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TensileMembraneActionofCompositeSlabsinFire
ArethecurrentmethodsreallyOK?
IanBurgess
UniversityofSheffield,UK
Cardington
Maxbeamtemperature~1150°Ccf.Codecriticaltemperature~ 680°C
2
Thebasisofallcurrentsimplifiedmethods:Hayes(1968)
3
Small-deflectionyield-linemechanism– slabonly
L=al
l
nL
gHoggingrotationsaboutedgesof
panel
Saggingrotationsaboutinternalyield
lines
Yield-linepatternisoptimized
forminimumconcreteslab
failureload.
4
Large-deflectionfailurecrack
sometimesobservedintests
andusedbyHayes.
Equilibrium1– nothrough-depthYLcracks- Hayes
kbKT0
bKT0
Rationale: Superpositionofrebar
tensionandconcretecompression
force/unitlength.
Tension
Compression
5
T0l/2
E
Criterion: Cracksfrom
intersection.Moment
equilibriumaboutE.Findsbandk.
Equilibrium2– somethrough-depthYLcracks- Hayes
kKT0
KT0
Tension
6
Rationale: Superpositionofrebar
tensionandconcretecompression
force/unitlength.
Compression
Bothb andk areconstant foreachofthe2cases.No
variationwithdeflection.
1 12
2
“Membraneforce”enhancements:
e1m Momentofmembraneforcesabout11/totalresistance
momentaboutx-axisatinitialYL.
e2m Momentofmembraneforcesabout22/totalresistance
momentabouty-axisatinitialYL.
Thesestartatzeroforzerodeflection
Resistancemomentenhancement(reduction)
e1b Proportionalchangeofresistancemomentaboutx-axisdue
tomembranecompression.
e2b Proportionalchangeofresistancemomentabouty-axisdue
tomembranecompression.
Partialenhancementfactors– bothcases- Hayes
7
8
Bending“enhancements”- Hayes
Wood’sequationforreductionofmomentcapacityofarectangularRCcross-
sectionduetoaxialcompression:
2
0 0 0
1M N NA BM T T
= +1.Long-spanreinforcement:
2
0 0 0
1 ' 'M N NA BM KT KT
= +2.Short-spanreinforcement:
Theseareintegratedinx- andy- directionsrespectivelyforthebending
momentsacrosstheyieldlinesforPortions1and2.
T0
T0
T0
T0N
9
!" = !"$ + !"&!' = !'$ + !'&
! = !" −!" − !'1 + 2+,'
Thesearenearlyalwaysunequal
(WHY?).Puttogetheras
Overallenhancementfactor
x
y
z
V
V
Verticalshear
resultantsacross
yieldlines
Thesedon’tincludeanyverticalshearbetween
thefacets.Iftheseareincludedthereisonly
oneenhancementfactor.
(TonyGillies2015)
Newenhancement
Factorequivalentto ! = !" −!" − !'
1 + 2+n,'
Forminganoverallenhancementfactor- Hayes
10
Anyproblemssofar?
• Themembranetractiondistributionisanassumption.
Itcorrespondstounfractured meshandeither:
• Nothrough-depthcracksalongyieldlines.
• Partialthrough-depthcracksalongyieldlines.
• Bothofthesedistributionsapplyonlytothecasewherea
lateralthrough-depthcrackhasformedacrosstheshort
spanthroughtheYLintersection.
• Distributionisfixedforeachcase.Enhancementfactor
startsbelow1.0– actuallyatzero.
• Internalforcesdon’tdependondeflection.
Structuralfireresistancemethodsforcompositefloors
11
BREMethod(Bailey2000)• AmendedversionofHayes’s
method.
• FireSafeDesign(SCIP288)checkedusingBRE-Baileydesign
method.
NewZealandSPM(Clifton2006)FRACOF(2011)• BasedonaEuropeanproject.
• AlmostidenticaltoBREmethod.
Afewchangestosafetyfactors,
extradeflectioncheck.
TypicaldesignstrategyforTMA
• Protectmemberson
columngridlines.
• Leaveintermediate
secondarybeams
unprotected.
• Designindividualpanels
withoutcontinuity.
12
BRE/FRACOFmethod
13
• Unprotectedcomposite
beamsathightemperature
carrysomeoftheloadas
simplysupported.
• Concreteslabcarriesremainingloadintensile
membraneaction.Needs
enoughdeflection.
