Civil Engineering Reference
In-Depth Information
Figure 8.18
Variation of normalized ultimate shear capacities of beams with
a/d
where
V
n
,
b
is the balanced shear force and
C
b
is the constant corresponding to the balanced
condition. The balanced condition occurs when the amount of shear reinforcement in a beam
is such that the yielding of the transverse steel occurs simultaneously with the web crushing
of the concrete. When
V
n
<
V
n
,
b
, the beam is defined as under-reinforced in shear, where the
transverse steel yields before the crushing of concrete. When
V
n
>
V
n
,
b
, the beam is defined
as over-reinforced in shear, where the concrete crushes without the yielding of steel.
The balanced constant
C
b
can be calibrated by comparing the over-reinforc
ed
beams ver-
sus the under-reinforced beams, as shown in Figure 8.18, which plots
V
n
/
f
c
b
w
d
against
a/d
. Over-reinforced beams have been tested by three groups of researchers (Bennett and
Balasooriya, 1971; Rangan, 1991; Ma
et al
., 2000). Under-reinforced beams have been
te
sted
by all the other researchers. It can be seen that all the over-reinforced beams had a
V
n
/
f
c
b
w
d
value abov
e 1
.5 for
f
c
in MPa (or 18 for
f
c
in psi) and almost all the under-reinforced beams
have
V
n
/
f
c
b
w
d
value below 1.5 for
f
c
in MPa (or 18 for
f
c
in psi). Therefore, the balanced
1.5 for
f
c
in MPa.
In order to provide some ductility in shear failure, the constant
C
1
for maximum shear
strength must be taken as less than
C
b
.
C
1
constant can be taken as
C
b
=
C
b
is also desirable because Rangan's over-
reinforced beams have large web stiffeners under the loads to prevent local compression
failures. In view of the fact that prestressed beams used in practice do not contain web
stiffeners, it was decided to choose a conservative
C
1
value of 1.33 for
f
c
in MPa (or 16 for
f
c
in psi) and
V
n,max
can thus be expressed as
<
33
f
c
(MPa)
b
w
d
or 16
f
c
(psi)
b
w
d
V
n
,
max
=
1
.
(8.24)
Equation (8.24) is the equation for UH maximum shear strength, and is also plotted in Figure
8.17. Comparison of the UH formula with available test data shows that Equation (8.24) is
the most reasonable for concrete strengths up to 100 MPa (14 500 psi). Obviously, more
