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developed in Chapter 6, Section 6.4 for a rectangular element with in-plane deformation.
Thus, introduce the bilinear shape functions
N
1
N
2
N
3
N
4
.
.
0
0
w
···
θ
x
···
θ
y
···
···
···
.
N
1
N
2
N
3
N
4
.
v
i
u
=
=
0
0
···
···
···
.
.
0
0
N
1
N
2
N
3
N
4
N
w
θ
0
0N
w
Nv
i
=
=
(13.100)
θ
where
N
w
=
[
N
1
N
2
N
3
N
4
]
N
1
N
2
N
3
N
4
0
N
θ
=
0
N
1
N
2
N
3
N
4
and
N
1
=
(
−
ξ)(
−
η)
=
ξ(
−
η)
1
1
N
2
1
N
3
=
ξη
N
4
=
η(
1
−
ξ)
C
0
These shape functions will lead to a first order
(
)
element, for which the displacement
w
and the rotations
y
will be continuous along the element boundaries.
The stiffness matrix will be established using the principle of virtual work, with
θ
x
and
θ
δ
W
i
given by (Eq. 13.97)
6
k
s
(
1
−
ν)
θ
T
k
B
θ
dA
u
T
k
V
A
δ
+
A
δ
u
dA
−
δ
W
i
=
K
(13.101)
t
2
I
II
θ
y
]
T
,
k
B
=
κ
D
T
E
B
D
,
and
k
V
=
γ
D
T
E
V
D
where
θ
=
are given in Eq. (13.97). Integral
I will lead to the stiffness matrix
k
B
for bending, and integral II will provide the stiffness
matrix
k
V
corresponding to shear deformation effects.
In the case of in-plane deformation, the internal virtual work (Eq. 13.18) takes a form
similar to Eq. (13.101)
[
θ
x
κ
γ
u
T
k
D
u
dA
−
δ
W
i
=
A
δ
(13.102)
[
u
x
u
y
]
T
and
k
D
u
D
T
ED
u
with
u
=
=
.
This leads to the stiffness matrix
k
for in-plane
deformation.
Since
D
κ
(Eq. 13.95)
K
=
D
u
(Eq. 13.1c) and
E
B
(Eq. 13.87)
=
D
E
(Eq. 13.10a), we conclude
K
D
k
t
2
12
k
It is apparent that, with the factor
t
2
that
k
B
12 the stiffness matrix for
bending can be formed directly from the stiffness matrix (Eq. (6.34), Chapter 6), for in-
plane deformation. Thus, if the displacement vector
θ
of
k
B
is expanded to include
w
,
=
=
.
/
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