the damaged interface portion, of length c ( also to be determined), and the still elastic interface portion, of length    d l a c ( Fig. 6). In these three distinct regions, the solution to the problem is given respectively by Eqs. (22)–(25), Eqs. (13)–(20) and Eqs. (4)–(11). Imposing the boundary conditions, ( ) 0, ( ) (0) (0), (0) (0), (0) (0), (0) (0), (0) 0 ( ) ( ), ( ) ( ), ( ) ( ), ( ) ( ), ( ) ( ), ( ) ( ) ( ) 0, ( ) 0, ( ) 0                           d d d s d s d s d s s s e s e s e s e s e s e e e e v a M a M v v M M T T N v c v c c c M c M c T c T c w c w c N c N c l a T l a w l a (38) yields the integration constants for stage 3, whose analytical expressions are given in the Appendix.