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Kinematical Relationships
The displacements in a circular plate with in-plane deformation
u
φ
]
T
u
=
[
u
r
(13.8a)
and the three strains
]
T
=
[
r
φ
γ
r
φ
(13.8b)
are related by
r
=
∂
u
r
∂
u
r
r
+
r
∂
1
u
1
r
∂
∂φ
+
∂
u
r
u
1
r
u
φ
∂φ
φ
φ
=
γ
r
φ
=
r
−
φ
r
∂
or
∂
r
0
u
r
r
φ
γ
=
1
/
r
(
1
/
r
)∂
φ
(13.8c)
u
(
1
/
r
)∂
φ
∂
r
−
1
/
r
r
φ
φ
D
u
u
These relations can be obtained by transforming the displacements
u
x
,
u
y
of the previous
section to polar coordinates. With respect to the Cartesian coordinates, the displacement
components from
P
to
P
of Fig. 13.3a are
u
x
and
u
y
, and with respect to the polar co-
ordinates, the displacement components are
u
r
and
u
φ
. It can be observed in Fig. 13.3a
that
=
u
x
=
u
r
cos
φ
−
u
sin
φ
φ
(13.9)
u
y
=
u
r
sin
φ
+
u
cos
φ
φ
Substitution of the first of these relations into Eq. (13.1c) leads to
x
=
∂
u
x
∂
x
=
∂
u
x
∂φ
∂φ
∂
x
+
∂
u
x
∂
∂
r
r
∂
x
1
r
−
∂
u
r
∂φ
φ
+
∂
u
φ
∂φ
=
cos
φ
+
u
r
sin
sin
φ
+
u
φ
cos
φ
sin
φ
∂
cos
u
r
∂
φ
−
∂
u
φ
∂
+
cos
sin
φ
φ
r
r
When
φ
→
0
,
x
→
r
, so that
=
∂
u
r
∂
=
lim
φ
→
0
r
x
r
The strains
φ
and
γ
r
φ
can be obtained in a similar fashion.
Material Law
For the circular plate of thickness
t
, the stress resultants
s
]
T
=
[
n
r
n
n
r
φ
are defined as
φ
t
/
2
t
/
2
t
/
2
=
2
σ
n
φ
=
2
σ
φ
dz
n
r
φ
=
2
τ
n
r
r
dz
dz
r
φ
−
t
/
−
t
/
−
t
/
The material relations of Eq. (13.3) remain valid. Thus, the relationships between these
stress resultants and the strains
]
T
=
[
φ
γ
are
r
r
φ
n
r
n
φ
n
r
φ
1
ν
0
r
φ
γ
Et
=
D
ν
10
00
1
−
ν
2
D
=
(13.10a)
1
−
ν
2
r
φ
=
s
E
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