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1
p
0
K
+
β
x
sinh
π
+
cosh
π
−
1
w
=
sinh
β
x
β
4
π
+
sinh
π
x
sin
π
+
β
x
(
cosh
π
−
1
)
−
π
−
sinh
π
y
L
y
cosh
β
π
+
sinh
π
13.14 For a single sine series solution, determine the transformed loading function
p
m
for a
rectangular plate with a constant loading in the
y
direction. Suppose the load begins
at
y
=
b
1
and ends at
y
=
b
2
.
Answer:
cos
m
2
p
0
(
x
)
b
1
L
y
−
π
cos
m
b
2
L
y
π
p
m
=
π
m
4
p
0
(
x
)
if
m
=
1
,
3
,
5
,
7
,
...
m
π
=
=
=
If
b
1
0
,
2
L
y
:
p
m
0if
m
=
2
,
4
,
6
,
8
,
...
13.15 Derive the first-order relations of Eq. (13.58a).
13.16 Use an analytical solution to determine the deflection of the rectangular plate shown
in Fig. P13.16 if the edges at
y
=
0 and
y
=
L
y
are:
(a) simply supported
(b) fixed
FIGURE P13.16
13.17 Find the natural frequencies
ω
mn
,m,n
=
1
,
2
,
3
,
...
,
for a simply supported rectan-
gular plate.
Answer:
K
ρ
1
/
2
C
mn
L
y
2
m
2
2
n
2
ω
=
,
C
mn
=
π
(
+
α
)
,
α
=
L
y
/
L
mn
13.18 Find the critical (buckling) in-plane force
(
n
y
)
cr
in the
y
direction of a rectangular
plate that is simply supported on all sides.
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