Risk Analysis for Engineering 6 pot

• A. J. Clark School of Engineering •Department of Civil and Environmental Engineering
CHAPTER
4b
CHAPMAN
HALL/CRC
Risk Analysis in Engineering and Economics
Risk Analysis for Engineering
Department of Civil and Environmental Engineering
University of Maryland, College Park
RELIABILITY ASSESSMENT
CHAPTER 4b. RELIABILITY ASSESSMENT
Slide No. 1
̈ Availability
– If the time to failure is characterized by its
mean, called mean time to failure (MTTF), and
the time to repair is characterized by its mean,
called mean time to repair (MTTR), a definition
of this probability of finding a given product in
a functioning state can be given by the
following ratio for availability (A):
Empirical Reliability Analysis
Using Life Data
MTTRMTTF
MTTF
A
+
=
(35)
CHAPTER 4b. RELIABILITY ASSESSMENT
Slide No. 2
̈ Reliability, Failure Rates, and Hazard

Functions
– As a random variable, the time to failure (TTF
or T for short) is completely defined by its
reliability function
, R(t).
– The reliability function
is defined as the
probability that a unit or a component does not
fail in the time interval (0,t] or, equivalently, the
probability that the unit or the component
survives the time interval (0, t], under a
specified environment.
Empirical Reliability Analysis
Using Life Data
CHAPTER 4b. RELIABILITY ASSESSMENT
Slide No. 3
̈ Reliability, Failure Rates, and Hazard
Functions (cont’d)
– The probability part of this definition of the TTF
can be expressed using the reliability function
R(t) as follows:
R(t) = Pr (T > t)
(35)
Pr = probability
T = time to failure
t = any time period
Empirical Reliability Analysis
Using Life Data
CHAPTER 4b. RELIABILITY ASSESSMENT
Slide No. 4

̈ Reliability, Failure Rates, and Hazard
Functions (cont’d)
– The reliability function is also called the
survivor
(or survivorship) function.
– Another function, that can completely define
any random variable (e.g., time to failure as
well as time to repair) is the cumulative
distribution function. This function is given as
Empirical Reliability Analysis
Using Life Data
F(t) = 1 - R(t) = Pr (T
≤
t)
(36)
CHAPTER 4b. RELIABILITY ASSESSMENT
Slide No. 5
̈ Reliability, Failure Rates, and Hazard
Functions (cont’d)
– The CDF is the probability that the product
does not survive the time interval (0, t].
– Assuming the TTF as a random variable to be
a continuous positively defined, and F(t) to be
differentiable, the CDF can be written as
Empirical Reliability Analysis
Using Life Data
0for )()(
0
>=
∫

tdxxftF
t
(37a)
CHAPTER 4b. RELIABILITY ASSESSMENT
Slide No. 6
̈ Reliability, Failure Rates, and Hazard
Functions (cont’d)
– Exponential Distribution
• The exponential distribution has a reliability
function R(t) as given by
Empirical Reliability Analysis
Using Life Data
R(t) = exp(-
λ
t)
(38)
λ = failure rate = constant
CHAPTER 4b. RELIABILITY ASSESSMENT
Slide No. 7
̈ Reliability, Failure Rates, and Hazard
Functions (cont’d)
– Weibull Distribution
• The reliability function of the two-parameter Weibull
distribution is
Empirical Reliability Analysis
Using Life Data
(39)
α = scale parameter
β = shape parameter
R(t) = exp[-(t/

α
)
β
]
CHAPTER 4b. RELIABILITY ASSESSMENT
Slide No. 8
̈ Reliability, Failure Rates, and Hazard
Functions (cont’d)
– Lognormal Distribution
• The reliability function of the lognormal distribution
is given by
Empirical Reliability Analysis
Using Life Data
(40)
µ = log mean
σ = log standard deviation
Φ(.) = standard normal cumulative distribution function






