20 2. RELIABILITY OF A COMPONENT UNDER CYCLIC LOAD
• the fatigue strength S
0
f
D 31:89 ksi at the fatigue life N D 6:5 10
5
(cycles) under a fully
reversed cyclic stress, that is, (S
0
f
D 31:89 ksi, N D 6:5 10
5
cycles).
2.6 THE FATIGUE STRESS CONCENTRATION FACTOR
e fatigue stress-concentration factor K
f
will be used to multiply the nominal stress amplitude
and can be treated as a normally distributed random variable. Its mean
K
f
can be calculated
by the following equation [7]:
K
f
D
K
t
1 C
2
p
r
K
t
1
K
t
p
a
; (2.22)
where K
t
is static stress concentration factor which can be obtained through any design hand-
book and some websites. r is the notch radius in the unit of inch.
p
a is defined as the Neuber
constant and can be calculated through the following equation:
p
a D
8
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
<
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
:
5
S
ut
For a transverse hole
4
S
ut
For a shoulder
5
S
ut
For a groove;
(2.23)
where S
ut
is material tensile ultimate strength in the unit of ksi.
e coefficient of variance of the fatigue stress concentration factor K
f
can be estimated
by the following equation:
K
f
D
8
ˆ
ˆ
<
ˆ
ˆ
:
0:11 For a transverse hole
0:08 For a shoulder
0:13 For a groove:
(2.24)
e standard deviation of the fatigue stress concentration factor K
f
will be:
K
f
D
K
f
K
f
: (2.25)
2.6. THE FATIGUE STRESS CONCENTRATION FACTOR 21
Example 2.5
A machined steel shaft with a shoulder, as shown in Figure 2.3 is subjected to cyclic bending
stress. e ultimate strength of the shaft material is 61.5 (ksi). Determine the fatigue stress
concentration factor K
f
in the shoulder of the shaft.
3
16
R "
Ø 2.00
Ø 3.00
Figure 2.3: Schematic of a segment of a shaft with a shoulder.
Solution:
e Neuber constant on the shaft shoulder per Equation (2.23) is:
p
a D
4
S
ut
D
4
61:5
D 0:06504: (a)
From design handbooks or some websites about static stress concentration factors, the theoretical
geometric stress concentration of this shoulder K
t
due to bending is
K
t
D 1:78: (b)
e mean
K
f
of the fatigue stress concentration factor K
f
in the shoulder of the shaft per
Equation (2.22) is
K
f
D
K
t
1 C
2
p
r
K
t
1
K
t
p
a
D
1:78
1 C
2
p
0:1875
1:78 1
1:78
0:06504
D 1:573: (c)
e standard deviation of the fatigue stress concentration factor K
f
per Equations (2.24) and
(2.25) is:
K
f
D
K
f
K
f
D 1:573 0:08 D 0:1258: (d)
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