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Pressure Relief ValvePressure Relief Valve
Pop Test DataPop Test Data
StatisticalStatistical
Replacement Interval CalculationReplacement Interval Calculationbyby
Frederick T. ElderFrederick T. ElderIRC Research and Technology ForumIRC Research and Technology Forum
February 9, 2007February 9, 2007
(c) Frederick T. Elder(c) Frederick T. Elder
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When to Replace per IIAR 110When to Replace per IIAR 110 After a known relief, and within a reasonable time,After a known relief, and within a reasonable time,
springspring--loaded relief valves shall be replaced by newloaded relief valves shall be replaced by new
or remanufactured certified valves. If reor remanufactured certified valves. If re--seating isseating isnot complete, replacement shall be immediate.not complete, replacement shall be immediate. When a component reliability program is in place toWhen a component reliability program is in place to
verify relief valve functionality and longevity byverify relief valve functionality and longevity byhistory, testing, disassembly and inspection, andhistory, testing, disassembly and inspection, and
periodic statistical review of these activities, reliefperiodic statistical review of these activities, reliefvalves may be replaced at any interval justified byvalves may be replaced at any interval justified bythe findings of such a program. In the absence ofthe findings of such a program. In the absence ofsuch a program, each relief valve shall be replacedsuch a program, each relief valve shall be replaced
at the frequency recommended by the relief valveat the frequency recommended by the relief valvemanufacturer. In the absence of both a componentmanufacturer. In the absence of both a componentreliability program and manufacturersreliability program and manufacturersrecommendations, relief valves shall be replacedrecommendations, relief valves shall be replacedevery five years if not indicated earlier at annualevery five years if not indicated earlier at annual
inspection.inspection.
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Why Test?Why Test?
Properly assess health of NH3Properly assess health of NH3
refrigeration safety systemrefrigeration safety system
OSHA has required it in priorOSHA has required it in prior
settlement agreementssettlement agreements It may save $$$It may save $$$
It may answer a PHA questionIt may answer a PHA question
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Maintenance GuidanceMaintenance Guidance
http://www.valvehttp://www.valve--world.net/srv/ShowPage.aspx?pageID=640world.net/srv/ShowPage.aspx?pageID=640
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OutlineOutline
Background andBackground and
Advantages of WeibullAdvantages of WeibullAnalysisAnalysis
Failure CriteriaFailure Criteria
Generating Weibull PlotGenerating Weibull Plot
Weibayes AnalysisWeibayes Analysis
ExamplesExamples
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BackgroundBackground Invented byInvented by WaloddiWaloddi WeibullWeibull
in 1937in 1937 he used it forhe used it forfatigue life estimationfatigue life estimation
Dr. Robert Abernethy theDr. Robert Abernethy the
modern Weibull Analysismodern Weibull Analysis
expertexpert
Weibull Analysis first usedWeibull Analysis first usedextensively in aerospaceextensively in aerospace
applicationsapplicationsWaloddi Weibull 1887-1979
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Advantages of Weibull AnalysisAdvantages of Weibull Analysis
Main advantage:Main advantage: Small sample sizeSmall sample size
Samples may be expensiveSamples may be expensive
Reduces time/cost of testingReduces time/cost of testing
May not have many recorded failuresMay not have many recorded failures
Weibull Analysis is displayed by anWeibull Analysis is displayed by an
easy to read graphical ploteasy to read graphical plot
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Pop Test Failure CriteriaPop Test Failure Criteria
Example 250 psig valveOpens at pressures < 242.5 psig failure
Opens at pressures > 262.5 psig -- failure
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Alternate Failure CriteriaAlternate Failure Criteria
Reduce the set pressure of reliefReduce the set pressure of reliefvalves when possiblevalves when possible then expandthen expand
failure definitionfailure definition
Do not consider low pressureDo not consider low pressureopening a failure for those valvesopening a failure for those valves
where that does not create a hazardwhere that does not create a hazard
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Weibull Analysis PlotWeibull Analysis Plot
Most Weibull Analysis done from plotMost Weibull Analysis done from plot
To Plot, you need:To Plot, you need:
Failure criteriaFailure criteria
Number of failures and timesNumber of failures and times Number of suspensions and timesNumber of suspensions and times
From Plot, you get:From Plot, you get:
Predicted failure ratePredicted failure rate
Failure mechanismFailure mechanism
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Plotting DataPlotting Data
Plot scalesPlot scales X axis: Age parameter (Units of Hours in Figure)X axis: Age parameter (Units of Hours in Figure)
Y axis: Cumulative Distribution Function (CDF)Y axis: Cumulative Distribution Function (CDF)
Defines percentage of units that will fail up to an ageDefines percentage of units that will fail up to an age..
