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.