Civil Engineering Reference
In-Depth Information
Again we can derive a higher order element by adding mid-side nodes on the element
sides. A quadratic element is shown in Figure 3.8 and the local coordinates of nodes are
shown in Table 3.1. The shape functions for the mid-side nodes are given by
1
2
N
1
[
1
K
K
for
n
5
n
n
2
(3.17)
1
2
N
1
K
1
[
[
for
n
6
n
n
2
The corner node functions are constructed in a similar way as for the one-dimensional
element (Figure 3.9)
1
1
1
N
1
[
1
K
N
N
1
5
8
4
2
2
1
1
1
N
1
[
1
K
N
N
2
5
6
4
2
2
1
1
1
(3.18)
N
1
[
1
K
N
N
3
6
7
4
2
2
1
1
1
N
1
[
1
K
N
N
4
7
8
4
2
2
By writing down the shape functions in this manner, it is possible to derive elements
with variable numbers of nodes by deleting appropriate terms. For example, for an
element with no midside node 5, a linear function is assumed between nodes 1 and 2 and
the shape functions are obtained by simply setting
N
5
= 0.
i=3
i=2
i=1
Ș
Ș
3
4
ȟ
7
j=2
6
j=3
8
9
ȟ
j=1
z
1
2
5
y
x
b )
a)
Figure 3.10
Quadratic Lagrange element in a) global and b) local coordinate system
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