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Weibayes AnalysisWeibayes Analysis
Weibayes is used when thereWeibayes is used when there
are no or very few failures:are no or very few failures: Finding the MTTF of a unit afterFinding the MTTF of a unit after
initial testing lead to no failuresinitial testing lead to no failures
Redesigned component, severalRedesigned component, severalunits tested without failure, isunits tested without failure, is
testing sufficient?testing sufficient?
Smaller sample sizes neededSmaller sample sizes needed
with Weibayes since previouswith Weibayes since previousfailure history is knownfailure history is known
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Weibayes AnalysisWeibayes Analysis Weibayes Analysis equation,Weibayes Analysis equation,
uses anuses an assumedassumed
Can be used whenCan be used when no failuresno failures
have occurredhave occurred
Need to have back groundNeed to have back ground
failure infofailure info
Company Weibull libraryCompany Weibull library
OtherOther WeibullWeibull librarieslibraries
Where:Where:
N =N = total number oftotal number ofsuspensions and failuressuspensions and failures
r =r = number of failed unitsnumber of failed units
== assumed slopeassumed slope
t =t = time or cyclestime or cycles
/1N
i
i=1
t
r
=
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Weibayes AnalysisWeibayes Analysis Relief valve failure dataRelief valve failure data
shows typicalshows typical value ofvalue of 11 http://www.barringer1.com/http://www.barringer1.com/
wdbase.htmwdbase.htm
Weibayes can be used toWeibayes can be used todetermine replacementdetermine replacementinterval timeinterval time
Can input data intoCan input data into
Weibull program orWeibull program orcalculate by hand usingcalculate by hand usingequationequation
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Weibayes AnalysisWeibayes Analysis
Determine New Replacement IntervalDetermine New Replacement Intervalfor Test with Zero Failuresfor Test with Zero Failures
1.1. Gather suspension dataGather suspension data2.2. FindFind (as described in next slides)(as described in next slides)
3.3. Find kFind k11--value from Onevalue from One--Failure PlanFailure Plantable for your assumedtable for your assumed andand
number of samples being testednumber of samples being tested
4.4. Replacement Interval =Replacement Interval =(k(k11))
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Weibayes: FindingWeibayes: Finding With No FailuresWith No FailuresHand CalculationHand Calculation
Confidence Limit Equation for Zero Failures:Confidence Limit Equation for Zero Failures:
Use:Use:
wherewhere r=# of failuresr=# of failures
TTii=Time of each replacement=Time of each replacement
: look up this value from Chi: look up this value from Chi--squaredsquaredtable for C confidence and 2r+2 degrees of freedomtable for C confidence and 2r+2 degrees of freedom
( ){ }2 ; 2 2C f +
( )1
22 ; 2 2 0
iT C r for r
= +
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Weibayes: FindingWeibayes: Finding With No FailuresWith No FailuresWinSMITH CalculationWinSMITH Calculation
Can Select SpecificCan Select Specific
ConfidenceConfidence Enter number of units, allEnter number of units, all
as suspensionsas suspensions
Select Weibayes methodSelect Weibayes method
Choose specificChoose specific
confidence,confidence, 63.2%63.2%
confidence equivalent toconfidence equivalent to
assuming 1 failure isassuming 1 failure isimminentimminent
FindFind from Weibayes plotfrom Weibayes plot
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Weibayes Example:Weibayes Example: No FailuresNo Failures
During TestingDuring Testing
Parameters: 30 relief valves used forParameters: 30 relief valves used for5 years, 0 failures, want to increase5 years, 0 failures, want to increase
Replacement IntervalReplacement Interval
Question:Question: How many years can theHow many years can thevalves be used and have at most onevalves be used and have at most one
failure with a 90% confidence?failure with a 90% confidence?
