Page 331 - Shigley's Mechanical Engineering Design

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306 Mechanical Engineering Design
Five criteria of failure are diagrammed in Fig. 6–27: the Soderberg, the modified
Goodman, the Gerber, the ASME-elliptic, and yielding. The diagram shows that only
the Soderberg criterion guards against any yielding, but is biased low.
Considering the modified Goodman line as a criterion, point A represents a limit-
ing point with an alternating strength S a and midrange strength S m. The slope of the load
line shown is defined as r = S a /S m .
The criterion equation for the Soderberg line is

S a S m
+ = 1 (6–40)
S e S y
Similarly, we find the modified Goodman relation to be

S a S m
+ = 1 (6–41)
S e S ut
Examination of Fig. 6–25 shows that both a parabola and an ellipse have a better
opportunity to pass among the midrange tension data and to permit quantification of the
probability of failure. The Gerber failure criterion is written as
2

S a S m
+ = 1 (6–42)
S e S ut
and the ASME-elliptic is written as
2 2

S a S m
+ = 1 (6–43)
S e S y
The Langer first-cycle-yielding criterion is used in connection with the fatigue
curve:
(6–44)
S a + S m = S y
The stresses nσ a and nσ m can replace S a and S m , where n is the design factor or factor
of safety. Then, Eq. (6–40), the Soderberg line, becomes

σ a σ m 1
Soderberg + = (6–45)
S e S y n
Equation (6–41), the modified Goodman line, becomes
σ a σ m 1
mod-Goodman + = (6–46)
S e S ut n
Equation (6–42), the Gerber line, becomes
2

nσ a nσ m
Gerber + = 1 (6–47)
S e S ut
Equation (6–43), the ASME-elliptic line, becomes
2 2

nσ a nσ m
ASME-elliptic + = 1 (6–48)
S e S y
We will emphasize the Gerber and ASME-elliptic for fatigue failure criterion and the
Langer for first-cycle yielding. However, conservative designers often use the modified
Goodman criterion, so we will continue to include it in our discussions. The design
equation for the Langer first-cycle-yielding is

S y
Langer static yield σ a + σ m = (6–49)
n