Civil Engineering Reference
In-Depth Information
This program allows analysis of axisymmetric solids subjected to non-axisymmetric
loads. Variations in displacements, and hence strains and stresses, tangentially are described
by Fourier series (Wilson, 1965; Zienkiewicz and Taylor, 1989). Although the analysis is
genuinely three dimensional, with 3 degrees of freedom at each node, it is only necessary
to discretise the problem in a radial plane. The integrals in radial planes are performed
using Gaussian quadrature in the usual way. Orthogonality relationships between typical
terms in the tangential direction enable the integrals in the third direction to be stated
explicitly. The problem therefore takes on the appearance of a two-dimensional analysis
with the obvious benefits in terms of storage requirements. The disadvantages of the method
over conventional three-dimensional finite element analysis are that, (1) the method is
restricted to axisymmetric solids, and (2) for complicated loading distributions, several
loading harmonic terms may be required, and a global stiffness matrix must be stored for
each. Several harmonic terms may be required for elastic-plastic analysis (Griffiths, 1986),
but for most elastic analyses such as the one described here, one harmonic will often be
sufficient.
It is important to realise that the basic stiffness relationships relate
amplitudes
of load to
amplitudes
of displacement. Once the amplitudes of a displacement are known, the actual
displacement at a particular circumferential location is easily found.
For simplicity, consider only the components of nodal load which are symmetric about
the
θ
=
0 axis of the axisymmetric body. In this case a general loading distribution may
be given by
2
R
o
+
R
1
cos
θ
+
R
2
cos 2
θ
+···
1
R
=
2
Z
o
+
Z
1
cos
θ
+
Z
2
cos 2
θ
+···
1
Z
=
(5.1)
=
T
1
sin
θ
+
T
2
sin 2
θ
+···
T
where
R
,
Z
and
T
represent the load per radian in the radial, depth, and tangential directions.
The bar terms with their superscripts, represent amplitudes of these quantities on the various
harmonics.
For antisymmetric loading, symmetrical about the
θ
=
π/
2 axis, these expressions
become
R
=
R
1
sin
θ
+
R
2
sin 2
θ
+···
Z
=
Z
1
sin
θ
+
Z
2
sin 2
θ
+···
(5.2)
2
T
o
+
T
1
cos
θ
+
T
2
cos 2
θ
+···
1
T
=
Corresponding to these quantities are amplitudes of displacement in the radial, depth, and
tangential directions. Since there are now three displacements per node, there are six strains
at any point taken in the order,
r
z
θ
γ
rz
γ
zθ
γ
θr
T
eps
=
and six corresponding stresses, thus the 6
6 stress-strain matrix
dee
(2.84) is formed by
the subroutine
deemat
as usual. Using the notation of equation (2.79), the [
A]
matrix now
×
