Civil Engineering Reference
In-Depth Information
Tests made on reinforced concrete beams show that they will not fail by the widening
of the diagonal tension cracks until the stirrups going across the cracks have been stressed to
their yield stresses. For the derivation to follow, it is assumed that a diagonal tension crack
has developed and has run up into the compression zone but not all the way to the top, as
shown in Figure 8.6. It is further assumed that the stirrups crossing the crack have yielded.
The nominal shear strength of the stirrups
V
s
crossing the crack can be calculated from
the following expression, where
n
is the number of stirrups crossing the crack and
A
v
is the
cross-sectional area each stirrup has crossing the crack. (If a
stirrup is used,
A
v
2 times
the cross-sectional area of the stirrup bar. If it is a
,
A
v
4 times the cross-sectional
area of the stirrup bar.)
V
s
A
v
f
y
n
If it is conservatively assumed that the horizontal projection of the crack equals the
effective depth
d
of the section (thus a 45
crack), the number of stirrups crossing the
crack can be determined from the expression to follow, in which
s
is the center-to-center
spacing of the stirrups:
d
n
s
Then
A
v
f
y
d
V
s
(ACI Equation 11-15)
s
From this expression the required spacing of vertical stirrups is
A
v
f
y
d
V
s
s
and the value of
V
s
can be determined as follows:
V
u
V
c
V
s
V
u
V
c
V
s
Going through a similar derivation, the following expression can be determined for the
required area for inclined stirrups, in which
is the angle between the stirrups and the lon-
gitudinal axis of the member. Inclined stirrups should be placed so they form an angle of at
least 45
with respect to the longitudinal bars, and they must be securely tied in place.
A
v
f
y
(sin
cos
)
d
V
s
(ACI Equation 11-16)
s
Figure 8.6
