Civil Engineering Reference
In-Depth Information
crack. Stage T4 ends at point TD (0,
f
cT4
) where the stress
f
cT4
=−
8
f
cT2
is taken
to be a function of the cracking stress,
f
cr
, and the maximum tensile stress,
f
cT2
1
.
5
f
cr
+
0
.
attained at
point TB.
As the load progresses in the compression region, the first compressive virgin loading, after
cracking, follows the envelope compression curves, stage C1 and stage C2, given by Equations
(6.100) and (6.101), respectively. As the load is reversed at point CB from the compressive
direction to the tensile direction, the response follows two straight lines, Stage C3 (from CB to
CC) and stage C4 (from CC to CD). The slope of stage C3 is 80% of
E
c
, the initial modulus of
elasticity of concrete, while the slope of stage C4 is 20% of
E
c
. A straight line is also assumed
in stage C5, when the strain continues to increase in the positive direction from point CD to
point TB.
If in stage C5 the direction of loading is reversed at point CE, the compressive reloading
response will follow stage C6 up to point CF. The slope of stage C6 was chosen to decrease
as the horizontal distance between points CC and point CE increases. The strain at point
CF is assumed to be
ε
cC6
=
ε
cCC
−
(
ε
cCC
−
ε
cC5
)
/
3, where
ε
cCC
is the compressive strain at
point CC and
ε
cC5
is the compressive strain at point CE. The strain at point CC is equal to
ε
cCC
=
ε
cC2
−
ε
cC2
, where
ε
cC2
=
f
cC2
/
ε
cC2
is the strain at point CB and
E
c
. Hence, the strain
ε
cC6
=
2
ε
cC2
−
ε
cC2
+
ε
cC5
/
at point CF becomes
3. It should be noted that the strain at
point CE (
ε
cC5
) should be taken as zero when the strain is in tension. The resulting strain would
be equal to
2
ε
cC2
−
ε
cC2
/
ε
cC6
=
3 which defines the strain at point CF'. As the compression
load continues to increase from CF, the response follows stage C7, passing through point CG
(0.85
f
cC2
,0
ε
cC2
) until reaching the envelope curve designated by C1 and C2.
Unloading for the second time from the envelope compression curve at point CB', the
response will follow the two straight lines defined by stage C3' and stage C4', respectively.
As the cyclic loading progresses from compression to tension, the response follows stage C5'
from point CD' in the compression region to the previous point TB in the tension region.
Further increase in the tensile load will cause the concrete response to follow the tensile
envelope curve (stage T2) given by Equation (6.99). As the load is reversed at point TB' from
the tensile direction to the compressive direction, the concrete response follows stage T3' and
stage T4' until point TD' is reached. As the load continues to progress in the compression
region, the response will now follow stage C6' and C7', passing by point CF' and CG',
respectively, until reaching the envelope compression curve designated by C1 and C2. After
that, the process of loading, unloading and reloading continues as described earlier.
The unloading and reloading curves are constructed by connecting consecutively a set of
points given in Figure 6.28. The linear expression between two points is given as:
.
98
c
c
i
σ
=
σ
+
E
cc
(¯
ε
−
¯
ε
i
)
(6.103)
i
and ¯
where
ε
i
are concrete stress and strain at the load reversal point 'i' or at the point where
the stages change;
E
cc
is the slope of the linear expression and is taken to be
σ
c
i
c
i
+
1
E
cc
=
σ
−
σ
(6.104)
ε
i
−
¯
ε
i
+
1
¯
c
i
+
where
σ
1
and ¯
ε
i
+
1
are the concrete stress and strain at the end of the stage under consideration.
