PSI- Issue 9

Ernesto Grande et al. / Procedia Structural Integrity 9 (2018) 257–264 Author name / Structural Integrity Procedia 00 (2018) 000–000

260 4

 

2 d s K s i e e

      2 i i e e s K s      

 

 

0







1

2 dx d s d s dx dx  2 2 2 i

       

(4)

e

0



2



where K 1 and K 2 are two constants equal to:

1

1

(5)

,

K

K





1

2

e

E t

E t

p p

c c

Starting from the system (4), aim of this paper is to numerically investigate the bond behavior of FRCMs externally applied on masonry substrates by particularly focusing the attention on the shear stresses transfer mechanism at the reinforcement/mortar interface level. For this aim, the explicit solution of the system (4) is derived by introducing different shear stress-slip laws characterizing the behavior of the reinforcement/mortar interface. 2.1. Approach 1: nonlinear behavior of the lower interface The first approach is based on the assumption of a linear-fragile behavior with a residual shear strength in the post peak stage only for the lower interface:

   

i   i

( ) ( ) i i i res s G s s s s     i i i

(6)

1

otherwise

where i res  is the residual value of the shear strength in the post-peak stage, and interface in the pre-peak stage. On the contrary, a linear elastic behavior is considered for the upper interface ( ) e e

i G is the shear stiffness of the lower

e e s G s   , where

e G is the shear

stiffness of the upper interface. In this case, after the attainment of the slip threshold value at the lower interface, the specimen is divided into two parts: part “1” where the upper mortar and the interfaces are both in the pre-peak stage and part “2” where the upper mortar and the upper interface are both in the pre-peak stage while the lower interface is de-bonded for a length a , representing an unknown of the problem. Thus, four differential equations govern the problem. The first two equations are derived by considering the equilibrium involving an infinitesimal portion of the strengthening system in the part “1”:

         

2 d s K s i

 

 

e

i s   

0



1

3 1

1

2

dx

(7)

0

x L a

  

  

2 d s d s dx dx  2 1 2 i

e

0

4 1   e K s

1

2

i

G G

where: . The other two equations are derived by considering the equilibrium involving an infinitesimal portion of the strengthening system in the part “2”: 3 1 4 2 , K K G K K G   , e e e  