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13.9 Use the Navier solutions of Problems 13.6 and 13.7 to find the response of a simply
supported rectangular plate subjected to a uniformly distributed load
p
0
. The plate is
of widths
L
and 2
L
in the
x
and
y
directions, respectively. Find the center deflection,
maximum moments and edge reactions.
0101
p
0
L
4
Answer:
Two terms provide sufficient accuracy. Maximum
moments occur at center. Four terms needed for reasonable accuracy.
w
0
.
/
K
.
max
∞
∞
n
2
+
(
−
ν)(
m
2
/
)
π
16
p
0
L
π
2
4
sin
m
y
2
L
V
x
=
0
=
m, n
=
1
,
3
,
5
,
...
3
(
n
2
+
m
2
/
)
2
m
4
n
=
1
m
=
1
p
0
sin
π
L
sin
π
L
,
find the maximum deflection, the peak bending moment per unit length, the reactions
along the edges, and the downward forces necessary to prevent the corners from
rising. All edges are simply supported. Plot the corner and edge forces. Verify that
the total downward force equals the total upward force.
13.10 For a square plate of side
L
subject to the applied loading
p
z
(
x, y
)
=
Answer:
p
0
L
4
4
K
sin
π
x
L
sin
π
y
L
p
0
L
4
4
w
=
,
w
max
=
/(
4
K
π
)
π
4
p
0
L
2
2
m
max
=
m
x
|
2
=
m
y
|
2
=
(
1
+
ν)/(
4
π
)
x
=
L
/
2
x
=
L
/
2
y
=
L
/
y
=
L
/
p
0
L
4
sin
π
y
L
2
p
0
L
2
2
V
along
x
=
0
=
π
(
3
−
ν)
,
Corner forces
=
(
1
−
ν)/(
4
π
)
2
p
0
L
2
2
Total upward force
=
(
3
−
ν)/π
13.11 A simply supported rectangular plate is twice as long
(
2
L
)
as it is wide
(
L
)
. For a
uniformly distributed load
p
0
per unit area, find the maximum deflection.
Answer:
p
0
16
L
4
sin
m
2
sin
n
2
p
0
16
L
4
(
−
)
(
n
+
m
−
2
)/
2
1
w
=
w
=
2
=
max
center
π
6
mn
(
n
2
+
m
2
/
)
π
6
mn
(
n
2
+
m
2
/
)
2
K
4
K
4
13.12 For a single sine series s
o
lution (a Levy or Levy-Nadai solution) with
w(
x, y
)
=
m
=
1
w
m
(
)
=
m
=
1
sin
m
π
y
L
y
sin
m
π
y
L
y
x
)
and
p
z
(
x, y
p
m
(
x
)
,
show that the governing
fourth order equation for the deflection is
2
m
2
m
4
L
y
L
y
p
m
K
m
i
w
m
+
w
−
w
=
m
d
dx
, which has the complementary solution
=
where '
C
1
cosh
m
x
L
y
+
π
C
2
sinh
m
x
L
y
+
π
m
π
x
cosh
m
x
L
y
+
π
m
x
L
y
π
sinh
m
x
L
y
π
w
m
=
C
3
C
4
L
y
p
0
sin
π
L
y
.
13.13 Determine the deflection of a square plate with the sinusoidal loading
p
z
=
The edges at
y
L
y
are simply supported, while the other two edges are
fixed. Use a single sine series solution.
=
0 and
y
=
Answer:
p
0
m
=
1
2
2
p
m
=
β
=
(π/
L
)
0
m
>
1
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