Civil Engineering Reference
In-Depth Information
N
2
1
N
3
0.8
0.6
0.4
N
1
0.2
0
x
0.2
Figure 6.3
Spatial variation of interpolation
functions for a three-node line element.
Substituting
s
=
0
, we obtain
C
1
0
(0
1
2
N
1
(
s
=
0)
=
1
=
−
−
1)
(6.22)
yielding
C
1
=
2
and
2
s
(
s
1
2
N
1
(
s
)
=
−
−
1)
(6.23)
Following similar logic and procedure shows that
N
2
(
s
)
=−
4
s
(
s
−
1)
(6.24)
2
s
s
1
2
N
3
(
s
)
=
−
(6.25)
Substituting
s
=
x
/
L
in Equations 6.23-6.25 and expanding shows that the
results are identical to those given in Equation 6.20. The monomial-based proce-
dure can be extended to line elements of any order as illustrated by the following
example.
EXAMPLE 6.1
Use the monomial method to obtain the interpolation functions for the four-node line
element shown in Figure 6.4.
L
3
L
3
L
3
x
1
2
3
4
Figure 6.4
Four-node line
element of Example 6.1.
