232 4. RELIABILITY OF A COMPONENT UNDER STATIC LOAD
4.12 REFERENCES
[1] Oberg, E., Jones, F. D., Horton, K. L., and Ryffel, H. H., Machinery’s Handbook, 30th
ed., South Norwalk, Industrial Press, Incorporated, 2016. 161
[2] Callister, W. D. and Rethwisch, D. R., Materials Science and Engineering: An Introduction,
9th ed., Joseph Wiley, Hoboken, NJ, 2014. 168
[3] Hu, Z. and Le, Xiaobin, Probabilistic Design Method of Mechanical Components, Shanghai
Jiao Tong University Publisher, Shanghai, China, September 1995. 168
[4] Haugen, E. B., Probabilistic Mechanical Design, John Wiley & Sons, Inc., 1980. 168, 174,
175
[5] Budynas, R. G. and Nisbett, J. K., Shigley’s Mechanical Engineering Design, 10th ed., Mc-
Graw Hill Education, New York, 2014. 215, 222
[6] Rao, S. S., Reliability Engineering, Pearson, 2015. 176, 177
[7] Pilkey, W. D., Formulas for Stress, Strain, and Structural Matrices, 2nd ed., John Wiley &
Sons, Inc., Hoboken, NJ, 2005. DOI: 10.1002/9780470172681. 177
4.13 EXERCISES
4.1. e length of a component is L D 3:25 ˙ 0:010
00
. Determine its mean and standard
deviation if it is treated as a normal distribution.
4.2. e cross-section of a rectangular shape is with a height h D 1:25 ˙ 0:008
00
and a width
b D 2:25 ˙ 0:010
00
. ese dimensions can be treated as normal distributions. Determine
their distribution parameters.
4.3. e concentrated load P on a beam is P D 1520 ˙ 200 (lb). Determine its distribution
parameters if it treated as a normal distribution.
4.4. e torque T of a shaft is a uniform distribution between 2100 (lb/in) and 2500 (lb/in).
Determine its PDF and distribution function.
4.5. e bending moment M on the free end of a cantilever beam is M D 2215 ˙ 300
(lb/in). Determine its mean and standard deviation if it is treated as a normal distri-
bution.
4.6. Conduct literature research to find the distribution parameters of yield strength or ul-
timate strength of two steel materials. e source, test method, and sample size should
be included in the summary.
4.13. EXERCISES 233
4.7. Conduct literature research to find distribution parameters of yield strength or ultimate
strength of two aluminum alloys. e source, test method, and sample size should be
included in the summary.
4.8. Conduct literature research to display distribution parameters of Young’s modulus of
steel. e source, test method, and sample size should be included in the summary.
4.9. Conduct literature research to display the distribution parameter of Young’s modulus
of an aluminum. e source, test method, and sample size should be included in the
summary.
4.10. e ultimate strength S
u
of material follows a normal distribution with a mean
S
u
D
36:4 (ksi) and a standard deviation
S
u
D 2:79 (ksi). Estimate distribution parameters
of its ultimate shear strength.
4.11. e yield strength S
y
of a material can be described by a normal distribution with a
mean
S
y
D 68:3 (ksi) and a standard deviation
S
u
D 7:12 (ksi). Estimate distribution
parameters of its shear yield strength.
4.12. e information of material shows that its yield strength will be 45.89–62.67 ksi. Esti-
mate its distribution parameters if it is treated as a normal distribution.
4.13. e table for a material shows that its ultimate strength is between 25.67 and 35.24 ksi.
Estimate its distribution parameters if it is treated as a normal distribution.
4.14. e stress concentration factor K
s
of a stepped shaft under torsion is 2.17 from a design
handbook table. Estimate its distribution parameters if it can be simplified as a normal
distribution.
4.15. e stress concentration factor K
t
of a stepped plate under bending is 1.78 from a table.
Estimate its distribution parameters if it follows a normal distribution.
4.16. From three sample tests, averages of the ultimate strength, yield strength, and Young’s
modulus are 19.8 ksi, 24.5 ksi, and 2:45 10
4
(ksi). If its mechanical properties are
treated as normal distributions, estimate distribution parameters of yield strength, shear
yield strength, ultimate strength, ultimate shear strength, Young’s modulus, and shear
Young’s modulus.
4.17. A two-bar supporter as shown in Figure 4.23 is subjected to a concentrated force F
C
at the joint C . e concentrated force is F
C
D 2:1 ˙ 0:20 (klb). e bar AC and BC
are pinned to a wall. e bar AC and BC have the same diameter d D 0:25 ˙ 0:005
00
.
e angle between the two bars is D 60 ˙ 1
ı
. 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). Calculate the reliability of the bar AC. (Note: the
angle needs to be converted into radian.)
234 4. RELIABILITY OF A COMPONENT UNDER STATIC LOAD
A
B
C
F
C
θ
Figure 4.23: A two-bar supporter.
4.18. A two-bar supporter as shown in Figure 4.23 is subjected to a concentrated force F
C
at
the joint C . e concentrated force F
C
can be described by a PMF:
p
.
