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From Eqs. (6.66), it follows that the natural coordinates are defined such that
L
1
=
1 f
i
falls on point 1
,
L
1
=
0 f
i
falls on point 2
L
2
=
1 f
i
falls on point 2
,
L
2
=
0 f
i
falls on point 1
It can be seen that
L
1
and
L
2
satisfy the properties of Eq. (6.63) and, hence, can be considered
to be forms of Lagrangian interpolation functions. The
ξ
coordinate of any point along the
line can be expressed as
ξ
=
L
1
ξ
+
L
2
ξ
(6.68)
1
2
a result which is verified by use of Eq. (6.66). Equations (6.67) and (6.68) placed together
appear as
L
1
L
2
1
ξ
11
ξ
=
ξ
1
2
These relationships, i.e., Eqs. (6.67) and (6.68), can be considered to be the definition of the
natural coordinates, since they lead to
L
1
L
2
ξ
2
1
ξ
1
(ξ
2
−
ξ
1
)
−
1
=
(6.69)
−
ξ
1
1
which is the same as Eq. (6.66)
A useful property of these natural coordinates is
ξ
2
L
1
L
2
d
1
1
ξ
=
12
2
(6.70)
ξ
1
where
2
.
The natural coordinates are useful in representing shape functions for a line divided into
segments. It is convenient to utilize a different node numbering scheme. Designate the left
end point as 0 and the right end point as
m
, i.e., these points are numbered 0
,
1
,
2
,
12
is the length from
ξ
1
to
ξ
,m
.
Using natural coordinates, each point is identified by its location relative to the two end
points of the line. Define numbers
p
and
q
, where
p
and
q
are the number of points to the
right and left, respectively, of the point under consideration. Observe Figs. 6.29a, b, and c
wherein the three nodes in Fig. 6.29c are denoted as 20, 11, and 02. Thus, the coordinate
...
ξ
2
ξ
ξ
ξ
of Eq. (6.66) is now
02
and
1
is
20
. The nodal degrees of freedom will now be labeled with
p, q,
i.e.,
v
pq
or in the case of Fig. 6.29c,
v
20
,
v
11
and
v
02
. The corresponding shape functions
are
N
pq
,
i.e.,
N
20
,N
11
,
and
N
02
.
Shape functions in terms of natural coordinates are given by
N
pq
(
L
1
,L
2
)
=
N
p
(
L
1
)
N
q
(
L
2
)
(6.71a)
where, from the Lagrangian interpolation formula,
N
p
(
L
1
)
and
N
q
(
L
2
)
are defined by
i
j
=
1
mL
k
−
j
+
1
,
for
i
≥
1
N
i
(
L
k
)
=
j
k
=
1
,
2
,
···
(6.71b)
1
for
i
=
0
where
m
is the number of segments.
EXAMPLE 6.8 Three-Node Element
For the element of Fig. 6.29c, the trial function for the displacement can be expressed as
=
v
+
v
+
v
02
u
N
20
N
11
N
02
20
11
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