Issue 22

S. Bennati et alii, Frattura ed Integrità Strutturale, 22 (2012) 39-55; DOI: 10.3221/IGF-ESIS.22.06 42 ( ) ( ) ( )     w s w s h s (1) In particular, we assume the following piecewise linear constitutive law (Fig. 3): 0 0 , 0 ( ) ( ), 0, ( elastic response) ( softening response) ( debonding)                         s u u u k w w w w k w w w w w w w (2) where k and s k are the interface elastic constants for the fields of elastic and softening response, respectively, and 0  w and  u w are the relative displacements corresponding, respectively, to the elastic limit and debonding of the interface. D IFFERENTIAL PROBLEM he behaviour of the FRP-strengthened beam varies according to the response field in which the interface is found. The adopted piecewise linear constitutive law enables distinguishing three cases, for which the differential problem is appropriately formulated and solved in the following. The resulting solutions will then be employed to reconstruct the overall response of the system in the different stages of behaviour. Case a) Interface with elastic response In the case of the interface response in the elastic field, from the equilibrium equations and the constitutive law we formulate the following set of governing differential equations (here and henceforth, primes indicate derivation with respect to the abscissa s ; the quantities in this case are indicated with the subscript e elastic  ): 2 ''''( ) ''( ) '( ) 0 ''( ) ( ) '( ) 0             f f e e e f f e e e f f f f kB h kB h v s v s w s EJ EJ kB kB h w s w s v s E A E A (3) By solving Eqs. (3), we obtain the general solution for the transverse displacement of the beam, 3 2 1 2 3 4 5 6 ( ) cosh sinh         e v s A s A s A s A s A s A (4) and for the axial displacement of the reinforcement, 2 1 2 3 4 3 5 ( ) ( sinh cosh ) 3 2 6          f f e f f f E A h EJ w s A s A s A hs A hs A A h E A h kB (5) where 2 2 (1 / / )    f f f kB E A h EJ . Thus, we can obtain the rotation of the beam cross section, 2 1 2 3 4 5 ( ) '( ) ( sinh cosh ) 3 2           e e s v s A s A s A s A s A (6) the relative displacement at the interface, 3 1 2 3 ( ) ( ) ( ) ( sinh cosh ) 6           f f e e e f f E A h EJ w s w s h s A s A s A kB h kB (7) the tangential stress at the interface, 3 1 2 3 ( ) ( ) ( sinh cosh ) 6          f f e e f f E A h EJ s k w s A s A s A B h B (8) the bending moment in the beam, T