F
C
/
D
(
0:8 when F
C
D 2:0 (klb)
0:2 when F
C
D 2:5 (klb):
e bar AC and BC are pinned to a wall. e bar AC and BC have the same diameter d D
0:375 ˙ 0:005
00
. e angle between the two bars is D 50 ˙ 1
ı
. e ultimate strength
S
u
of the bar of a brittle material follows a normal distribution with a mean
S
u
D 82:3
(ksi) and a standard deviation
S
u
D 6:32 (ksi). Calculate the reliability of the bar AC.
(Notes: the angle needs to be converted into radian.)
4.19. A bar connected to the supporter at section A is subjected to two concentrated loads F
B
and F
C
, as shown in Figure 4.24. e axial loads are: F
B
D 1500 ˙ 120 (lb) and F
C
D
1000 ˙90 (lb). e geometric dimensions are: diameter d D 0:375 ˙ 0:005
00
, length of
AB segment L
1
D 8:00 ˙ 0:003
00
, and the length of BC segment L
1
D 10; 00 ˙ 0:003
00
.
e Young’s modulus E of the bar material 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). Calculate the
reliability of this bar when the maximum allowable deflection of the bar is 0:008
00
.
4.20. A bar, as shown in Figure 4.24, is subjected to two axial loads. e axial loads are:
F
B
D 2:40 ˙ 0:21 (klb) and F
C
D 1:10 ˙ 0:08 (klb). e diameter of the bar is d D
4.13. EXERCISES 235
B
F
B
A
C
F
C
L
2
L
1
Figure 4.24: A bar under two axial loadings.
0:25 ˙0:005
00
. e ultimate strength S
u
of the bar of a brittle material follows a normal
distribution with a mean
S
u
D 82:3 (ksi) and a standard deviation
S
u
D 6:32 (ksi).
Calculate the reliability of the bar ABC.
4.21. e pin at point B, as shown in Figure 4.23, is a double shear pin. e concentrated
load F
C
at the joint C is F
C
D 2200 ˙ 300 (lb). e diameter of the pin is d D 3=16 ˙
0:005
00
. e angle between the two bars is D 60 ˙ 1
ı
. e shear yield strength S
sy
of
the pin of a ductile material follows a normal distribution with a mean
S
sy
D 32,200
(psi) and a standard deviation
S
sy
D 3630 (psi). Calculate the reliability of the double-
shear pin at point B. (Note: the angle needs to be converted into radian.)
4.22. e pin at point A, as shown in Figure 4.24, is a single shear pin. e diameter of the pin
is d D 0:250 ˙ 0:005
00
. e axial loads are: F
B
D 0:600 ˙ 0:08 (klb) and F
C
D 0:400 ˙
0:04 (klb). e ultimate shear strength S
su
(ksi) of the pin of a ductile material follows
a lognormal distribution with a log mean
ln S
su
D 3:25 and a log standard deviation
ln S
su
D 0:181. Calculate the reliability of the single-shear pin at point A.
4.23. A constant cross-section shaft with a diameter d D 0:875 ˙0:005
00
is subjected to a
torque T . e torque T (lb/in) can be described by a lognormal distribution with a log
mean
ln T
D 7:76 and a log 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
32200 (psi) and a standard deviation
S
sy
D 3630 (psi). Calculate the reliability of the
shaft.
4.24. Schematic of a segment of a shaft at its critical cross-section as shown in Figure 4.25 is
subjected to a torque T D 1350 ˙ 95 (lb/in). e smaller diameter d
1
, the fillet radius
236 4. RELIABILITY OF A COMPONENT UNDER STATIC LOAD
r and the larger diameter d
2
are: d
1
D 0:750 ˙ 0:005
00
, r D 1=32
00
, and d
2
D 1:000 ˙
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
D32,200 (psi) and a standard deviation
S
sy
D 3630 (psi).
Calculate the reliability of the shaft.
T
d
1
d
2
r
T
Figure 4.25: Schematic of a segment of a shaft.
4.25. A constant cross-section shaft with a diameter d D 1:125 ˙ 0:005
00
and a length L D
15:00 ˙ 0:032
00
is subjected to torque 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 deviation
G
D 2:793 10
5
. If the design requirement is the angle of twist
between two ends is less than 1
ı
, calculate the reliability of the shaft.
4.26. A simple support beam is subjected to a concentrated force in the middle, as shown in
Figure 4.26. e concentrated force is P D 1500 ˙ 180 (lb). e span of the beam is
L D 22 ˙ 0:065
00
. e diameter of this beam is d D 1:25 ˙ 0:010
00
. e yield strength
S
y
of the beam’s material follows a normal distribution with a mean
S
y
D 34,500 (psi)
and a standard deviation
S
y
D 3120 (psi). Calculate the reliability of the beam.
L
0.5L
P
Figure 4.26: A simple support beam.
4.27. A simple support beam, as shown in Figure 4.27, is subjected to a uniformly distributed
load and a concentrated load 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 height h D 1:50 ˙ 0:010
00
and
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