Static Stress Distributions
in Parallel Cylindrical Contact
Hertzian
stress analysis is for static loading, but is also applied to pure rolling
contact. The stress distributions within the material are similar to those
shown in Figure 12-15 (p. 358) for the sphere-on-sphere case. Two cases are
possible:
plane stress,
where the cylinders are very short axially as
in some cam roller-followers, and
plane
strain,
where the cylinders
are long axially such as in squeeze-rollers. In the plane-stress case, one of
the principal stresses is zero. In plane strain, all three principal stresses
may be nonzero.
Figure
12-17 shows the principal and maximum shear stress distributions across the
patch width at the surface and along the
z
axis (where
they are largest) for two cylinders in static or pure rolling contact. The
normal stresses are all compressive and are maximal at the surface. They
diminish rapidly with depth into the material and also diminish away from the
centerline, as shown in the figure.
At
the surface on the centerline, the maximum applied normal stresses are
These
stresses are principal, since there is no applied shear stress. The maximum
shear stress
13 on the
z
axis that
results from the combination of stresses is beneath the surface as it was in
the spherical-contact case. For two steel cylinders in static contact, the peak
value and location of the maximum shear stress are[16]
However,
note in Figure 12-17 that, on the
z
axis, the shear stress is not
zero but is 0.22
pmax
at the surface and does not vary greatly over
the depth 0 <
z
< 2a
.
EXAMPLE 12-2
Stresses
in Cylindrical Contact.
Problem:
A
cylindrical roller follower runs against a radial cam. What is the size of the
contact patch between cam and roller and what are the stresses at the point of
minimum radius of curvature? What is the depth of the maximum shear stress?
Given:
The minimum
positive radius of curvature of the cam is 1.25 in. The roller diameter is 1
in. The cam is 0.875-in thick and the roller is 1 in long. Both parts are
steel. The radial load at that point is 500 lb.
Assumptions:
The
roller axis is exactly parallel to the cam surface.
Solution:
1
First determine the size of the contact patch, for which the geometry constant
and material constants are found from equations 12.15a (p. 360) and 12.9a (p.
355).
Note
both materials are the same in this example. The material and geometry
constants can now be used in equation 12.15b (p. 360).
where
a
is the half-width of the contact patch. The
rectangular contact-patch area is
2
The average and maximum contact pressure are found from equations 12.14b and c
(p. 359).
3
The maximum normal stresses in the center of the contact patch at the surface
are then
found using equations 12.17a (p. 360).
4
The maximum shear stress and its location (depth) are found from equations
12.17b (p.
361).
5
All the stresses found exist on the
z
axis and the normal stresses
are principal. These
stresses
apply to the cam and roller follower, as both are steel.
Copyright 2004, Industrial
Press, Inc., New York, NY