Civil Engineering Reference
In-Depth Information
2.3.1.1 Transverse Torsional Steel
Assuming yielding of the steel, the symbols
n
t
in Equation (2.48) become
n
t
=
A
t
f
yt
/
s
, and
the symbol
T
becomes
T
n
=
T
u
/φ
. The torsional transverse steel can be directly designed
according to Equation (2.48):
A
t
s
=
T
u
(2.99)
φ
2
A
o
f
yt
cot
θ
60
◦
in order to control
cracking. It will be shown in Section 5.2.4 and Figure 5.7 (Chapter 5) that crack width increases
very rapidly when
is limited to a range of 30
◦
<θ<
In Equation (2.99) the angle
θ
45
◦
is recommended
for reinforced concrete because this angle represents the best crack control. For prestressed
concrete though, the ACI code uses an angle for crack control of 37.5
◦
. This value is based
on the past research of MacGregor and Ghoneim (1995). However, current research (Laskar,
2009) indicates that, even for prestressed concrete,
θ
moves away from this range. An angle of
θ
=
45
◦
provides optimal crack control.
This topic is discussed further in Section 8.3.2.4 and Figure 8.13 (Chapter 8).
The lever arm area
A
o
in Eq. (2.99) depends on the thickness of the shear flow zone
t
d
,
which, in turn, is a function of the applied torsional moment
T
n
. The larger the torsional
moment
T
n
, the larger the shear flow zone
t
d
and the smaller the lever arm area
A
o
. These
relationships can be derived theoretically from the warping compatibility condition of the wall
as shown in Section 7.1.2 (Chapter 7). For design practice, simplified expressions are given
for
t
d
and
A
o
by Equations (7.82) and (7.74), respectively:
θ
=
4
T
u
t
d
=
(2.100)
φ
f
c
A
cp
t
d
2
A
o
=
A
cp
−
p
cp
(2.101)
where
A
cp
is the area enclosed by the outside perimeter of concrete cross-section, and
p
cp
is
the outside perimeter of the concrete cross-section. Substituting
t
d
from Equation (2.100) into
Equation (2.101),
A
o
becomes:
2
T
u
p
cp
φ
A
o
=
A
cp
−
(2.102)
f
c
A
cp
For structural members commonly employed in buildings (Figure 2.16a), the ACI Code
suggests a simpler but less accurate expression:
A
o
=
0
.
85
A
oh
(2.103)
where
A
oh
is the area enclosed by the centerline of the outermost closed transverse torsional
reinforcement. Equation (2.103) may underestimate the torsional strength of lightly reinforced
small members by up to 40% and overestimate the torsional strength of heavily reinforced
large members by up to 20%.
The transverse torsional bars required by Equations (2.99) and (2.102) or (2.103) should be
in the form of hoops or closed stirrups. They should meet the maximum spacing requirement
of
s
≤
p
h
/
8 or 300 mm (12 in.).
