3.6. REFERENCES 203
3.6 REFERENCES
[1] Le, Xiaobin, Reliability-Based Mechanical Design, Volume 1: Component under Static Load,
Morgan & Claypool Publishers, San Rafael, CA, 2020. 121, 123, 124, 125, 129, 131,
136, 137, 138, 145, 152, 154, 157, 161, 164, 165, 166, 167, 168, 170
[2] Budynas, R. G. and Nisbett, J. K., Shigley’s Mechanical Engineering Design, 10th ed., Mc-
Graw Hill Education, New York, 2014. 122, 165, 167
[3] Le, Xiaobin, e research on reliability fatigue design of mechanical components, Doc-
toral dissertation, Shanghai Jiao Tong University, Shanghai, 1993. 129
[4] Zong, W. H. and Le, Xiaobin, Probabilistic Design Method of Mechanical Components,
Shanghai Jiao Tong University Publisher, Shanghai, China, September 1995.
[5] Le, Xiaobin and Johan, R., A Probabilistic Approach for Determining the Compo-
nent’s Dimension under Fatigue Loadings, ASME International Design Engineering
Technical Conferences (IDETC), San Diego, CA, August 31–September 2, 2009. DOI:
10.1115/detc2009-86334. 129, 136
[6] Le, Xiaobin, A probabilistic fatigue damage model for describing the entire set of fatigue
test data of the same material, ASME International Mechanical Engineering Congress and
Exposition, IMECE–10224, Salt Lake City, UT, November 8–14, 2019.
3.7 EXERCISES
3.1. Describe and explain one example in your design where the shape of the component has
roughly defined the sketch of assembly.
3.2. Describe and explain one example where the radius of the fillet in the stress concentra-
tion area is determined by the purchased component.
3.3. Describe and explain one example where the radius of the fillet in the stress concentra-
tion area can be treated as a well-rounded fillet.
3.4. A rectangular bar is subjected to an axial loading F
a
D 3:1 ˙ 0:60 (klb). e width of the
bar b is 1:500 ˙ 0:005
00
. e yield strength S
y
of the bar of a ductile material follows
a normal distribution with a mean
S
y
D 34:5 (ksi) and a standard deviation
S
y
D
3:12 (ksi). Use the FORM method to determine the height h of the bar with a reliability
0.95 when the height h has a dimension tolerance ˙0:005
00
.
3.5. A constant round bar is subjected to an axial loading F
a
D 2:1 ˙ 0:20 (klb). e yield
strength S
y
of the bar of a ductile material follows a normal distribution with a mean
S
y
D 34:5 (ksi) and the standard deviation
S
y
D 3:12 (ksi). Use the FOSM method
204 3. THE DIMENSION OF A COMPONENT WITH REQUIRED RELIABILITY
to determine the diameter of the bar with a reliability 0.99 when the diameter has a
dimension tolerance ˙0:005
00
.
3.6. Use the modified H-L method to do Problem 3.5.
3.7. A bar connected to the supporter at point A is subjected to two concentrated loads
F
B
and F
C
as shown in Figure 3.15. e axial loads are: F
B
D 1500 ˙ 120 (lb) and
F
C
D 1000 ˙ 90 (lb). e length of the AB segment L
1
D 8:00 ˙ 0:003
00
and the
length of BC segment L
2
D 10; 00 ˙ 0:003
00
. e Young’s modulus E of the bar ma-
terial follows a normal distribution with a mean
E
D 2:73 10
7
(psi) and a standard
deviation
E
D 1:30 10
6
(psi). e maximum allowable deflection of the bar is 0.008
00
.
Use the modified H-L method to determine the diameter of the bar with a reliability
0.99 when its dimension tolerance is ˙0:005
00
.
L
1
L
2
F
B
F
C
B
A
C
Figure 3.15: A bar under two axial loading.
3.8. Use the modified Monte Carlo method to do Problem 3.7.
3.9. e pin at point A, as shown in Figure 3.15, is a double shear pin. e shear yield
strength S
sy
of the pin of a ductile material follows a normal distribution with a mean
S
sy
D 32:2 (ksi) and a standard deviation
S
sy
D 3:63 (ksi). Use the H-L method to
determine the diameter of the pin with a reliability 0.99 when its dimension tolerance
is ˙0:005
00
.
3.10. Use the modified Monte Carlo method to do Problem 3.9.
3.11. e double shear pin is subjected to a shearing force V (klb) which can be described by
a lognormal distribution with a log mean
ln V
D 0:721 and a log standard deviation
ln V
D 0:137. e shear yield strength S
sy
of the pin of a ductile material follows a
normal distribution with a mean
S
sy
D 32:2 (ksi) and the standard deviation
S
sy
D
3:63 (ksi). Use the modified R-F method to determine the diameter of the pin with a
reliability 0.99 when its dimension tolerance is ˙0:005
00
.
3.7. EXERCISES 205
3.12. Use the modified Monte Carlo method to do Problem 3.11.
3.13. A constant cross-section shaft is subjected to a torque T . e torque T (lb.in) can be
described by a lognormal distribution with a mean
ln T
D 7:76 and a standard deviation
ln T
D 0:194. e shear yield strength S
sy
of the shaft of a ductile material follows a
normal distribution with a mean
S
sy
D 32,200 (psi) and the standard deviation
S
sy
D
3630 (psi). Use the modified R-F method to determine the diameter of the shaft with
a reliability 0.99 if its dimension tolerance is ˙0:005
00
.