−
−Φ=







−
Φ−=
σ
µ
σ
µ
)ln()ln(
1)(
tt
tR
dx
x
y
y








−=Φ
∫
∞
−
2
exp
2

1
)(
2
π
(41)
CHAPTER 4b. RELIABILITY ASSESSMENT
Slide No. 9
̈ Hazard Functions
– The conditional probability
Pr(t < T ≤ t + ∆t | T > t)
is the failure probability of a product unit in the
time interval (t, t + ∆t], with the condition that
the unit is functioning at time t, for small ∆t.
– This conditional probability can be used as a
basis for defining the hazard function for the
unit by expressing the conditional probability
as
Empirical Reliability Analysis
Using Life Data
CHAPTER 4b. RELIABILITY ASSESSMENT
Slide No. 10
̈ Hazard Functions (cont’d)
Empirical Reliability Analysis
Using Life Data
tth
t
tR
tf
tTtt
∆=

∆=>∆+≤<
)(
)(
)(
) T (t Pr
(42)
)(
)(
)(t
tR
tf
h =
h(t) = hazard (or failure) rate function
(43)
CHAPTER 4b. RELIABILITY ASSESSMENT
Slide No. 11
̈ Hazard Functions (cont’d)
– The CDF, F(t), for the time to failure, F(t), and
the reliability function, R(t), can always be
expressed in terms of the so-called cumulative
hazard rate function (CHRF), H(t), as follows:
Empirical Reliability Analysis
Using Life Data
))(exp(1)( tHtF
−
−
=
)](exp[)( tHtR
−
=

(44)
(45)
CHAPTER 4b. RELIABILITY ASSESSMENT
Slide No. 12
̈ Hazard Functions (cont’d)
– Based on Eq. 45, the CHRF can be expressed
through the respective reliability function as
– It can be shown that the cumulative hazard
rate function and the hazard (failure) rate
function are related to each other as
Empirical Reliability Analysis
Using Life Data
)](ln[)( tRtH
−
=
(46)
d
t
tdH
th
)(
)( =
(47)
CHAPTER 4b. RELIABILITY ASSESSMENT
Slide No. 13
̈ Hazard Functions (cont’d)
– The cumulative hazard rate function and its
estimates must satisfy the following
conditions:
H(t) = non-decreasing function that can be

expressed as
Empirical Reliability Analysis
Using Life Data
0)0(
=
H
(
)
∞
=
∞→
)(tHLim
t
0)(
)(
≥= th
dt
tdH
(48a)
(48b)
(48c)
CHAPTER 4b. RELIABILITY ASSESSMENT
Slide No. 14
̈ Hazard Functions (cont’d)
– For the exponential distribution, the hazard
(failure) rate function is constant, and is given
by
and the exponential cumulative hazard rate
function is
Empirical Reliability Analysis

Using Life Data
h(t) =
λ
H(t) =
λ
t
(49)
(50)
CHAPTER 4b. RELIABILITY ASSESSMENT
Slide No. 15
̈ Hazard Functions (cont’d)
– The Weibull hazard (failure) rate function is a
power law function, which can be written as
– and the respective Weibull cumulative hazard
rate function is
Empirical Reliability Analysis
Using Life Data
1
)(
−






=
β
αα
β

t
th
(51)
H(t) = (t/
α
)
β
(52)
CHAPTER 4b. RELIABILITY ASSESSMENT
Slide No. 16
̈ Hazard Functions (cont’d)
– For the lognormal distribution, the cumulative
hazard (failure) rate function can be obtained,
using Eqs. 46 and 40, as
Empirical Reliability Analysis
Using Life Data















−
−Φ−=
σ
µ
)ln(
ln)(
t
tH
(53)
µ = log mean
σ = log standard deviation
Φ(.) = standard normal cumulative distribution function
CHAPTER 4b. RELIABILITY ASSESSMENT
Slide No. 17
̈ Hazard Functions (cont’d)
– The lognormal hazard (failure) rate function
can be obtained as the derivative of the
corresponding CHRF:
Empirical Reliability Analysis
Using Life Data















−
Φ














−
=
=
σ
µ
σ
µ
φ
σ
)ln(

)ln(1
)(
)(
t
t
t
td
tdH
th
(54)
CHAPTER 4b. RELIABILITY ASSESSMENT
Slide No. 18
̈ Selection and Fitting Reliability Models
– The best lifetime distribution for a given
product is the one based on the probabilistic
physical model of the product.
– Unfortunately, such models might not be
available.
– Nevertheless, the choice of the appropriate
distribution should not be absolutely arbitrary,
and at least some physical requirements must
be satisfied.
Empirical Reliability Analysis
Using Life Data
CHAPTER 4b. RELIABILITY ASSESSMENT
Slide No. 19
̈ Selection and Fitting Reliability Models
– Complete Data, Without Censoring
• If the available data are complete, i.e., without
censoring, the following empirical reliability