+
Small-deflectionyield-linemechanism– slabonly
BRE/FRACOF
L=al
l
nL
gHoggingrotationsaboutedgesofpanel
Saggingrotationsaboutinternalyield
lines
Theanalysisisbasedonthe
optimalyield-line patternfor
theconcreteslabwithout
consideringthesteelbeams.
14
Tensilecrackacross
shortmid-span
Large-deflectionfailurecrack
observedintestsandusedin
Bailey/BRE,FRACOFandNZ
SPM.
Forceequilibrium– nothrough-depthYLcracks– BREetc
kbKT0
bKT0
15
E
Criterion: Crackacross
mid-long-span.Moment
equilibriumaboutE.
Findsb andk.
(Ultimatestrength
ofreinforcement
acrossFracture)1.1T0l/2
Thisistheonlymechanism–
noseparationofconcrete
alongtheyieldlines.
0.0
0.5
1.0
1.5
2.0
2.5
0 1 2 3 4 5 6 7 8
Enhancementfactor
d/d1
BRE1.0 Gillies1.0
BRE1.5 Gillies1.5
BRE2.0 Gillies2.0
BRE3.0 Gillies3.0
16
TMAenhancementcalculations– BRE/FRACOF
SimilarlytoHayes:• Horizontalforceequilibrium
assumingmid-spancrack.(But
onlythelinearmembrane
tractiondistribution).
• Separate“membrane”
enhancementse1mande2m by
momentsaboutlongandshort
edges.
• Add“bending”enhancements
e1bande2b tomakee1ande2.• Overallenhancementfactor
! = !" −./0.1"2'341
• …orGillies! = !" −./0.1
"2'3641
• Cutoff atenhancement1.0for
aspectratios>1.0.
17
78 =9.;<=>?=
@A1
B�
7D =E F' − F" G'
16ℎ
wE wq
Limitingdeflection(centralcracking) criterion
+
Backtobasics
18
Increasingdeflectionofyield-linemechanism
Yield-linemechanismisaplasticbendingmechanismat
smalldeflections.Yieldlines
areessentiallydiscretecracks.
Asdeflectionsstarttoincreasetheyield-linepatternincreases
therotationsofitsflatfacets,
withtherebaryieldinguntilit
fractures.
Sotheinitiallarge-deflectionmechanismisthisone.
“MechanismB”
19
h
x
µtt
s
yx
zf
qy
x
DxDy
TOPSURFACEOFSLABCRACK OPENING AT REBAR LEVEL
x
Asdeflectionsstarttoincreasetheyield-linepatternincreasestherotationsofitsflatfacets,and
rebaryieldsacrosscracksuntilitfractures.
Geometryofyield-linecrack opening
20
ForceequilibriumofMechanismB
S
C1T1T2 C2
M1 M2
M3
Shapeofconcretecompressionblocksisdictatedbycompatibilityandequilibrium:• Initiallytensionandcompressionateverypointofyield-
lines.
• Asdeflectionincreasesconcretecompressionblocks
concentratetowardsslabcorners,rebarfractureswhen
itsstrainexceedsitsductility.
• Notensionwithincompressionblocks
21
Shapeofconcretecompressionblocksisdictatedbycompatibilityandequilibrium:• Initiallytensionandcompressionateverypointofyield-
lines.
• Asdeflectionincreasesconcretecompressionblocks
concentratetowardsslabcorners,rebarfractureswhen
itsstrainexceedsitsductility.
• Notensionwithincompressionblocks
Changeofstressblocks– ductiley-reinforcement
Compression
Tension
z
y
x
y
a1
22
Shapeofconcretecompressionblocksisdictatedbycompatibilityandequilibrium:• Initiallytensionandcompressionateverypointofyield-
lines.
• Asdeflectionincreasesconcretecompressionblocks
concentratetowardsslabcorners,rebarfractureswhen
itsstrainexceedsitsductility.
• Notensionwithincompressionblocks
Changeofstressblocks – ductiley-reinforcement
Compression
Tension
z
y
x
y
a1
23
Shapeofconcretecompressionblocksisdictatedbycompatibilityandequilibrium:• Initiallytensionandcompressionateverypointofyield-
lines.
• Asdeflectionincreasesconcretecompressionblocks
concentratetowardsslabcorners,rebarfractureswhen
itsstrainexceedsitsductility.
• Notensionwithincompressionblocks
Changeofstressblocks – ductiley-reinforcement
Compression
Tension
z
y
x
y
a1
24
Shapeofconcretecompressionblocksisdictatedbycompatibilityandequilibrium:• Initiallytensionandcompressionateverypointofyield-
lines.
