Civil Engineering Reference
In-Depth Information
where
a
,
b
,
c
,
d
are positive integers and
V
is total element volume, appear in
element formulation for various physical problems. As with area coordinates,
integration in volume coordinates is straightforward [7], and we have the inte-
gration formula
a
!
b
!
c
!
d
!
L
1
L
2
L
3
L
4
d
V
(6.66)
=
3)!
(6
V
)
(
a
+
b
+
c
+
d
+
V
which is the three-dimensional analogy to Equation 6.49.
As another analogy with the two-dimensional triangular elements, the tetra-
hedral element is most useful in modeling irregular geometries. However, the
tetrahedral element is not particularly amenable to use in conjunction with other
element types, strictly as a result of the nodal configurations. This incompati-
bility is discussed in the following sections. As a final comment on the four-
node tetrahedral element, we note that the field variable representation, as given
by Equation 6.64, is a linear function of the Cartesian coordinates. Therefore,
all the first partial derivatives of the field variable are constant. In structural
applications, the tetrahedral element is a constant strain element; in general, the
element exhibits constant gradients of the field variable in the coordinate
directions.
Other elements of the tetrahedral family are depicted in Figure 6.18. The
interpolation functions for the depicted elements are readily written in volume
coordinates, as for higher-order two-dimensional triangular elements. Note par-
ticularly that the second element of the family has 10 nodes and a cubic form for
the field variable and interpolation functions. A quadratic tetrahedral element
cannot be constructed to exhibit geometric isotropy even if internal nodes are
included.
(a)
(b)
Figure 6.18
Higher-order tetrahedral elements:
(a) 10 node. (b) 20 node.
