Issue 54

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

 

d  

 

  

G Y d

0

2

2

 2 1 M U W G d M d M U W G d M         2 2 2 2 2 i b i i i j



i

i

i

  

 

i



G Y d 

d   

0

i    

i

i

(26)

  j

 

d

G Y d

0

 

j

j

b

  

  j





j

2

G Y d 

d   

0



d

2 1



j

j

j

j

j

where G i and G j are the damage driving moments for the hinges i and j , respectively, and Y ( d i ) and Y ( d j ) are the cracking resistance functions. The cracking resistance function for a hinge i was obtained by means of experiments on reinforced concrete beams [23]:

 ln 1 1





d

  Y d Y q  

i

(27)

i

0



d

i

being Y 0 and q model parameters. In order to generalise the previous equation, Alva and El Debs [27] proposed the following relation:





d

ln 1

  Y d Y q



  

i

exp 1 

    

d

(28)



 

i

i

0

d

1

i

where γ is an empiric parameter introduced as an attempt to ensure a better fitting between numerical and experimental responses for service loads. Plastic evolution law The plastic evolution laws for each hinge of the element are given by:

0                     0 0 0 0 0 p i  i p i i  p j j p j  f f f f    j

M f

p

i

 

0 i  C k   

0

i



d

1

0

i

(29)

M

j

p

0 j  C k   

f  

0

j



d

1

0

j

being f i and f j the yield functions for the hinges i and j , respectively, and C and k 0 model parameters.

M ODEL ASSOCIATION WITH REINFORCED CONCRETE THEORY

A

n evident advantage of the lumped damage models for reinforced concrete structures is that the model parameters can be easily associated to the classic reinforced concrete theory. An engineer in practice knows how to calculate four key quantities for any reinforced concrete element: first cracking moment ( M r ), ultimate moment ( M u ), plastic moment ( M p ) and ultimate plastic rotation (  u p ) [43]. Considering an inelastic hinge i , cracks start to nucleate when the first cracking moment is reached i.e.

2 2 M U        2 0 r b r i M 2

M M d

Y

(30)

i

0

i

Then, the parameter Y 0 is defined as the first cracking resistance.

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