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Weibayes Example:Weibayes Example: No FailuresNo Failures
During TestingDuring Testing
Data entered in WinSMITHData entered in WinSMITH
30 suspensions, 5 year time30 suspensions, 5 year time Weibayes method,Weibayes method, =1, 90% Confidence=1, 90% Confidence
=65.14
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Weibayes Example:Weibayes Example: No FailuresNo Failures
During TestingDuring Testing
Table of KTable of K11--values For Onevalues For One--Failure Test Plans,Failure Test Plans, =1=1
Read N=30, KRead N=30, K11=0.132=0.132
Complete table and equation to derive KComplete table and equation to derive K11--valuesvalues
included in Appendix Bincluded in Appendix B
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Weibayes Example:Weibayes Example: No FailuresNo Failures
During TestingDuring Testing
Replacement Interval: 65.14(0.132)= 8.6Replacement Interval: 65.14(0.132)= 8.6yearsyears
So with a 90% confidence, you can replaceSo with a 90% confidence, you can replace
the relief valves every 8.6 years and havethe relief valves every 8.6 years and haveat most one failure during that periodat most one failure during that period
Reasonable approach: 8.6 years minus 5Reasonable approach: 8.6 years minus 5
years = 3.6 years/2=1.8 years, so add 1.8years = 3.6 years/2=1.8 years, so add 1.8years to 5 year zero failure plan to haveyears to 5 year zero failure plan to have
reasonable probability of no failuresreasonable probability of no failures
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Weibayes AnalysisWeibayes AnalysisDetermine New Replacement Interval forDetermine New Replacement Interval for
One or More FailuresOne or More Failures During TestingDuring Testing
Most CommonMost Common
Typically, there will be failuresTypically, there will be failures
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Weibayes AnalysisWeibayes Analysis
Determine New Replacement Interval forDetermine New Replacement Interval forOne or More FailuresOne or More Failures During TestingDuring Testing
1.1. Gather failure and suspensionGather failure and suspensiondatadata
2.2. FindFind (as described in next slides)(as described in next slides)
3.3. Find kFind k00--value from Zerovalue from Zero--Failure Plan table for yourFailure Plan table for yourassumedassumed and number ofand number ofsamples being testedsamples being tested
4.4. Replacement Interval =Replacement Interval =(k(k00))
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Weibayes: FindingWeibayes: Finding With FailuresWith FailuresHand CalculationHand Calculation
Use Weibayes equation to findUse Weibayes equation to find
Use:Use:
to get ato get a specific confidencespecific confidence, where f=# of, where f=# offailuresfailures
: look up this value from: look up this value from
ChiChi--squared table for C confidence andsquared table for C confidence and2f+2 degrees of freedom2f+2 degrees of freedom
( ){ }
1/
2
2
; 2 2c
f
C f
=
+
( ){ }2 ; 2 2C f +
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Weibayes: FindingWeibayes: Finding With FailuresWith FailuresHand CalculationHand Calculation
ChiChi--Squared Table, C: 90% ConfidenceSquared Table, C: 90% Confidence
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Weibayes: FindingWeibayes: Finding With FailuresWith Failures
WinSMITH CalculationWinSMITH Calculation
Enter number of failures, allEnter number of failures, allwith the assumed time ofwith the assumed time of
half the usage timehalf the usage time
Enter number of suspensionsEnter number of suspensions Choose the specificChoose the specific
confidenceconfidence
FindFind from the Weibayesfrom the Weibayesplotplot
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Weibayes Example:Weibayes Example: One or MoreOne or More
FailuresFailures During TestingDuring Testing
Parameters: 30 relief valves used forParameters: 30 relief valves used for5 years, 2 failures, dont know when5 years, 2 failures, dont know when
failures occurredfailures occurred
Question:Question: How many years can theHow many years can thevalves be used and have zerovalves be used and have zero
failures with a 90% confidence?failures with a 90% confidence?
