Risk Analysis for Engineering 6 pot

Clark School of Engineering •Department of Civil and Environmental Engineering4b CHAPMAN HALL/CRC Risk Analysis for Engineering Department of Civil and Environmental Engineering Universi

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• A J Clark School of Engineering •Department of Civil and Environmental Engineering

4b

CHAPMAN

HALL/CRC

Risk Analysis for Engineering

Department of Civil and Environmental Engineering University of Maryland, College ParkRELIABILITY ASSESSMENT

̈ 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

MTTR MTTF

MTTF A

+

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̈ 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

Using Life Data

t = any time period

Using Life Data

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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

Using Life Data

F(t) = 1 - R(t) = Pr (T ≤ t) (36)

– 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

Using Life Data

0for )()

(0

>

=∫ f x dx t t

F

t

(37a)

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λ = failure rate = constant

– Weibull Distribution

• The reliability function of the two-parameter Weibull distribution is

Using Life Data

(39)

α = scale parameter

β = shape parameter

R(t) = exp[-(t/ α )β]

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σ = log standard deviation

Φ(.) = standard normal cumulative distribution function

) ln(

1 )

R

dx x y

1 ) (

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

Using Life Data

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̈ Hazard Functions (cont’d)

Using Life Data

t t h

t t R

t f t

T t t

≤

<

) (

) ( )

T (t

Pr

(42)

) (

) ( )

(t

t R

t f

h(t) = hazard (or failure) rate function

(43)

– 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:

Using Life Data

)) ( exp(

1 )

)]

( exp[

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– 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

Using Life Data

)]

(ln[

– The cumulative hazard rate function and its estimates must satisfy the following

conditions:

H(t) = non-decreasing function that can be

expressed as

Using Life Data

0)0

dt

t dH

(48a) (48b)

(48c)

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– For the exponential distribution, the hazard (failure) rate function is constant, and is given

– 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

Using Life Data

1)

t

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– For the lognormal distribution, the cumulative hazard (failure) rate function can be obtained, using Eqs 46 and 40, as

Using Life Data

ln)

µ = log mean

σ = log standard deviation

Φ(.) = standard normal cumulative distribution function

– The lognormal hazard (failure) rate function can be obtained as the derivative of the

corresponding CHRF:

Using Life Data

µφσ

) ln(

1

) ( )

(

t

t t

t d

t dH t

h

(54)

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̈ 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

Using Life Data

– 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:

Using Life Data

n i

t t t n

i n

t t t

S

n

i i n

0

1 , , 2 , 1 and

0 1 )

1

(55)

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– 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

Using Life Data

values (order statistics) as t1< t2< < t k

k = the number of failures, and n is the sample size

– In the case of Type I censoring, the time

is the test (or observation) duration

– In the case of Type II censoring, the

largest observed failure time

the empirical survivor function

Using Life Data

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– Complete Data, Without Censoring (cont’d)

• Based on 55, an estimate of the CDF of TTF can

be obtained as

Using Life Data

) ( 1

)

̈ 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

Table 4

Using Life Data

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Using Life Data

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̈ 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 censored data

right-Using Life Data

̈ 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.

TTC TTC TTC TTC TTF TTF TTF TTF TTF TTF TTF TTF

TTF or TTC

51 51 51 51 46 40 37 31 18 15 14 7

Time (Years)

12 11 10 9 8 7 6 5 4 3 2 1

Time Order

Number

Table 1 Example of Type IRight Censored Data (in Years) for Equipment

TTF = time to failure, and TTC = time to censoring

Using Life Data

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Empirical Survivor Function, S n (t),

Based on Data Given in Table 1

– 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

Using Life Data

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Using Life Data

̈ Example 7: Single Failure Mode, Large

– The table shows only a portion of data since the simulation process was carried out for

20,000 simulation cycles

Using Life Data

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Using Life Data

M M

0.850500 179

67 2004

0.859450 177

66 2003

0.868300 172

65 2002

0.876900 170

64 2001

1.000000 0

0 1937

Survivor Function

Number

of Failures

TTF

(Years) Year

Table 6 Example 7 Data and Empirical Survivor Function, S n (t)

Using Life Data

999750

0 000

, 20

5 000 , 20

=

−

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Figure 8 Empirical Survivor Function for Example 7

– 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.

