Civil Engineering Reference
In-Depth Information
The sequence of operations described by (3.47) to (3.48) results in
deriv
, the required
gradients with respect to (
x,y,z
) and the Jacobian determinant
det
.
For a three-dimensional elastic solid the element stiffness is given by
[
B
]
T
[
D
][
B
] d
x
d
y
d
z
[
k
m
]
=
(3.74)
where
bee
and
dee
are formed by the subroutines
beemat
and
deemat
as usual, but
with
nst
, the number of components of stress and strain now 6.
The numerical integration summation follows the same course as described previously
for 2D elements in equations (3.51).
The 8-node cuboidal element will usually be integrated used “2-point” Gaussian integra-
tion (
nip=8
). For higher order cuboid elements the number of integrating points can expand
rapidly. For example, “exact” integration of the 20-node cuboid element (see Appendix B)
requires “3-point” integration, or
nip=27
. As with the 8-node plane element, “reduced”
integration of the 20-node element (
nip=8
) can often improve its performance; however
Smith and Kidger (1991) show that full
nip=27
is essential with this element if spurious
“zero energy” eigenmodes are to be avoided in the element stiffness. In addition to the
conventional Gaussian rules, subroutine
sample
allows Irons's (1971) 14 and 15 point
rules to be used for cuboid elements, and the reader is invited to experiment with these
different integration strategies.
For 3D steady laminar fluid flow, the element conductivity or “stiffness” matrix is
given by
[
T
]
T
[
K
][
T
] d
x
d
y
d
z
[
k
c
]
=
(3.75)
where the property matrix is formed as
k
x
00
0
k
y
0
00
k
z
[
K
]
=
(3.76)
Similarly, the “mass” matrix for fluid flow is
[
N
]
T
[
N
] d
x
d
y
d
z
[
m
m
]
=
(3.77)
In both cases, an identical sequence of operations as described previously for planar
flow in equations (3.63) and (3.65) delivers the numerically integrated conductivity matrix
kc
and “mass” matrix
mm
.
A 14-node hexahedral element
The 20-node element mentioned previously is rather cumbersome and its stiffness can be
expensive to compute in non-linear analyses, especially if employing
nip=27
. Furthermore,
the 20-node brick element stiffness matrix exhibits “zero energy modes” when integrated
using
nip=8
(Smith and Kidger, 1991) which could be problematic.