(survivor) function, i.e., estimate of the reliability
function, can be used:
Empirical Reliability Analysis
Using Life Data









∞<≤
−=<≤
−
<≤
=
+
tt
nittt
n
in
tt
tS
n
iin
0
1, ,2,1 and
01

)(
1
1
(55)
CHAPTER 4b. RELIABILITY ASSESSMENT
Slide No. 20
̈ Selection and Fitting Reliability Models
– In the case of complete data with distinct
failures, k = n.
– The estimate can also be applied to the Type I
and II right-censored data.
Empirical Reliability Analysis
Using Life Data
t
i
= the i
th
failure time denoted according to their ordered
values (order statistics) as t
1
< t
2
< < t
k
k = the number of failures, and n is the sample size
CHAPTER 4b. RELIABILITY ASSESSMENT
Slide No. 21
̈ Selection and Fitting Reliability Models
– In the case of Type I censoring, the time
interval of S

n
(t) estimation is (0, T], where T = t
0
is the test (or observation) duration.
– In the case of Type II
censoring, the
respective time interval is (0, t
r
], where t
r
is the
largest observed failure time.
– This commonly-used estimate, S
n
(t), is called
the empirical survivor function.
Empirical Reliability Analysis
Using Life Data
CHAPTER 4b. RELIABILITY ASSESSMENT
Slide No. 22
̈ Selection and Fitting Reliability Models
– Complete Data, Without Censoring (cont’d)
• Based on 55, an estimate of the CDF of TTF can
be obtained as
Empirical Reliability Analysis
Using Life Data
)(1)( tStF
nn
−
=

(56)
F
n
(t) = estimate of the CDF of time to failure
CHAPTER 4b. RELIABILITY ASSESSMENT
Slide No. 23
̈ Example 5: Single Failure Mode, Small
Sample Without Censoring
– The single failure mode, non-censored data
presented in Example 2 are used to illustrate
the estimation of an empirical reliability
function using Eq. 55.
– The sample size n in this case is 19.
– The TTFs and the results of calculations of the
empirical survivor function S
n
(t) are given in
Table 4.
Empirical Reliability Analysis
Using Life Data
CHAPTER 4b. RELIABILITY ASSESSMENT
Slide No. 24
Time Order
Number
TTF
(Years)
Empirical Survivor
Function
0 0 19/19 = 1
1 26 18/19 = 0.947368

2 27 17/19 = 0.894737
3 28 16/19 = 0.842105
4 29 15/19 = 0.789474
5 30 14/19 = 140.736842
6 31 13/19 = 0.684211
7 32 12/19 = 0.631579
8 33 11/19 = 0.578947
9 34 10/19 = 0.526316
10 35 9/19 = 0.473684
11 36 8/19 = 0.421053
12 37 7/19 = 0.368421
13 38 6/19 = 0.315789
14 39 5/19 = 0.263158
15 40 4/19 = 0.210526
16 42 3/19 = 0.157895
17 43 2/19 = 0.105263
18 50 1/19 = 0.052632
19 56 0/19 = 0

Empirical Reliability Analysis
Using Life Data
Table 4
Empirical Survivor Function
,
S
n
(t), Based on Data of
Example 2
CHAPTER 4b. RELIABILITY ASSESSMENT
Slide No. 25

Empirical Reliability Analysis
Using Life Data
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 102030405060
Time to Failure (Years)
Survivorship Value
Figure 6. . Survivorship Function for Single Failure Mode Without
Censoring of Example 5
CHAPTER 4b. RELIABILITY ASSESSMENT
Slide No. 26
̈ Example 6: Single Failure Mode, Small
Sample, Type I Right Censored Data
– Equation 55 can be applied to Type I and II
right censored data as was previously stated,
which is illustrated in this example.
– The data for this example are given in Table 1
as based on single failure mode, Type I right-
censored data.
Empirical Reliability Analysis
Using Life Data

