Civil Engineering Reference
In-Depth Information
Figure 8.9
Stress distribution at support for
y
=2
c.
Hence with Eqns (8.4) and (8.5),
V
=
T
(
d
-
T
/2
b
), which expresses moment
equilibrium at the loaded section. The classical flexural failure load
V
=
V
F
is
found by putting
T
=
T
Y
and
s
s
=f
c
*
=
vf
c
, and introducing
F
=
T
Y
/
bhf
c
:
(8.8)
However, the flexural solution fails to account for the transfer of forces from
load to support, which requires a closer examination of the stress
distribution. It appears that the highest load is obtained with the maximum
compressive stress in the concrete (
=f
c
*
) whereas it is not always optimal to
have maximum force in the reinforcement (
T
=
T
Y
).
Inspection of
Figure 8.7
shows that if the parameters
l, h, c
and
t
are
given, one of the quantities
s
and
y
is necessary and sufficient to define the
stress distribution. Thus the lower bound is determined either by the strength
T
y
of the reinforcement or by the length
s
of the load platen. In the former
case we have:
s
whereas in the latter case
y
<
y
o
, which means that the reinforcement is not
yielding. The strut inclination
q
satisfies the geometrical relation:
cot
q
=
y/x
=(
a
+
x
)/(
h
-
y
)
(8.9)
which also expresses moment equilibrium of the strut. Solving for
x
and
using Eqns (8.4) and (8.5), we find the lower bound solution:
