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6.5.5
Polynomial Shape Functions
Polynomials are commonly employed to approximate unknown functions. In the finite
element method, the interpolation functions N i are almost always polynomials. This class
of functions is considered to be desirable because of the ease of manipulation. Normally,
the higher the degree of the polynomial the better the approximation, and a refined mesh
usually improves convergence. Moreover, polynomials are easy to differentiate. Thus, if
polynomials approximate the displacements of the structure, the strains can be evaluated
with ease.
It is useful to sketch the derivation of interpolation functions, even though similar deriva-
tions are given in Chapter 4 and earlier in this chapter. Begin with a two-dimensional trial
function for the displacement u
,
η)
in the form
2
=
+
ξ +
η +
+
N u
u
u 1
u 2
u 3
u 4
ξ
u 5
ξη +··· =
u
(6.57)
2
with N u
=
[1
ξηξ
ξη ···
] and the generalized displacements or parameters are
u 1
u 2
.
u
=
The number n of generalized parameters
u i corresponds to the number of nodal DOF.
The generalized parameters
u i are replaced by the mechanically meaningful nodal vari-
ables v , using
= N u
v
u
(6.58)
n matrix N u contains the discrete values resulting from evaluating Eq. (6.57),
or its derivatives, at the nodes. Then
where the n
×
= N 1
u
=
u
v
Gv
and
N u N 1
u
=
v
=
Nv
(6.59)
u
where, in two dimensions,
N u N 1
N
,
η) =
=
N u G
(6.60)
u
is the desired interpolation function.
EXAMPLE 6.6 Interpolation Functions
Suppose the polynomial
]
u 1
u
=
u 1 +
u 2 ξ =
[1
ξ
=
N u
u
(1)
u 2
is to be employed for a two-node element with end displacements as DOF (see Fig. 6.25a).
To replace the constants
u 1 and
u 2 by the coordinates
v a and
v b , use
=
+
u
=
0
) = v
u 1
u 2
·
0
a
(2)
=
) = v
=
+
·
u
1
u 1
u 2
1
b
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