CHAPTER 4b. RELIABILITY ASSESSMENT
Slide No. 27
̈ Example 6 (cont’d)
– The TTFs and the calculation results of the
empirical survivor function based on Eq. 55
are given in Table 5.
– The sample size n case is 12.
TTCTTCTTCTTCTTFTTFTTFTTFTTFTTFTTFTTFTTF or TTC
51515151464037311815147Time (Years)
121110987654321
Time Order
Number
Table 1. Example of Type I Right Censored Data (in Years) for Equipment
TTF = time to failure, and TTC = time to censoring
Empirical Reliability Analysis
Using Life Data
CHAPTER 4b. RELIABILITY ASSESSMENT
Slide No. 28
̈ Example 6 (cont’d)
Empirical Reliability Analysis
Using Life Data
Time Order
Number
Time to
Failure, TTF
(Years)
Time to
Censoring, TTF
(Years)
Empirical Survivor

Function
0 0 1.000000
1 7 0.916667
2 14 0.833333
3 15 0.750000
4 18 0.666667
5 31 0.583333
6 37 0.500000
7 40 0.416667
8 46 0.333333
9 51 0.333333
10 51 0.333333
11 51 0.333333
12 51 0.333333

Table 5
Empirical Survivor Function, S
n
(t),
Based on Data Given in Table 1
CHAPTER 4b. RELIABILITY ASSESSMENT
Slide No. 29
̈ Example 6 (cont’d)
– Censoring was performed at the end, i.e.,
without any censoring in between failures.
– The empirical survivor function in the case of
right censoring does not reach the zero value
on the right, i.e., at the longest TTF observed.
– The results are plotted in Figure 7 as
individual points.

Empirical Reliability Analysis
Using Life Data
CHAPTER 4b. RELIABILITY ASSESSMENT
Slide No. 30
Empirical Reliability Analysis
Using Life Data
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0 102030405060
Time to Failure (Years)
Survivorship
value
Figure 7. Survivorship Function for Single Failure Mode With Censoring
of Example 6
CHAPTER 4b. RELIABILITY ASSESSMENT
Slide No. 31
̈ Example 7: Single Failure Mode, Large
Sample Data
– The data in this example are based on Monte
Carlo simulation.
– The TTFs and the estimation results of the

empirical survivor function based on Eq. 55
are given in Table 6.
– The table shows only a portion of data since
the simulation process was carried out for
20,000 simulation cycles.
Empirical Reliability Analysis
Using Life Data
CHAPTER 4b. RELIABILITY ASSESSMENT
Slide No. 32
̈ Example 7 (cont’d)
Empirical Reliability Analysis
Using Life Data
MMMM
0.850500179672004
0.859450177662003
0.868300172652002
0.876900170642001
1.000000001937
Survivor
Function
Number
of Failures
TTF
(Years)
Year
Table 6. Example 7 Data and Empirical Survivor Function, S
n
(t)
CHAPTER 4b. RELIABILITY ASSESSMENT
Slide No. 33

̈ Example 7 (cont’d)
– The complete data set covers years from 1937
to 2060.
– For example, the survivorship value at the
year 1974 is computed as
– The empirical survivorship values are shown
in Figure 8.
Empirical Reliability Analysis
Using Life Data
(
)
999750.0
000,20
5000,20
=
−
CHAPTER 4b. RELIABILITY ASSESSMENT
Slide No. 34
̈ Example 7 (cont’d)
Empirical Reliability Analysis
Using Life Data
0.84
0.86
0.88
0.9
0.92
0.94
0.96
0.98
1

35 40 45 50 55 60 65 70
Time To Failure (Ye ars)
Fitted
Data
Survivorship value
Figure 8. Empirical Survivor Function for Example 7
CHAPTER 4b. RELIABILITY ASSESSMENT
Slide No. 35
̈ Selection and Fitting Reliability Models
– Samples With Censoring
• In the this case, the Kaplan-Meier (or product-limit)
estimation procedure can be applied to obtain the
survivor function that accounts for both TTFs and
TTCs.
• The Kaplan-Meier estimation procedure is based
on a sample of n items, among which only k values
are distinct failure times with r observed failures.
• Therefore, r minus k (i.e., r-k) repeated (non-
distinct) failure times exist.
Empirical Reliability Analysis
Using Life Data
CHAPTER 4b. RELIABILITY ASSESSMENT
Slide No. 36
̈ Selection and Fitting Reliability Models
– Samples With Censoring (cont’d)
• The failure times are denoted similar to Eqs. 33a
and 33b, according to their ordered values:
t
1
< t

