Civil Engineering Reference
In-Depth Information
yielding), the over-prediction of torsional strength by Rausch's equation using
A
1
exceeds
30%. This large error has two causes. First, the thickness of the shear flow zone
t
d
may be very
large, of the order of one-quarter of the outer cross-sectional dimension, due to the softening
of concrete. Second, in contrast to the bending strength
M
n
which is linearly proportional to
the lever arm
jd
, the torsional strength
T
n
is proportional to the lever arm area
A
o
, which, in
turn, is proportional to the square of the lever arm
r
.Theleverarm
r
is shown in Figure 2.6(a)
or Figure 7.2(a).
In order to reduce the nonconservative effect of using
A
1
in Rausch's equation, Lampert and
Thurlimann (1968) have proposed that
A
o
be defined as the area within the polygon connecting
the centers of the corner longitudinal bars. This lever arm area is commonly denoted as
A
2
and
has appeared first in the CEB-FIP Model Code (1978). In terms of the bending analogy, this
definition is equivalent to assuming that the lever arm
jd
is defined as the distance between
the centroid of the tension bars and the centroid of the longitudinal compression bars. The
introduction of
A
2
has reduced the nonnconservative nature of Rausch's equation for high
steel percentages. However, the assumption of a constant lever arm area (not a function of the
thickness of shear flow zone) continues to be unsatisfactory.
Another way of modifying Rausch's equation has been suggested by Hsu (1968a, 1968b)
and adopted by the early version of 1971 ACI Building Code (ACI 318-71).
A
t
f
ty
s
T
n
=
T
c
+
α
t
A
1
)
(
(7.71)
where
α
t
=
:
0
.
66
+
0
.
33
y
1
/
x
1
≤
1
.
5;
x
1
=
shorter center-to-center dimension of a closed stirrup;
y
1
=
longer center-to-center dimension of a closed stirrup;
T
c
=
nominal
torsion
al strength contributed by concrete
8
x
2
y
f
c
(
psi
) where
x
and
y
are the shorter and longer sides, respectively, of a
rectangular section.
=
0
.
Two modifications of Rausch's equation are made in Equation (7.71) based on tests. First,
a smaller lever arm area (
α
t
varies from 1 to 1.5. Second, a
new term
T
c
is added. This term represents the vertical intercept of a straight line in the
T
n
−
α
t
/
2)
A
1
is specified, where
s
)(
A
1
) diagram (Figure 7.10).
These definitions of the lever arm areas
A
1
,
A
2
or (
(
A
t
f
ty
/
2)
A
1
all have a common weakness.
They are not related to the thickness of the shear flow zone or the applied torque. A logical
way to define
A
o
must start with the determination of the thickness
t
d
of shear flow zone.
α
t
/
7.2.3 Thickness t
d
of Shear Flow Zone for Design
The thickness of the shear flow zone
t
d
, given in Equation (7.55) or [19] is suitable for the
analysis of torsional strength. It is, however, not useful for the design of torsional members.
In design, the thickness
t
d
should be expressed in terms of the torsional strength
T
n
.This
approach will now be introduced.
