Civil Engineering Reference
In-Depth Information
A
i,l
is defined in equation (3.12) and
KK
KK
m
B
if j
z
m
jm
(3.20)
j
m
B
1
if j
m
jm
where
i
and
j
are the column and row numbers of the nodes. This numbering is defined
in Figure 3.10. The nodes are given by
n
(1,1) = 1
n
(2,1) = 2
n
(3,1) = 5
n
(1,2) = 4
n
(2,2) = 3
n
(3,2) = 7
n
(1,3) = 8
n
(2,3) = 6
n
(3,3) = 9
The Serendipity and Lagrange shape functions are compared in Figure 3.11
The Lagrange element has an additional 'bubble mode' and is, therefore, able to describe
complicated shapes more accurately. Triangular elements can be formed from
quadrilateral elements, by assigning the same global node number to two or three corner
nodes. Such degenerate elements are shown in Figure 3.12.
Ș
Ș
4
3
4
7
3
ȟ
8
ȟ
6
2
2
1
1
5
Figure 3.12
Linear and quadratic degenerate elements
Alternatively triangular elements may be defined using the iso-parametric concept. In
Figure 3.13 we show a triangular element in the global and local coordinate system. The
shape functions for the transformation are defined as
4
N
N
N
(, ) 1
(, )
(, )
[K [ K
[K [
[K K
1
(3.21)
2
3
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