Civil Engineering Reference
In-Depth Information
y
y_coords(1)
1
x
7
4
10
1
3
5
nxe=3
nye=2
y_coords(2)
2
8
5
11
dir='y'
6
2
4
y_coords(3)
3
6
9
12
x_coords(1)
x_coords(2)
x_coords(3)
x_coords(4)
Figure 5.9 Global node and element numbering for mesh of 4-node quadrilaterals num-
bered in the '
y
' direction
g(4)
g(6)
2
3
g(
3)
g(5)
i
=
1
i=2
g(2)
g(8)
i=3
i=4
g(
1)
g(7)
1
4
Figure 5.10 Local node, freedom and Gauss point numbering for the 4-node quadrilateral
(nip
=
4)
and the vertical stress at the centroid of the element immediately beneath the load gives
σ
y
=−
1
.
332 kN/m
2
. Comparison with closed form or other numerical solutions will show
that, with such a coarse mesh of these elements, these results can be quite inaccurate. Such
discretisation errors are inevitable in finite element work, and it is the user's responsibil-
ity to experiment with mesh designs to help discover whether the numerical solution is
adequate.
The fourth example, shown in Figure 5.13, illustrates the use of a higher-order ele-
ment, namely the 8-noded quadrilateral, with nodes numbered in the
x
-direction. The local
node and freedom numbering for this element as shown in Figure 5.14 indicate as usual,
that node 1 is assigned to a corner and the rest follow in a clockwise sense. The general
8-node quadrilateral element stiffness matrix contains fourth order polynomial terms and
thus requires
nip
to be 9 for “exact” integration. It is often the case, however, that the
