Page 364 - Shigley's Mechanical Engineering Design

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Fatigue Failure Resulting from Variable Loading 339
curves are most likely to be used with a radial load line we will use the equation given
in Table 6–7, p. 307, expressed in terms of the strength means as
⎡ ⎤
2
2 ¯ 2
r S ut 2S e
¯
¯
S a = ⎣ −1 + 1 + ⎦ (6–80)
¯ ¯
2S e rS ut
¯ ¯ ¯
Because of the positive correlation between S e and S ut , we increment S e by C Se S e , S ut
by C Sut S ut , and S a by C Sa S a , substitute into Eq. (6–80), and solve for C Sa to obtain
¯
¯
¯
⎧ ⎫

2S e (1 + C Se )

⎨ ¯ 2 ⎬
−1 + 1 +
¯
(1 + C Sut ) 2 ⎩ rS ut (1 + C Sut ) ⎭
C Sa = − 1 (6–81)
⎡
⎤
1 + C Se ¯ 2
2S e
−1 + 1 +
⎣ ⎦
¯
rS ut
Equation (6–81) can be viewed as an interpolation formula for C Sa , which falls between
C Se and C Sut depending on load line slope r. Note that S a = S a LN(1, C Sa ).
¯
Similarly, the ASME-elliptic criterion of Table 6–8, p. 308, expressed in terms of
its means is
¯ ¯
rS y S e
¯ (6–82)
S a = '
2 ¯ 2
r S + S ¯ 2 e
y
¯
¯
¯
¯
¯
Similarly, we increment S e by C Se S e , S y by C Sy S y , and S a by C Sa S a , substitute into
¯
Eq. (6–82), and solve for C Sa :
,
- 2 ¯ 2 ¯ 2
r S + S
- y e
C Sa = (1 + C Sy )(1 + C Se ) . − 1 (6–83)
2
2 ¯ 2
¯ 2
r S (1 + C Sy ) + S (1 + C Se ) 2
y
e
Many brittle materials follow a Smith-Dolan failure criterion, written deterministi-
cally as
nσ a 1 − nσ m /S ut
= (6–84)
S e 1 + nσ m /S ut
Expressed in terms of its means,
¯ ¯ ¯
S a 1 − S m /S ut
= (6–85)
¯ ¯ ¯
S e 1 + S m /S ut
¯
For a radial load line slope of r, we substitute S a /r for S m and solve for S a , obtaining
¯
¯
⎡ ⎤

¯ ¯ ¯ ¯
rS ut + S e 4rS ut S e
¯
S a = ⎣ −1 + 1 + ⎦ (6–86)
2 (rS ut + S e ) 2
¯
¯
and the expression for C Sa is
¯
¯
rS ut (1 + C Sut ) + S e (1 + C Se )
C Sa =
¯
2S a
(6–87)

#
¯ ¯
4rS ut S e (1 + C Se )(1 + C Sut )
· −1 + 1 + − 1
¯
[rS ut (1 + C Sut ) + S e (1 + C Se )] 2
¯