172 3. THE DIMENSION OF A COMPONENT WITH REQUIRED RELIABILITY
3.4 DIMENSION OF A COMPONENT WITH REQUIRED
RELIABILITY UNDER CYCLIC LOADING SPECTRUM
3.4.1 INTRODUCTION
Two probabilistic fatigue theories, the P-S-N curves approach and the K-D probabilistic fa-
tigue damage model, have been discussed in Chapter 2 for establishing limit state function and
calculating the reliability of a component under different cyclic loading spectrum.
Since component stress will be the function of the component dimension, it is an unknown
parameter before the dimension design has been completed. erefore, it is difficult to use the
P-N-S curves approach for determining the dimension with the required reliability under cyclic
loading spectrum.
e K-D probabilistic fatigue damage model can be used to establish the limit state func-
tion of a component under any cyclic loading spectrum. erefore, it can be used to conduct the
component dimension design.
After the limit state function of a component under a cyclic loading spectrum is estab-
lished, the FOSM, modified H-L method, modified R-F method, and/or modified Monte
Carlo method discussed in Section 3.2 can be used to conduct component dimension design,
that is, to determine the dimension with the required reliability.
In this section, we first discuss how to conduct component dimension design with infinite
life. en, we will use the K-D probabilistic model to demonstrate examples to show how to
run dimension design under axial cyclic loading, cyclic direct shearing loading, cyclic torsion
loading, and cyclic bending moment loading. Finally, we will discuss the dimension design of a
rotating shaft under cyclic combined loadings.
3.4.2 COMPONENT WITH AN INFINITE FATIGUE LIFE
e limit state function and its reliability calculation of a component under cyclic loading spec-
trum with an infinite life have been discussed in detail in Section 2.7. After the limit state
function of a component for an infinite life is established, we can run the dimension design with
the required reliability. Now we will show how to conduct component dimension design with
the required reliability and an infinite life under cyclic loading spectrum.
Example 3.18
A machined constant circular bar is subjected a cyclic axial loading. e mean axial loading
F
m
is equal to 12 (klb). e loading amplitude F
a
follows a normal distribution with a mean
F
a
D 9:8 (klb) and a standard deviation
F
a
D 1:1 (klb). e ultimate material strength is 61.5
(ksi). Its endurance limit S
0
e
follows a normal distribution with a mean
S
0
e
D 24:7 (ksi) and a
standard deviation
S
0
e
D 2:14 (ksi), which are based on the fully reversed bending specimen
tests. is bar is designed to have an infinite life. Determine the diameter d of the bar with the
required reliability 0.99 when it has a dimension tolerance ˙0:005
00
.
3.4. DIMENSION OF A COMPONENT UNDER CYCLIC LOADING SPECTRUM 173
Solution:
(1) Preliminary design for the size modification factor k
b
.
Since the loading is a cyclic axial loading, the size modification factor k
b
will be equal to
1. So, in this problem, there is no dimension-dependent parameter.
(2) Establish the limit state function of this problem.
e mean stress
m
and the stress amplitude
a
of the cyclic axial stress due to the cyclic
axial loading can be calculated by the following equations:
m
D
F
m
d
2
=4
D
4F
m
d
2
.ksi/ (a)
a
D
F
a
d
2
=4
D
4F
a
d
2
.ksi/: (b)
Since the cyclic axial stress is not a fully reversed cyclic stress, we need to use Equation (2.21)
to consider the effect of mean stress and converted it into a fully reversed cyclic stress with an
equivalent stress amplitude
aeq
:
aeq
D
a
S
ut
S
ut
m
D
4F
a
d
2
0
B
@
S
ut
S
ut
4F
m
d
2
1
C
A
D
4F
a
S
ut
S
ut
d
2
4F
m
: (c)
e limit state function of this bar per Equation (2.26) is
g
S
0
e
; k
a
; k
c
; F
a
; d
D k
a
k
c
S
0
e
4F
a
S
ut
S
ut
d
2
4F
m
D
8
ˆ
ˆ
<
ˆ
ˆ
:
> 0 Safe
0 Limit state
< 0 Failure:
(d)
ere are five normally distributed variables in this limit state function. e mean and the
standard deviation of the surface finish modification factor k
a
can be determined per Equa-
tions (2.14), (2.15), and (2.16). e mean and the standard deviation of the load modification
factor k
c
can be determined per Equations (2.18), (2.19), and (2.20). e mean
d
and the stan-
dard deviation
d
of the diameter, d can be determined per Equation (1.1). eir distribution
parameters for the limit state function (
d) are listed in Table 3.42.
(3) Use the modified H-L method to determine the diameter.
All random variable in the limit state function (d) are normal distributions. We can fol-
low the procedure of the modified H-L method discussed in Section 3.2.3 and the program
flowchart shown in Figure 3.2 to compile a MATLAB program. e iterative results are listed
in Table 3.43.
