Page 472 - Shigley's Mechanical Engineering Design

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Screws, Fasteners, and the Design of Nonpermanent Joints 447
Goodman:

S a S m
+ = 1 (8–42)
S e S ut
Gerber:
2

S a S m
+ = 1 (8–43)
S e S ut
ASME-elliptic:
2 2

S a S m
+ = 1 (8–44)
S e S p
Now if we intersect Eq. (8–41) and each of Eqs. (8–42) to (8–44) to solve for S a , and
apply Eq. (8–37), we obtain fatigue factors of safety for each failure criteria in a repeated
loading situation.
Goodman:

S e (S ut − σ i )
n f = (8–45)
σ a (S ut + S e )
Gerber:
1
2
2
n f = S ut S + 4S e (S e + σ i ) − S − 2σ i S e (8–46)
ut
ut
2σ a S e
ASME-elliptic:

S e 2
2
2
n f = S p S + S − σ − σ i S e (8–47)
p
e
2
i
2
σ a (S + S )
p
e
Note that Eqs. (8–45) to (8–47) are only applicable for repeated loads. Be sure to use
K f for both σ a and σ m . Otherwise, the slope of the load line will not remain 1 to 1.
If desired, σ a from Eq. (8–39) and σ i = F i /A t can be directly substituted into
any of Eqs. (8–45) to (8–47). If we do so for the Goodman criteria in Eq. (8–45), we
obtain
2S e (S ut A t − F i )
n f = (8–48)
CP(S ut + S e )
when preload F i is present. With no preload, C = 1, F i = 0, and Eq. (8–48) becomes
2S e S ut A t
n f 0 = (8–49)
P(S ut + S e )
Preload is beneficial for resisting fatigue when n f /n f 0 is greater than unity. For
Goodman, Eqs. (8–48) and (8–49) with n f /n f 0 ≥ 1 puts an upper bound on the pre-
load F i of
F i ≤ (1 − C)S ut A t (8–50)
If this cannot be achieved, and n f is unsatisfactory, use the Gerber or ASME-elliptic
criterion to obtain a less conservative assessment. If the design is still not satisfactory,
additional bolts and/or a different size bolt may be called for.