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Note that
B
u
does not involve
ξ
and
η
. The differential area is transformed as
dx dy
=
dA
=|
J
|
d
ξ
d
η
=
2
Ad
ξ
d
η
The stiffness matrix for the slave element is expressed as
t
1
0
1
1
4
A
2
B
u
t
2
B
u
k
i
=
EB
u
2
Ad
ξ
d
η
=
EB
u
(6.146)
0
It is seen that the formation of the stiffness matrix for the three-node triangular element
does not require numerical integration. For triangular elements with more than three nodes,
B
u
involves
ξ
η
, and the stiffness matrix tends to be complicated to compute. As a
consequence, explicit calculation of
k
i
and
is not feasible and numerical integration is normally
employed.
Recall that a linear trial displacement was used here. Higher order trial displacements
would lead to the development of elements with curved boundaries.
6.7.2
Four-Sided Isoparametric Element
Suppose the master element of Fig. 6.43a is to be mapped into the slave element of Fig. 6.43b.
For the coordinate system, use an internal origin at the center for the master and slave
elements with vertices defined to be at
The same coor-
dinate system was employed in Fig. 6.39. Represent the shape functions, with interpolation
polynomials
N
i
,
and the element coordinates of the isoparametric element as
(
−
1
,
−
1
)
,
(
−
1
,
1
)
,
(
1
,
−
1
)
,
and
(
1
,
1
).
g
g
u
x
=
N
i
u
xi
u
y
=
N
i
u
yi
i
=
1
i
=
1
g
g
x
=
N
i
x
i
y
=
N
i
y
i
i
=
1
i
=
1
where
g
is the number of nodes. With cubic terms, the interpolation functions are given by
i
=
5
i
=
6
i
=
7
i
=
8
1
1
1
N
1
=
4
(
1
−
η)(
1
−
ξ)
···−
2
N
5
···
··· −
2
N
8
1
4
1
2
N
5
1
2
N
6
=
(
+
ξ)(
−
η)
···−
−
N
2
1
1
1
1
1
N
3
=
4
(
1
+
η)(
1
+
ξ)
···
−
2
N
6
−
2
N
7
1
1
1
N
4
=
4
(
1
−
ξ)(
1
+
η)
···
··· −
2
N
7
−
2
N
8
(6.147)
1
2
N
5
=
2
(
1
−
ξ
)(
1
−
η)
1
2
2
N
6
=
(
1
−
η
)(
1
+
ξ)
1
2
N
7
=
2
(
1
−
ξ
)(
1
+
η)
1
2
=
(
−
η
2
)(
−
ξ)
N
8
1
1
The functions
N
5
,N
6
,N
7
,
and
N
8
are the interpolation functions associated with nodes 5,
6, 7, and 8 in Fig. 6.43. If these four nodes are present in the element, the element shape
functions are Eq. (6.147). If fewer nodes are present, the element shape functions include
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