Civil Engineering Reference
In-Depth Information
nearest plane of the pyramid is “Pascal's triangle”, which can be used to select shape
function terms for 2D elements. Smith and Kidger (1992) tried six permutations which
were called “Types 1 to 6”. For example, Type 1 contained all 10 polynomials down to
the second “plane” of the pyramid plus the terms
ξηζ
,
ξ
2
η
,
η
2
ζ
,and
ζ
2
ξ
from the third
“plane”. Type 6 selectively contained terms as far down as the fifth “plane” and this is the
version available in library subroutines
shape_fun
and
shape_der
. Computer algebra
was essential when deriving the shape functions for the “Type 6” element which are listed
in Appendix B.
Tetrahedral elements
An alternative element for 3D analysis is the tetrahedron, the simplest of which has 4 corner
nodes and is called the “constant strain” tetrahedron. The local coordinate system involves
mapping a general tetrahedron onto a right-angled tetrahedron with three orthogonal sides
of unit length as shown in Figure 3.15 and Appendix B. This approach can be shown to
be identical to “volume coordinates”. For example, point P can be identified uniquely by
the coordinates (
L
1
,L
2
,L
3
). As with triangles, an additional coordinate
L
4
given by
L
4
=
−
L
1
−
L
2
−
L
3
1
(3.78)
is sometimes retained for algebraic convenience.
The shape functions for the “constant strain” tetrahedron are
L
1
L
2
L
3
L
4
fun
=
(3.79)
and these, together with their derivatives with respect to
L
1
,
L
2
and
L
3
are formed by the
usual subroutines
shape_fun
and
shape_der
with
ndim=3
and
nod=4
. The sequence
4
z
y
x
2
1
L
2
3
L
1
L
3
Figure 3.15 A 4-node tetrahedron element
