121
Element Fields in Linear
Problems
This chapter presents interpolation models in physical coordinates for the most part,
for the sake of simplicity and brevity. However, in ﬁnite-element codes, the physical
coordinates are replaced by natural coordinates using relations similar to interpola-
tion models. Natural coordinates allow use of Gaussian quadrature for integration
and, to some extent, reduce the sensitivity of the elements to geometric details in
the physical mesh. Several examples of the use of natural coordinates are given.
9.1 INTERPOLATION MODELS
9.1.1 O
NE
-D
IMENSIONAL
M
EMBERS
9.1.1.1
Rods
The governing equation for the displacements in rods (also bars, tendons, and
shafts) is
(9.1)
in which
u
(
x
,
t
) denotes the radial displacement, E,
Α
and
ρ
are constants,
x
is the
spatial coordinate, and
t
denotes time. Since the displacement is governed by a
second-order differential equation, in the spatial domain, it requires two (time-
dependent) constants of integration. Applied to an element, the two constants can
be supplied implicitly using two nodal displacements as functions of time. We now
approximate
u
(
x
,
t
) using its values at
x
e
and
x
e
+
1
, as shown in Figure 9.1.
The lowest-order interpolation model consistent with two integration constants
is linear, in the form
(9.2)
We seek to identify
Φ
m1
in terms of the nodal values of
u
. Letting
u
e
=
u
(
x
e
) and
u
e
+
1
=
u
(
x
e
+
1
), furnishes
(9.3)
9
E
A
u
x
A
u
t
∂
∂
=
∂
∂
2
2
2
2
,
uxt
x
t
t
ut
xx
m
T
mm
m
e
e
m
T
(,)
()
,
() (
)
.
==
=
+
ϕ
Φ
γ
γ
ϕ
11
1
1
1
1
1
,
x
t
u t
x
t
e
e mm
e
e
)
,
.
++
1
1 11
Φγ
© 2003 by CRC CRC Press LLC