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can be performed in the
(ξ
,
η)
coordinate system. From Eq. (1.24),
∂
0
x
=
0
∂
y
D
u
∂
y
∂
x
J
11
∂
ξ
+
J
12
∂
η
,
J
21
∂
ξ
+
J
22
∂
η
. Thus,
and from Eq. (6.136),
∂
=
∂
=
x
y
J
11
∂
ξ
+
J
12
∂
η
0
J
21
∂
ξ
+
J
22
∂
η
=
0
D
u
(6.139)
J
21
∂
ξ
+
J
22
∂
η
J
11
∂
ξ
+
J
12
∂
η
The stiffness matrix is obtained by substituting this
D
u
and Eq. (6.137) into Eq. (6.138). The
required integration can be performed in the
system.
A similar procedure can be used in three-dimensional situations. From the coordinate
transformation, elements of different shapes can be obtained from a master element. Master
elements of different order define different transformations and generate slave elements
which have the same order as their master elements and may have more complex shapes.
Figures 6.40 and 6.41 show various possible transformations. Note from these figures that
(ξ
,
η)
FIGURE 6.40
Triangular elements and their isoparametric forms.
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