Issue 54

T. I. J. Brito et alii, Frattura ed Integrità Strutturale, 54 (2020) 1-20; DOI: 10.3221/IGF-ESIS.54.01

Note that the analysed cross section (hinge i ) has a load bearing capacity, therefore the ultimate moment ( M u ) is associated to a certain cracking level, which is quantified by the ultimate damage ( d u ). Therewith, the generalised Griffith criterion is expressed as:





2

2

d

ln 1

M U 



  

u

u

b

exp 1 

   

Y q     

M M d d

d

(31)



 

i

u

i

u

u

0

 2 1



2

2 M d  

d

1

u

i

u

where q and d u are the unknown variables. Since the load bearing point ( d u , M u ) is the local maximum point of the moment-damage curve, another equation may be defined as:



M

    0 0 2 1          1 ln 1 u u Y d q d  

  1 ln 1 d







i

exp 1 



d    

   

d

0

(32)





u

u

u



d

i M M i d d  

u

i

u

Then, with Eqns. (31) and (32) the variables q and d u can be calculated. Note that the parameter q is related to an additional cracking resistance due to the reinforcement. Analogously to the load bearing point, the plastic moment ( M p ) is associated to the plastic damage ( d p ). Thus, d p is calculated by the generalised Griffith criterion i.e.





d

2 p

ln 1

2 M U 



  

p



b

   

Y q     

exp 1 

M M d d

d

(33)

 





i

p

i

p

p

0

2

2 M d  

d

1

2 1

p

i

p

At this stage, the reinforcement starts yielding, therefore:

M

p

p

0         0 f k

M M

(34)

i

p

i

i

0



d

1

p

where k 0 is defined as the effective plastic moment. Finally, for the load bearing capacity the plastic evolution law is expressed by:



 

M

1

p

p

u

i 

u 

      

0    k

M M

f

C

0

(35)

i

p

i

p

u d 

u 

1

being C then defined.

E XAMPLES

Cantilever beam [43] lórez-López et al. [43] presents the test of a cantilever reinforced concrete beam subject to a lateral concentrated load as illustrated in Fig. 4. The beam properties are shown in Tab. 1 and the yield and rupture tensions of the steel bars are 412 MPa and 520 MPa, respectively. The numerical analyses carried out in this example vary the value of γ . The comparison among experimental and numerical responses is shown in Fig. 5. F

Beam

f c ' (MPa)

A s

A' s

h (cm)

b (cm)

1

26.0 20 Table 1: Geometric and material proprieties of the beam [43] 3 Ø 12.5mm 3 Ø 12.5mm 20

10