Information Technology Reference
In-Depth Information
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
(ξ
,
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
(ξ
,
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.
Search WWH ::
Custom Search