**
Stress Distributions in
General Contact
**

The
stress distributions within the material are similar to those shown in Figure
12-17 for the cylinder-on-cylinder case. The normal stresses are all
compressive and are maximal at the surface. They diminish rapidly with depth
into the material and away from the centerline. At the surface on the centerline,
the maximum normal stresses are:[16]

These
applied stresses are also the principal stresses. The maximum shear stress at
the surface associated with these stresses can be found from equation 12.6. The
largest shear stress occurs slightly below the surface, with that distance
dependent on the ratio of the semiaxes of the contact ellipse. For
*
b / a
*
= 1.0, the largest shear stress occurs at
*
z
*
=
0.63
*
a
*
, and for
*
b / a
*
= 0.34 at
*
z
*
=
0.24
*
a
*
. Its peak magnitude is approximately 0.34
*
pmax
*
.[16]

At
the ends of the major axis of the contact ellipse, the shear stress at the
surface is

At
the ends of the minor axis of the contact ellipse, the shear stress at the
surface is

The
location of the largest surface shear stress will vary with the ellipse ratio
*
k
*
3.
For some cases it is as shown in equation 12.21c, but in others it moves to the
center of the ellipse and is found from the principal stresses in equation
12.21a, using equation 12.6 (p. 349).

**
**

**
EXAMPLE 12-3
**

Stresses
in a Crowned Cam Follower.

**
**

**
Problem:
**
**
**
A crowned
cam roller-follower has a gentle radius transverse to its rolling direction to
eliminate the need for critical alignment of its axis with that of the cam. The
cam’s radius of curvature and dynamic load vary around its circumference. What
is the size of the contact patch between cam and follower and what are the
worst-case stresses?

**
**

**
Given:
**
**
**
The roller
radius is 1 in with a 20-in crown radius at 90
°
to the roller radius.
The cam’s radius of curvature at the point of maximum force is 3.46 in and it
is flat axially. (The minimum radius of curvature is 1.72 in.) The rotational
axes of the cam and roller are parallel, which makes the angle between the two
bodies zero. The maximum force is 250 lb, normal to the contact plane.

**
**

**
Assumptions:
**
**
**
Materials
are steel. The relative motion is rolling with <1% sliding.

**
**

**
Solution:
**

1
Find the material constants from equation 12.9a.

2
Two geometry constants are needed from equations 12.19a and b.

*
*

The
angle
ö
is found from their ratio (equation 12.19c),

and
used in Table 12-2 (p. 364) to find the factors
*
ka
*
and
*
kb
*
. Cubic interpolation* for
*
ka
*

and
linear interpolation* for
*
kb
*
gives

* The different interpolation methods are used to best
fit the functions, one of which is linear and the other nonlinear. Plot the
values in Table 12-2 to see this.

*
*

3
The material and geometry constants can now be used in equation 12.19d (p.
364).

*
*

where
*
a
*
is the half-width of the major axis, and
*
b
*
is
the half-width of the minor axis of the

contact patch. The contact-patch area is then:

4
The average and maximum contact pressure can be found from equations 12.18b and
c (p.

363).

5
The maximum normal stresses in the center of the contact patch at the surface
are then found using equations 12.21a (p. 365).

These
stresses are principal:
1 =
*
x
*
,
2 =
*
y
*
,
3 =
*
z
*
.
The maximum shear stress associated with them at the surface will be (from
equation 12.6, p. 349):

6
The largest shear stress under the surface on the
*
z
*
axis
is approximately:

7
All the stresses found so far exist on the centerline of the patch. At the edge
of the patch, at the surface, there will also be a shear stress. Two constants
are found from equation 12.21b (p. 365) for this calculation.

*
*

These
constants are used in equations 12.21c and
*
d
*
(p. 365) to
find the shear stresses on the surface at the ends of the major and minor axes.

Copyright 2004, Industrial
Press, Inc., New York, NY