Page 354 - Shigley's Mechanical Engineering Design

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Fatigue Failure Resulting from Variable Loading 329
cycles associated with the first tangible evidence of fatigue. In gear studies a similar
K factor is used:
K 1
K g = sin φ (6–65)
4
2
2
where φ is the tooth pressure angle, and the term [(1 − ν )/E 1 + (1 − ν )/E 2 ] is
2
1
2
defined as 1/(πC ), so that
P

F 1 1
S C = C P + (6–66)
w r 1 r 2
8
Buckingham and others reported K 1 for 10 cycles and nothing else. This gives only one
point on the S C N curve. For cast metals this may be sufficient, but for wrought steels, heat-
8
treated, some idea of the slope is useful in meeting design goals of other than 10 cycles.
Experiments show that K 1 versus N, K g versus N, and S C versus N data are recti-
fied by log-log transformation. This suggests that
K 1 = α 1 N β 1 K g = aN b S C = αN β
The three exponents are given by
log(K 1 /K 2 ) log(K g1 /K g2 ) log(S C1 /S C2 )
β 1 = b = β = (6–67)
log(N 1 /N 2 ) log(N 1 /N 2 ) log(N 1 /N 2 )
Data on induction-hardened steel on steel give (S C ) 10 = 271 kpsi and (S C ) 10 =
7
8
239 kpsi, so β, from Eq. (6–67), is
log(271/239)
β = =−0.055
7
8
log(10 /10 )
It may be of interest that the American Gear Manufacturers Association (AGMA) uses
4
β 0.056 between 10 < N < 10 10 if the designer has no data to the contrary
7
beyond 10 cycles.
8
A longstanding correlation in steels between S C and H B at 10 cycles is
0.4H B − 10 kpsi

(S C ) 10 8 = (6–68)
2.76H B − 70 MPa
AGMA uses
0.99 (S C ) 10 = 0.327H B + 26 kpsi (6–69)
7
Equation (6–66) can be used in design to find an allowable surface stress by using
a design factor. Since this equation is nonlinear in its stress-load transformation, the
designer must decide if loss of function denotes inability to carry the load. If so, then
to find the allowable stress, one divides the load F by the design factor n d :

F 1 1 C P F 1 1 S C
σ C = C P + = √ + = √
wn d r 1 r 2 n d w r 1 r 2 n d
2
and n d = (S C /σ C ) . If the loss of function is focused on stress, then n d = S C /σ C . It is
recommended that an engineer
• Decide whether loss of function is failure to carry load or stress.
• Define the design factor and factor of safety accordingly.
• Announce what he or she is using and why.
• Be prepared to defend his or her position.