48 2. RELIABILITY OF A COMPONENT UNDER CYCLIC LOAD
Example 2.12
A component is subjected to a fully reversed cyclic bending stress with a constant stress amplitude
a
D 26:6 (ksi). e number of cycles n
L
of this cyclic loading can be described by a normal dis-
tribution with a mean
n
L
D 4:25 10
4
(cycles) and a standard deviation
n
L
D 3253 (cycles).
e fatigue life of this component N
c
at the fatigue strength S
0
f
D
a
D 26:6 (ksi) follows a log-
normal distribution with a log-mean
ln N
c
D 11:01 and the standard deviation
ln N
c
D 0:158.
Calculate the reliability of the component.
Solution:
Since the component fatigue life N
c
of this component at the cyclic stress level
a
D 26:6 (ksi)
is given, we can use Equation (2.41) to build the limit state function:
g
.
N
c
; n
L
/
D N
c
n
L
D
8
ˆ
ˆ
<
ˆ
ˆ
:
> 0 Safe
0 Limit state
< 0 Failure:
(a)
e distribution parameters of the limit state function (a) are listed in Table 2.17.
e limit state function (a) contains one normally distributed random variable and one
lognormal distribution. We can use the R-F method to calculate its reliability, which is presented
in Appendix A.2, to compile a MATLAB program for this example. e iterative results are
listed in Table 2.18. From the iterative results, the reliability index ˇ and corresponding reliability
R of the component in this example are:
ˇ D 2:02131 R D ˆ
.
2:02131
/
D 0:9784:
Table 2.17: e distribution parameters of random variables in Equation (a)
N
C
(lognormal distribution) n
L
(normal distribution)
μ
lnN
C
σ
lnN
C
μ
n
L
σ
n
L
11.01 0.158 42500 3253
2.8.5 RELIABILITY OF A COMPONENT UNDER MODEL #3 CYCLIC
LOADING SPECTRUM
Model #3 cyclic loading spectrum is one constant number of cycles n
L
with a distributed cyclic
stress amplitude
a
. When a component is subjected to model #3 cyclic loading and if the com-
ponent fatigue strength S
Cf
at the fatigue life N D n
L
is provided, we will have the following
2.8. RELIABILITY OF A COMPONENT BY THE P-S-N CURVES APPROACH 49
Table 2.18: e iterative results of Example 2.12 by the R-F method
Iterative #
N
c
*
n
L
*
β
*
|∆β
*
|
1 61,235.48 61,235.48 1.760759
2 44,322.18 44,322.18 2.02009 0.259331
3 45,263.78 45,263.78 2.021308 0.001218
4 45,217.85 45,217.85 2.021311 2.82E-06
limit state function:
g
S
Cf
;
a
D S
Cf
a
D
8
ˆ
ˆ
<
ˆ
ˆ
:
> 0 Safe
0 Limit state
< 0 Failure;
(2.43)
where S
Cf
is the component fatigue strength at the fatigue life N , which is equal to the number
of cycles n
L
of model #3 cyclic loading spectrum.
If the component fatigue strength S
Cf
at the fatigue life N D n
L
is obtained per Equa-
tion (2.39), the limit state function of a component under the model #3 cyclic loading spectrum
will be:
g
k
a
; k
c
; K
f
; S
0
f
;
a
D S
Cf
a
D
k
a
k
b
k
c
K
f
S
0
f
a
D
8
ˆ
ˆ
<
ˆ
ˆ
:
> 0 Safe
0 Limit state
< 0 Failure;
(2.44)
where k
a
, k
b
, k
c
, and K
f
are the surface finish modification factor, the size modification factor,
the loading modification factor, and the fatigue stress concentration factor, respectively. k
a
, k
c
,
and K
f
will be normally distributed random variables. k
b
will be treated as a deterministic con-
stant. S
0
f
is a distributed material fatigue strength at the fatigue life N D n
L
.
a
is a distributed
fully reversed cyclic stress amplitude of the model #3 cyclic loading with the given constant
number of cycles n
L
.
Both limit state functions (2.43) and (2.44) can be used to calculate the reliability of a
component under model #3 cyclic loading with the H-L, R-F, or Monte Carlo method. We
will use Equation (2.44) to run one example.