174 3. THE DIMENSION OF A COMPONENT WITH REQUIRED RELIABILITY
Table 3.42: e distribution parameters of random variables in Equation (d)
S'
e
(ksi)
k
a
k
c
F
a
(klb) d (in)
μ
S'
e
σ
S'
e
μ
k
a
σ
k
a
μ
k
c
σ
k
c
μ
F
a
σ
F
a
μ
d
σ
d
24.7 2.14 0.905 0.0543 0.774 0.1262 9.8 1.1
μ
d
0.00125
Table 3.43: e iterative results of Example 3.18 by the modified H-L method
Iterative #
S'
e
*
k
a
*
k
c
*
F
a
*
d
*
|∆d
*
|
1 24.7 0.905 0.774 11.8 1.056792
2 22.69742 0.869811 0.551753 10.69245 1.224026 0.167235
3 22.99391 0.876337 0.529924 10.84312 1.241508 0.017482
4 23.05687 0.877242 0.526051 10.87936 1.245111 0.003603
5 23.06867 0.877395 0.52534 10.88653 1.245804 0.000693
6 23.07085 0.877422 0.525209 10.88788 1.245934 0.00013
7 23.07125 0.877427 0.525184 10.88813 1.245958 2.43E-05
According to the result obtained from the program, the mean of the diameter with a
reliability 0.99 is
d
D 1:246
00
: (e)
erefore, the diameter of the bar with the required reliability 0.99 under the specified loading
is
d D 1:246 ˙ 0:005
00
:
Example 3.19
e critical section for a machined rotating shaft is on the shoulder section, as shown in Fig-
ure 3.10. e bending moment M on the shoulder section can be described by a uniform distri-
bution between 1.2 (klb.in) and 1.6 (klb.in). e shaft material’s ultimate strength is 61.5 (ksi).
Its endurance limit S
0
e
follows a normal distribution with a mean
S
0
e
D 24:7 (ksi) and a stan-
dard deviation
S
0
e
D 2:14 (ksi), which are based on the fully reversed bending fatigue specimen
tests. is shaft is designed to have an infinite life. Determine the diameter d of the shaft with
required reliability 0.99 when it has a dimension tolerance ˙0:005
00
.
Solution:
(1) Preliminary design for dimension-dependent parameters K
t
, K
f
, and k
b
.
3.4. DIMENSION OF A COMPONENT UNDER CYCLIC LOADING SPECTRUM 175
1
16
R "
d Ø 1.500±.005
Figure 3.10: Schematic of a shoulder section of a shaft.
According to the preliminary design discussed in Section 3.2.1, we can assume that the
fillet, in this case, will be a sharp-fillet. So, the static stress concentration factor K
t
is
K
t
D 2:7: (a)
e fatigue stress concentration factor
K
f
can be calculated per Equations (
2.22), (2.23), (2.24),
and (2.25):
K
f
D 2:0337;
K
f
D 0:1627: (b)
According to the preliminary design discussed in Section 3.2.1, the size modification factor k
b
can be
k
b
D 0:87: (c)
(2) Establish the limit state function of the rotating shaft.
For a rotating shaft, the bending moment will induce a fully reversed cyclic bending stress.
Its stress amplitude
a
of the fully reversed cyclic bending stress will be:
a
D
M d=2
I
D
M d=2
d
4
=64
D
32M
d
3
: (d)
Per Equation (2.12), the component endurance limit will be:
S
e
D k
a
k
b
k
c
S
0
e
: (e)
e surface finish modification factor k
a
follows a normal distribution. Its mean and standard
deviation can be determined per Equations (2.14), (2.15), and (2.16). k
b
is treated as determin-
istic value and can be calculated per Equation (2.17). It will be updated in each iterative step.
e load modification factor k
c
will be 1 because the shaft is subjected to cyclic bending stress.
176 3. THE DIMENSION OF A COMPONENT WITH REQUIRED RELIABILITY
e limit state function of this problem per Equation (2.26) is
g
M; S
0
e
; k
a
; K
f
; d
D k
a
k
b
S
0
e
K
f
32M
d
3
D
8
ˆ
ˆ
<
ˆ
ˆ
:
> 0 Safe
0 Limit state
< 0 Failure:
(f)
ere are five random variables in this limit state function. e bending moment M is a uniform
distribution. e mean
d
and the standard deviation
d
of d can be determined per Equa-
tion (1.1). e distribution parameters for the limit state function (f) are listed in Table 3.44. In
the table the fatigue stress concentration factor K
f
will be updated in each iterative process.
Table 3.44: e distribution parameters for the limit state function (f)
M (klb.in)
S'
e
(ksi)
k
a
K
f
d (in)
a b μ
S'
e
σ
S'
e
μ
K
a
σ
K
a
μ
K
f
σ
K
f
μ
d
σ
d
1.2 1.6 24.7 2.14 0.905 0.0543 2.0337 0.1627
μ
d
0.0125
(3) Use the modified R-F method to determine the diameter d of the shaft.
e limit state function (f) contains four normally distributed random variable and one
uniform distribution. We can follow the procedure of the modified R-F method discussed in
Section 3.2.4 and the program flowchart shown in Figure 3.3 to compile a MATLAB program.
e iterative results are listed in Table 3.45.
Table 3.45: e iterative results of Example 3.19 by the modified R-F method
Iterative #
M
*
S'
e
*
k
a
*
K
f
*
d
*
k
b
*
|∆d
*
|
1 1.4 24.7 0.905 2.0337 1.142484 0.87
2 1.418724 24.50914 0.901646 3.373608 1.36365 0.842356 0.221167
3 1.495111 23.72887 0.888004 1.598743 1.111104 0.86213
0.252546
4 1.471456 23.66889 0.887261 1.567885 1.09082 0.863932
0.020284
5 1.475578 23.71912 0.888153 1.570136 1.090464 0.863964
0.000357
6 1.474928 23.71568 0.888075 1.570485 1.090456 0.863964
8.25E-06
According to the result obtained from the program, the mean of the diameter with the
reliability 0.99 is
d
D 1:091
00
: (g)
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