• Asdeflectionincreasesconcretecompressionblocks
concentratetowardsslabcorners,rebarfractureswhen
itsstrainexceedsitsductility.
• Notensionwithincompressionblocks
Changeofstressblocks – ductiley-reinforcement
Compression
Tension
z
y
x
y
b1
25
Shapeofconcretecompressionblocksisdictatedbycompatibilityandequilibrium:• Initiallytensionandcompressionateverypointofyield-
lines.
• Asdeflectionincreasesconcretecompressionblocks
concentratetowardsslabcorners,rebarfractureswhen
itsstrainexceedsitsductility.
• Notensionwithincompressionblocks
Changeofstressblocks – ductiley-reinforcement
Compression
Tension
z
y
x
y
b1
26
Shapeofconcretecompressionblocksisdictatedbycompatibilityandequilibrium:• Initiallytensionandcompressionateverypointofyield-
lines.
• Asdeflectionincreasesconcretecompressionblocks
concentratetowardsslabcorners,rebarfractureswhen
itsstrainexceedsitsductility.
• Notensionwithincompressionblocks
Changeofstressblocks – ductiley-reinforcement
Compression
Tension
z
y
x
y
b2
27
Shapeofconcretecompressionblocksisdictatedbycompatibilityandequilibrium:• Initiallytensionandcompressionateverypointofyield-
lines.
• Asdeflectionincreasesconcretecompressionblocks
concentratetowardsslabcorners,rebarfractureswhen
itsstrainexceedsitsductility.
• Notensionwithincompressionblocks
Changeofstressblocks – ductiley-reinforcement
Compression
Tension
z
y
x
y
c
28
Withdifferentrebar
ductility,meshcan
eitherfractureabruptly
orprogressivelyatany
stage.
Changeofstressblocks– possibilitieswithlessductility
Tension
z
y
x
y
Unfractured
MidYLfractured
Diagonalsunzipping
29
Withdifferentrebar
ductility,meshcan
eitherfractureabruptly
orprogressivelyatany
stage.
Changeofstressblocks– possibilitieswithlessductility
Tension
x
y
Unfractured
MidYLfractured
Diagonalsunzipping
z
y
30
Withdifferentrebar
ductility,meshcan
eitherfractureabruptly
orprogressivelyatany
stage.
Changeofstressblocks– possibilitieswithlessductility
x
y
Unfractured
MidYLfractured
Diagonalsunzipping
z
y
Tension
31
a1x
b1x
b1x'
b1x**
b1x***
0.0
0.5
1.0
1.5
2.0
2.5
0 1 2 3 4 5 6 7 8
Enhancementfactor
d/d1
Garstontestcomparison
• A142mesh(142mm2 permetre,580MPa
steelat200mmspacinginxandy
directions)at69mmeffectivedepth;
• Meshductility12%.
• Slabaspectratio1.4706(6.360mx
9.353).
• 120mmthick,52MPaconcrete;
• Edgesverticallysupported.
32
0.0
0.5
1.0
1.5
2.0
2.5
0 1 2 3 4 5 6 7 8
Enhancementfactor
d/d1
1.0 BRE1.0
1.4706 BRE1.4706
2.0 BRE2.0
3.0 BRE3.0
Garstoncomparisonfordifferentaspectratios
33
0
1
2
3
4
5
6
7
0 1 2 3 4 5 6 7 8
Bendingstress(MPa)
d/d1
StressQ
StressR
StressS
Garston– applytensilestrengthtochangemechanism
Q R
S
EC2tensile
strengthforC52
[0.3fc0.67]
34
b
DyDx
Dxy x
y
2
1
1
20x
xu z
v
=
=
22
2
2
2
2
2
2
y
x
y
u yyv x z
x Gap
y Gap
xy z
yx z
=
= +
=
=
+ +
+ +42
4
2
2
4
2 2 2 2
y
y
lu
lv x z
ly Gap x z
=
= +
= + +
32
3 2 2
2 2
y
xy
u yyrlv z
x Gap y
=
= +
=
3' 2 xu y= +
Newmechanism– centralthrough-depthcrack
35
Changeofstressblocks – ductiley-reinforcement
z
y
x
y
36
Changeofstressblocks – ductiley-reinforcement
z
y
x
y
37
Changeofstressblocks – ductiley-reinforcement
z
y
x
y
38
0.0
0.5
1.0
1.5
2.0
2.5
0 1 2 3 4 5 6
Enhancementfactor
d/d1
Frombasicmechanismtocentrallycracked
39
Withattachedsteelbeams…
…theyield-linemechanismchanges.
