Civil Engineering Reference
In-Depth Information
Table 7.1
k
1
as a function of
ζ
and
ε
ds
for softened concrete (
ε
0
=
0.002)
ε
ds
0.0005
0.001
0.0015
0.002
0.0025
0.003
0.0035
ζ
0.10
0.8654
0.9215
0.9218
0.8994
0.8610
0.8089
0.7439
0.20
0.7333
0.8611
0.8883
0.8806
0.8513
0.8048
0.7429
0.30
0.6018
0.7980
0.8526
0.8604
0.8409
0.8005
0.7419
0.40
0.4948
0.7333
0.8147
0.8385
0.8294
0.7956
0.7407
0.50
0.4167
0.6667
0.7747
0.8148
0.8167
0.7901
0.7394
0.60
0.3588
0.6019
0.7325
0.7891
0.8026
0.7840
0.7379
0.70
0.3146
0.5442
0.6889
0.7613
0.7870
0.7771
0.7362
0.80
0.2799
0.4948
0.6445
0.7314
0.7698
0.7693
0.7342
0.90
0.2521
0.4527
0.6018
0.6997
0.7506
0.7603
0.7319
1.00
0.2292
0.4167
0.5625
0.6667
0.7292
0.7500
0.7292
Integration and simplification of Equation (7.39) result in
1
1
1
p
p
ε
ε
1
3
ε
p
ε
o
−
ε
p
2
ε
ds
1
3
ε
ds
k
1
=
−
−
+
−
(7.40)
2
ε
o
−
ε
p
2
2
ε
ds
ε
p
ε
p
Expressing
k
1
in terms of
ζ
by letting
ε
p
=
ζε
o
gives, when
ε
ds
/ζ ε
o
>
1:
1
1
1
2
2
ζ
1
3
ζε
o
ζ
)
2
ε
ds
1
3
ε
ds
k
1
=
−
−
+
−
(7.41) or [13
b
]
)
2
ε
ds
ζε
o
ζε
o
(2
−
ζ
(2
−
ζ
Values of the coefficient
k
1
as expressed by Equations [13
a
] and [13
b
] are tabulated in
Table 7.1 as a function of
ε
ds
and
ζ
, while
ε
o
is taken as 0.002.
7.1.3.3 Location of Centerline of Shear Flow
Similar to the calculation of coefficient
k
1
, coefficient
k
2
for the location of resultant
C
can
be obtained from Equation (7.35) using the softened stress-strain curve of Figure 5.12 and
expressed analytically by Equations (5.100), (5.101) and (5.102). These calculations show that
k
2
varies generally in the range 0.40-0.45. The resultant
C
, therefore, should lie approximately
in the range 0
45
t
d
from the extreme compression fiber.
The location of the centerline of the shear flow is a more tricky problem to determine.
Considering the concrete struts, it should be theoretically located at the position of the resultant
C
, which is a distance 0
.
40
t
d
−
0
.
45
t
d
from the extreme compression fiber. However, because
the shear flow is constituted from the truss action of both the concrete and the steel, the
location of the steel bars (or the thickness of the concrete cover) should also have an effect
on the location of the shear flow. Fortunately, tests (Hsu and Mo, 1985b) have shown that this
effect of the steel bar location is small.
It could also be argued that the average strain
.
40
t
d
−
0
.
ε
d
be defined at the location of the resultant
C
,
rather than at the mid-depth of the thickness
t
d
. Such treatment, of course, will considerably
complicate the calculation without any convincing theoretical and experimental justifications.
A simple solution to this tricky problem is to assume that the centerline of shear flow lies at
the mid-depth of the thickness
t
d
, the same location where the average strain has been defined.
