3.10. REFERENCES 157
all statistically independent random variables with only the known means and the known
standard deviations, we can still use the FOSM method to estimate the reliability of the
component.
4. When the limit state function of a component is a nonlinear function of all statistically
independently normally distributed random variables, we can use the H-L method to cal-
culate the reliability of the component, which is discussed in detail in Section 3.6. e
H-L method is an improved version of the FOSM method.
5. When the limit state function of a component contains one or more non-normally dis-
tributed random variables, we can use the R-F method to calculate the reliability of the
component, which is discussed in detail in Section 3.7. e R-F method is an improved
H-L method.
6. e Monte Carlo method can be used to calculate the reliability of a component for a limit
state function with any type of random variables. e Monte Carlo method is discussed
in Section 3.8 and must be implemented through a computer program because of a huge
amount of trial numbers in this virtual experiment.
3.10 REFERENCES
[1] Hasofer, A. M. and Lind, N., An exact and invariant first-order reliability format, Journal
of Engineering Mechanics, ASCE, vol. 100, no. EM1, pp. 111–121, February 1974. 124,
133
[2] Nowak, A. S. and Collins, K. R., Reliability of Structures, 2nd ed., CRC Press, 2013.
DOI: 10.1201/b12913. 124, 130
[3] Rackwitz, R. and Fiessler, B., Structural reliability under combined random load se-
quences, Computers and Structures, vol. 9, pp. 489–494, 1978. DOI: 10.1016/0045-
7949(78)90046-9. 140
[4] Shooman, M. L., Probabilistic Reliability: An Engineering Approach, McGraw-Hill, New
York, 1968. 152
[5] Rao, S. S., Reliability Engineering, Pearson, 2015. 152
3.11 EXERCISES
3.1. A double-shear pin with a diameter d is subjected to direct shearing force V . e shear-
ing yield strength of this material is S
sy
. e failure mode is that when the direct shear
stress is more than the shearing yield strength, the pin is treated as a failure. Establish
the limit state function of this pin.
158 3. COMPUTATIONAL METHODS FOR THE RELIABILITY OF A COMPONENT
3.2. A simple support beam with a span L is subjected to a concentrated force P at the
middle of the beam, as shown in Figure 3.11. e beam is a constant round bar with a
diameter d . e yield strength of the beam material is S
y
. It is assumed that the shear
stress at the critical section induced by the sheer force is negligible. It is assumed that
the beam will be treated as a failure when the maximum normal stress is more than the
yield strength. Establish the limit state function of this beam.
P
L /2
Figure 3.11: A simple support beam under a concentrated force at the middle of the beam.
3.3. e design specification of a simple support beam, as shown in Figure 3.11 is that the
maximum deflection of the beam is not allowed to be more than . e Young’s modulus
of the beam material is E. Establish the limit state function of this beam.
3.4. e failure mode of a pin is a static failure at which the direct shear stress
D
is more
than the yield shearing strength S
sy
. e yield shearing strength S
sy
and the direct shear
stress
D
are uniform distributions. eir PDFs are:
f
S
sy
s
sy
D
8
<
:
1
18:57
21:21 s
sy
39:67
0 otherwise:
f
D
.
D
/
D
8
<
:
1
7:73
15:67 s
sy
23:4
0 otherwise:
Use the interference theory to calculate the reliability of the pin.
3.5. e failure mode of a bar under axial loading is a static failure at which the maximum
normal stress
max
at the critical section is more than the ultimate material strength
S
u
. e ultimate material strength S
u
follows a normal distribution with a mean
S
u
D
78:2 (ksi) and standard deviation
S
u
D 6:9 (ksi). e maximum normal stress
max
at the critical section is a normally distributed random variable with a mean
max
D
57:4 (ksi) and standard deviation
max
D 10:21 (ksi). Use the interference theory to
calculate the reliability of the bar.
3.11. EXERCISES 159
3.6. e maximum bending stress
max
(ksi) at the critical section of a beam follows a log-
normal distribution with a mean
ln
max
D 3:22 and standard deviation
ln
max
D 0:52.
e material yield strength S
y
(ksi) follows a log-normal distribution with a mean
ln S
y
D
4:56
and standard deviation
ln S
y
D
0:17
. It is assumed that the component
is a failure when the maximum bending stress
max
is more than the material yield
strength S
y
. Use the interference theory to calculate the reliability of the component.
