Issue 53

M. C. Oliveira et alii, Frattura ed Integrità Strutturale, 53 (2020) 13-25; DOI: 10.3221/IGF-ESIS.53.02 15     T j i MM  M (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]:       M M Φ                         LGA LGA LGA LGA EI L EI L EI L EI L e 1 1 1 1 3 6 6 3 (4) 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]:         M Φ                j j i i d d EI Ld d EI Ld 1 3 0 0 1 3 (5) {  d  is the matrix that represents the beam distortion caused by diagonal shear cracks through the damage variable d s , expressed as [4]:             M γ                  s s s s s s s s d d LGA d d LGA d d LGA d d LGA d 1 1 1 1 (6)