Issue 22

S. Bennati et alii, Frattura ed Integrità Strutturale, 22 (2012) 39-55; DOI: 10.3221/IGF-ESIS.22.06 44 2 1 2 3 4 ( ) '( ) ( cos sin ) 2 (3 )          s s f f EJ N s EAw s B s B s E A h B s B h (20) In Eqs. (13)–(20), 1 2 6 , , ,  B B B are integration constants to be determined by imposing boundary conditions. Case c) Debonded interface (or lacking reinforcement) In the case that the interface is debonded (or the reinforcement is absent right from the start), the differential equation for the beam is simply (the quantities in this case are indicated by the subscript  d debonding ): ''''( ) 0  d EJv s (21) The transverse displacement of the beam has the following expression 3 2 1 2 3 4 ( )     d v s C s C s C s C (22) Thus, we can obtain the rotation of the beam cross section, 2 1 2 3 ( ) '( ) 3 2      d d s v s C s C s C (23) the bending moment in the beam, 1 2 ( ) ''( ) (6 2 )      d d M s EJv s EJ C s C (24) and lastly, the shear force in the beam, 1 ( ) '''( ) 6     d d T s EJv s C EJ (25) In Eqs. (22)–(25), 1 2 3 4 , , , C C C C are integration constants to be determined by imposing boundary conditions. S OLUTION FOR THE FRP- STRENGTHENED BEAM IN THE DIFFERENT STAGES OF BEHAVIOUR Stage 1) Entirely elastic interface or small values of the applied couple M , the system exhibits a linearly elastic response. In particular, the entire interface is within the initial stage of elastic behaviour. The strengthened beam can be divided into two parts: the portion lacking reinforcement, of length 0 a , and that with the reinforcement, of length 0 0   b l a ( Fig. 4). In these two distinct regions, the solution to the problem is given by Eqs. (22)–(25) and Eqs. (4)–(11), respectively. Now, imposing the boundary conditions, 0 0 0 0 0 ( ) 0, ( ) (0) (0), (0) (0), (0) (0), (0) (0), (0) 0 ( ) 0, ( ) 0, ( ) 0               d d d e d e d e d e e e e e v a M a M v v M M T T N b T b w b (26) yields the integration constants for stage 1, whose analytical expressions are given in the Appendix. Figure 4 : Stage 1) Entirely elastic interface. F