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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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