This preview shows pages 1–4. Sign up to view the full content.
Section 4.3
Solid Mechanics Part II
Kelly
66
4.3
Plane Axisymmetric Problems
In this section are considered
plane
axisymmetric problems
.
These are problems in
which both the geometry
and
loading are axisymmetric.
4.3.1
Plane Axisymmetric Problems
Some three dimensional (not necessarily plane) examples of axisymmetric problems
would be the thickwalled (hollow) cylinder under internal pressure, a disk rotating about
its axis
1
, and the two examples shown in Fig. 4.3.1; the first is a complex component
loaded in a complex way, but exhibits axisymmetry in both geometry and loading;
the
second is a sphere loaded by concentrated forces along a diameter.
Figure 4.3.1: axisymmetric problems
A twodimensional (plane) example would be one plane of the thickwalled cylinder
under internal pressure, illustrated in Fig. 4.3.2
2
.
Figure 4.3.2: a cross section of an internally pressurised cylinder
It should be noted that many problems involve axisymmetric geometries but non
axisymmetric loadings, and
vice versa
.
These problems are
not
axisymmetric.
An
example is shown in Fig. 4.3.3 (the problem involves a plane axisymmetric geometry).
1
the rotation induces a stress in the disk
2
the rest of the cylinder is coming out of, and into, the page
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document Section 4.3
Solid Mechanics Part II
Kelly
67
Figure 4.3.3: An axially symmetric geometry but with a nonaxisymmetric loading
The important characteristic of these axisymmetric problems is that all quantities, be they
stress, displacement, strain, or anything else associated with the problem,
must be
independent
of the circumferential variable
θ
.
As a consequence, any term in the
differential equations of §4.2 involving the derivatives
2
2
/
,
/
∂
∂
∂
∂
, etc. can be
immediately set to zero.
4.3.2
Governing Equations for Plane Axisymmetric Problems
The twodimensional straindisplacement relations are given by Eqns. 4.2.4 and these
simplify in the axisymmetric case to
⎟
⎠
⎞
⎜
⎝
⎛
−
∂
∂
=
=
∂
∂
=
r
u
r
u
r
u
r
u
r
r
r
rr
θθ
ε
2
1
(4.3.1)
Here, it will be assumed that the displacement
0
=
u
.
Cases where
0
≠
u
but where the
stresses and strains are still independent of
are termed
quasiaxisymmetric problems
;
these will be examined in a later section.
Then 4.3.1 reduces to
0
,
,
=
=
∂
∂
=
r
r
r
rr
r
u
r
u
(4.3.2)
It follows from Hooke’s law that
0
=
σ
r
.
The nonzero stresses are illustrated in Fig.
4.3.4.
axisymmetric plane
representative of
feature
Section 4.3
Solid Mechanics Part II
Kelly
68
Figure 4.3.4: stress components in plane axisymmetric problems
4.3.3
Plane Stress and Plane Strain
Two cases arise with plane axisymmetric problems: in the plane stress problem, the
feature is very thin and unloaded on its larger freesurfaces, for example a thin disk under
external pressure, as shown in Fig. 4.3.5.
Only two stress components remain, and
Hooke’s law 4.2.5a reads
[]
rr
rr
rr
E
E
νσ
σ
ε
θθ
−
=
−
=
1
1
or
rr
rr
rr
E
E
νε
ν
+
−
=
+
−
=
2
2
1
1
(4.3.3)
with
()
0
,
=
=
+
−
=
θ
z
zr
rr
zz
E
and
0
=
zz
.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview. Sign up
to
access the rest of the document.
This note was uploaded on 01/20/2012 for the course ENGINEERIN 2 taught by Professor Staff during the Fall '11 term at Auckland.
 Fall '11
 Staff

Click to edit the document details