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a
2
r
+
ν)
p
0
tE
a
2
b
2
Answer:
The radial displacement is
u
r
=
(
1
−
ν)
+
r
(
1
b
2
−
13.4 Find the radial displacement and internal forces in a circular disk with no center hole.
The disk rotates at an angular velocity
and is subject to a radially directed external
pressure of magnitude
p
0
on the periphery at radius
b
.
2
r
p
0
r
(
1
−
ν)
Et
p
(
1
−
ν)
b
2
r
2
]
Answer:
u
r
=−
+
[
(
3
+
ν)
−
(
1
+
ν)
8
E
13.5 Thick cylinders are usually modeled as being in a state of plane strain in the axial
direction, i.e., the axial strain in the cylinder is zero. The in-plane element equations of
this chapter apply to the thick cylinder if the plane stress relations of this chapter are
transformed to plane strain, using the plane-stress to plane-strain conversion factors
of Chapter 1. Find the radial stress and displacement due to thermal loading
T
(
r
)
in a thick, long cylinder of radius
r
=
a
L
with no center hole.
r
a
L
0
r
2
0
ξ
E
a
a
L
E
Answer:
σ
r
=
ξ
T
(ξ)
d
ξ
−
T
(ξ)
d
ξ
1
−
ν
1
−
ν
Rectangular Plates
13.6 Consider a rectangular plate with dimensions
L
x
and
L
y
, which is simply supported
on all sides. Expand the loading and response variables in double sine series, e.g.
∞
∞
p
mn
sin
n
x
L
x
π
sin
m
π
y
p
z
(
x, y
)
=
L
y
m
=
1
n
=
1
∞
∞
mn
sin
n
π
x
sin
m
π
y
w(
x, y
)
=
1
w
L
x
L
y
m
=
1
n
=
This is referred to as a Navier solution. Show that, in general,
p
mn
K
4
n
2
2
m
2
L
y
w
mn
=
π
L
x
+
L
x
L
y
4
L
x
L
y
sin
n
x
L
x
π
sin
m
π
y
where
p
mn
=
p
z
(
x, y
)
dx dy
L
y
0
0
Hint:
Use orthogonality conditions of the sort employed in deriving Eq. (13.63) from
Eq. (13.62).
13.7 For the Navier solution of Problem 13.6, find the deflection, slope, moment, and shear
force distributions for a uniform loading of magnitude
p
0
.
Answer:
2
mn
p
mn
=
16
p
0
/(π
)
for
m
,
n
odd integers,
p
mn
=
0if
m
or
n
or both are
even.
6
K
16
p
0
/(π
)
w
mn
=
for
m
and
n
odd
nm
n
2
L
y
2
m
2
L
x
+
13.8 Find the deflection in a rectangu
la
r plate simply supported on all boundaries due to
a transverse concentrated force
P
applied at
x
=
c, y
=
d
.
Answer:
Use the expression for
w
mn
of Problem 13.6, with
4
P
L
x
L
y
sin
n
c
L
x
π
sin
m
π
d
p
mn
=
L
y
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