Civil Engineering Reference
In-Depth Information
'
cc
f
-
= concrete strength in biaxial compression for
;
VV
/
1 and
W
0
xx
yy
xy
I
= first invariant of the stress tensor;
-
J
-
= second invariant of the stress deviator.
2
The parameters of criterion
a
and
b
, functions of
D
and
E
, are derived from
three tests: uniaxial compression
f
'
c
, uniaxial traction g
t
'
and biaxial compression
f'
cc
,
(
VV
and
/
x
W
).
1
0
xx
yy
7.3.3.2.
Cracked concrete
When the failure area in the traction domain has been reached, the behavior of
the material is considered as uniaxial in each orthotropic direction, given by the
directions parallel and perpendicular to the crack. The direction of the crack is
perpendicular to the direction of the traction maximum main stress (Figure 7.9); it is
traditionally determined owing to axis rotating matrices.
Y
y
x
)
V
V
X
)
Figure 7.9.
Axes linked to the crack
During the loading increment that causes cracking in a given material point,
passing from a biaxial elastoplastic model to an uncoupled orthotropic model
requires specific treatment, and several connecting rules are used to ensure
continuity between both models. The behavior of cracked concrete is then described
by a cyclic uniaxial law in all directions. The stress tensor within the local reference
axis is complemented by the shear stress and elastically calculated with a restricted
modulus of transverse elasticity to account for the meshing effect of the crack that
corresponds to friction at both surfaces of the crack lips. This effect is implicitly
taken into account in the law by arbitrarily decreasing the initial modulus of
elasticity in shear using a shear transfer factor
K
. Best values depend on the kind of
problem, but an initial value of factor
K
ranging from 0.2 to 0.4 seems capable of
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