2.8. RELIABILITY OF A COMPONENT BY THE P-S-N CURVES APPROACH 29
Table 2.7: e distribution parameters of random variables in Equation (j)
k
a
S
'
e
(ksi) h
(in) b
(in) M
a
(klb)
μ
k
a
σ
k
a
μ
S
'
e
σ
S
'
e
μ
h
σ
h
μ
b
σ
d
μ
M
a
σ
M
a
0.772 0.0757 24.7 2.14 2.0 0.00025 2.0 0.00025 11.5 1.5
(2) Use the Monte Carlo method to calculate the reliability of the beam.
e limit state function (j) contains five normally distributed random variable and is a
nonlinear function. We can follow the procedure and the flowchart of the Monte Carlo method
presented in Appendix A.3 to compile a MATLAB program. It is displayed in Appendix B as
“B.3: e Monte Carlo method for Example 2.8.” e program results are:
e reliability of the component: R D 15280173=15998400 D 0:9551.
e failure probability of the component: F D 1 R D 0:00459.
e relative error of the failure probability: " D 0:00231.
e error range of the failure probability: F D F ˙"F D 0:00459 ˙ 0:0001.
e reliability range with a 95% confidence level: R D 0:9551 ˙ 0:0001.
2.8 RELIABILITY OF A COMPONENT BY THE P-S-N
CURVES APPROACH
2.8.1 THE MATERIAL P-S-N CURVES
e fatigue failure of a component is mainly due to the “defects” such as manufactured scratches
on the surfaces of a component, “dislocations,” “impurity,” “micro-cracks,” or “micro-cavities”
inside a component. e “defects” or the “uncertainty” are randomly scattered on the surface
of or inside the component. erefore, fatigue strengths of the component under cyclic loading
due to “uncertainty” are random variables. Fatigue tests usually cost lots of time and human
resources. If there are only a few fatigue tests, the traditional S-N is used for fatigue design,
which has been briefly in Section 2.3. When there are at least three different stress amplitude
levels with enough number (at least 30) of fatigue tests at each cyclic stress level, the P-S-N
curves can be used to describe material fatigue strength. e P-S-N curves approach is a well-
known approach for describing the uncertainty in fatigue strength. e P-S-N curves approach,
as shown in Figure 2.5, is a common probabilistic fatigue design theory for components under
cyclic loading spectrum.
e P-S-N curves contain two sets of distribution functions: (1) the distribution functions
of the failure cycle number at different constant cyclic stress amplitudes. ese functions are also
called as the P-N distributions (Probabilistic-Number of the cycle) at the given fatigue strength
S
0
f
; and (2) the distribution functions of cyclic stress amplitudes at different constant cycle num-
bers. ese functions are also named as the P-S distribution (Probabilistic-fatigue Strength) at
30 2. RELIABILITY OF A COMPONENT UNDER CYCLIC LOAD
S
1
S
2
S
3
S
e
Fatigue Life Distribution
Fatigue Strength Distribution
Number of Cycle to Failure
Fatigue Strength
ln(S)
ln(N)
Figure 2.5: A schematic sketch of the P-S-N curve.
the given fatigue life N (the number of cycles to failure). ese two sets of distribution functions
can be used for presenting fatigue test data and estimating the reliability of a component under
cyclic loading.
e distribution functions of the failure cycle number at different constant cyclic stress
amplitudes can be directly obtained through the fatigue tests under the same cyclic stress level
(fatigue strength) and can usually be described by a lognormal distribution [7, 9].
e distribution functions of fatigue strength at different constant fatigue life can usually
be described by the three parameters Weibull distribution or normal distribution or lognormal
distribution, and its distribution parameters are derived by the statistical S-N envelope as shown
in Figure 2.5 [7, 9].
Following, we will show two examples of the distributions of the fatigue life at the given
fatigue strength, that is, the P-N distributions at the given fatigue strength S
0
f
.
In 1969, Dr. Dimitri B. Kececioglu and his colleagues presented P-S-N curves of several
fatigue test data [10]. One set of fatigue date is the P-N curves of a carbon cold-drawn steel
wire specimen under fully reversed cyclic bending loading, as shown in Table 2.8 [10, 11]. e
numbers of fatigue tests in the first two stress levels are more than 30. e number of fatigue
tests in the third level is almost 30. e number of fatigue tests at rests of stress levels are less
than 30. Table 2.8 shows five lognormal distributed material fatigue life at the corresponding
material fatigue strength (fully reversed stress amplitude) of the P-N curves. For example, at
the given fatigue strength S
0
f
D 66 (ksi), the corresponding material fatigue life N is a lognor-
mal distributed random variable with a log mean
log N
D 5:587 and a log standard deviation
log N
D 0:108. is distribution can be used for reliability-based fatigue design.
2.8. RELIABILITY OF A COMPONENT BY THE P-S-N CURVES APPROACH 31
Table 2.8: P-S-N curves of a steel wire specimen under fully reversed cyclic bending stress [10,
11]
Fatigue Strength
(fully reversed
stress amplitude
ksi)
Number of
Specimens
Lognormal Distributed the Number of Cycles to
Failure
Mean μ
logN
Standard Deviation σ
logN
66 50 5.587 0.108
76 37 5.140 0.094
86 26 4.715 0.068
96 17 4.394 0.052
106 10 4.102 0.073
During 2016–2018, we used the hydraulic Instron 8801 fatigue test machine to conduct
a total of 195 fatigue tests under cyclic axial loading [12]. e hydraulic Instron 8801 fatigue
test machine, as shown in Figure 2.6, is a standard device manufactured by Instron for running
fatigue tests under cyclic axial loading. e fatigue test specimen is clamped by the upper grip and
the lower grip. e upper grip is connected to the load cell. e lower grip is directly connected
with a hydraulic actuator. It forces the lower grip to have a cyclic up-and-down motion.
