Issue 38

S. Bennati et alii, Frattura ed Integrità Strutturale, 38 (2016) 377-391; DOI: 10.3221/IGF-ESIS.38.47

f Q s k w s , ( )   

 





b Q w s w s , , ( ) ( )  b P

f P w s ,

( *)

( *)

(9)



Fig. 6 shows a free-body diagram of an elementary segment of the strengthened beam included between the cross sections at s and s + ds . From static equilibrium, the following equations are deduced:

b Q dN s ds dV s ds dM s ds N s , , , b Q b Q f Q ,

( )

 

b s 

( )

f

( )

 

q

(10)

( )

f b b h s 1 ( ) ( ) 2 





b Q V s ,

 

b Q N s ,

( )

( )

where N b,Q is the axial force in the laminate due to the imposed load. By solving the differential problem defined by Eqs. (9) and (10) – as explained in the Appendix – the following expressions are obtained for the interfacial shear stress,   s q l s l s ( ) exp( )        (11) , V b,Q , and M b,Q respectively are the axial force, shear force, and bending moment in the beam; N f , Q

and internal forces in the beam,

l

1 2

 

  

 

  





f qb s s l   

s 



1 exp(    

b Q N s ,

( )

)



b Q V s q l s , ( ) (  

(12)

)

l

1 2

1 2

1 2

 

  

 

  





f b qb h s s l   

s 

q a s l a s ( )(2      )

1 exp(    

b Q M s ,

( )

)



where   and  are constant parameters defined by Eq. (A10) in the Appendix.

Figure 6 : Free-body diagram of an elementary beam segment.

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