Using Life Data

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– Samples With Censoring (cont’d)

• The failure times are denoted similar to Eqs 33a and 33b, according to their ordered values:

t1< t2< < t k , and t0is equal to zero, i.e., t0= 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:

Using Life Data

j j

• Under these conditions, the product-limit estimate

of the reliability function, S n (t), is given by

Using Life Data

k i

t t t n

d n

t t

t S

k

i i i

j j n

0

1 , , 2 , 1 for

0 1

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• For cases where d j = 1, i.e., one failure at time t j,

k i

t t t n n

t t t

S

k

i i i

j n

0

1 , , 2 , 1 for 1

0 1

)

1

• 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:

Using Life Data

j n

n

n t

j n

n

n t

n n n

j n

n

n t

n n n n n

n

n t

n i n

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̈ 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

Using Life Data

associated with failure modes other than FMi

as times to censoring (TTC)

Using Life Data

t1(FM1) < t2(FM1) < t3(FM2) < t4(FM1) < < t k(FM2)

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– 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

Using Life Data

0.208333 0

1 51.7

11

0.416667 0

1 51.0

10

0.625000 0

1 49.6

9

0.833333 1

0 21.3

8

0.833333 0

1 16.2

7

1.000000 1

0 11.7

6

1.000000 1

0 9.0

5

1.000000 1

0 6.2

4

1.000000 1

0 1.9

3

1.000000 1

0 1.1

2

1.000000 1

0 0.1

1

1.000000 0

0

Empirical Survivor Function for Failure Mode1 (Strength)

Number of Occurrences

of Failure Mode 2 (Fatigue)

Number of Occurrences

of Failure Mode 1 (Strength)

Using Life Data

Table 7a Example 8 Small Sample Data and Respective Empirical Survivor Function for Failure Mode 1 S n (t)

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– Table 7b provides the computational details of the empirical survivorship values for failure

number of items censored at time j.

– At time order 7 of Tables 7a and 7b,

– Similarly at the time order number 9 of these

tables,

Using Life Data

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

Table 7b Example 8 Computational Details for Empirical Survivor Function

for Failure Mode 1 S n (t)

Using Life Data

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̈ 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

Using Life Data

0.997984 67

1 19

2003

0.998036 73

1 18

2002

0.998087 64

2 17

2001

0.998190 55

1 16

2000

0.998241 44

2 15

1999

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)

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– The figure also shows the fitted reliability

function using loglinear transformation and regression as discussed in Example 11

Using Life Data

Figure 9 Empirical Survivor Function for Example 9

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– Parametric Reliability Functions

• Besides the traditional distribution estimation

methods, such as the method of moments and

maximum likelihood described in Appendix A, the empirical survivor functions can be used to fit

analytical reliability functions.

• After evaluating an empirical reliability function, an analytical parametric hazard rate functions, such as given by Eqs 45 and 47, can be fitted using the empirical survivorship function obtained from life data.

Using Life Data

– Parametric Reliability Functions (cont’d)

• The Weibull reliability function was used in studies performed for the U S Army Corps of Engineers

as provided in Eq 39 including the exponential reliability function as its specific case.

• Also, the reliability function having a polynomial cumulative hazard function was used as follows:

Using Life Data

R(t) = exp(-H(t)) H(t) = a0 + a1 t + a2t2

(60a) (60b)

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– Parametric Reliability Functions (cont’d)

• Therefore, the hazard function is given by

• For the special case where the parameters a0and

a2are equal to zero, Eq 60b reduces to the

exponential distribution.

• For the special case where the parameters a0and

a1are zeros, the Eq 60b reduces to the specific case of the Weibull distribution with the shape

parameter of 2 (Rayleigh distribution)

Using Life Data

(60c)

h(t) = a1+ 2 a2t

– Parameter Estimation Using Loglinear

Transformation

• Eqs 60a to 60c provide exponential models with

parameters a0, a1, and a2 The logarithmic

transformation of Eqs 60a to 60c leads to

Using Life Data

- ln(R(t)) = a0+ a1t

- ln(R(t)) = a0+ a1t + a2t2

(61a)

(61b)

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