34 2. RELIABILITY OF A COMPONENT UNDER CYCLIC LOAD
Table 2.11: Fatigue test data under cyclic axial loading with a stress ratio S
r
D 0 [11]
F
a
(lb) F
m
(lb) σ
a
(ksi) σ
m
(ksi)
Frequency Samples
1250 1250 20.833 20.833 20 (Hz) 50
N
92171, 70272, 81285, 122628, 117940, 108584, 105626, 108478, 91504, 124341,
82764, 72315, 61440, 70179, 111914, 105461, 63454, 77811, 112188, 97636,
88818, 125856, 125108, 128583, 83833, 131692, 129422, 91333, 65294, 88960,
112093, 106840, 107792, 76654, 80709, 144728, 90767, 109190, 119484, 107040,
76279, 134353, 61451, 91384, 112944, 78865, 90895, 59056, 133075, 107914
F
a
(lb) F
m
(lb) σ
a
(ksi) σ
m
(ksi)
Frequency Samples
1325 1325 22.083 22.083 20 (Hz) 55
N
75819, 65144, 74538, 57606, 56898, 83130, 70734, 81732, 59123,34609, 51723,
67689, 90634, 92805, 80232, 80146, 50395, 58294, 58306, 77666, 45867, 53352,
45295, 46894, 74711, 66332, 67625, 45168, 65383, 52614, 59240, 81547, 51968,
53924, 63214, 82222, 67763, 71004, 79390, 56301, 72795, 86831, 45676, 59665,
64815, 47176, 63627, 61775, 31474, 62380, 76340, 62633, 55449, 64220, 62850,
F
a
(lb) F
m
(lb) σ
a
(ksi) σ
m
(ksi) Frequency Samples
1350 1350 22.5 22.5 20 (Hz) 30
N
55740, 30958, 61312, 50246, 69952, 69652, 44155, 75026, 69953, 76280, 42858,
81745, 54140, 62791, 83492, 58030, 67268, 64992, 54181, 60379, 41778, 71907,
68426, 52853, 46879, 56964, 51692, 62875, 37836, 40000
F
a
(lb) F
m
(lb) σ
a
(ksi) σ
m
(ksi) Frequency Samples
1375 1375 22.917 22.917 20 (Hz) 30
N
60639, 56861, 45306, 50836, 46038, 56269, 54201, 50630, 67016, 46932, 31499,
67836, 33708, 53025, 52895, 42112, 49586, 30872, 46168, 55442, 51839, 40660,
31760, 45480, 41600, 50742, 64874, 34599, 67186, 53949
F
a
(lb) F
m
(lb) σ
a
(ksi) σ
m
(ksi) Frequency Samples
1400 1400 23.333 23.333 20 (Hz) 30
N
57008, 35051, 32983, 30485, 39212, 68979, 33110, 60411, 54700,50617, 33079,
57704, 44198, 37292, 47345, 45435, 40340, 64051, 51288, 50667, 46380,55792,
45103, 44285, 37200, 31247, 30815, 48816, 50164, 66793
2.8. RELIABILITY OF A COMPONENT BY THE P-S-N CURVES APPROACH 35
Table 2.12: Some statistical values about the number of cycles to failure N
F
a
(lb) F
m
(lb) N
min
N
max
μ
N
σ
N
1,250 1,250 59,056 144,728 98,768 22,527
1,325 1,325 31,474 92,805 63,904 13,693
1,350 1,350 30,958 83,492 58,812 13,354
1,375 1,375 30,872 67,836 49,352 10,574
1,400 1,400 30,485 68,979 46,352 11,093
14
12
10
8
6
4
2
0
5 6 7 8 9 10 11 12 13 14 15
×10
4
Figure 2.9: Histogram for the cyclic stress level:
a
D 22:083 (ksi) and
m
D 22:083 (ksi).
9
8
7
6
5
4
3
2
1
0
3 4 5 6 7 8
×10
4
Figure 2.10: Histogram for the cyclic stress level:
a
D 22:5 (ksi) and
m
D 22:5 (ksi).
stress level F
a
D F
m
D 1325 (lb). Figure 2.10 is the histogram of N with a sample size 30 at the
third stress level F
a
D F
m
D 1350 (lb). Figure 2.11 is the histogram of N with a sample size 30
at the fourth stress level F
a
D F
m
D 1375 (lb). Figure 2.12 is the histogram of N with a sample
size 30 at the fifth stress level F
a
D F
m
D 1400 (lb).
36 2. RELIABILITY OF A COMPONENT UNDER CYCLIC LOAD
9
8
7
6
5
4
3
2
1
0
3 3.5 4 4.5 5 5.5 6 6.5 7
×10
4
Figure 2.11: Histogram for the cyclic stress level:
a
D 22:917 (ksi) and
m
D 22:917 (ksi).
10
9
8
7
6
5
4
3
2
1
0
3 3.5 4 4.5 5 5.5 6 6.5 7
×10
4
Figure 2.12: Histogram for the cyclic stress level:
a
D 23:333 (ksi) and
m
D 23:333 (ksi).
e histograms shown in Figures 2.8, 2.9, and 2.10 indicate that the number of cycles to
failures N might follow a lognormal distribution. e histograms shown in Figures 2.11 and
2.12 do not suggest any distribution. However, they are not symmetric. So, they might also be
a lognormal distribution.
We can use the Chi-Square goodness-of-fit test [7, 8, 14] to check whether they are
a lognormal distribution. In the MATLAB program, the function “chi2gof ” can be used to
conduct the Chi-Square goodness-of-fit test, which has been discussed in Section 2.13 of the
book [8]. e Chi-Square goodness-of-fit test on these data shows that the number of cycles
to failures in each loading level can be treated as a lognormal distribution. e P-N curves for
this set of fatigue test data with corresponding distribution parameters are listed in Table 2.13.
In Table 2.13,
ln N
and
ln N
are the log mean and log standard deviation of the lognormal
distributed number of cycles to failures N .
Dr. E. B. Haugen in 1980 provided a set of P-S distributions of several materials under
fully reversed bending loading or fully reversed axial loading [9]. Table 2.14 lists the P-S distri-
butions at given fatigue life of three different materials. In Table 2.14, the fatigue strength S
0
f
has the unit of ksi. For three-parameter Weibull Distribution of S
0
f
, is the location parameter,
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