Civil Engineering Reference
In-Depth Information
Applying the general procedure outlined previously in the context of the beam
element, we apply the nodal (boundary) conditions to obtain
10 0
1
2
3
a
0
a
1
a
2
L
2
4
L
2
=
1
(6.19)
2
1
L
from which the interpolation functions are obtained via the following sequence
1
0
0
a
0
a
1
a
2
1
2
3
3
L
4
L
1
L
−
−
=
(6.20a)
2
L
2
4
L
2
2
L
2
−
1
0
0
1
2
3
3
L
4
L
1
L
−
−
xx
2
]
(
x
)
=
[1
2
L
2
4
L
2
2
L
2
−
N
3
]
1
=
[
N
1
N
2
2
3
(6.20b)
3
L
x
2
L
2
x
2
N
1
(
x
)
=
1
−
+
1
4
x
L
x
L
N
2
(
x
)
=
−
(6.20c)
2
x
L
−
1
x
L
=
N
3
(
x
)
Note that each interpolation function varies quadratically in
x
and has value
of unity at its associated node and value zero at the other two nodes, as illustrated
in Figure 6.3. These observations lead to a shortcut method of concocting the
interpolation functions for a
C
0
line element as products of monomials as fol-
lows. Let
s
=
x
/
L
such that
s
1
=
0,
s
2
=
1
/
2,
s
3
=
1
are the nondimensional
coordinates of nodes 1, 2, and 3, respectively. Instead of following the formal
procedure used previously, we hypothesize, for example,
N
1
(
s
)
s
3
)
(6.21)
where
C
1
is a constant. The first monomial term ensures that
N
1
has a value of zero
at node 2 and the second monomial term ensures the same at node 3. Therefore,
we need to determine only the value of
C
1
to provide unity value at node 1.
=
C
1
(
s
−
s
2
)(
s
−
