Civil Engineering Reference
In-Depth Information
z
y
6
7
2
3
8
5
1
4
x
Figure 3.12 General linear hexahedron “brick” element
As was the case with changes of plane element types, changes of element dimensions
are readily made. For example, the 8-node hexahedral “brick” element in Figure 3.12 is
the three-dimensional extension of the 4-noded quadrilateral.
The coordinate matrix becomes
x
1
y
1
z
1
x
2
y
2
z
2
x
3
y
3
z
3
x
4
y
4
z
4
x
5
y
5
z
5
x
6
y
6
z
6
x
7
y
7
z
7
x
8
y
8
z
8
coord
=
(3.72)
and using the three-dimensional local coordinate system (
ξ,η,ζ
), the shape functions
become
1
8
(
1
−
ξ)(
1
−
η)(
1
−
ζ)
1
8
(
1
−
ξ)(
1
−
η)(
1
+
ζ)
1
+
ξ)(
1
−
η)(
1
+
ζ)
8
(
1
1
8
(
1
+
ξ)(
1
−
η)(
1
−
ζ)
fun
=
(3.73)
1
8
(
1
−
ξ)(
1
+
η)(
1
−
ζ)
1
8
(
1
−
ξ)(
1
+
η)(
1
+
ζ)
1
+
ξ)(
1
+
η)(
1
+
ζ)
8
(
1
1
8
(
1
+
ξ)(
1
+
η)(
1
−
ζ)
which together with their derivatives with respect to local coordinates, are as usual formed
by the subroutines
shape_fun
and
shape_der
with
ndim=3
and
nod=8
.