Figure 2.6: A photo of Instron 8801 fatigue test machine.
e fatigue test specimen was made from Aluminum 6061-T6 10 Gauge sheet. e chem-
ical compositions of this 10 gauge 6061-T6 sheet provided by the material supplier are displayed
32 2. RELIABILITY OF A COMPONENT UNDER CYCLIC LOAD
in Table 2.9. e mechanical properties of this sheet provided by the material supplier are listed
in Table 2.10.
Table 2.9: Chemical composition of the 6016-T6 sheet
Element Si Fe Cu Mn Mg Cr Zn Ti Others Al
% 0.629 0.306 0.276 0.020 1.030 0.185 0.004 0.021 0.15 Remain
Table 2.10: Mechanical properties of the sheet
Size
0.100˝X48˝X144˝
Tensile
strength
51.2 ksi
Yield
strength
41.8 ksi Elongation % 16
e fatigue specimen was a rectangular sheet-type flat fatigue test specimen, shown in
Figure 2.8, and was designed according to ASTM STM E466–15, Standard Practice for Con-
ducting Force Controlled Constant Amplitude Axial Fatigue Tests of Metallic Materials [13].
e manufacturing route for this specimen is: (1) use a sheet shearing machine to shear the sheet
into a rectangular plate 11:75
00
2:375
00
along the longitudinal direction; (2) use a milling ma-
chine to mill the sheared rectangular plate into a rectangular plate 11:375
00
2
00
; and (3) clamp
the milled rectangular plate in the designed fixture and then use the compiled CNC program to
mill the plate per the drawing shown in Figure 2.7.
(
2.250
)
TYP.
R6"
TYP.
.600 ± .005
1.250 ± .010
11.375 ± .020
2.000 ± .010
(10 Gage)
Figure 2.7: e dimensions of the sheet-type flat fatigue specimen.
Since the thickness of the sheet-type fatigue specimen is only 0:100
00
, the fatigue test
specimen under cyclic axial loading is very easy to be buckling. To avoid the buckling during the
cyclic axial loading, the cyclic axial loading for the fatigue test was with a loading ratio S
r
D 0,
that is, the axial loading varying between 0 to maximum tensile loading. e fatigue tests under
five different stress levels with a total 195 of fatigue tests have been conducted. e fatigue test
procedure is: (1) visually check the fatigue test specimen before testing. Any abnormal fatigue
specimen such as a bent specimen or visual big crack or scratch on the outer surfaces of the
2.8. RELIABILITY OF A COMPONENT BY THE P-S-N CURVES APPROACH 33
14
12
10
8
6
4
2
0
5 6 7 8 9 10 11 12 13 14 15
×10
4
Figure 2.8: Histogram for the cyclic stress level:
a
D 20:833 (ksi) and
m
D 20:833 (ksi).
specimen will be discarded; (2) measure the actual width and thickness of the specimen and
record them in a test log; (3) install specimen and make sure that the specimen is installed
vertically and centrally in the upper and lower grips; (4) use the compiled WaveMatrix program
to run the cyclic axial loading fatigue test, which is the control program for the Instron 8801
fatigue test machine, until the specimen is fractured; and (5) record the number of cycles at
failure and some special notes in the test log.
For all following fatigue tests, the dimensions of the middle section of the specimen are
the width b D 0:600 ˙ 0:005
00
and the thickness t D 0:100 ˙ 0:005
00
. e loading frequency is
20 (Hz); that is, the cyclic loading cycles will be repeated 20 times in a second. e stress ratio
S
r
D
F
min
F
max
D 0. e test conditions and results of these five different stress levels are listed in
Table 2.11. Stresses in Table 2.11 are calculated by using the nominal dimensions, that is, the
width b D 0:600
00
and the thickness t D 0:100
00
.
In Table 2.11, F
a
and F
m
are the loading amplitude and the loading mean of the cyclic
axial loading.
a
and
m
are the stress amplitude and the stress mean of the cyclic axial stress
under the cyclic axial loading with F
a
and F
m
. N is the number of cycles to failure of each
fatigue specimen test under the corresponding specified cyclic loading.
e minimum N
min
, the maximum N
max
, the mean
N
, and the standard deviation
N
of the number of cycles to failure under five different stress levels are listed in Table 2.12. From
Table 2.12, the number of cycles to failure at the same stress level has a very big variation. For
example, at the cycling loading level F
a
D 1325 (lb) and F
m
D 1325 (lb), the maximum cycle of
numbers to failure is almost three times of the minimum cycle of the number to failure.
e number of fatigue tests in each stress level listed in Table 2.11 is more than 30.
Histograms of the number of cycles to failure N in each stress level are displayed in Fig-
ures 2.8–2.12. Figure 2.8 is the histogram of N with a sample size 50 at the first stress level
F
a
D F
m
D 1250 (lb). Figure 2.9 is the histogram of N with a sample size 55 at the second
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