Issue 38

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

Moving on to consider the elastic limit state of the laminate, we observe that the maximum axial force, N f max , occurs at the mid-span cross section of the beam. By recalling Eqs. (3), (10), and (12), we obtain

2

   

    

l

l





P f P Q f Q N N l , , ( )   

  

Q f P q b  

l 



1 exp(     

N

)

(19)

f

P

max



2

where  P

= 1.0 is the partial factor for pre-stressing actions [19, 21]. By putting N f max = f fd A f

, the corresponding imposed

load is obtained,

fd f f A P   P

1 1



q

(20)

f

2





b l

l





Q f

l 

1 exp(   

)



2

Lastly, we consider the elastic limit state of the beam. To this aim, the normal stresses at the upper and lower surfaces of the mid-span cross section of the beam,  2   and   2  , respectively, are computed. By recalling Eqs. (2), (3), and (12), we find

G b G P b P N l N l , , ( ) ( )   



G b G P b P M l M l , , ( ) ( )   







N l

M l

( )

( )

Q b Q ,

Q b Q ,











2

A

W

b

b

(21)

2

  

  







g

g

h

l

L

1 2

1

1

1 1

1 2

  

 

  

  

2

G

G

b

1 1

2 2

 

 

Q f P q b l   

 

l     )





L

q

exp(

P

Q





W

A W 2

W

2

b

b

b

b

where  G

(equal to  G 1

, and  G 1

= 1.3 for the self-weight, g 1

= 1.5 for the other dead loads, g 2

) represents the partial factor

for permanent actions. Hence, by putting  2  = f yd , respectively, and taking the minimum between the two resulting values of the imposed load, we obtain the imposed load corresponding to the elastic limit state of the beam,  =  f yd and  2

     

  

  







g

g

h

1 2

1

1

2

G

G

b

1 1

2 2

L P  





f

yd

P

W

A W 2

1 min

b

b

b



q

;

b

2



 

   

h

L  

l

1 2

1

1

1 1 exp(

  

Q

b



l 



 



f b l

)





b W A W g g 2 2 2 1     b G G 1 1

2





b

(22)

     

  

  

h

1

1

2

b

L P  





f

yd

P

W

A W 2

2

b

b

b

2

  

   

h

L

l

1 2

1

1

1 1 exp(

  

b



l 





 



b l

)

f





W A W 2

2





b

b

b

Tab. 4 summarises the results obtained by applying Eqs. (18), (20), and (22). The weakest element turns out to be always the steel beam, whose yielding stress determines the load corresponding to the elastic limit state of the system. Evaluation of increased bearing capacity To evaluate the increase in the load bearing capacity given by the strengthening system, we first calculate the load bearing capacity of the unstrengthened steel beam. The imposed load, q u , corresponding to the complete plasticisation of the mid span cross section of the beam is calculated as follows:

1 2

1 2 

  

2





q L M q q )    

 







p M g 1 1  G



(23)

g

g

g

(

 

G

G

Q

p

u

G

1 1

2 2

2 2

2



L

Q

= f yd Z b is the plastic moment of the steel cross section.

where M p

386