Information Technology Reference
In-Depth Information
with
1
r
a
−
(
1
+
ν)
2
(
r
2
−
a
2
)
Et
(
1
1
−
ν
2
)
(
r
2
−
a
2
)
U
uu
=
U
un
=
r
2
2
r
Et
2
ar
2
(
−
(
1
−
ν)
2
(
r
2
−
a
2
)
r
2
a
2
U
nu
=
−
)
U
nn
=
1
r
2
r
2
−
a
2
1
−
ν
2
ρ
2
)
ρ
2
t
+
ν)
+
(
1
−
ν)
2
(
r
2
+
a
2
)
u
r
=−
n
r
=−
(
r
2
a
2
−
(
1
r
8
E
4
r
2
The i
n
itial parameters
u
r
and
n
r
at
r
=
a
can be determined from the relation
z
b
=
U
i
z
a
+
z
i
. Note that
n
r
|
r
=
b
=
n
r
|
r
=
a
=
0
.
Thus, (1) can be written as
u
r
0
U
uu
u
r
0
u
r
n
r
U
un
b
=
a
+
U
nu
U
nn
(2)
r
=
r
=
r
=
a
z
i
U
i
z
b
=
(
b
)
z
a
+
Then
r
=
b
=
n
r
U
nu
ab
2
2
b
2
a
2
u
r
|
r
=
a
=
−
ρ
+
ν)
+
(
1
−
ν)
2
+
(
1
(3)
2
E
b
2
With the initial parameters known, the response of the disk can be determined from (1).
The corresponding loading terms and stiffness matrices are given in Table 13.1.
13.1.3 Variational (Global) Relationships
The principle of virtual work for these in-plane deformation problems becomes (Chapter
2, Example 2.5)
u
T
u
D
T
s
u
T
δ
W
=−
A
δ
(
−
p
V
)
dA
+
S
p
δ
p
ds
=
0
(13.15)
where
ds
is an infinitesima
l l
ength along a perimeter boundary of the flat element. Upon
substitution of
u
,
s
,
D
,
and
p
V
for a rectangular element,
x
∂
p
Vx
p
Vy
u
y
]
p
x
p
y
ds
n
x
n
y
n
xy
0
∂
y
−
δ
W
=−
A
δ
[
u
x
u
y
]
dA
+
S
p
δ
[
u
x
=
0
0
∂
∂
y
x
(13.16)
p
y
]
T
where
p
=
[
p
x
are the boundary loads. Remember that
x
∂
and
y
∂
mean that the
derivatives
∂
x
and
∂
y
are taken on the variables to the left of
x
∂
and
y
∂.
For circular disks,
r
∂
p
Vr
p
V
φ
n
r
n
φ
n
r
φ
1
/
r
(
1
/
r
)
φ
∂
−
δ
W
=−
A
δ
[
u
r
u
φ
]
dA
0
(
1
/
r
)
φ
∂
∂
−
1
/
r
r
u
p
]
p
r
p
ds
+
δ
=
[
u
r
0
(13.17)
φ
S
p
where
p
r
and
p
are the applied boundary loads.
In terms of displacements, the principle of virtual work appears as
φ
u
T
k
D
u
u
T
p
ds
δ
W
=−
A
δ
(
−
p
V
)
+
S
p
δ
=
dA
0
(13.18)
Search WWH ::
Custom Search