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would have replaced
x
and
y
. The interpolation functions for
u
x
and
u
y
are
u
x
=
Nv
x
(6.91)
=
u
y
Nv
y
where
u
x
1
u
x
2
u
x
3
u
y
1
u
y
2
u
y
3
v
x
=
v
y
=
and
N
=
[
L
1
L
2
L
3
]
in which
u
xi
,u
yi
,i
=
1
,
2
,
3
,
are the values of
u
x
and
u
y
at the nodes, and
L
i
,i
=
1
,
2
,
3are
the shape functions defined in Eq. (6.75). It is readily shown that
N
=
[
L
1
L
2
L
3
]. From
Eq. (6.76) with
m
=
1
,
u
x
=
N
100
v
+
N
010
v
+
N
001
v
=
N
100
u
x
1
+
N
010
u
x
2
+
N
001
u
x
3
100
010
001
where
N
pqr
,
with
p
+
q
+
r
=
1, is defined in Eq. (6.77). From Eq. (6.71b),
N
0
(
L
1
)
=
N
0
(
L
2
)
=
N
0
(
L
3
)
=
1
,N
1
(
L
1
)
=
L
1
,N
1
(
L
2
)
=
L
2
,N
1
(
L
3
)
=
L
3
. Finally,
N
1
=
N
100
=
L
1
,N
2
=
N
010
=
L
2
,N
3
L
3
.
The displacement vector is
=
N
001
=
u
x
u
y
N0
0N
v
x
v
y
N0
0N
v
i
u
=
=
=
(6.92)
The principle of virtual work expression is [Eqs. (6.20) and (6.28)]
V
δ
u
T
u
D
T
ED
u
u
dV
u
T
u
T
p
dS
δ
W
=
−
V
δ
p
V
dV
−
S
p
δ
T
v
iT
t
T
N0
0N
N0
0N
dA
v
i
∂
0
∂
x
0
M
x
=
1
δ
0
∂
E
0
∂
y
y
A
∂
∂
∂
y
∂
x
i
=
y
x
p
dS
t
N0
0N
T
N0
0N
T
−
p
V
dV
+
V
S
p
M
v
iT
k
i
v
i
p
i
=
1
δ
(
−
)
=
0
(6.93)
=
i
where
p
is the external load on the element boundary
S
p
and
t
is the thickness of the plate.
E
is given in Eq. (1.39), and
D
u
is taken from Eq. (1.24). The stiffness matrix and loading
vector for each element are given by
T
T
t
N0
0N
∂
0
∂
x
0
N0
0N
dA
x
k
i
=
0
∂
E
0
∂
y
(6.94a)
y
A
∂
∂
∂
y
∂
x
y
x
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