Issue 53

M. C. Oliveira et alii, Frattura ed Integrità Strutturale, 53 (2020) 13-25; DOI: 10.3221/IGF-ESIS.53.02

   j i MM  M  T

(2)

Figure 1: Lumped Damage beam: (a) deformed shape, (b) finite element and (c) inelastic variables.

According to the deformation equivalence hypothesis [4], the matrix of generalised deformations can be expressed as:             p p d d e γ Φ γ ΦΦΦ      (3)

where {  e  is the matrix of elastic deformations, given by [4]:

EI L

EI L

1

1

    

    

    

    



  Φ e

LGA LGA LGA LGA 1 1

3

6

  M

  M





(4)

EI L

EI L



6

3

being E the Young’s modulus, G the shear modulus, I the inertia moment and A the cross section area; {  d  is the matrix that represents the deformation of the beam due to bending cracking in concrete by means of damage variables ( d i e d j ) in each hinge, given by [4]:

Ld

     

     

i

0





 1 3

d EI

  Φ d

  M

i



(5)

Ld

j

0





 1 3

d EI

j

{  d  is the matrix that represents the beam distortion caused by diagonal shear cracks through the damage variable d s , expressed as [4]:

d

d

     

     

s

s









  γ d





LGA

d

LGA

d

1

1

  M



(6)

s

s

d

d

s

s













LGA

d

LGA

d

1

1

s

s

15