Civil Engineering Reference
In-Depth Information
is sufficiently long (
t
=
t
1
), otherwise the solution is either trivial
or
governed by the analysis below.
If the support platen is shorter than required (
s
<
s
l
) then the depth y of the
concrete compression is determined by Eqn (8.6). From
Figure 8.7
we find
the geometrical relation: cot
q
=
y
/
x
=(
a
+
s
)/(
h
-2
c
). Solving for y, inserting into
Eqns (8.6) and using Eqn (8.4) we find the lower bound solution:
(8.15)
By Eqns (8.11) and (8.15) the lower bound solution is given in terms of the
clearance
a.
In most design situations, however, it is the distance
l
between
load and reaction which is given. Exceptions are formed e.g. by cases of
indirect loading and built-in support.
When the capacity is governed by the reinforcement the load is found in
terms of the span
l
from Eqn (8.8):
(8.16)
With
y
o
=
h
F
/
v
£
h
/2. The relationship between
l
and
a
is (Figure 8.7):
l
=
a
+
s
/
2+
t
/2. Inserting
s
=
s
l
from Eqn (8.14) and
t
=
t
l
from Eqn (8.12), we find:
(8.17)
Eqn (8.16) is identical with Eqn (8.11) by virtue of Eqn (8.17).
The limiting size of the load platen
t
l
=
x
=
V
/
bvf
c
as a function of
l
is found
from Eqn (8.16):
t
l
=
x
=(2
h
-2
c
-
y
o
)
y
o
/2
l
Inserting into Eqn (8.13) we find the minimum size
s
l
of the load platen:
(8.19)
When the capacity is governed by the support length the load is found in
terms of
l
by solving Eqns (8.7) and (8.6) for
x
=
V/bvf
c
. Elimination of
y
yields the cubic equation:
x
3
-2
x
2
(2
l
+
s
)+
x
[(2
l
+
s
)
2
+4(
h
-
c
)(
h
-2
c
)]
-4(
h
-2
c
) [2
lc
+
s
(
h
-
c
)]=0
(8.20)
Eqn (8.20) has one real root, which may be expressed analytically, but the
result is not particularly illuminating, and Eqn (8.20) is most easily solved
by iteration.