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Weibayes Example:Weibayes Example: One or MoreOne or More
FailuresFailures During TestingDuring Testing Data entered in WinSMITHData entered in WinSMITH
28 suspensions, 5 year time28 suspensions, 5 year time
2 failures, assumed half of 5 years, or 2.5 years2 failures, assumed half of 5 years, or 2.5 years
Weibayes method,Weibayes method, =1, 90% Confidence=1, 90% Confidence
27.26=
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Weibayes Example:Weibayes Example: One or MoreOne or More
FailuresFailures During TestingDuring Testing Table of KTable of K00--values For Zerovalues For Zero--Failure Test Plans,Failure Test Plans, =1=1
Read N=30, KRead N=30, K00=0.077=0.077
Complete table and equation to derive KComplete table and equation to derive K00--valuesvalues
included in Appendix Bincluded in Appendix B
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Weibayes Example:Weibayes Example: One or MoreOne or More
FailuresFailures During TestingDuring Testing
Replacement Interval:Replacement Interval:
27.26(0.077)= 2.1 years27.26(0.077)= 2.1 years
So with a 90% confidence, you canSo with a 90% confidence, you can
replace the relief valves every 2.1 yearsreplace the relief valves every 2.1 years
and have no failures during the intervaland have no failures during the interval
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RememberRemember
MI of pipes and vessels is also of high priorityMI of pipes and vessels is also of high priority
Relief Valves not to be placed back in serviceRelief Valves not to be placed back in service
after testingafter testing Need judgment to extend the replacement/testNeed judgment to extend the replacement/test
intervalinterval
Failed relief valve may never be neededFailed relief valve may never be needed
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Where to Buy Weibull MaterialWhere to Buy Weibull Material The New Weibull Handbook and theThe New Weibull Handbook and the
WinSMITH software packages can beWinSMITH software packages can bepurchased at:purchased at:
http://http://www.weibullnews.com/contents.hwww.weibullnews.com/contents.h
tm#Pricestm#Prices
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SourcesSources Engineering Safety Relief SystemsEngineering Safety Relief Systems, March 2006., March 2006.
by Reindl, D.T, Jekel, T.B., Available from theby Reindl, D.T, Jekel, T.B., Available from the
Industrial Refrigeration ConsortiumIndustrial Refrigeration Consortium The New Weibull HandbookThe New Weibull Handbook, Fourth Edition, 2000,, Fourth Edition, 2000,
by Robert Abernethyby Robert Abernethy
The New Weibull HandbookThe New Weibull Handbook, Fifth Edition, 2006,, Fifth Edition, 2006,by Robert Abernethyby Robert Abernethy
Fitness for Service of Pressure Relieving SystemsFitness for Service of Pressure Relieving Systems,,by W. E. Short II, presented at The 2003 ASMEby W. E. Short II, presented at The 2003 ASME
Pressure Vessels and Piping ConferencePressure Vessels and Piping Conference Reliability Testing of Relief ValvesReliability Testing of Relief Valves, by Robert E., by Robert E.