3.7. A bar is under three statistically independent axial loadings P
1
; P
2
, and P
3
. ese three
axial loadings follow normal distributions. eir means and standard deviations are:
P
1
D 1:51 (kip),
P
1
D 0:121 (kip;
P
2
D 3:79 (kip),
P
2
D 0:292 (kip) and
P
3
D
14:12 (kip),
P
3
D 1:32 (kip). e maximum normal stress at the critical section is
the sum of the normal stress induced by these three axial-loadings, that is,
A
D
2:264P
1
C 2:264P
2
C 2:264P
3
(ksi). e yield strength S
y
of the material is a normally
distributed random variable with the mean
S
y
D 61:5 (ksi) and the standard deviation
S
y
D 5:91 (ksi). Use the FOSM method to calculate the reliability of the bar.
3.8. e bar is under axial loading. e diameter d of the bar can be treated as a normal distri-
bution with a mean
d
D 0:75 (in) and a standard deviation
d
D 0:002 (in). e axial
loading P is a normal distribution with a mean
P
D 8:12 (kip) and a standard deviation
P
D 2:45 (kip). e yield strength S
y
of the material is a normally distributed random
variable with a mean
S
y
D 61:5 (ksi) and a standard deviation
S
y
D 5:91 (ksi). Use
the FOSM method to calculate the reliability of the bar.
3.9. A shaft is under a torsion. e diameter d of the shaft can be treated as a normal
distribution with a mean
d
D 1:125 (in) and a standard deviation
d
D 0:002 (in).
e torsion T is a normal distribution with the mean
T
D 2:89 (kip) and the stan-
dard deviation
T
D 0:24 (kip). e shear yield strength S
sy
of the material is a normal
distributed random variable with a mean
S
sy
D 31:5 (ksi) and a standard deviation
S
sy
D 2:98 (ksi). Use the FOSM method to calculate the reliability of the shaft.
3.10. A beam with a constant round cross-section is under a bending moment. e diameter
d of the shaft can be treated as a normal distribution with a mean
d
D 1:25 (in) and
a standard deviation
d
D 0:002 (in). e bending moment M on the critical section
is a normal distribution with a mean
M
D 2:25 (kip) and a standard deviation
M
D
0:19 (kip). e yield strength S
y
of the material is a normal distributed random variable
with a mean
S
y
D 61:5 (ksi) and a standard deviation
S
y
D 5:91 (ksi). Use the FOSM
method to calculate the reliability of the beam.
3.11. Use the H-L method to compile the MATLAB program for calculating the reliability
of Problem 3.9.
3.12. Use the H-L method to compile the MATLAB program for calculating the reliability
of Problem 3.10.
160 3. COMPUTATIONAL METHODS FOR THE RELIABILITY OF A COMPONENT
3.13. A component is under axial tensile loading with a limit state function g
S
y
;
N
D
S
y
N
. S
y
is the material yield strength, which follows a normal distribution with the
mean
S
y
D 61:5 (ksi) and the standard deviation
S
y
D 5:91 (ksi). e normal stress
N
(ksi) of the component at the critical section is a log-normal distributed variable
with the log-mean
ln
N
D 3:9 and the log-standard deviation
ln
N
D 0:159. Use
the R-F method to compile the MATLAB program for calculating the reliability of
this component.
3.14. A component is under axial tensile loading. e component strength index is the ulti-
mate tensile strength S
u
. e ultimate tensile strength S
u
(ksi) follows a Weibull distri-
bution with D 47 and ˇ D 1:5. e axial loading P at the critical section is a normal
distributed random variable with a mean 18.6 (klb) and a standard deviation 2.1 (klb).
e diameter d of the critical section follows a normally distributed variable with a mean
1:750
00
and a standard deviation 0:002
00
. Build the limit state function and then use the
R-F method to determine the reliability of the component.
3.15. Calculate the trial number N with a 95% confidence level for a critical component with
a pre-specified relative error 1% of the probability of component failure.
3.16. If the component is treated as a key component, use the Monte Carlo method to cal-
culate the reliability of Problem 3.13. Estimate the range of reliability with a 95% con-
fidence level.
3.17. If the component is treated as a key component, use the Monte Carlo method to cal-
culate the reliability of Problem 3.14. Estimate the range of the reliability with a 95%
confidence level.
3.18. Use Monte Carlo method to calculate the reliability of Problem 3.9.
3.19. Use Monte Carlo method to calculate the reliability of Problem 3.10.
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