The Finite Element Method - Mechanics of Structures: Variational and Computational Methods - page 355

TABLE 6.1
Numbering of the Global DOF
Assigned Global
DOF Number

Global Node
Number
(Fig. 6.14b)

u x u y

1 1 2

2 3 4

3 5 6

4 7 8

5 9 10

6 11 12

7 13 14

8 15 16

9 17 18

TABLE 6.2
Element Topology

Element Node number Element Node Numbers

Number 1 2 3 4 Fig. 6.10

1 1 2 5 4

"

Global Node

2 2 3 6 5

Numbers

3 4 5 8 7

#

Fig. 6.14b

4 5 6 9 8

relationship between the element nodal numbers and the global node numbers for element
1is

Element Node Number Global Node Number

1 −→ 1

2 −→ 2

3 −→ 5

4 −→ 4

The next step of the assembly is to fit the element stiffness matrices into the global matrix
so that the element matrices can be superimposed (added) to form the global stiffness
matrix. To accomplish this, a relationship between each element node number and a global
DOF number has to be established. This can be achieved by the combination of Tables
6.1 and 6.2, i.e., the incidence table of Table 6.2 is expanded to include the global DOF
numbers. Each global node number of Table 6.2 is associated with two corresponding DOF
as provided in Table 6.1. The result is Table 6.3, which relates the element node numbers
and the numbers of the global DOF. From this table, the entries in the element stiffness
matrices of (2) can be located in the global stiffness matrix.
Figure 6.15a shows the distribution of the entries of the element stiffness matrix of ele-
ment 1 in the global stiffness matrix. For example for element 1, it follows from Table 6.3 that
element node number 2 corresponds to global node number 2 and global DOF numbers 3
and 4. Hence, k i 3 ,k i 4 ,k 3 i , and k 4 i ,i , 8, of (2) correspond to node 2 in the element

node numbering system, which is node 2 in the global numbering system with global DOF
3 and 4. Thus, these entries

= 1 , 2 , ...

(

k i 3 ,k i 4 ,k 3 i , and k 4 i )

are placed in the 3rd and 4th rows and