Biomedical Engineering Reference
In-Depth Information
Hazard function (failure rate)
curve:
λ
(
x
)
f
(
x
*)
λ
(
x
*)
R
(
x
*)
Probability density
function curve:
f
(
x
)
R
(
x
*)
F
(
x
*)
σ
*
σ
FIGURE 6.31
Relationships for probability density function (
f
(
x
)) and hazard function (
λ
(
x
)) curves of reliability engineering
statistics. Shaded area represents reliability function
R
(
x
*), and cumulative failure probability
F
(
x
*) is defined
as 1 −
R
(
x
*) at a specific strength
σ
*.
x
*
+
∫
∆
x
(
)
=
( )
P x
*
x
x
*
∆
x
f x
d
x
≤ ≤
+
(6.27)
r
x
*
Then the absolute value for the probability density function of failure at
x
* (
f
(
x
*)) can be
presented as follows:
(
)
P x
*
≤ ≤
x
x
*
+
∆
x
(
)
=
r
f x
*
lim
(6.28)
∆
x
∆
x
→
0
As shown in Figure 6.31, the area under curve of
f
(
x
) is called the cumulative distribution
function (CDF), which is denoted by
F
(
x
). Note that
f
(
x
) is just the derivative of
F
(
x
).
F
(
x
)
describes where the population failures lie. When the fracture occurred at a certain value
x
*, Equation 6.29 displays the cumulative failure probability
F
(
x
*), which is the integrated
area under the PDF curve to the right of a specific value
x
*, as the value is varied from 0
to
x
*. The cumulative failure probability
F
(
x
*) can also be recognized as the unreliability
function.
x x
=
∫
*
(
)
=
(
)
−
( )
=
( )
F x
*
P x
*
P
0
f x
d
x
(6.29)
x
=
0
Suppose that the total sample space is 1.0, therefore, the complementary event of the cumu-
lative failure probability is defined as the survival probability, which is also recognized
as the survivor function and denoted by
R
(
x
) [239]. Therefore, the reliability of a material
larger than
x
*(
R
(
x
*)) with a relation of
R
(
x
*) = 1 −
F
(
x
*) can be shown in Equation 6.30.
x
=∞
∫
(
)
= −
(
)
=
( )
R x
*
1
F x
*
f x
d
x
(6.30)
x x
=
*
Search WWH ::
Custom Search