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Hint:
Assume u x ,u y , and u z to be of the form
u 1
+
u 2
ξ +
u 3
η +
u 4
ξη +
u 5
ζ +
u 6
ξζ +
u 7
ηζ +
u 8
ξηζ
1
Answer:
N i =
8 (
1
+ ξξ i ) + (
1
+ ηη i )(
1
+ ζζ i )
i
=
1 , 2 ,
...
, 8
6.11 Find a set of interpolation functions N of u
Nv for the eight-node, 16-DOF plane
stress element of Fig. P6.11. Also, apply Lagrangian interpolation to the same problem.
=
FIGURE P6.11
Answer:
For a coordinate system at the center of the element and
ξ =
2 x
/
a,
η =
2
2
2 y
/
b, with u x or u y assumed to be u x
,
η) =
u 1
+
u 2
ξ +
u 3
η +
u 4
ξ
+
u 5
ξη +
u 6
η
+
2
η +
2 , the interpolation function u would be
u 7
ξ
u 8
ξη
1
4 (
u
,
η) =−
1
ξ)(
1
η)(
1
+ ξ + η)
u x 1
1
4 (
1
4 (
1
+ ξ)(
1
η)(
1
ξ + η)
u x 2
1
+ ξ)(
1
+ η)(
1
ξ η)
u x 3
1
4 (
1
2 (
2
1
ξ)(
1
+ η)(
1
+ ξ η)
u x 4 +
1
ξ
)(
1
η)
u x 5
1
2 (
1
2 (
2
2
+
1
η
)(
1
+ ξ)
u x 6
+
1
ξ
)(
1
+ η)
u x 7
1
2 (
2
+
1
η
)(
1
ξ)
u x 8
6.12 Find an interpolation function for the one-dimensional element shown in Fig. P6.12.
Use a natural coordinate system.
FIGURE P6.12
1
16
3
Answer:
N 1 =
3 L 1 (
4 L 1
1
)(
2 L 1
1
)(
4 L 1
3
)
,N 2 =
L 1 (
4 L 1
1
)(
2 L 1
1
)
L 2 ,
16
3
1
3 L 2
N 3
=
4 L 1 L 2
(
4 L 1
1
)(
4 L 2
1
)
,N 4
=
L 1 L 2
(
4 L 2
1
)(
2 L 2
1
)
,N 5
=
(
4 L 2
1
)
(
2 L 2
1
)(
4 L 2
3
)
6.13 Find the interpolation function corresponding to nodes 1 and 2 of the triangular
element shown in Fig. P6.13. Use a two-dimensional natural coordinate system.
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