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

! = ,-@- − ,2@21.RunthemodelforISO834exposuretillfailure 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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