Example 2.13
e critical section of a machined stepped plate is at its stepped section, as shown in Figure 2.15.
e plate has a thickness t D 0:375 ˙ 0:010
00
. e plate is subjected to a fully reversed axial
loading amplitude F
a
which is a normal distribution with a mean
F
a
D 5:2 (klb) and a standard
deviation
F
a
D 0:61 (klb). e number of cycles of this fully reversed axial loading is n
L
D
50 2. RELIABILITY OF A COMPONENT UNDER CYCLIC LOAD
3 10
5
(cycles). e ultimate strength of this material is 61.5 (ksi). e material fatigue strength
S
0
f
at the fatigue life N D 3:5 10
5
from fully reversed rotating bending stress tests follows a
normal distribution with a mean
S
0
f
D 26:52 (ksi) and standard deviation
S
0
f
D 1:98 (ksi).
Use the Monte Carlo method to calculate the reliability of this plate and its range with a 95%
confidence level.
1
8
R "
W = 3.000±.010
h = 2.000±.010
Figure 2.15: Schematic of the stepped section of a plate.
Solution:
(1) Build the limit state function at the stepped section of the plate.
We will use Equation (2.44) to build the limit state function at the stepped section of the
plate. Per Equation (2.39), we can have the expression of the component fatigue strength S
cf
at
the fatigue life N D 3:5 10
5
:
S
cf
D
k
a
k
b
k
c
K
f
S
0
f
at N D 3:5 10
5
: (a)
Per Equations (2.14)–(2.16), for a machined component, the mean and standard deviation of
the surface finish modification factor k
a
are:
k
a
D 0:9053I
k
a
D 0:05432: (b)
Per Equation (2.17), the size modification factor k
b
for cyclic axial loading is
k
b
D 1: (c)
Per Equations (2.18)–(2.20), for cyclic axial loading, the mean and the standard deviation of the
loading modification factor are:
k
c
D 0:774I
k
c
D 0:1262: (d)
In this example, K
t
D 2:14 and r D 0:25
00
. Per Equations (2.22)–(2.25), we can calculate the
mean and standard deviation of the fatigue stress concentration factor K
f
:
K
f
D 1:880I
K
f
D 0:1504: (e)
2.8. RELIABILITY OF A COMPONENT BY THE P-S-N CURVES APPROACH 51
So, the component fatigue strength S
cf
for this example is
S
cf
D
k
a
K
f
S
0
f
at N D 3:5 10
5
: (f)
e fully reversed stress amplitude
a
due to the fully reversed axial loading F
a
in this example
is:
a
D
F
a
h t
: (g)
e mean and standard deviation of geometric dimensions h per Equation (1.1) are:
h
D 2
00
;
h
D 0:0025
00
: (h)
e mean and standard deviation of geometric dimensions t per Equation (1.1) are:
t
D 0:375
00
;
h
D 0:0025
00
: (i)
So, the limit state function at the stepped section of this plate per Equation (2.44) is
g
k
a
; k
c
; K
f
; S
0
f
; h; t; F
a
D S
Cf
a
D
k
a
k
c
K
f
S
0
f
F
a
h t
D
8
ˆ
ˆ
<
ˆ
ˆ
:
> 0 Safe
0 Limit state
< 0 Failure:
(j)
e distribution parameters of every random variable in Equation (j) are all normal distributions.
eir distribution parameters are listed in Table 2.19.
Table 2.19: e distribution parameters of random variables for Example 2.13 in Equation (j)
k
a
k
c
K
f
S
'
f
(ksi) h t F
a
(klb)
μ
k
a
σ
k
a
μ
k
c
σ
k
c
μ
K
f
σ
K
f
μ
S
'
f
σ
S
'
f
μ
h
σ
h
μ
t
σ
t
μ
F
a
σ
F
a
0.9503 0.05432 0.774 0.1262 1.880 0.1504 26.52 1.98 2 0.0025 0.375 0.0025 5.2 0.61
(2) Reliability of the stepped plate and its range with a 95% confidence level.
We can use the Monte Carlo simulation method to calculate the reliability of this example.
We can follow the Monte Carlo method and the program flowchart in Appendix A.3 to create
a MATLAB program. e estimated reliability of this component R, the probability of failure
F , and the relative error " of F are:
R D 0:9464; F D 0:0536; " D 0:0066:
So, the range of the probability of failure with a 95% confidence level will be:
F D 0:0536 ˙ 0:0536 0:0066 D 0:0536 ˙ 0:0004:
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