Gross, presented at The 2004 ASME PressureGross, presented at The 2004 ASME PressureVessels and Piping ConferenceVessels and Piping Conference
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SourcesSources Plant Guidelines for Technical Management ofPlant Guidelines for Technical Management of
Chemical Process SafetyChemical Process Safety
, pp 169, pp 169
--172, by Center172, by Center
for Chemical Process Safety, 1992for Chemical Process Safety, 1992
Armor SwiftArmor Swift EckrichEckrich OSHA settlementOSHA settlement
agreement of October 9, 1997agreement of October 9, 1997
IBPIBP OSHA settlement agreement of 2001OSHA settlement agreement of 2001
Code Requirements for Safety Relief SystemsCode Requirements for Safety Relief Systems,,
Todd Jekel, 2005 Research and TechnologyTodd Jekel, 2005 Research and Technology
Forum, January 20, 2005Forum, January 20, 2005
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SourcesSources Center for Chemical Process Safety (1998).Center for Chemical Process Safety (1998).
Guidelines for Pressure Relief and EffluentGuidelines for Pressure Relief and Effluent
Handling SystemsHandling Systems. (pp. 104. (pp. 104--107). Center for107). Center for
Chemical Process Safety/Chemical Process Safety/AIChEAIChE. Online version. Online version
available at: http://www.knovel.com/knovel2/available at: http://www.knovel.com/knovel2/
Toc.jsp?BookIDToc.jsp?BookID=831&VerticalID=0=831&VerticalID=0 Center for Chemical Process Safety (1989).Center for Chemical Process Safety (1989).
Process Equipment Reliability Data with DataProcess Equipment Reliability Data with Data
TablesTables. P 212. P 212
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Appendix A:Appendix A:Weibull Analysis BackgroundWeibull Analysis Background
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Advantages of Weibull AnalysisAdvantages of Weibull Analysis Weibull AnalysisWeibull Analysis
can be used for:can be used for: Failure DistributionFailure Distribution
Failure ForecastsFailure Forecasts
and Predictionsand Predictions
MaintenanceMaintenancePlanningPlanning
Effectiveness of aEffectiveness of aRedesignRedesign
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Weibull Analysis SoftwareWeibull Analysis Software WinSMITH Weibull from FultonWinSMITH Weibull from Fulton
FindingsFindings http://www.barringer1.com/wins.htmhttp://www.barringer1.com/wins.htm
Created by Wes Fulton and Dr. BobCreated by Wes Fulton and Dr. Bob
AbernetheyAbernethey
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Weibull Analysis SoftwareWeibull Analysis Software Enter age data, suspensions and failuresEnter age data, suspensions and failures
Software will:Software will: Plot DataPlot Data
CalculateCalculate EtaEta, Beta, and PVE numbers, Beta, and PVE numbers
Run a distribution analysisRun a distribution analysis
Generate a results reportGenerate a results report
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Plotting DataPlotting Data Age must be known for dataAge must be known for data
Standard Life Data: exact age of partsStandard Life Data: exact age of partsknownknown
Interval Data: Age of parts not exactlyInterval Data: Age of parts not exactly
know, so parts are groupedknow, so parts are grouped Could be from weekly, monthly, etcCould be from weekly, monthly, etc
inspectionsinspections
Age may be operating time,Age may be operating time,starts/stops, etc.starts/stops, etc.
