Civil Engineering Reference
In-Depth Information
vectors
x_coords
and
y_coords
). For each element, the subroutine works out the nodal
coordinates (held in array
coord
) and the nodal numbering (held in vector
num
). Both the
coordinates and the node numbering are generated in an order consistent with the local node
numbering of the element. In the case of a 4-node quadrilateral, this would be in the order
1-2-3-4 as shown in Figure 3.1 (see Appendix B for local numbering of all elements used
in this topic). For example, element E in Figure 3.16(b) has the node numbering vector
num =
[871011]
T
(3.80)
Once the element node numbering is found, the “steering vector”
g
which holds the freedom
numbers for the element can be found by comparing
num
with the “node freedom array”
nf
. This operation is performed by the subroutine
num_to_g
, which, again for element E
would give
g=
[11 12 9 10 0 13 0 14]
T
(3.81)
In its turn
g
is used to assemble the coefficients of the element property matrices such
as
km
,
kc
,and
mm
into the appropriate places in the overall global coefficient matrix. This
is done according to one of the schemes given in Table 3.7.
A simple three-dimensional mesh is shown in Figure 3.17 and the system coefficients
can again be assembled using the same building blocks.
Although the user of these subroutines does not strictly need to know how the storage is
carried out, examples are given in Figure 3.18 of the most commonly used storage strategies
generated by the subroutines in Table 3.7.
Plane
restrained
in
x-direction
29,30,31
0,27,28
20
21
19
32,33,34
z
10
11
12
nxe
=
2
nye
=
2
nze
=
2
nels
=
8
neq
=
42
nn
=
27
nr
=
19
nod
=
8
nodof
=
3
ndof
=
24
nband
=
29
16,17,18
13,14,15
0,11,12
24
40,41,42
2
3
1
15
0,0,1
2,0,3
4,0,5
Plane
restrained
in
y-direction
24,25,26
27
0,0,0
6
5
4
18
0,0,6
9,0,10
7,0,8
0,0,0
y
7
9
8
x
0,0,0
0,0,0
0,0,0
Fixed base
Figure 3.17 Numbering system and data for a regular 3D mesh with three degrees of
freedom per node
