Civil Engineering Reference
In-Depth Information
In other words, the centerline of the shear flow is located at a distance 0
.
5
t
d
from the extreme
compression fiber. This assumption has three advantages:
1. By using
k
2
=
0.5, a constant, the tedious process of integration to calculate the coefficient
k
2
is avoided. The centerline of shear flow also coincides with the location of the average
strain
ε
d
, Figure 7.4(b).
2. The calculations of
A
o
and
p
o
are considerably simplified (see Section 7.1.3.4). This
simplification is slightly on the conservative side, because the constant 0.5 is somewhat
greater than the actual value of the coefficient
k
2
(about 0.4-0.45).
3. In calculating the torque
T
q
(2
A
o
), the slight conservatism in
A
o
is actually desirable to
counteract the unconservatism involved in the use of the expression
q
=
=
(
A
t
/
s
)
f
t
tan
α
r
.
In the expression for
q
, the ratio
A
t
/
s
has been nonconservatively used to describe the
transverse steel areas per unit length. (The nonconservative nature of
A
t
/
s
is caused by the
discrete spacing of the transverse steel bars (explained in detail in Section 4.4.3.1 of Hsu,
1984). Tests have shown (Hsu and Mo, 1985a) that this simplifying assumption of
k
2
=
0.5
results in good agreement between theory and tests.
7.1.3.4 Formulas for
A
o
and
p
o
Now that the centerline of the shear flow is assumed to coincide with the centerline of the
shear flow zone, the formulas for calculating
A
o
and
p
o
can be derived. The formulas must be
sufficiently accurate for a thick tube, because the thickness of the shear flow zone
t
d
is usually
quite large with respect to the overall dimensions of the cross-section when the softening of
concrete is taken into account. For a rectangular cross-section of height
a
and width
b
as
shown in Figure 7.6(a),
A
o
and
p
o
are:
1
2
p
c
t
d
+
t
d
t
d
A
o
=
(
a
−
t
d
)(
b
−
t
d
)
=
ab
−
(
a
+
b
)
t
d
+
=
A
c
−
(7.42)
p
o
=
p
c
−
4
t
d
(7.43)
t
d
t
d
2
t
d
2
t
d
d
a
b
(a) RECTANGULAR SECTION
(b) CIRCULAR SECTION
Figure 7.6
Calculation of A
0
and P
0
(dotted curves represent center lines of shear flow)
