Page 105 - Handbook of Materials Failure Analysis

100 CHAPTER 5 Failure analysis of reinforced concrete structures

and strains, Cauchy’s tensor in the current configuration coincides with the second
Piola-Kirchhoff tensor of the last equilibrium configuration. Thus, the update pro-
cess for stress occurs by adding the extra stress of the current step to the last step
stress values, as follows:
x ¼ x a + Δx
(5.15)
y ¼ y a + Δy

σ xx ¼ σ xxa + Δσ xx
(5.16)
τ xy ¼ τ xya + Δτ xy
in which x a and y a are nodes positions along x and y directions of the last step, Δx and
Δy are displacements calculated on the current step, σ xxa and τ xya are normal and
shear stresses of the last step, and σ xx and τ xy extra stresses determined in the
current step.

4 SHEAR STRENGTH MODEL
There is still a lack of knowledge in the modeling of shear effects based on finite
beam elements applied to the representation of mechanical behavior of reinforced
concrete structures. Therefore, the improvements presented in this item compose
one contribution of this study.
Figure 5.3 illustrates a cracked reinforced concrete member and the shear trans-
fer mechanisms considered by the proposed model. The mechanical shear resis-
tance is composed by concrete and reinforcements contributions. The concrete
shear-resistant component, V C , is composed by V i and V a parts, which are related
to intact concrete and interlock aggregate phenomenon, respectively. The resistant
shear component due to longitudinal and transversal reinforcements is composed
by dowel action phenomenon, V d , and pure shear reinforcements parts, V sw ,
respectively.

V i

V sw
V a

V d

FIGURE 5.3
Cracked reinforced concrete member and shear transfer mechanisms.