Civil Engineering Reference
In-Depth Information
The geometric shape of the compression stress block in Figure 7.4(c) will be identical to
that of the stress-strain curve of softened concrete (Figure 7.4d), if two assumptions are made.
First, the strain distribution in Figure 7.4(b) is assumed to be linear, and the strain
ε
at a
distance
x
from the neutral axis is related to
x
by similar triangles as follows:
t
d
ε
ds
ε
x
=
(7.29)
or
t
d
ε
ds
d
d
x
=
ε
(7.30)
Second, the strain gradient in the stress block is assumed to have no effect on the stress-strain
curve. Substituting d
x
from Equation (7.30) into (7.28) and changing the integration limit from
the distance
t
d
to the strain
ε
ds
,wehave
ε
ds
t
d
ε
ds
C
=
k
1
σ
p
t
d
=
σ
(
ε
)d
ε
(7.31)
o
and
ε
ds
1
σ
p
ε
ds
k
1
=
σ
ε
ε
(
)d
(7.32)
o
The coefficient
k
2
can be determined by taking the equilibrium of moments about the neutral
axis:
t
d
C
(1
−
k
2
)
t
d
=
σ
(
x
)
x
d
x
(7.33)
o
Substituting
x
from Equations (7.29) and d
x
from Equation (7.30) into (7.33) while changing
the integration limit, we obtain
ε
ds
t
d
ε
C
(1
−
k
2
)
t
d
=
σ
(
ε
)
ε
d
ε
(7.34)
2
ds
o
Substituting
C
from Equation (7.31) into (7.34) gives
ε
ds
o
σ
ε
ε
ε
1
ε
ds
(
)
d
k
2
=
1
−
ε
ds
o
(7.35)
σ
(
ε
)d
ε
The coefficients
k
1
and
k
2
can be calculated from Equations (7.32) and (7.35), respectively,
if the stress-strain curve
σ
−
ε
is given mathematically.
7.1.3.2 Coefficient
k
1
for Average Compression Stress
Coefficient
k
1
can be obtained from Equation (7.32) using the softened stress-strain curve
given in Figure 5.12 and expressed analytically by Equation (5.100) in the ascending branch
and by Equation (5.101) in the descending branch. The softened coefficient
in these two
equations is given in Figure 5.13 and is expressed by Equation (5.102). The two curves,
one ending in the ascending branch and one ending in the descending branch, are sketched
ζ
