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into the unknown nodal displacement vector
w
a
=−
w
a
θ
a
v
=
(4.42)
w
b
=−
w
b
θ
b
The assumed series is then referred to as an
interpolation
or
shape function
. The derivative of
w
, which is needed to form Eq. (4.42), is given by
w
1
w
2
w
3
w
4
=
w
=
3
x
2
]
N
u
w
T
N
u
)
T
[012
x
w
=
(
(4.43)
θ
=−
w
Now, evaluate
w
and
at
x
=
a
and
x
=
b,
giving
w
a
θ
a
w
b
θ
b
)
−
w
(
w(
0
10
0
0
w
1
w
2
w
3
w
4
=
=
0
)
0
−
10
0
(4.44)
2
3
w()
−
w
()
1
2
0
−
1
−
2
−
3
N
u
v
=
w
The constants
w
are found in terms of the (unknown) displacements at
a
and
b
by using
the inverse of
N
u
,
=
N
−
1
u
w
v
=
Gv
(4.45)
where
1
0
0
0
0
−
1
0
0
=
N
−
1
u
G
=
−
3
/
2
2
/
3
/
2
1
/
/
3
−
/
2
−
/
3
−
/
2
2
1
2
1
The relationship between
w
and
v
, i.e., between the deflection
w
and the values of dis-
placements
w
and
θ
at the ends of the element, is
w
=
N
u
w
=
N
u
Gv
=
Nv
(4.46a)
or
1
3
x
2
x
2
−
3
x
2
2
x
3
2
x
2
x
3
2
x
3
x
3
w(
x
)
=
−
2
+
w
a
+
−
x
+
−
θ
a
+
2
−
w
b
+
θ
b
3
2
3
2
(4.46b)
where
N
This expression is the desired form of the assumed series, where the
matrix
N
, which is often called a
shape function matrix
, characterizes the “interpolation”
or “shape” between the nodes. The components of this expression are often referred to as
shape, basis
,or
interpolation functions
.
=
N
u
G
.
Interpolation Functions Based on a Normalized Coordinate
Some mathematical handbooks tabulate various interpolation functions. The Hermitian
2
interpolation polynomials of Fig. 4.8 can be employed when derivatives of the displace-
ments at the nodes are involved, as with beams. The Hermitian polynomials can be derived
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