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Plotting DataPlotting Data FailuresFailures
Establish failure modeEstablish failure mode
Every part displayingEvery part displayingthis mode constitutes athis mode constitutes a
failurefailure
SuspensionsSuspensions
Parts that failed via aParts that failed via adifferent modedifferent mode
Parts that have not yetParts that have not yet
failedfailed
Early Suspension: AgeEarly Suspension: Agebelow age of firstbelow age of first
failurefailure
Late Suspension: AgeLate Suspension: Age
above age of lastabove age of last
failurefailure
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Plotting DataPlotting Data Plot scalesPlot scales X axis: Age parameter (Units of Hours in Figure)X axis: Age parameter (Units of Hours in Figure)
Y axis: Cumulative Distribution Function (CDF)Y axis: Cumulative Distribution Function (CDF) Defines proportion of units that will fail up to an age as aDefines proportion of units that will fail up to an age as a
percentagepercentage
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Plotting DataPlotting Data
WinSMITH Data PlotWinSMITH Data Plot
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Plotting DataPlotting Data TwoTwo--parameter most widely used Weibullparameter most widely used Weibull
distributiondistribution
CDFCDF (Cumulative Distribution Function):(Cumulative Distribution Function):
F(tF(t) = 1) = 1-- ee--(t(t//))
F(tF(t) = fraction failing up) = fraction failing up--toto--time ttime t t= failure timet= failure time
= characteristic life= characteristic life
e = 2.718281, the base for natural logarithmse = 2.718281, the base for natural logarithms = slope parameter= slope parameter
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Plotting Data on Weibull PaperPlotting Data on Weibull Paper Arrange failures and suspensions in time ascendingArrange failures and suspensions in time ascending
orderorder
Set up the following table:Set up the following table:
Fill in Rank and Reverse Rank, and in Time column,Fill in Rank and Reverse Rank, and in Time column,include whether it was a Suspension or Failureinclude whether it was a Suspension or Failure
If two data points have the same time to failure, theyIf two data points have the same time to failure, theyare both presented in the column, and they will bothare both presented in the column, and they will both
get median rank valuesget median rank values
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Plotting Data on Weibull PaperPlotting Data on Weibull Paper Use equation to get Adjusted Rank (A.R.):Use equation to get Adjusted Rank (A.R.):
A.R.=[Reverse Rank X Previous Rank + (N+1)] / [Reverse Rank + 1]A.R.=[Reverse Rank X Previous Rank + (N+1)] / [Reverse Rank + 1]
UseUse BenardsBenards Median Rank formula to getMedian Rank formula to getnew Median Rank (since adjusted rank isnew Median Rank (since adjusted rank isnot an integer):not an integer):
BenardsBenards M.R.=(iM.R.=(i--0.3) X 100 / (N+0.4)0.3) X 100 / (N+0.4)
Fill out previous table, and plot:Fill out previous table, and plot: BenardsBenards M. R. on the yM. R. on the y--axisaxis
Time on xTime on x--axisaxis Draw a best fit line through the points,Draw a best fit line through the points,
make sure it is 1:1 Weibull papermake sure it is 1:1 Weibull paper
W ib ll E lW ib ll E l P iP i
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Weibull Example:Weibull Example: PreparingPreparing
Weibull Plot by HandWeibull Plot by Hand
Parameters: You are given theParameters: You are given the
following data, 8 total parts, 5following data, 8 total parts, 5failures at 49,82,96,30, and 90 hoursfailures at 49,82,96,30, and 90 hours
and, 3 suspensions at 45,10, andand, 3 suspensions at 45,10, and
100 hours100 hours Question:Question:At how many hours canAt how many hours can
you expect approximately 50% ofyou expect approximately 50% of
the parts to fail?the parts to fail?
