Thomas Industrial Library

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.