Issue 54

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

Figure 3: Elastic circular arch element with two inelastic hinges.

Model reduction to straight elements A particular characteristic of the presented lumped damage model is that the circular arch element can degenerate to a straight element [37]. Therewith, if R b tends to infinity then χ b tends to zero, R b sin χ b tends to the length of the element L b and β b becomes its orientation. Then, the transformation and flexibility matrices are given as follows:

b 

b 

b 

b 

sin

cos

sin

cos

        

        





1

0

L

L

L L

b

b

b

b

b 

b 

b 

b 

sin

cos

sin

cos

  B

 



lim

0

1

b

L

L

L L



R

b

b

b

b

b

b 

sin 0 cos  

b 

sin 0 



cos

b

b

(24)

L

L

1

 

        

b

b



0

1 3 

d EI

EI

6

i



L

L

1





 

  

b

b

d d

lim , F

0



i

j

1 3 

EI

d EI

6



R

b

   

b

j

b L AE

0

0

Damage evolution law The complementary energy of the element is given by:        1 1 , T T p i j W d d      M Φ Φ M F

   M

(25)





b

b

b

2

2

b

b

Since the lumped damage variable accounts for the concrete cracking, its evolution law is given by the generalised Griffith criterion for each hinge i.e.

8