Civil Engineering Reference
In-Depth Information
The compressive stress-strain curve of concrete in a 2-D element subjected to shear exhibits
three characteristics, as shown in Figure 6.7. First, the peak point is reduced or 'softened' in
both stress and strain. The locus of the peak point follows a straight line from the nonsoftened
peak to the origin. Second, the pre-peak ascending curve is found to be parabolic. Third, the
post-peak descending curve is also a parabolic curve, but the gently sloping curve intersects
the horizontal axis at a large strain of 4
ε
o
.
The
ascending branch
of the softened stress-strain curve can be expressed as:
2
¯
2
¯
ε
2
ζε
o
ε
2
ζε
o
c
2
f
c
σ
=
ζ
−
ε
2
/ζ ε
o
≤
¯
1
(6.42)
is the softened coefficient. Notice that peak concrete stress
f
c
is multiplied by
where
ζ
ζ
to achieve the effect of stress softening, and the strain
to
achieve the effect of strain softening. Proportional softening is accomplished by using the same
ζ
ε
o
at peak point is multiplied by
ζ
for both stress and strain softening. For a family of such curves with decreasing softening
coefficient
, the locus of the peak points traces a straight line passing through the origin.
The
descending branch
of the softened stress-strain curve is derived as follows. The de-
scending portion of the solid curve in Figure 6.7 is assumed to be a parabolic curve from the
peak point to the point of 4
ζ
ε
o
on the horizontal axis. The vertical distance from the parabolic
curve to the peak stress level is designated as
. This vertical distance
is located at a
ε
p
from the
peak point, however, the vertical distance from the parabolic curve to the peak stress level is
σ
p
. The ratio of
ε
2
-
ε
p
from the peak point. At a horizontal distance of 4
ε
o
-
horizontal distance ¯
/σ
p
can be obtained from the geometry of parabolic shape:
¯
2
σ
p
=
ε
2
−
ε
p
4
(6.43)
ε
o
−
ε
p
2
at the location of ¯
Then, the stress
σ
ε
2
is
1
2
¯
ε
2
−
ε
p
4
2
σ
=
σ
p
−
=
σ
p
−
(6.44)
ε
o
−
ε
p
or more convenient for calculation:
1
2
¯
ε
2
/ζ ε
o
−
1
c
2
f
c
σ
=
ζ
−
ε
2
/ζ ε
o
≥
¯
1
(6.45)
4
/ζ
−
1
c
2
f
c
, based
The compressive stress
σ
calculated by Equation (6.45) should be limited to 0
.
5
ζ
on test results.
6.1.7 Softening Coefficient
ζ
The softening coefficient
in Equations (6.42) and (6.45) is the most important parameter
affecting the compressive stress-strain relationship of cracked reinforced concrete.
ζ
ζ
is a
function of three variables: the uniaxial tensile strain ¯
ε
1
in the perpendicular direction, the
