Page 108 - Handbook of Materials Failure Analysis

4 Shear Strength Model 103

3
V d ¼ E s I s λ Δ s V du (5.20)
The parameters involved in the evaluation of Equation 5.20 are given by:
p ffiffiffiffi
ffiffiffiffiffiffiffiffiffiffiffiffiffi
r
πφ 4 k c φ 127c f c
I s ¼ s , λ ¼ 4 s , k c ¼ p ffiffiffiffiffi (5.21)
64 4E s I s 3 φ 2
s
in which E s is the longitudinal elasticity modulus of steel, I s is the inertia moment of a
circular cross-section, φ s is the reinforcement diameter, λ is a parameter that com-
pares the stiffness of surrounded concrete with reinforcement stiffness, Δ s is the
dowel displacement, k c represents the stiffness coefficient of the surrounding con-
crete, f c is the concrete compressive strength and c is an experimental parameter that
reflects the spacing between reinforcements. Values between 0.6 and 1.0 may be
assumed, according to He and Kwan [39].
The dowel strength is limited by the reinforcement shear resistance, which is
determined as follows:
V du ¼ 1:27φ 2 p ffiffiffiffi p ffiffiffiffiffiffi (5.22)
f c σ sy
s
in which σ sy represents the yield stress of longitudinal reinforcements.
The dowel displacement of a finite element is determined by averaging the strain
values assessed for each integration point, as proposed by Enright and Frangopol
[27]. Then:

n ht
X n o
π 2
ε 1 cos αðÞsin αðÞ + γ cos αðÞ
xy
λ i
i¼1
Δ s ¼ (5.23)
n ht
in which α is the main tensile direction defined over the horizontal plane and n ht is the
number of integration points along the cross-section of a finite element.
4.3 SHEAR REINFORCEMENTS STRENGTH CONTRIBUTION
Traditional structural modeling based on one-dimensional finite beam elements does
not account the mechanical strength contribution of transversal reinforcements,
named as stirrups. In this regard, to improve the mechanical representation provided
by the numerical approach, it is necessary to introduce into the proposed nonlinear
FEM formulation the presence of transversal reinforcements.
For bended finite beam elements with high span-to-depth ratio, the bidimensional
stress state causes an increase on material damage, which is assessed based on shear
and normal stresses intensities. In such cases, the concrete lost faster its mechanical
stiffness. Therefore, the presence of shear reinforcements is required to guarantee the
resistant capacity of cross-sections, especially on shear solicitations.
According to Belarbi and Hsu [41], transversal reinforcements are subjected to
significant strains only after the beginning of diagonal concrete cracking. Before
of such cracking, concrete resistance is composed by intact concrete, over the
non-damaged region, and aggregate interlock mechanism, over the damaged regions.