112 3. COMPUTATIONAL METHODS FOR THE RELIABILITY OF A COMPONENT
3.2 LIMIT STATE FUNCTION
A mechanical component of a ductile material will yield when the maximum component stress
is more than the material yield strength. Such status of the component is a failure. When the
maximum component normal stress is less than the yield strength, the status of the component is
safe. e critical status between the safe and the failure will be a limit state when the maximum
component normal stressis equal to the material yield strength.
Limit state function. For a general case, let S represent component strength index, which is
a permissible or allowable parameter of a component in the safe status such as yield strength,
ultimate strength, allowable deflection, and fatigue strength. Let Q represent component stress
index, which is a component parameter induced by loading such as maximum normal stress,
maximum shear stress, maximum Von-Mises stress, maximum deflection, and fatigue damage.
e limit state function g.S; Q/ of a component is defined as:
g
.
S; Q
/
D S Q D
8
ˆ
ˆ
<
ˆ
ˆ
:
> 0 Safe
D 0 Limit state
< 0 Failure:
(3.1)
A limit state function is a mathematic function and will be the function of other design parame-
ters such as material mechanical properties, component geometric dimensions, and loading. For
a limit state function, it can describe three possible states of a component, as shown in Equa-
tion (3.1). When the value of a limit state function is more than zero, the component is safe.
When the value of a limit state function is less than zero, the component is a failure. When the
value of a limit state function is equal to zero, the component is in a limit state between the safe
and the failure or can be called as on the surface of the limit state function: g
.
S; Q
/
D 0.
Per the definition of a limit state function, the reliability of a component will be:
R D P
Œ
g
.
S; Q
/
0
D P
.
S Q 0
/
: (3.2)
e probability of failure of a component will be:
F D 1 R D P
Œ
g
.
S; Q
/
< 0
D P
.
S Q < 0
/
: (3.3)
e limit state function of a component will be established by the failure theories of the problem
under consideration. Corresponding failure theories for a component under static loading will be
discussed in Chapter 4. Failure theories for a component under cyclic loading will be discussed in
Chapter 1 in Reliability-Based Mechanical Design, Volume 2. Here, we will give several examples
of limit state functions.
3.2. LIMIT STATE FUNCTION 113
Example 3.1
A shaft with a diameter d (in) is subjected to a pure torque T (lb.in). e material is ductile
material with a shear yield strength S
sy
. If the maximum shear stress failure theory is used,
establish the limit state function of this shaft.
Solution:
Per the maximum shear stress theory, the component strength index in this example will be the
shear yield strength S
y
, that is:
S D S
sy
:
e component stress index in this example will be the shear stress induced by the pure torque
and can be calculated by the following equation:
Q D
max
D
T d=2
d
4
=32
D
16T
d
3
:
When the maximum shear stress failure theory is used, the limit state function of the shaft is:
g
.
S; Q
/
D
S
Q
D
g
S
y
; d; T
D S
sy
16T
d
3
D
8
ˆ
ˆ
<
ˆ
ˆ
:
> 0 Safe
D 0 Limit state
< 0 Failure:
e limit state function g
S
sy
; d; T
is the function of three design parameters S
sy
, d; and
T . Typically, these parameters will be random variables. erefore, the limit state function
g
S
sy
; d; T
will also be a random variable.
Example 3.2
A cantilever beam, as shown in Figure 3.1, is subjected to a concentrated load P at the free
end. e beam is a constant rectangular bar with a length L and a height h and a width b. e
material’s Young’s modulus is E. e design specification is that the allowable deflection at the
free end B will be . Establish the limit state function of this problem.
A B
P
L
Figure 3.1: Schematic of a cantilever beam.
114 3. COMPUTATIONAL METHODS FOR THE RELIABILITY OF A COMPONENT
Solution:
In this example, the component strength index is the allowable deflection , that is,
S D :
e component stress index will be the maximum deflection at the free end B. For a cantilever
beam under a concentrated load at the free end, the maximum deflection at the free end is:
Q D ı D
PL
3
3EI
D
PL
3
3E
1
12
h
3
b
D
4PL
3
Eh
3
b
:
According to the design requirement of this example, the limit state function will be:
g
.
S; Q
/
D S Q D g
.
E; P; h; b; L
/
D
4PL
3
Eh
3
b
D
8
ˆ
ˆ
<
ˆ
ˆ
:
> 0 Safe
D 0 Limit state
< 0 Failure:
e limit state function g
.
E; P; h; b ; L
/
is the function of five design parameters E; P , h, b; and
L.
Example 3.3
A beam structure is designed with an infinite fatigue life. e material fatigue endurance limit
of the beam at the critical cross-section is S
e
. e critical rectangular cross-section with a height
h and a width b is subjected to a fully reversed cyclic bending moment with a magnitude M .
Determine the limit state function of this beam structure at the critical cross-section.
Solution:
For this example, the component strength index will be the material fatigue endurance limit S
e
.
So, we have:
S D S
e
:
e component stress index will be the stress amplitude of the fully reversed cyclic stress induced
by a fully reversed cyclic bending moment. e following equation can calculate the component
maximum bending stress:
Q D
a
D
M
h
2
I
D
M
h
2
1
12
h
3
b
D
6M
h
2
b
:
According to the fatigue theory, the limit state function of this example will be:
g
.
S; Q
/
D S Q D g
.
S
e
; h; b; M
/
D S
e
6M
h
2
b
D
8
ˆ
ˆ
<
ˆ
ˆ
:
> 0 Safe
D 0 Limit state
< 0 Failure:
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