Civil Engineering Reference
In-Depth Information
h
y
5
4
3
x
6
2
7
1
8
x
Figure 3.10 General quadratic quadrilateral element
quadrilateral element with mid-side nodes shown in Figure 3.10. The coordinate matrix
becomes
x
1
y
1
x
2
y
2
x
3
y
3
x
4
y
4
x
5
y
5
x
6
y
6
x
7
y
7
x
8
y
8
coord
=
(3.68)
and using the same local coordinate system for quadrilaterals, the shape functions [
N
]
T
are
now
1
−
ξ)(
1
−
η)(
−
ξ
−
η
−
4
(
1
1
)
1
−
η
2
)
2
(
1
−
ξ)(
1
1
4
(
1
−
ξ)(
1
+
η)(
−
ξ
+
η
−
1
)
1
−
ξ
2
)(
1
2
(
1
+
η)
fun
=
(3.69)
1
4
(
1
+
ξ)(
1
+
η)(ξ
+
η
−
1
)
1
−
η
2
)
2
(
1
+
ξ)(
1
1
4
(
1
+
ξ)(
1
−
η)(ξ
−
η
−
1
)
1
−
ξ
2
)(
1
2
(
1
−
η)
formed by subroutine
shape_fun
. The number of nodes (
nod=8
), and the dimensionality
of the problem (
ndim=2
), serve to uniquely identify the required element, and hence the
appropriate values of
fun
. Their derivatives with respect to local coordinates,
der
,are
again formed by the subroutine
shape_der
.
The sequence of operations described by (3.47) to (3.51) obtains the terms needed for
the element stiffness matrix integration.
