Reliability and probability of failure in structural fire...
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Transcript of Reliability and probability of failure in structural fire...
Reliability and probability of failure in structural fire safety engineeringAn introduction
Ruben Van Coile
Structures in Fire Forum, April 12, 2016
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I.Whyuseprobabilisticcalculations?
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I.Whyuseprobabilisticcalculations?
Perfectsafetydoesnotexist
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I.Whyuseprobabilisticcalculations?
It’sthebasisoftheEurocodes andBS
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I.Whyuseprobabilisticcalculations?
Perfectsafetydoesnotexist It’sthebasisoftheEurocodes andBS
Decisionmakingandcostoptimization
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I.Whyuseprobabilisticcalculations?
II.Basicconceptsforcalculatingfailureprobabilities
III.Discussion
Perfectsafetydoesnotexist It’sthebasisoftheEurocodes andBS
Decisionmakingandcostoptimization
PerfectsafetydoesnotexistEverystructureorstructuralelement
hasaprobabilityoffailure
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Theloadeffectexhibitsrandomvariations
(Baldwinetal.,1970)
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Materialpropertiesexhibitrandomvariations
Figure II.2: Observed mean minus specified strength and standard deviation of standard cube strength of 88 production units of concrete grade C35 (Rackwitz 1983)
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(Caspeele,2010)
Engineeringmodelsandcalculationtoolsareimperfect
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(Soetens,2015)
Load E Resistance R
Situations with failure
Thus:Evenforagooddesign,theloadEmaybelargerthantheresistanceR
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Boththe loadeffectE and the resistance R
exhibit randomvariations
Load E Resistance Rprior to fireResistance R
during fire
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Thus:Evenforagooddesign,theloadEmaybelargerthantheresistanceR
Situations with failure
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LimitingtheprobabilityoffailurebydesignThebasisoftheEurocodes
TheEurocodespecifiesatargetsafetylevel/failureprobability
Targetreliabilityindexinfunctionoftheconsequencesofstructuralfailure(normaldesignconditions)
• Eurocodepartialsafetyfactorsderivedfromthetarget
safetylevel
• ApplicationofEurocodedesignrulesresultsinasafety
levelof3.8
(generallyslightlyhigherbecauseofconservatism)
Unfortunately,notargetisspecifiedforstructuralfiresafety
• TargetreliabilitiesintheEuropean
NaturalFireSafetyConcept
(backgrounddocuments)
• Back-calculatingtheBStarget
reliabilityindexforspecificcases
Butoptions doexist
• Cost-optimization:
Whatistheoptimumlevelof
structuralfireresistance?
(VanCoile, 2015)
Isthereanoptimumlevelofinvestmentinstructuralfiresafety?Decisionmakingand
costoptimization
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Isthereanoptimumlevelofinvestmentinstructuralfiresafety?
Utility function: ( ) ( ) ( ) ( )Y p B p C p D p= − −
Benefitfunction
Initial construction
cost
Expected costs due to failure
and partial damage
p =theoptimizationparameter
Includes• [Annual]Probabilityofafullydeveloped fire(λ*)
• Probabilityoffailure givenafullydeveloped fire(Pf,fi)
• Failurecostsandrepair/reconstructioncostsincaseofpartialdamage(ξ)
Optimizationcriterion:MaximizeY
Isthereanoptimumlevelofinvestmentinstructuralfiresafety?
Safetyinvestmentcostfactor
ISO834fire exposure,given λ* 120minISO834fireexposure
Butcost-optimizationisnosilverbullet
• Parametersofcost-optimizationareuncertain
• Stakeholdersmaynotallagreeoneachinputparameter
• Casespecificevaluationsarenotalwaysfeasible needforgeneralrules
DetermineanAcceptableRangeforthestructuralfireresistancetime
Basedonresultsofcost-optimization,i.e.
• failureprobabilities• fireignition frequencies
• failurecosts…
ξ =350
Anacceptablerangeforthestructuralfireresistancetime
ξ failurecostratio
Avoidingoverinvestment
Avoidingunderinvestment
Acceptable Range20
BasicconceptsforcalculatingfailureprobabilitiesAconceptualintroduction
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ThelimitstatefunctionZHowdoyoudefinefailure?
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! = # − %Generalformula ! = # − % < 0Failure
! = # − % ≥ 0Safe/Success
“Load” E “Resistance” R
Situations with failure
UseMonteCarlomethods
Orevaluate/knowthePDF
)* = + ! = # − % < 0
ThelimitstatefunctionZApplicationtofire-exposedstructuralmembers:cross-sectionbased
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! = # − %Generalformula PureBending
! = ,-.-,*0,1 − ,2 .3 +.5
10000MonteCarlo
andMixedLNapproximation
ThelimitstatefunctionZApplicationtofire-exposedstructuralmembers:advancedmodels(2nd ordereffects)
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! = # − %Generalformula
6)ButthemomentmE depends
onthedeflection…
Doomedtocomputationally
expensiveMonteCarlo?
Adetourtoafeasiblecalculationmethod…Evenforverycomplexmodels,thePDFofascalarmodeloutputY canbeapproximated“quickly”
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Simpleexample
7 = 29:9;9<With:
X1 :LN(3;0.3)
X2 :LN(4;0.5)
X3 :LN(2;0.2)
Y:LN(12.48;0.646)
10000MCS
ThelimitstatefunctionZApplicationtofire-exposedstructuralmembers:advancedmodels(2nd ordereffects)
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! = # − %Generalformula
6)
Asthereascalarmodeloutput
whichisrepresentativeforthe
resistanceR?
Foreverycolumnrealizationthereisa
maximumloadPmax
(takingintoaccount2nd ordereffects)
=runmodeltofailure10000MCS
ThelimitstatefunctionZApplicationtofire-exposedstructuralmembers:advancedmodels(2nd ordereffects)
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! = # − %Generalformula
6)
Eccentricloadingincl.2nd order
! = ,-)=>?,*0,1 − ,2 )3 + )5
Failureprobabilityevaluationfeasible
withPDFofPmax approximated
DiscussionHowtodefine“failure”forstructuralsystemsexposedtofire?
4.Other…
NeedtodefinealimitstatefunctionZAnddeterminearepresentativescalarmodeloutputY
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! = # − %Generalformula Options
! = ,-@- − ,[email protected] tR
! = ,-A=>? − ,2A22.Runthemodelforincreasingfireloadtillfailure qmax
! = ,-B=>? − ,2BC0=013.Determinearepresentativefailureindicator e.g vmax
Toapplyreliabilityconceptstostructuralsystemsexposedto
fire,weneeda“performanceindicator/criterion”
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Summary/Conclusions
StandardreliabilitycalculationsarebasedonalimitstatefunctionZ
Perfectsafety doesnot exist.
Everystructure orstructural elementhasaprobability offailure
Needtodefineperformanceindicators/criteriaforstructuralfire
Limiting the probability offailureisthe very goalofthe Eurocodes
Probabilisticcalculationsforcost-optimizationanddecisionmaking
Thank you for your attention !
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