40
l
rlnxl
gx
Forcesonthex-alignedmechanism
SC1Cy2
Tx1
Ty1
Ty2
Tb, t°
Tb, t°
A B
Unprotectedbeamsat
hightemperature
41
SC1
Cx2
Tx1
Ty1
Tx2
Tb, t°
Tb, t°
A
B
nyl
rl
l
gy
Forcesonthey-alignedmechanism
Unprotectedbeamsat
hightemperature42
Combinationsofcompressionblockandrebarfracture
CompressionblockReinforcementmeshfracturelevel(x-alignedmechanism)
None Centraly Diag.xCentral+
Diag.y
Central+
Diag.x
Central+
Diag.x,y
Full abovemesh a1 a1’ a1* a1** a1*’ a1***
belowmesh a2 a2’ a2* a2** a2*’ a2***
Triangular abovemesh b1 b1’ b1* b1** b1*’ b1***
belowmesh b2 b2’ b2* b2** b2*’ b2***
Trapezoidal c1 c1’ c1* c1** c1*’ c1***
y
x
43
Exampleofapplication:9mx6mcompositeslab
9m
6m
• 130mmthickslab,30MPa
concrete;
• A142mesh(142mm2 per
metre,500MPasteelat
200mmspacinginxandy
directions)at60mmeffective
depth;
• Mesheffectiveductility(over
200mmlength)1%:fracture
crack-width2mm;
• Onecentraldownstandsteel
beam,305x165UKB40,Grade
S275- unprotectedagainst
fire;
• Edgesverticallysupported.
305
60
130
16510
6
44
ny
0.6
0.4
0.2
0 2 4 6 8 10 12
0.8
n xorn y
Loadcapacity(kN/m2)
nx
Initialyield-lineparameterfordifferentloadcapacities
nx
ny
45
x
Criticaltemperature(°C) 900
600
300
0
Loadcapacity(kN/m2)
2 4 6 8 10 12
1200
y-mechanism
x-mechanism
Criticaltemperaturevariationwithloadcapacity
46
x
y
1200
1000
800
600
400
200
0 20 40 60 80 100
1400Unprotectedbeamtemperature(°C)
Slabdeflection(mm)
10
8
6
5
3
2.0
2.772.4
2.2
(Nofailureofnon-compositeslab)
(Ambient-temperaturefailureofcompositeslab)
4
Enhancementofcriticalsteeltemperaturewithdeflection
47
800
700
0 20 40 60 80 100
900
Unprotectedbeamtemperature(°C)
Slabdeflection(mm)
3kN/m2
Enhancementofcriticalsteeltemperaturewithdeflection
a1y
b1y’
b1y
b1y***b1y’*
48
MaximumsteeltemperatureenhancementsTemperatureenhancement(°C)
120
80
40
0 2 4 6 8 10
160
y-mechanismx-mechanism
b1y’
B1y*
b1y’*
b1y’*
Loadcapacity(kN/m2)
b1x’
B1x**
49
MaximumtensilestressatsectionA
5
Stress(M
Pa)
3
2
1
0 20 40 60 80 100
Slabdeflection(mm)
A
5kN/m2
A
B
50
Insummary…
Existingsimplifiedmethods:Forconcreteslabs:
• Fixedmembranetractiondistribution– independentofslab
deflection.
• Membranetractiondistributiononlyvalidwhileconcretehas
compressionalongwholeyieldlines.
• Assumescentralcrackfullyformed.Rebaratultimatestrength
(+10%)
• Enhancementfactorstartsbelow1.0.
Forcompositeslabsinfire:
• Yield-linepatternbasedonnon-compositeslab.
• Superposeshigh-temperaturecompositebeamcapacityand
deflection-controlledslabenhancement.
• Criterionformid-spanthrough-depthcrackismeaningless.
51
Insummary…
Thenewapproach:Forallslabs:
• Basedonthekinematicsofdeflectingflatfacetsofthesmall-deflection
yieldlinemechanism,togetherwithin-planeequilibriumofthe
concreteandsteelforces.
• Allowsconcretestressblockstomoveandmeshtofractureacross
yieldlines.
Forcompositeslabsinfire:
• Keepsloadconstant,allowsbeamstemperaturetoincreaseuntilyield
linemechanismforms.
• Enhancementofsteelbeamtemperaturewithdeflection.
Biggestproblemstobesolved:• Fractureductilityofrebaracrossdiscretecracks- yieldlinesor
through-depthmid-spancrack.
• Concretetensilestresstoinitiatethemid-span(orintersection)crack.
52
Thankyou
53