220 4. RELIABILITY OF A COMPONENT UNDER STATIC LOAD
3. Reliability of the thin-cylindrical vessel under combined stress.
e limit state function (c) is a nonlinear equation with four normal distributed ran-
dom variable. We can follow the H-L method discussed in Section
3.6 and the program
flowchart in Figure 3.6 to create a MATLAB program. e iterative results are listed in
Table 4.43. e reliability index ˇ and corresponding reliability R of this vessel are:
ˇ D 3:56441 R D ˆ
.
3:56441
/
D 0:99982:
Table 4.43: e iterative results of Example 4.21 by the H-L method
Iterative #
S
y
*
t
*
p
*
d
*
β
*
|∆β
*
|
1 34500 0.375 350 73.17857 3.371331
2 25083.84 0.35518 355.3634 49.4314 3.568117 0.196786
3 24077.35 0.358227 354.2512 47.97873 3.56444 0.003677
4 24029.32 0.358973 354.1123 48.00033 3.564409 3.12E-05
Example 4.22
A plane stress element of a component at the critical point is shown in Figure 4.20. e normal
stress
x
(ksi), the normal compression stress
y
(ksi), and the shear stress
xy
(ksi) follows normal
distributions. eir distribution parameters are listed in Table 4.44.
Table 4.44: e distribution parameters of a plane stress element
σ
x
(ksi) σ
y
(ksi) τ
xy
(ksi)
μ
σ
x
σ
σ
x
μ
σ
y
σ
σ
y
μ
τ
xy
σ
τ
xy
15.2 1.12 17.5 2.1 10.5 0.98
e yield strength S
y
of the shaft’s material follows a normal distribution with a mean
S
y
D 50:5 (ksi) and a standard deviation
S
y
D 4:12 (ksi). Calculate the reliability of this com-
ponent by using the MSS theory.
1. e maximum shear stress.
As shown in Figure 4.20,
x
is normal tensile stress,
y
is normal compressive stress, and
xy
is positive shear stress. For this plane stress case,
A
and
B
can be calculated per Equa-
4.10. RELIABILITY OF A COMPONENT UNDER COMBINED STRESSES 221
x
y
τ
xy
σ
y
σ
x
Figure 4.20: A plane stress element.
tion (4.43).
A
D
x
y
2
C
s
x
C
y
2
2
C
2
xy
B
D
x
y
2
s
x
C
y
2
2
C
2
xy
:
(a)
From Equation (a), it is clear that
A
> 0 and
B
< 0. erefore, the maximum shear stress
in this case per Equation (4.44) is
max
D
A
B
2
D
s
x
C
y
2
2
C
2
xy
: (b)
2. e limit state function of this example.
Per Equation (4.41), the limit state function of this example by using the MSS theory is:
g
S
y
;
x
;
y
;
xy
D
S
y
2
s
x
C
y
2
2
C
2
xy
: (c)
ere are four normally distributed random variables in Equation (c). eir distribution
parameters are listed in Table 4.45.
Table 4.45: e distribution parameters of random variables in Equation (c)
S
y
(ksi) σ
x
(ksi) σ
y
(ksi) τ
xy
(ksi)
μ
S
y
σ
S
y
μ
σ
x
σ
σ
x
μ
σ
y
σ
σ
y
μ
τ
xy
σ
τ
xy
50.5 4.12 15.2 1.12 17.5 2.1 10.5 0.98
3. Reliability of the component in this example.
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