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2. Reliability measuresObjectives:

• Learn how to quantify reliability of a system

• Understand and learn how to compute the following measures– Reliability function– Expected life– Failure rate and hazard function

• Learn some common probability density functions of time to failure and learn when to apply them

– Exponential– Normal– Weibull

• Learn how to estimate hazard functions from data

• Learn how to select a reliability function for a given problem

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Reliability function• Assumption: New equipment• T=Failure time, random variable because we do

not know when a system will fail• Probability density function of failure time, fT(t).

Units: # of failures per unit time• Reliability function, R(t)= probability that system

will work properly at time t• Failure distribution function, FT(t)= probability

that a system will fail by time t

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Notation

• Probability density function, fT(t)

• T random variable (in this case it is the component life)

• t value that the random variable assumes

• fT(t)=limt0 P(t<T t+ t)/ t

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• Expected life of a component or system, E(T)

t

fT(t)

E(T)

Expected life

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Hazard function

• Hazard function: h(t)=probability that, given that a system has survived until time t, it will fail between times t and t+t, divided by t. Units of h(t): 1/unit time

• h(t)= fT(t)/R(t)

• Example start with N=1000 light bulbs, at T=1000 hrs, 300 light bulbs are still working. After 10 hrs 5 more bulbs fail. The hazard function is approximately: h(1005)=5/(300*10)

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Shape of hazard function of most real-life systems: bathtub

function

t

h(t)

Debugging,or infantmortality

Aging

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Relation between reliability measures

)(tfT )(tFT )(tR )(th

)(tfT -dt

tdFT )(dttdR)(

tdtth

ethR 0')'(

)()0(

)(tFT t

T dttf0

')'( - )(1 tR

tdtth

eR 0')'(

)0(1

)(tR t

T dttf0

')'(1 )(1 tFT - t

dtth

eR 0')'(

)0(

)(th

t

T

T

dttf

tf

0')'(1

)(

)(1

)(

tFdt

tdF

T

T

)(

)(

tRdttdR -

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Reliability and hazard functions for well known distributions

• Exponential– Good choice for systems or components whose

strength does not change with time and which are subjected to extreme disturbances occurring completely at random and independently.

– fT(t)=1/*exp(-t/ )

– R(t)= exp(-t/ )– h(t)=1/

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Shape of exponential distribution

t

fT(t)

1/

E(T)=θ

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Normal distribution

t

fT(t)

Standard deviation,

Two parameter distribution

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• No closed form analytical expression for cumulative distribution

• Cumulative distribution of standard normal, (z), has been tabulated. We also have excellent polynomial approximations. Standard normal has zero mean and unit standard deviation.

• Very easy to do reliability computations with normal distributions

• Finding FT(t) if T is normal. Transform T into standard normal.

• FT(t)=P(Tt)=P[(T-)/ (t-)/]= (z)

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Cumulative distribution of standard normal variable

z (z)

0 0.5

-1 0.16

-2 0.02

-3 0.001

-4 310-5

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If we model the time to failure using a normal distribution then there is small probability of the time to failure being negative. This does not make sense. Always check that the probability of the time being negative is small compared to the probabilities we are calculating in the problem at hand. For example, if the we are working with systems whose failure probabilities are about 10-3, then the probability of the time to

failure being negative should be about 10-5 or less.

t

fT(t)

The area under the curve to the left of zero is the probability of t being negative

0

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Weibull distribution• Good choice for systems or components whose strength

deteriorates with time and which are subjected to extreme disturbances occurring completely at random and independently.

• Consider a building in Greece that is expected to be sustain a very strong earthquake (say above 6.5 in the Richter scale) once every ten years. Like any real life system, the strength of the building deteriorates with time. A Weibull distribution is a good candidate for modeling the time to failure (or length of the life) of the building.

• Very popular for describing strength and life length

• Generalizes the exponential distribution

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Reliability function

– for t greater than – Three parameter distribution:

• use shape parameter, , to control shape is the scale parameter, affects dispersion

• use location parameter, , to shift the mean value

• shape parameter=1, Weibull reduces to the exponential distribution

)()(

t

etR

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Shape of Weibull probability density functionif shape parameter less than 3.6, density is skewed to the right

if shape parameter is greater than 4, density is skewed to the left .

0 2 4 60

2

4

6

8

theta=0.5theta=1theta=4

beta=0.5

t

f(t)

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Shape of Weibull probability density function.

0 2 4 60

1

2

theta=0.5theta=1theta=4

beta=1

t

f(t)

18

.

0 2 4 60

1

2

3

4

theta=0.5theta=1theta=4

beta=4

t

f(t)

Shape of Weibull probability density function

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Shape of Weibull probability density function

0 2 4 60

2

4

6

8

theta=0.5theta=1theta=4

beta=10

t

f(t)

.

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Effect of shape parameter

Consider building exposed to earthquakes:The larger the value of the shape parameter, the larger the rate of deterioration in strengthIf the shape parameter is one then there

is no deterioration in the strength

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Statistics

3665.0

222

)(ˆ

])}1

1({)2

1([)(

)1

1()()(

eTmedian

TE

Median: 50% probability lower than median, 50% higher than median

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Other common distributions

• Lognormal; If x is normal then exp(x) is lognormal

• Gamma: quite similar to Weibull

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Estimating hazard function, failure density function and reliability function

from dataCase 1: Large sample of data about failures (N

greater than 30)• Start with N systems.

tNttNtN

tf

ttNttNtN

th

N(t)N

R(t)

tN

)()()(

)()()(

)(

tat time,ly successful operate that systems ofnumber ),(

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Case 2: Small samplesStudy homework 3

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Selecting a probability distribution on the basis of knowledge of the particular

physical situation causing failures

• In most real life problems, we do not have enough data to estimate probability distributions. Therefore, we rely on experience or on analytically obtained associations of physical situations causing failure and probability distributions to select type of probability distribution to failure.

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Weibull and exponential models• Extreme disturbances occurring completely at random

and independently. Example: time of occurrence and intensity of a strong earthquake does not affect the time of occurrence and intensity of the next.

• Probability of occurrence of one earthquake during [t, t+dt] is dt. Average rate of occurrence of extreme disturbances is disturbances/unit time

• Probability of a system failing because of a disturbance, p(t)

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Earthquake intensity versus time

t

IntensitySevere earthquakes

severe earthquakes per yearreturn period, 1/

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Reliability

ptT

pt

tPT

dptP

petf

etR

etptf

eetR

t

)(

)(

:constant is p(t) If

)()(

)(

)(

)()( 0

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Suggested reading

• Fox, E., “The Role of Statistical Testing in NDA,” Engineering Design Reliability Handbook, CRC press, 2004, p. 26-1.