Page 327 - Shigley's Mechanical Engineering Design

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302 Mechanical Engineering Design
In addition to Eq. (6–36), the stress ratio

σ min
R = (6–37)
σ max
and the amplitude ratio

σ a
A = (6–38)
σ m
are also defined and used in connection with fluctuating stresses.
Equations (6–36) utilize symbols σ a and σ m as the stress components at the loca-
tion under scrutiny. This means, in the absence of a notch, σ a and σ m are equal to the
nominal stresses σ ao and σ mo induced by loads F a and F m , respectively; in the presence
of a notch they are K f σ ao and K f σ mo , respectively, as long as the material remains
without plastic strain. In other words, the fatigue stress-concentration factor K f is
applied to both components.
When the steady stress component is high enough to induce localized notch yield-
ing, the designer has a problem. The first-cycle local yielding produces plastic strain
and strain-strengthening. This is occurring at the location where fatigue crack nucle-
ation and growth are most likely. The material properties (S y and S ut ) are new and
difficult to quantify. The prudent engineer controls the concept, material and condition
of use, and geometry so that no plastic strain occurs. There are discussions concerning
possible ways of quantifying what is occurring under localized and general yielding
in the presence of a notch, referred to as the nominal mean stress method, residual
20
stress method, and the like. The nominal mean stress method (set σ a = K f σ ao and
σ m = σ mo ) gives roughly comparable results to the residual stress method, but both are
approximations.
21
There is the method of Dowling for ductile materials, which, for materials with a
pronounced yield point and approximated by an elastic–perfectly plastic behavior
model, quantitatively expresses the steady stress component stress-concentration factor
K fm as

K fm = K f K f |σ max,o | < S y
S y − K f σ ao
K fm = K f |σ max,o | > S y (6–39)
|σ mo |
K fm = 0 K f |σ max,o − σ min,o | > 2S y

For the purposes of this book, for ductile materials in fatigue,
• Avoid localized plastic strain at a notch. Set σ a = K f σ a,o and σ m = K f σ mo .
• When plastic strain at a notch cannot be avoided, use Eqs. (6–39); or conservatively,
set σ a = K f σ ao and use K fm = 1, that is, σ m = σ mo .

20 R. C. Juvinall, Stress, Strain, and Strength, McGraw-Hill, New York, 1967, articles 14.9–14.12; R. C.
Juvinall and K. M. Marshek, Fundamentals of Machine Component Design, 4th ed., Wiley, New York, 2006,
Sec. 8.11; M. E. Dowling, Mechanical Behavior of Materials, 2nd ed., Prentice Hall, Englewood Cliffs,
N.J., 1999, Secs. 10.3–10.5.
21 Dowling, op. cit., pp. 437–438.