Page 307 - Marks Calculation for Machine Design
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P1: Shashi
January 4, 2005
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Brown˙C07
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FATIGUE AND DYNAMIC DESIGN
Yield line 289
Alternating stress (s a ) S e Soderberg line Gerber line
S
y
Goodman line
0
0 S y S ut
Mean stress (s )
m
FIGURE 7.11 Goodman, Soderberg, and Gerber lines.
The mathematical expression for the Soderberg theory is given in Eq. (7.21),
S a S m
+ = 1 (7.21)
S e S y
the mathematical expression for the Goodman theory is given in Eq. (7.22),
S a S m
+ = 1 (7.22)
S e S ut
and the mathematical expression for the Gerber theory is given in Eq. (7.23),
2
S a S m
+ = 1 (7.23)
S e S ut
where (S m ) is the mean strength and (S a ) is the alternating strength.
Factors-of-safety (n) can be established for each of these three theories by substituting
the actual mean stress (σ m ) for the mean strength (S m ), substituting the actual alternating
stress (σ a ) for the alternating strength (S a ), and substituting (1/n) for 1.
For the Soderberg theory the factor-of-safety (n) is found from Eq. (7.24),
σ a σ m 1
+ = (7.24)
S e S y n
for the Goodman theory the factor-of-safety (n) is found from Eq. (7.25),
σ a σ m 1
+ = (7.25)
S e S ut n
and for the Gerber theory the factor-of-safety is found from Eq. (7.26),
2
nσ a nσ m
+ = 1 (7.26)
S e S ut
As the Goodman theory is the most commonly used, it is shown by itself in Fig. 7.12
where the mean stress (σ m ) and alternating stress (σ a ) are plotted.