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2. Reliability measures Objectives: 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 failureand 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, f T (t). Units: # of failures per unit time Reliability function , R(t)= probability that system
will work properly at time t
Failure distribution function, F T (t)= probabilitythat a system will fail by time t
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Expected life of a component or system,E(T)
t
f T(t)
E(T)
Expected life
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Hazard function
Hazard function: h(t)=probability that, given that asystem has survived until time t, it will fail
between times t and t+ t, divided by t. Units ofh(t): 1/unit time h(t)= f T(t)/R(t) Example start with N=1000 light bulbs, at T=1000
hrs, 300 light bulbs are still working. After 10 hrs5 more bulbs fail. The hazard function isapproximately: h(1005)=5/(300*10)
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Shape of hazard function of mostreal-life systems: bathtub
function
t
h(t)
Debugging,or infantmortality
Aging
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Relation between reliability measures
)(t f T )(t F T )(t R )(t h
)(t f T - dt t dF T )(
dt t dR )( t
dt t h
et h R 0
')'(
)()0(
)(t F T t
T dt t f 0
')'( - )(1 t R t
dt t he R 0
')'()0(1
)(t Rt
T dt t f 0
')'(1 )(1 t F T - t
dt t h
e R 0
')'(
)0(
)(t h t T
T
dt t f
t f
0')'(1
)(
)(1
)(
t F dt
t dF
T
T
)(
)(
t Rdt
t dR -
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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 whichare subjected to extreme disturbances occurringcompletely at random and independently.
f T(t)=1/ *exp(-t/ )
R(t)= exp(-t/ ) h(t)=1/
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Shape of exponential distribution
t
f T(t)
1/
E(T)=
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Normal distribution
t
f T(t)
Standard deviation,
Two parameter distribution
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No closed form analytical expression forcumulative distribution
Cumulative distribution of standard normal, (z),has been tabulated. We also have excellent
polynomial approximations. Standard normal haszero mean and unit standard deviation. Very easy to do reliability computations with
normal distributions
Finding F T(t) if T is normal. Transform T intostandard normal.
FT(t)=P(T t)=P[(T- )/ (t- )/ ]= (z)
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Cumulative distribution of standardnormal variable
z (z)
0 0.5
-1 0.16
-2 0.02
-3 0.001
-4 3 10 -5
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If we model the time to failure using a normal distribution thenthere is small probability of the time to failure being negative.This does not make sense. Always check that the probabilityof the time being negative is small compared to the
probabilities we are calculating in the problem at hand. Forexample, 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
f T(t)
The area under thecurve to the left of zerois the probabilityof t being negative
0
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Weibull distribution Good choice for systems or components whose strengthdeteriorates with time and which are subjected to extreme
disturbances occurring completely at random andindependently.
Consider a building in Greece that is expected to be sustaina very strong earthquake (say above 6.5 in the Richterscale) once every ten years. Like any real life system, thestrength of the building deteriorates with time. A Weibulldistribution is a good candidate for modeling the time tofailure (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
et R
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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=0.5
theta=1
theta=4
bet a=0.5
t
f ( t )
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Shape of Weibull probability density function.
0 2 4 60
1
2
theta=0.5
theta=1
theta=4
bet a=1
t
f ( t )
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.
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
bet a=1 0
t
f ( t )
.
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Effect of shape parameter
Consider building exposed toearthquakes:The larger the value of the shape
parameter, the larger the rate ofdeterioration in strengthIf the shape parameter is one then thereis no deterioration in the strength
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Statistics
3665.0
222
)(
])}1
1({)2
1([)(
)1
1()()(
eT median
T E
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) islognormal
Gamma: quite similar to Weibull
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Estimating hazard function, failuredensity function and reliability function
from dataCase 1: Large sample of data about failures (N
greater than 30)
Start with N systems.
t N t t N t N
t f
t t N t t N t N t h
N (t) N
R(t)
t N
)()()(
)()()()(
tat time,lysuccessfuloperatethatsystemsof number),(
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Case 2: Small samples
Study 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 enoughdata to estimate probability distributions.Therefore, we rely on experience or onanalytically obtained associations of physical
situations causing failure and probabilitydistributions to select type of probabilitydistribution to failure.
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Weibull and exponential models Extreme disturbances occurring completely at
random and independently. Example: time ofoccurrence and intensity of a strong earthquake
does not affect the time of occurrence andintensity of the next. Probability of occurrence of one earthquake
during [t, t+dt] is dt. Average rate of occurrenceof extreme disturbances is disturbances/unit time
Probability of a system failing because of adisturbance, p(t)
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Earthquake intensity versus time
t
IntensitySevere earthquakes
severe earthquakes per yearreturn period, 1/
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Reliability
pt T
pt
t P T
d pt P
pet f
et R
et pt f
eet R
t
)(
)(
:constantis 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.