Page 475 - Shigley's Mechanical Engineering Design

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450 Mechanical Engineering Design
of the load line L with the respective failure lines at points C, D, and E defines a set of
strengths S a and S m at each intersection. Point B represents the stress state σ a, σ m . Point A
is the preload stress σ i . Therefore the load line begins at A and makes an angle having a
unit slope. This angle is 45° only when both stress axes have the same scale.
The factors of safety are found by dividing the distances AC, AD, and AE by
the distance AB. Note that this is the same as dividing S a for each theory by σ a .
The quantities shown in the caption of Fig. 8–22 are obtained as follows:
Point A
14.4
F i
σ i = = = 63.72 kpsi
A t 0.226
Point B
CP 0.280(5)
σ a = = = 3.10 kpsi
2A t 2(0.226)
σ m = σ a + σ i = 3.10 + 63.72 = 66.82 kpsi

Point C
This is the modified Goodman criteria. From Table 8–17, we find S e = 18.6 kpsi.
Then, using Eq. (8–45), the factor of safety is found to be
S e (S ut − σ i ) 18.6(120 − 63.72)
Answer n f = = = 2.44
σ a (S ut + S e ) 3.10(120 + 18.6)
Point D
This is on the proof-strength line where
(1)
S m + S a = S p
In addition, the horizontal projection of the load line AD is
(2)
S m = σ i + S a
Solving Eqs. (1) and (2) simultaneously results in

S p − σ i 85 − 63.72
S a = = = 10.64 kpsi
2 2
The factor of safety resulting from this is

Answer n p = S a = 10.64 = 3.43
σ a 3.10
which, of course, is identical to the result previously obtained by using Eq. (8–29).
A similar analysis of a fatigue diagram could have been done using yield strength
instead of proof strength. Though the two strengths are somewhat related, proof
strength is a much better and more positive indicator of a fully loaded bolt than is the
yield strength. It is also worth remembering that proof-strength values are specified
in design codes; yield strengths are not.
We found n f = 2.44 on the basis of fatigue and the modified Goodman line, and
n p = 3.43 on the basis of proof strength. Thus the danger of failure is by fatigue, not
by overproof loading. These two factors should always be compared to determine
where the greatest danger lies.