W ib ll E lW ib ll E l P iP i
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Weibull Example:Weibull Example: PreparingPreparing
Weibull Plot by HandWeibull Plot by Hand Set up and fill in table:Set up and fill in table:
Plot the points on 1:1 Weibull PaperPlot the points on 1:1 Weibull Paper
Draw a best fit line through the points andDraw a best fit line through the points and
draw a line across from the 50% mark anddraw a line across from the 50% mark and
down to the time axisdown to the time axis
W ib ll E lWeib ll E ample P iPreparing
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Weibull Example:Weibull Example: PreparingPreparing
Weibull Plot by HandWeibull Plot by Hand
50% fail by 77 hours50% fail by 77 hours
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Interpreting the PlotInterpreting the Plot
PVE %: Goodness of fit indicator forPVE %: Goodness of fit indicator for
Weibull lineWeibull line 10% is acceptable, 50% is average10% is acceptable, 50% is average
N/S: Total number of data points/ NumberN/S: Total number of data points/ Number
of Suspensionsof Suspensions
Eta, Beta, PVE,and N/S
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Interpreting the PlotInterpreting the Plot EtaEta -- Characteristic life: Age at whichCharacteristic life: Age at which
63.2% of parts will fail63.2% of parts will fail Parameter most effected by suspensionsParameter most effected by suspensions
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Interpreting the PlotInterpreting the Plot
BetaBeta Slope of Weibull line: Failure ModeSlope of Weibull line: Failure Mode
Beta < 1.0 indicates infant mortalityBeta < 1.0 indicates infant mortality Beta = 1.0 indicates random failures that areBeta = 1.0 indicates random failures that are
independent of ageindependent of age
Beta > 1.0 indicates wear out failuresBeta > 1.0 indicates wear out failures
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Interpreting the PlotInterpreting the Plot Use PVE number to evaluate fit ofUse PVE number to evaluate fit of
lineline
Use Beta to evaluate failure methodUse Beta to evaluate failure method
Look for Bad Weibull characteristicsLook for Bad Weibull characteristics
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Bad WeibullBad Weibull Curved Weibull dataCurved Weibull data
Origin not at t=0, mustOrigin not at t=0, must
use threeuse three--parameterparameterWeibullWeibull
Outlying data pointsOutlying data points Look at engineeringLook at engineering
aspects of dataaspects of datarecording, test records,recording, test records,calibrations, etc.calibrations, etc.
Two different slopes ofTwo different slopes ofWeibull dataWeibull data More than one failureMore than one failure
mode represented bymode represented bydata, try to separatedata, try to separatedatadata
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Bad WeibullBad Weibull Close Serial NumbersClose Serial Numbers
Batch problemBatch problem
If PVE number is unacceptableIf PVE number is unacceptable
Look to different distributions, LogLook to different distributions, Lognormal, Threenormal, Three--parameter Weibullparameter Weibull
Careful, few data points leads to highCareful, few data points leads to highPVE numberPVE number
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Failure ForecastingFailure Forecasting Expected number ofExpected number of
failures that mayfailures that may
occur in a specificoccur in a specificperiod of timeperiod of time
Predicts:Predicts: Future failures whenFuture failures when
failed units are replacedfailed units are replaced
Future failures whenFuture failures when
failed units are notfailed units are notreplacedreplaced
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Failure ForecastingFailure Forecasting Additional input needed:Additional input needed:
Age of components in serviceAge of components in service Usage rateUsage rate
Introduction rate of new unitsIntroduction rate of new units
Failed parts replacement infoFailed parts replacement info
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Appendix B: WeibayesAppendix B: WeibayesExtrasExtras
Weibayes: FindingWeibayes: Finding With No FailuresWith No Failures
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Weibayes: FindingWeibayes: Finding With No FailuresWith No Failures
Hand CalculationHand Calculation
Assume at Least One Failure is Imminent:Assume at Least One Failure is Imminent:
Use Weibayes equation to findUse Weibayes equation to find
Assume 1 failure (r=1) since a failure isAssume 1 failure (r=1) since a failure is
imminent (yields 63% confidence)imminent (yields 63% confidence)
Use the following table to achieve differentUse the following table to achieve different
confidences:confidences:
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Zero Failure Plan Table,Zero Failure Plan Table, = 1= 1
K=[(K=[(--1/N)*ln(0.1)]1/N)*ln(0.1)](1/(1/))
Chi Squared Table for Use WithChi Squared Table for Use With
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Chi Squared Table for Use WithChi Squared Table for Use With
Weibayes Hand CalculationsWeibayes Hand Calculations
Use 0.10 column for 90% Lower Bound,Use 0.10 column for 90% Lower Bound,
0.05 for 95% Lower Bound, etc.0.05 for 95% Lower Bound, etc.
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OneOne--Failure Test Plan Table,Failure Test Plan Table, = 1= 1
(1(1
--Confidence)=(Confidence)=(
ee--(k(k))
))NN
+N(e+N(e--(k(k))
))NN--11
(1(1
--
ee--(k(k))
))