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h = y/b
u
y
3
u
y
4
x =
0
h =
1
x =
1
h =
1
u
x
4
u
x
3
4
3
Element node number
Element number
b
x =
0
h =
0
u
y
2
x =
1
h =
0
u
y
1
u
x
1
u
x
2
x = x/a
1
2
a
FIGURE 6.10
Notation for a rectangular plane element. Local coordinates are shown. The node are numbered counterclockwise.
form of the polynomial
u
y
(ξ
,
η)
=
N
uy
u
y
=
u
5
+
u
6
ξ
+
u
7
ξη
+
u
8
η
(6.12)
Thus, the complete assumed displacements are
u
=
N
u
u
u
1
.
ξξηη.
u
x
u
y
1
0
=
(6.13)
.
ξξηη
0
1
u
8
This assumed distribution of displacements is called a bilinear approximation.
The unknown constants
u
8
,
which have no direct mechanical sig-
nificance are referred to as
generalized displacements
. With the finite element method, the
generalized displacements are often replaced by mechanically meaningful unknowns. For
this in-plane element, it is convenient to choose as unknowns the eight displacements
u
x
1
,u
x
2
,u
x
3
,u
x
4
and
u
y
1
,u
y
2
,u
y
3
,u
y
4
at the nodes of the rectangular element. To effect
this replacement, begin at node 1 where
u
, i.e.,
u
1
,
u
2
,
···
,
ξ
=
η
=
0
.
Consider only
u
x
,
which at this node
takes the value
u
1
1000
u
2
u
x
1
=
(6.14)
u
3
u
4
where
u
x
1
is the displacement of node 1 in the
x
direction. For node 2, use
Let
v
x
be the vector of nodal displacements
u
x
1
,u
x
2
,u
x
3
,
and
u
x
4
. If the relations for the four
ξ
=
1
,
η
=
0
.
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