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In-Depth Information
In Chapter 5 care was taken to distinguish between the “ends” of an element and the
“nodes” of a system. In the study of multidimensional finite elements, it is traditional and
useful to relax this distinction and to refer to the corners of elements as “nodes”.
It is convenient at this stage to treat a specific example and to assign numerical values to
some of the variables of the problem. Choose
a
m
2
,
=
b
=
1
.
0m
,
t
=
0
.
2m
,
E
=
30 GN
/
ν
=
0
.
0
(1)
These lead to
α
=
b
/
a
=
β
=
a
/
b
=
1
.
0
Element Stiffness Matrix
The stiffness matrix of Eq. (6.34), which is identical for all four elements, is
12
3
−
6
−
3
−
6
−
303
u
x
1
u
y
1
u
x
2
u
y
2
u
x
3
u
y
3
u
x
4
u
y
4
3 230
−
3
−
6
−
3
−
6
−
63 2
−
30
−
3
−
63
−
30
−
3 2 3
−
63
−
6
k
i
v
i
=
−
6
−
303 23
−
6
−
3
−
3
−
6
−
3
−
63 230
0
−
3
−
63
−
63 2
−
3
3
−
63
−
6
−
30
−
3 2
k
11
k
i
12
k
i
13
k
i
17
k
i
18
···
u
x
1
u
y
1
u
x
2
u
y
2
u
x
3
u
y
3
u
x
4
u
y
4
k
i
21
k
i
22
k
i
28
···
k
i
31
k
i
32
k
i
38
k
i
41
k
i
48
···
=
(2)
k
i
51
k
i
58
···
k
i
61
k
i
68
···
k
i
71
k
i
78
···
k
i
81
k
i
82
k
i
87
k
i
88
where the displacements at each node have been placed together.
Assembly of the System Stiffness Matrix
The variational principle of Eq. (6.21) establishes a relationship involving a sum over all
element stiffness matrices and loading vectors. This is, of course, the same assembly pro-
cedure developed in Chapter 5 for the displacement method applied to the solution of bar
and beam systems. The element matrices must be summed such that the nodes common to
more than one element are properly taken into account. This is accomplished by identifying
each element in terms of the global node numbering system.
The assembly of the global stiffness matrix of this solid with two-dimensional elements
involves, as is to be expected, more bookkeeping than that of the examples of Chapter 5,
where the elements are simpler. Begin by numbering the global DOF at each node. Table 6.1
gives the assigned numbers for the global DOF for each of the nine global node numbers
of Fig. 6.14b. Since each node has two DOF, there are eighteen global DOF.
An incidence table, Table 6.2, is used to relate the numbering system for each element to
the global topology, i.e., the global node numbers. Note that the nodes of each element are
numbered in the same direction, counterclockwise. The entries of Table 6.2 are obtained by
comparing the element of Fig. 6.10 with each of the elements of Fig. 6.14b. For example, the
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