Civil Engineering Reference
In-Depth Information
x
=
0to
x
=
L
as follows:
L
1
L
ε
1
=
¯
¯
ε
s
=
ε
s
(
x
)d
x
¯
(6.59)
0
Before the first yielding of the steel, the linear relationship ¯
ε
s
(
x
)
=
f
s
(
x
)/
E
s
is valid at any
cross-section
x
. Substituting ¯
ε
s
(
x
) into Equation (6.59) gives
1
L
f
s
(
x
)d
x
L
1
E
s
ε
1
=
¯
ε
s
=
¯
(6.60)
0
The term in the parenthesis of Equation (6.60) is defined as the
smeared stress of steel f
s
,
i.e.
1
L
f
s
(
x
)d
x
L
f
s
=
(6.61)
0
Substituting Equation (6.61) into (6.60) gives a linear relationship between the smeared
stress of steel
f
s
and the smeared strain of steel ¯
ε
1
or ¯
ε
s
:
1
E
s
ε
1
=
¯
ε
s
=
¯
f
s
(6.62)
Using this concept of averaging, we can now average the steel stresses
f
s
(
x
) and the concrete
stresses
σ
c
(
x
) in Equation (6.58) by integrating these stresses from
x
=
0to
x
=
L
and divided
by the length
L
:
1
L
f
s
(
x
)d
x
1
L
σ
c
(
x
)d
x
L
L
1
ρ
f
so
=
+
(6.63)
0
0
The first term on the right-hand side of Equation (6.63) is obviously the smeared stress of
steel
f
s
, as defined in Equation (6.61). The quantity enclosed by the parenthesis in the second
term is the smeared tensile stress of concrete
c
1
σ
or
σ
c
:
1
L
σ
c
(
x
)d
x
L
c
1
σ
=
σ
c
=
(6.64)
0
Then Equation (6.63) becomes
1
ρ
σ
c
f
so
=
f
s
+
(6.65)
ε
s
from Equation (6.62) into (6.65), we derive the relationship between
the smeared tensile stress of concrete
Substituting
f
s
=
E
s
¯
1
(
σ
σ
c
) and the smeared tensile strain of concrete ¯
ε
1
(¯
ε
s
):
P
A
s
−
ε
1
P
A
c
−
ρ
1
σ
=
ρ
(
f
so
−
E
s
¯
ε
1
)
=
ρ
E
s
¯
=
E
s
¯
ε
1
(6.66)
For each load stage in a panel test, the load
P
was measured by load cells, and the smeared
tensile strain ¯
ε
s
) was measured by LVDTs over a length of 0.8 m (31.5 in.). A smeared
stress of concrete
ε
1
(¯
c
c
σ
1
(
σ
c
) can then be calculated by Equation (6.66). Plotting
σ
1
/
f
cr
against
¯
ε
1
(¯
ε
s
) consecutively for all load stages will give an experimental tensile stress-strain curve for
