Civil Engineering Reference
In-Depth Information
g(8)
g(6)
g(10)
g(7
)
g(5)
g(9)
3
4
5
g(12)
g(4)
g(
3)
g(11)
2
6
g(2)
g(14)
1
8
7
g(13)
g(1
)
g(16)
g(15)
Figure 5.14 Local node and freedom numbering for the 8-node quadrilateral
The fifth example and data shown in Figure 5.17 illustrates an axisymmetric foundation
analysis (
type 2d='axisymmetric'
) as opposed to the plane strain analyses used in
the previous examples. The mesh, while still “rectangular”, involves 4-node quadrilateral
elements of variable size. Node and element numbering is in the “depth” or
'z'
direction.
The mesh size data
nxe
and
nye
in an axisymmetric context, should be interpreted as the
number of “columns” in the radial direction and the number of rows in the depth direc-
tion respectively. Axisymmetric integration is never “exact” using conventional Gaussian
quadrature in elastic analysis, due to the 1
/r
terms that appear in the integrand of the
element stiffness matrix. This example uses
nip=9
, but slightly different results can be
expected as
nip
is increased. This example introduces variable properties in which
E
and
ν
are allowed to assume different values in each horizontal layer of elements. In this case
there are two property groups, so
np types
is read as 2, and two lots of properties are
read into the array
prop
.Since
np types
is greater than 1, then
etype
data is needed
and takes the form of integers 1 or 2 for each element, remembering that the mesh elements
are numbered in the
'z'
direction.
It should be noted that in axisymmetry, four components of strain and stress are required,
so the main program sets
nst
to 4, and the appropriate
dee
matrix (2.77) is returned by
subroutine
deemat
. Furthermore, axisymmetric conditions require the
bee
matrix to have
a fourth row (2.76), and integration (2.74) involves the radius of each integrating point held
in
gc(1)
. The main program checks whether
type 2d
is equal to
'axisymmetry'
and makes these adjustments as necessary.
The nodal loads imply a uniform stress of 1 kN/m
2
is to be applied to a one radian area
of radius 10m (see Appendix A). The computed results for this problem, including stresses
at the element centroids, are given in Figure 5.18. Thus the centreline
z
-displacement is
computed to be
10
−
1
m and the vertical central stress in the depth direction
−
0
.
3176
×
1
.
073 kN/m
2
.
within element 1 is
σ
z
=−
