Issue 58

R.N. da Cunha et alii, Frattura ed Integrità Strutturale, 58 (2021) 21-32; DOI: 10.3221/IGF-ESIS.58.02

being d i and d j the damage variables at hinges i and j , respectively, which account for concrete cracking. The terms F ij 0 are described in the Appendix for frame and arch elements.

Figure 3: Lumped damage modelling for reinforced concrete members [26-29].

In order to complete the constitutive law, the evolution laws of internal variables (damage and plastic rotation) must be considered. Thus, consider the complementary energy of the finite element:       T p W M Φ Φ (13)

1 2

b

b

The evolution law of the damage variables are obtained throughout the Griffith criterion i.e.                                                 2 0 11 2 2 0 22 2 0 0 2 1 0 0 2 1 i i i i i i i i i i j j j j j j j j j j d G Y d M F G W d G Y d d d d G Y d M F G W d G Y d d d

(14)

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 [28]. The cracking resistance function for reinforced concrete elements [29] is given by:

 ln 1 1

  d

  Y d Y q  

i

(15)

i

0



d

i

being Y 0 and q model parameters. 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 f f f f    j   j

 M C k d p i i

  f

   

0

i

0

1

0

i

(16)

M

j

p

  f

    C k

0

j

j

0



d

1

0

j

25