Civil Engineering Reference
In-Depth Information
Figure 8.6
Yield line in crossed by reinforcing bar.
Where
k
is the length of the chord AB, and the magnitude
v
and inclination
a
of the displacement rate are measured at the midpoint of the chord
(
Figure 8.5
).
The centre of rotation O must be outside the circle with diameter AB,
otherwise the hyperbola is replaced by straight yield lines OA and OB, one
with pure separation (
/2)
.
If O is at
infinity the yield line reduces to the straight line AB, with constant
v
and
a
=
p
/2) the other with pure crushing (
a
=-
p
.
Suppose a reinforcement stringer intersects a yield line at the angle
a
g
where 0
£g
=
p
and
g
=0 corresponds to the same direction as
a
=0 (Figure 8.6).
). The rate of internal work is
determined by the flow rule and the yield condition
(
Figure 8.3
)
:
W
s
=
vT
y
cos (
The strain rate
e
s
is then:
e
s
=(
v
/
D
) sin
g
cos (
g
-
a
g
-
a
)
for
g
-
a£p
/2
(8.2)
/2
If the yield line is intersected by a band of smeared reinforcement the
contribution to the rate of internal work per unit length of the yield line is:
D
s
=
bvrf
y
cos (
W
s
=0
for
g
-
a‡p
g
-
a
) sin
g
for
g
-
a£p
/2
(8.3)
D
s
=0
for
g
-
a‡p
/2
The factor sin
takes account of the fact that the reinforcement ratio
r
is
defined per unit area perpendicular to the direction of the reinforcing bars.
g
8.4
Shear strength of deep beams
Consider a rectangular beam of width
b
and depth
h,
subjected to a point
load
V
. The shear span
l
is defined as the distance between the point load
and the support reaction. The term
a
denotes the clearance between the
support and load platens, the lengths of which are
s
and
t
respectively
(
Figure 8.7
)
.
