Issue 22

S. Bennati et alii, Frattura ed Integrità Strutturale, 22 (2012) 39-55; DOI: 10.3221/IGF-ESIS.22.06

2 [ ( cosh  EJ A s A s A s A sinh ) 6 2 ]        ''( ) EJv s  

M s

( )

(9)

e

e

1

2

3

4

the shear force in the beam,

2

e T s

'''( ) EJv s hB s  

 A EJ E A h

( )

( ) 6 (  

)

 

(10)

e

f e

f

f

3

and, lastly, the axial force in the reinforcement,

EJ

2

( ) N s EAw s '( ) 

( cosh A s A s E A h A s A sinh ) 2   (3   

)





(11)

e

e

f

f

1

2

3

4

h

1 2 6 , , ,  A A A are integration constants to be determined by imposing boundary conditions.

In Eqs. (4)–(11),

Case b) Interface with softening response In the case of the interface response in the softening field of behaviour, the equilibrium equations and the constitutive law enable us to formulate the following set of governing differential equations (the quantities in this case are indicated by the subscript  s softening ):

    

2

s f k B h

s f k B h

v s

v s

w s

''''( )

''( )

'( ) 0 





s

s

s

EJ

EJ

(12)

k B

s f k B h

k B

s f

   s f w

w s

w s

v s

''( )

( )

'( )

0







s

s

s

u

E A

E A

E A

f

f

f

f

f

f

Solving Eqs. (12) yields the general solution for the transverse displacement of the beam,

3     s v s B s B s B s B s B s B 2 1 2 3 4 5 ( ) cos sin    

(13)

6

and the axial displacement of the reinforcement,

E A h

EJ

f

f

2 B s B s B hs B hs B sin cos ) 3   2 6    

w s

B h w

( )

(







  

(14)

s

u

1

2

3

4

3

5

E A h

k B

f

f

s f

2  

2

(1 / f k B E A h EJ . Thus, we can now obtain the rotation of the beam cross section, / )  s f f

where

2    B s B s B s B s B sin cos ) 3  2  

s s v s

( )

'( )    ( 

 s

(15)

1

2

3

4

5

the relative displacement at the interface,

E A h

EJ

f

f

3

( )    w s w s h s ( ) ( )

( sin B s B s B cos ) 6     

w

 s





 

(16)

s

s

u

1

2

3

s f k B h

k B

s f

the tangential stress at the interface,

E A h

EJ

f

f

3

( ) s k w w s [

( sin B s B s B cos ) 6    

( )]     

 s



(17)

s

u

s

1

2

3

B h

B

f

f

the bending moment in the beam,

2 [ ( cos  EJv s EJ B s B s B s B ''( ) sin ) 6 2 ]        

M s

( )

(18)

s

s

1

2

3

4

the shear force in the beam,

2

T s

'''( ) EJv s hB s  

 B EJ E A h

( )

( ) 6 (  

)

 

(19)

s

s

f s

f

f

3

and, lastly, the axial force in the reinforcement,

43