2
< < t
k
, and t
0
is equal to zero, i.e., t
0
= 0.
• The number of items under observation
(censoring) just before t
j
is denoted by n
j
.
• The number of failures at t
j
is denoted by d
j
. Then,
the following relationship holds:
Empirical Reliability Analysis
Using Life Data
jjj
dnn
−
=
+1
(57)
CHAPTER 4b. RELIABILITY ASSESSMENT
Slide No. 37

̈ Selection and Fitting Reliability Models
– Samples With Censoring (cont’d)
• Under these conditions, the product-limit estimate
of the reliability function, S
n
(t), is given by
Empirical Reliability Analysis
Using Life Data











∞<≤
−=<≤








−

<≤
=
+
=
∏
tt
kittt
n
dn
tt
tS
k
ii
i
j
j
jj
n
0
1, ,2,1for
01
)(
1
1
1
(58)
t = time to failure of an equipment
CHAPTER 4b. RELIABILITY ASSESSMENT
Slide No. 38
̈ Selection and Fitting Reliability Models

– Samples With Censoring (cont’d)
• For cases where d
j
= 1, i.e., one failure at time t
j
,
Eq. 58 becomes
Empirical Reliability Analysis
Using Life Data
(59)











∞<≤
−=<≤









−
<≤
=
+
=
∏
tt
kittt
n
n
tt
tS
k
ii
i
j
j
j
n
0
1, ,2,1for
1
01
)(
1
1
1
CHAPTER 4b. RELIABILITY ASSESSMENT
Slide No. 39

̈ Selection and Fitting Reliability Models
– Samples With Censoring (cont’d)
• For uncensored (complete) samples with d
j
= 1, the
product-limit estimate coincides with the empirical
S
n
(t) given by Eq. 55 as follows:
Empirical Reliability Analysis
Using Life Data
For i = 1:
∏
=








−
=
1
1
1
)(
j
j

j
n
n
n
tS






−
=
n
n 1







−
=
n
n 1

For i = 2:
∏
=









−
=
2
1
1
)(
j
j
j
n
n
n
tS






−
−







−
=
1
21
n
n
n
n







−
=
n
n 2

For i = 3:
∏
=









−
=
3
1
1
)(
j
j
j
n
n
n
tS






−
−







−
−






−
=
2
3
1
21
n
n
n
n
n
n






−

=
n
n 3

M M

M
Therefore for any i:
∏
=








−
=
j
j
j
j
n
n
n
tS
1
1

)(







−
=
n
in

CHAPTER 4b. RELIABILITY ASSESSMENT
Slide No. 40
̈ Example 8: A Small Sample with Two
Failure Modes
– In this example, life data consist of times to
failure related to multiple failure modes (FMs).
– The reliability function corresponding to each
FM needs to be estimated using Eq. 58.
– As an example, two FMs, i.e., FM1 and FM2,
are considered herein.
Empirical Reliability Analysis
Using Life Data
CHAPTER 4b. RELIABILITY ASSESSMENT
Slide No. 41
̈ Example 8 (cont’d)
– Such TTF sample can be represented, for
example, as follows:

– For cases involving more than two FMs in a
sample, the reliability function for a specific
FMi can be estimated by treating the TTFs
associated with failure modes other than FMi
as times to censoring (TTC).
Empirical Reliability Analysis
Using Life Data
t
1
(FM1) < t
2
(FM1) < t
3
(FM2) < t
4
(FM1) < < t
k
(FM2)
CHAPTER 4b. RELIABILITY ASSESSMENT
Slide No. 42
̈ Example 8 (cont’d)
– It should be noted that censoring means that an
item survived up to the time of censoring and
the item was removed from testing or service.
– A sample of 12 TTFs associated with two failure
modes, strength (FM1) and fatigue (FM2), are
shown in Table 7a.
– The calculations of the empirical survivor
function based on Eq. 58 are given in Table 7a.
Empirical Reliability Analysis

