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TABLE 6.3
Global DOF Numbers for Each Element
Element
Element Node Numbers
Number
1
2
3
4
1
2
5
4
---
1
1
2
3
4
9
10
7
8
2
3
6
5
---
2
3
4
5
6
11
12
9
10
4
5
8
7
-
--
3
7
8
9
10
15
16
13
14
5
6
9
8
---
4
9
10
11
12
17
18
15
16
Global Node Numbers
of an Element
Global DOF
Numbers
columns of the global stiffness matrix. Similarly, in Table 6.3 element node number 3 of ele-
ment 1 corresponds to global node number 5 and global DOF numbers 9 and 10. The entries
of the element stiffness matrix of (2) corresponding to element node 3 are
k
5
i
,k
6
i
,k
i
5
and
k
i
6
,
i
,
8. These should be placed in the 9th and 10th rows and columns of the
global stiffness matrix. Figure 6.15a displays the location of
k
i
3
,k
i
4
,k
3
i
,k
4
i
,k
5
i
,k
6
i
,k
i
5
,
and
k
i
6
,i
=
1
,
2
,
...
,
8, in the global stiffness matrix, as well as the rest of the stiffness coefficients
for element 1. Figure 6.15b shows the layout of the stiffness matrix of element 2 within the
global stiffness matrix.
The global stiffness matrix is formed by adding entries from the various element stiffness
matrices that occur in the same place in the global matrix. This can be represented as
=
1
,
2
,
...
M
k
i
jk
K
jk
=
(3)
i
=
1
where
M
is the number of elements. If this relationship is to be employed, the subscripts of
the entries in Figs. 6.15a and b should be replaced by the appropriate global DOF numbers.
Figures 6.15a and b would then appear as in Figs. 6.15c and d, respectively. The global
stiffness matrix, formed by superposition of the element stiffness matrices, takes the form
shown in Fig. 6.16a.
After the global stiffness matrix is formed, the boundary conditions, which are shown
in Fig. 6.14b, should be imposed. For example, nodal displacements such as the horizontal
displacement at global node 6 are set equal to zero. Comparison of Fig. 6.14b with Table 6.1
shows that DOF 2, 5, 11, and 17 are constrained. That is, the corresponding displacements
are set equal to zero. These zero displacements eliminate the columns in the global stiffness
matrix for DOF 2, 5, 11, and 17 and also the equivalent rows, as they represent the unknown
reactions at the constrained DOF. The final global stiffness matrix is shown in Fig. 6.16b. It
is seen that after the boundary conditions are imposed, the number of DOF in the global
stiffness matrix is reduced from eighteen to fourteen and the global stiffness matrix is a
square, banded, and symmetric matrix.
Formation of the System Loading Vector
The upper edge of the structure is subjected to a vertical, linearly varying load as shown in
Fig. 6.14. This loading is applied to elements 3 and 4 (Fig. 6.17). Equations (6.42) and (6.43)
provide the loading vector for each element. These vectors must be assembled into a global
vector corresponding to the system stiffness matrix. The loading vectors are reordered such
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