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First Order Governing Differential Equations
The first order governing differential equations for the in-plane loaded circular disk with
axially symmetrical deformation can be obtained from the first relations of Eqs. (13.10b)
and (13.11a). The variation of th
e
displa
ce
ments and internal forces are independent of
φ
for
axially symmetric loading. Let
p
r
(
p
Vr
denote the applied radial forces (force/length
2
)
in the plane of the disk, including radial pressure, centrifugal, and thermal forces, all of
which are assumed to be independent of
r
)
=
φ
. Because of the symmetry,
γ
r
φ
=
n
r
φ
=
0
.
Also,
∂
φ
()
=
0
.
From Eq. (13.8c),
φ
=
u
r
/
r,
and from Eq. (13.10b),
φ
=
(
−
ν
n
r
+
n
φ
)/
Et
or
Et
r
n
φ
=
u
r
+
ν
n
r
=
∂
=
(
−
ν
n
φ
)/
∂
=
(
−
ν
n
φ
)/
=
From Eqs. (13.8c) and (13.10b),
r
u
r
and
n
r
Et
,or
r
u
r
n
r
Et
−
ν
Et
r
n
r
]
r
r
+
ν
/
.
[
n
r
From this relation and the first condition of equilibrium of
Eq. (13.11a), the first order governing equations are
u
r
Et
2
du
r
dr
=−
ν
u
r
r
+
1
−
ν
n
r
(13.14a)
Et
dn
r
dr
=
Et
r
2
u
r
+
ν
−
1
n
r
−
p
r
(
r
)
r
or
u
r
n
r
−
ν/
u
r
n
r
0
−
2
d
dr
r
(
1
−
ν
)/
Et
=
+
(13.14b)
r
2
p
r
Et
/
(ν
−
1
)/
r
/
=
+
dz
dr
A
z
P
Note that
A
is not constant as
A
=
A
(
r
).
If inertia fo
rc
es a
re
included, change the ordinary
d
dr
∂
∂
2
u
r
t
2
,
where
derivatives
to partial derivatives
and replace
p
r
by
p
r
=−
ρ∂
/∂
ρ
is the
r
mass per unit area.
In higher order form, for a disk with constant thickness and material properties,
1
r
d
2
u
r
dr
2
+
2
1
r
du
r
dr
−
1
r
2
u
r
=
d
dr
d
dr
(
1
−
ν
ru
r
)
=−
p
r
(13.14c)
Et
Integration of this relationship gives
u
r
, which can then be placed in the first of Eqs. (13.14a)
to find
n
r
.
The transfer and stiffness matrices from the solution of Eq. (13.14) are given in Table 13.1.
EXAMPLE 13.1 Rotating Disk
Find the displacement
u
r
and the internal force
n
r
of a rotating circular disk with inner and
outer radii of
a
and
b
. The angular speed of the disk is
.
Assume
th
at the material of the disk is homogeneous. The disk is subjected to a symmetric
loading of
p
r
=
ρ
2
,
where
is the mass density (mass/area). The displacement
u
r
and
internal force
n
r
are independent of
r
ρ
. The governing equations for the disk are given by
Eq. (13.14), which are readily integrated. We find
φ
u
r
n
r
U
uu
u
r
n
r
u
r
n
r
U
un
r
=
a
+
U
nu
U
nn
(1)
r
=
U
i
z
i
z
=
z
a
+
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