188 4. RELIABILITY OF A COMPONENT UNDER STATIC LOAD
e limit state functions (4.18) or (4.20) or (4.21) can be used to calculate the reliability
of a rod under axial loading for a deformation issue.
When axial loading is expressed by a PMF, as shown in Equation (4.14), we can use
the total probability theorem Equation (2.24) in Chapter 2 to calculate the reliability of the
components. e reliability of a rod under axial loading for a deformation issue will be the same
as Equation (4.16) and is shown here again:
R D
n
X
iD1
.
p
i
R
i
/
: (4.16)
In this case, R
i
can be calculated by the limit state functions (4.18), (4.20), or (4.21) when the
axial loading F
a
is equal to F
ai
. p
i
is the PMF when the axial loading is equal to F
ai
.
We will use two examples to demonstrate how to calculate the reliability of a rod under
axial loading for a deformation issue. Example 4.10 will be the reliability of a rod under axial
loading, which is described by a PDF. Example 4.11 will be the reliability of a rod under axial
loading, which is described by a PMF.
Example 4.10
A key component: the stepped round-solid bar ABC is subjected to two axial loadings, as shown
in Figure 4.7. e bar ABC is fixed at the right end. ere is another component (does not
show) near the left end of the bar. e dimensions and loadings for this example are listed in
Table 4.21. e Young’s modulus of the bar material follows a normal distribution with a mean
E
D 2:76 10
7
(psi) and a standard deviation
E
D 6:89 10
5
(psi). e design specification
is that the gap between the deformation of the entire bar and the envelope dimension must be
at least 0:003
00
. Use the Monte Carlo method to calculate the reliability of the bar and its range
with a 95% confidence level.
Table 4.21: e dimensions and loading for the Example 4.10
d
1
(in) d
2
(in) L
1
(in) L
2
(in) L
e
P
1
(lb) P
2
(lb)
0.750 ± 0.002 0.500 ± 0.002 5.000 ± 0.005 3.000 ± 0.005 8.010 ± 0.005 1300 ± 120 500 ± 50
Solution:
1. Axial loading and geometric parameters in each segment.
e stepped bar ABC can be divided into two segments. e loading and corresponding
geometric parameters for each segment are listed in Table 4.22.
e deformation of the entire bar per Equation (4.19) will be:
ı D
.P
1
C P
2
/L
1
Ed
2
1
=4
C
P
2
L
2
Ed
2
2
=4
D
4.P
1
C P
2
/L
1
Ed
2
1
C
4P
2
L
2
Ed
2
2
: (a)