Using Life Data
CHAPTER 4b. RELIABILITY ASSESSMENT
Slide No. 43
0.0000000168.3
12
0.2083330151.7
11
0.4166670151.0
10
0.6250000149.6
9
0.8333331021.3
8
0.8333330116.2
7
1.0000001011.7
6
1.000000109.0
5
1.000000106.2
4
1.000000101.9
3
1.000000101.1
2
1.000000100.1
1
1.0000000
0
Empirical Survivor Function

for Failure Mode1
(Strength)
Number of Occurrences
of Failure Mode 2
(Fatigue)
Number of Occurrences
of Failure Mode 1
(Strength)
Time to
Failure
(Years)
Time Order
Number
Empirical Reliability Analysis
Using Life Data
Table 7a. Example 8 Small Sample Data and Respective Empirical Survivor Function for Failure Mode 1 S
n
(t)
CHAPTER 4b. RELIABILITY ASSESSMENT
Slide No. 44
̈ Example 8 (cont’d)
– Table 7b provides the computational details of
the empirical survivorship values for failure
mode 1, with the sample size n = 12 and c
j
=
number of items censored at time j.
– At time order 7 of Tables 7a and 7b,
S
n

(16.2) = 1-1/6 = 0.8333
– Similarly at the time order number 9 of these
tables,
S
n
(49.6) = (1-1/6)(1-1/4) = 0.625
Empirical Reliability Analysis
Using Life Data
CHAPTER 4b. RELIABILITY ASSESSMENT
Slide No. 45
Time
Order
Number
j
Time to
Failure
(Years)
t
j

Number of
Failures for
Mode 1
d
j

Number of
Censorings
for Mode 1
c

j



n
j
=
n – d
j-1
- c
j-1




(1-d
j
/n
j
)
Empirical
Survivor
Function for
Mode1
0 0 1.000000
1 0.1 0 1 12 1.000000
2 1.1 0 1 11 1.000000
3 1.9 0 1 10 1.000000
4 6.2 0 1 9 1.000000
5 9.0 0 1 8 1.000000

6 11.7 0 1 7 1.000000
7 16.2 1 0 6 1-1/6 0.833333
8 21.3 0 1 5 0.833333
9 49.6 1 0 4 1-1/4 0.625000
10 51.0 1 0 3 1-1/3 0.416667
11 51.7 1 0 2 1-1/2 0.208333
12 68.3 1 0 1 0 0.000000

Table 7b. Example 8 Computational Details for Empirical Survivor Function
for Failure Mode 1 S
n
(t)
Empirical Reliability Analysis
Using Life Data
CHAPTER 4b. RELIABILITY ASSESSMENT
Slide No. 46
̈ Example 9: Large Sample with Two Failure
Modes
– Two failure modes, strength (FM1) and fatigue
(FM2), are simulated in this example.
– A portion of these data related to one
component is examined herein.
– The full sample size is 20,000,
– The TTFs and the results of calculations of the
empirical survivor function based on Eq. 58
are given in Table 8.
Empirical Reliability Analysis
Using Life Data
CHAPTER 4b. RELIABILITY ASSESSMENT
Slide No. 47

̈ Example 9 (cont’d)
Empirical Reliability Analysis
Using Life Data
0.997984671192003
0.998036731182002
0.998087642172001
0.998190551162000
0.998241442151999
Survivor
Function for
Failure Mode1
(Strength)
Number of
Occurrences of
Failure Mode 2
(Fatigue)
Number of
Occurrences of
Failure Mode 1
(Strength)
Time to
Failure
(Years)
Year
Table 8. Data and Empirical Survivor Function for Failure Mode 1 S
n
(t)
CHAPTER 4b. RELIABILITY ASSESSMENT
Slide No. 48
̈ Example 9 (cont’d)

– The complete data set covers years from 1984
till 2060.
– The results are plotted in Figure 9 as a step
function.
– The figure also shows the fitted reliability
function using loglinear transformation and
regression as discussed in Example 11.
Empirical Reliability Analysis
Using Life Data
CHAPTER 4b. RELIABILITY ASSESSMENT
Slide No. 49
̈ Example 9 (cont’d)
0.9975
0.998
0.9985
0.999
0.9995
1
0 5 10 15 20
Time to Failure (Years)
Survivorship value
Data
Fitted
Empirical Reliability Analysis
Using Life Data
Figure 9. Empirical Survivor Function for Example 9