Page 147 - Shigley's Mechanical Engineering Design

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122 Mechanical Engineering Design
.
This approximation is good for a large curvature where e is small with r n = r c .
Substituting Eq. (3–66) into Eq. (3–64), with r n − y = r, gives

. My r c
σ = (3–67)
I r
.
If r n = r c , which it should be to use Eq. (3–67), then it is only necessary to calculate r c ,
and to measure y from this axis. Determining r c for a complex cross section can be done
easily by most CAD programs or numerically as shown in the before-mentioned refer-
ence. Observe that as the curvature increases, r → r c , and Eq. (3–67) becomes the
straight-beam formulation, Eq. (3–24). Note that the negative sign is missing because y
in Fig. 3–34 is vertically downward, opposite that for the straight-beam equation.

EXAMPLE 3–16 Consider the circular section in Table 3–4 with r c = 3 in and R = 1 in. Determine e by
using the formula from the table and approximately by using Eq. (3–66). Compare the
results of the two solutions.

Solution Using the formula from Table 3–4 gives

R 2 1 2
√ = 2.914 21 in
r n = 2 2 = 2
2 r c − r − R 2 3 − 3 − 1
c
This gives an eccentricity of
Answer e = r c − r n = 3 − 2.914 21 = 0.085 79 in

The approximate method, using Eq. (3–66), yields
4
. I π R/4 R 2 1 2
Answer e = = = = = 0.083 33 in
2
r c A r c (π R ) 4r c 4(3)
This differs from the exact solution by −2.9 percent.

3–19 Contact Stresses

When two bodies having curved surfaces are pressed together, point or line contact
changes to area contact, and the stresses developed in the two bodies are three-
dimensional. Contact-stress problems arise in the contact of a wheel and a rail, in auto-
motive valve cams and tappets, in mating gear teeth, and in the action of rolling
bearings. Typical failures are seen as cracks, pits, or flaking in the surface material.
The most general case of contact stress occurs when each contacting body has a
double radius of curvature; that is, when the radius in the plane of rolling is different
from the radius in a perpendicular plane, both planes taken through the axis of the con-
tacting force. Here we shall consider only the two special cases of contacting spheres
15
and contacting cylinders. The results presented here are due to Hertz and so are fre-
quently known as Hertzian stresses.

15 A more comprehensive presentation of contact stresses may be found in Arthur P. Boresi and Richard
J. Schmidt, Advanced Mechanics of Materials, 6th ed., Wiley, New York, 2003, pp. 589–623.