3.14. Schematic of a segment of a shaft at its critical cross-section as shown in Figure 3.16 is
subjected to a torque T D 1350 ˙ 95 (lb.in). e fillet radius r and the larger diameter
d
2
are r D 1=32
00
, and d
2
D 1:500 ˙ 0:005
00
. e shear yield strength S
sy
of the shaft of
a ductile material follows a normal distribution with a mean
S
sy
D 32,200 (psi) and
the standard deviation
S
sy
D 3630 (psi). Use the modified H-L method to determine
the diameter d
2
with a reliability 0.99 when its dimension tolerance is ˙0:005
00
.
T
d
1
d
2
r
T
Figure 3.16: Schematic of a segment of a shaft.
3.15. Use a modified Monte Carlo method to do Problem 3.14.
3.16. A constant cross-section shaft with a length L D 15:00 ˙ 0:032
00
is subjected to a pair
of opposite torques at both ends T D 1800 ˙120 (lb.in). e shear Young’s modulus
follows a normal distribution with a mean
G
D 1:117 10
7
(psi) and a standard de-
viation
G
D 2:793 10
5
. If the design requirement is the angle of twist between two
ends is less than 1
ı
, use the modified H-L method to determine the diameter of the
shaft with a reliability 0.99 if its dimension tolerance is ˙0:005
00
.
3.17. Use the modified Monte Carlo method to do Problem 3.16.
3.18. A simple support circular beam is subjected to a concentrated force in the middle, as
shown in Figure 3.17. e concentrated force is P D 1500 ˙ 180 (lb). e span of the
beam is L D 22 ˙ 0:065
00
. e yield strength S
y
of the material follows a normal distri-
bution with the mean
S
y
D 34500 (psi) and the standard deviation
S
y
D 3120 (psi).
206 3. THE DIMENSION OF A COMPONENT WITH REQUIRED RELIABILITY
Use the modified H-L method to determine the diameter of the beam with a reliability
0.99 if its dimension tolerance is ˙0:005
00
.
L
0.5L
P
Figure 3.17: A simple support beam.
3.19. Use the modified Monte Carlo method to do Problem 3.18.
3.20. A simple support beam, as shown in Figure 3.18, is subjected to a uniform distributed
loading on the beam and a concentrated loading in the middle of the beam. e span
of the beam is L D 15:00 ˙ 0:032
00
. e beam has a rectangular shape with a width b D
1:00 ˙ 0:010
00
. e uniform distributed loading is w D 100 (lb/in). e concentrated
force in the middle is P D 1500 ˙ 180 (lb). e yield strength S
y
of the material follows
a normal distribution with the mean
S
y
D 34,500 (psi) and the standard deviation
S
y
D 3120 (psi). Use the modified H-L method to determine the height of the beam
with a reliability 0.99 if its dimension tolerance is ˙0:010
00
.
L
0.5L
w
P
Figure 3.18: A simple support beam.
3.21. Use the modified Monte Carlo method to do Problem 3.20.
3.22. A circular cantilever beam as shown in Figure 3.19 is subjected to a concentrated force
at the free end P D 150 ˙ 80 (lb). e length of the beam is L D 20:0 ˙0:032
00
. e
Young’s modulus of the beam material follows a normal distribution with the mean
E
D 2:76 10
7
(psi) and the standard deviation
E
D 6:89 10
5
(psi). e maxi-
mum allowable deflection of the beam is D 0:022
00
. Use the modified H-L method
to determine the diameter of the beam with a reliability 0.99 if its dimension tolerance
is ˙0:005
00
.
3.7. EXERCISES 207
L
P
Figure 3.19: A cantilever beam.
3.23. Use the modified Monte Carlo method to do Problem 3.22.
3.24. e critical section of a constant circular shaft is subjected to a torque T D 3000 ˙
150 (lb.in) and a bending moment M D 9000 ˙600 (lb.in). e yield strength S
y
of
the material follows a normal distribution with the mean
S
y
D 34500 (psi) and the
standard deviation
S
y
D 3120 (psi). e diameter of the shaft has a dimension toler-
ance ˙0:005
00
. e required reliability of the shaft is 0.99.
(a) Use the modified H-L method to determine the diameter of the shaft by using the
MSS stress theory.
(b) Use the modified H-L method to determine the diameter of the shaft by using the
DE theory.
(c) Use the modified Monte Carlo method to determine the diameter of the shaft by
using the MSS stress theory.
3.25. A thin-wall cylindrical vessel has an inner diameter d D 40:0 ˙ 0:125
00
. e internal
pressure of the fluid is p D 300 ˙ 50 (psi). e yield strength S
y
of the material follows
a normal distribution with the mean
S
y
D 34,500 (psi) and the standard deviation
S
y
D 3120 (psi). e thickness the thick vessel has a dimension tolerance ˙0:015
00
.
Use the modified H-L method to determine the thickness of the thin vessel with the
reliability 0.999 by using the MSS stress theory.
3.26. Use the modified Monte Carlo method to do Problem 3.25.
3.27. A machined constant circular bar is subjected a cyclic axial loading. e mean ax-
ial loading F
m
is a constant F
m
D 4:9 (klb). e loading amplitude F
a
is a constant
F
a
D 3:84 (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. Use the modified H-L method to determine the di-
ameter d of the bar with the required reliability 0.99 when it has a dimension tolerance
˙0:005
00
.
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