4.10. RELIABILITY OF A COMPONENT UNDER COMBINED STRESSES 227
x
y
τ
xy
σ
y
σ
x
Figure 4.22: Schematic of a plane stress element at the critical point.
of these stresses. A negative sign will identify the negative stress before the stress. For this
plane stress case,
A
and
B
can be calculated per Equation (4.42):
A
D
x
C
y
2
C
r
x
y
2
2
C
xy
2
B
D
x
C
y
2
r
x
y
2
2
C
xy
2
:
(a)
From Equation (a), it is clear that
A
> 0 and
B
< 0. For example, we plug the mean
values of
x
,
y
, and
xy
in Equation (a) and have:
A
D 7:625 (ksi) and
B
D 36:825
(ksi).
2. e limit state function of this example.
Since
A
> 0 >
B
in this example, the limit state function of this component by using
the BCM theory per Equation (4.50) is:
g
S
ut
; S
uc
;
x
;
y
;
xy
D 1
0
B
@
x
C
y
2
C
q
x
y
2
2
C
xy
2
S
ut
C
x
C
y
2
C
q
x
y
2
2
C
xy
2
S
uc
1
C
A
D
8
ˆ
<
ˆ
:
> 0 Safe
0 Limit state
< 0 Failure:
(b)
ere are five random variables in the limit state function (b). All these random variables
are normal distributions, and their distribution parameters are provided and displayed in
Table 4.49.
3. Reliability of the component in this example.
We can use the Monte Carlo method to calculate the reliability of this example based on
the limit state function (b). We can follow the Monte Carlo method in Section 3.8 and
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