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Answer:
m
m
2
α
+
α
n
2
2
K
L
2
C
π
(
n
y
)
cr
=
,
C
=
n
y
α
m
2
,
α
=
L
y
/
L
n
x
1
+
n, m
=
number of half waves in the
x
and
y
directions
13.19 Determine the critical in-plane load
(
n
y
)
cr
in a simply supported plate for which
L
y
is much less than
L
. Also,
n
x
=
0
.
Hint:
In the answer of Problem 13.18, set
n
x
=
0 and
α
1. For
n
=
1
,
the minimum
value of
C
[and
(
n
y
)
cr
] occurs for
m
=
1
Answer:
L
2
1
2
2
K
π
(
n
y
)
cr
=
α
+
α
Circular Plates
13.20 Find the deflect
ion
of a circular plate with no center hole and loaded with a concen-
trated ring load
P
. This is a line load (force/length) extending symmetrically around
the plate at radius
r
=
.
=
a
1
The outer rim at
r
b
is simply supported.
Answer:
The initial parameters are
a
1
1
1
b
2
a
1
Pa
1
4
K
b
a
1
−
−
ν
w
0
=−
ln
+
ν)
+
−
2
(
1
1
a
1
b
2
Pa
1
2
b
a
1
+
1
+
ν
2
M
0
=
(
1
+
ν)
ln
−
θ
=
V
0
=
0
0
13.21 Determine an expression for the deflection of the circular plate shown in Fig. 13.21.
Hint
:
Reduce
th
e
sol
ution of Problem 13.20 to the case of a plate with a total center
load
P
total
. Set
P
=
P
total
/(
2
π
a
1
)
and take the limit of
w
as
a
1
→
0.
Answer:
P
total
16
3
+
ν
2
r
2
ln
b
r
b
2
r
2
w
=
(
−
)
+
ν
−
π
K
1
FIGURE P13.21
13.22 Derive an equation for the deflection of the circular plate with the symmetrical load-
ing shown in Fig. P13.22. Also, find the deflection if the outer rim is simply supported.
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