228 4. RELIABILITY OF A COMPONENT UNDER STATIC LOAD
Table 4.49: e distribution parameters of random variables in Equations (b)
S
ut
(ksi) S
uc
(ksi) σ
x
(ksi) σ
y
(ksi) τ
xy
(ksi)
μ
S
ut
σ
S
ut
μ
S
uc
σ
S
uc
μ
σ
x
σ
σ
x
μ
σ
y
σ
σ
y
μ
τ
xy
σ
τ
xy
22.00 1.80 82.00 10.50 31.2 2.51 1.80 0.085 15.0 2.31
the program flowchart in Figure 3.8 to create a MATLAB program. Since this problem is
no very big and complicated, we will use the trial number N D 1,598,400 from Table 3.2
in Section 3.8. e estimated reliability R of this component at the critical point is:
R D 0:9417:
4.11 SUMMARY
In reliability-based mechanical design, all design parameters, including geometric dimensions,
loadings, and material strengths, are treated as random variables. ese statements are true in
reality but require most information for their descriptions. Reliability links all design parameters
through a limit state function and is a measure of components’ safety status. e physical mean-
ing of reliability is a relative percentage of safe components in the sample space of the whole
same component.
When loading conditions, geometric dimensions, and the type of material are fully speci-
fied, we can calculate its reliability. Failure theories under static loadings in mechanics of mate-
rials and the reliability-based design are the same. ese failure theories discussed in the tradi-
tional design theory can be used to build the limit state function. After the limit state function is
established, the methods discussed in Chapter 3 including the H-L method, R-F method, and
Monte Carlo method can be used to calculate the reliability of a component under any design
case. e following is the summary of typical limit state functions.
For a strength issue, the typical limit state functions of a component under simple typical
loading are summarized as follows.
• For a rod of brittle material under axial loading,
g
.
S
u
; K
t
; A; F
a
/
D S
u
K
t
F
a
A
D
8
ˆ
<
ˆ
:
> 0 Safe
D 0 Limit state
< 0 Failure:
(4.12)
4.11. SUMMARY 229
• For a rod of ductile material under axial loading,
g
S
y
; K
t
; A; F
a
D S
y
K
t
F
a
A
D
8
ˆ
<
ˆ
:
> 0 Safe
D 0 Limit state
< 0 Failure:
(4.13)
• For a component of ductile material under direct shearing,
g
S
sy
; A; V
D S
sy
V
A
D
8
ˆ
<
ˆ
:
> 0 Safe
D 0 Limit state
< 0 Failure:
(4.22)
• For a component of brittle material under direct shearing,
g
.
S
su
; A; V
/
D S
su
V
A
D
8
ˆ
<
ˆ
:
> 0 Safe
D 0 Limit state
< 0 Failure:
(4.23)
• For a solid round shaft under torque,
g
S
sy
; K
s
; d
o
; T
D S
sy
K
s
16T
d
3
o
8
ˆ
<
ˆ
:
> 0 Safe
D 0 Limit state
< 0 Failure:
(4.27)
• For a hollow round shaft under torque,
g
S
sy
; K
s
; d
o
; d
i
; T
D S
sy
K
s
16T d
o
d
4
o
d
4
i
D
8
ˆ
<
ˆ
:
> 0 Safe
D 0 Limit state
< 0 Failure:
(4.28)
• For a beam of brittle material under bending:
g
.
S
u
; K
t
; Z; M
/
D S
u
K
t
M
Z
D
8
ˆ
<
ˆ
:
> 0 Safe
D 0 Limit state
< 0 Failure:
(4.35)
• For a beam of ductile material under bending,
g
S
y
; K
t
; Z; M
D S
y
K
t
M
Z
D
8
ˆ
<
ˆ
:
> 0 Safe
D 0 Limit state
< 0 Failure:
(4.36)
230 4. RELIABILITY OF A COMPONENT UNDER STATIC LOAD
For a strength issue, the typical limit state functions of a component under combined
stressed are summarized as follows.
• For ductile material, the limit state function of a component by the MSS theory,
g
S
y
;
max
D
S
y
2
max
D
8
ˆ
<
ˆ
:
> 0 Safe
D 0 Limit state
< 0 Failure:
(4.41)
• For ductile material, the limit state function of a component by the DE theory,
g
S
y
;
von
D S
y
von
D
8
ˆ
<
ˆ
:
> 0 Safe
D 0 Limit state
< 0 Failure:
(4.45)
• For brittle material, the limit state function of a component by the MNS theory,
when
1
2
3
0,
g
.
S
ut
;
1
/
D S
ut
1
D
8
ˆ
<
ˆ
:
> 0 Safe
D 0 Limit state
< 0 FailureI
(4.48)
when 0
1
2
3
,
g
.
S
uc
;
3
/
D
S
ut
j
3
j
D
8
ˆ
<
ˆ
:
> 0 Safe
D 0 Limit state
< 0 Failure:
(4.49)
• For brittle materials, the limit state function of a component under plane stress by the
BCM theory,
when
A
B
0,
g
.
S
ut
;
A
/
D S
ut
A
D
8
ˆ
<
ˆ
:
> 0 Safe
D 0 Limit state
< 0 FailureI
(4.50)
when
A
0
B
,
g
.
S
ut
; S
uc
;
A
;
b
/
D 1
A
S
ut
C
j
B
j
S
uc
D
8
ˆ
<
ˆ
:
> 0 Safe
D 0 Limit state
< 0 FailureI
(4.51)
4.11. SUMMARY 231
when 0
A
B
,
g
.
S
uc
;
B
/
D S
uc
j
B
j
D
8
ˆ
<
ˆ
:
> 0 Safe
D 0 Limit state
< 0 Failure:
(
4.52)
For a deformation issue, the typical limit state function of a component under simple
typical loading are summarized as follows.
• For a rod under axial loadings,
g
.
E; A
i
; L; F
ai
/
D
X
F
ai
L
i
EA
i
D
8
ˆ
<
ˆ
:
> 0 Safe
D 0 Limit state
< 0 Failure:
(4.20)
• For a shaft under torque,
g
.
G; L
i
; J
i
; T
i
/
D
X
T
i
L
i
GJ
i
D
8
ˆ
<
ˆ
:
> 0 Safe
D 0 Limit state
< 0 Failure:
(4.33)
• For a beam under bending and lateral force
g
.
y
max
/
D y
max
D
8
ˆ
<
ˆ
:
> 0 Safe
D 0 Limit state
< 0 Failure:
(4.38)
• For a simple support beam under a concentrated load in the middle,
g
.
E; I; L; P
/
D y
max
D
PL
3
48EI
: (4.39)
• For a cantilever beam under a concentrated load on the free end,
g
.
E; I; L; P
/
D y
max
D
PL
3
3EI
: (4.40)
When the limit state function of a component under static loading is established, we can
use the H-L method, R-F method, or Monte Carlo method to calculate its reliability.
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