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CE437/537,Spring2011 PretensionedBeamExample 1/15
Pretensionedbeamsaretypicallymanufacturedbyavendorinaprestressingyard. Structuralengineers
selectappropriatebeams(forexamplehollowcoreslabsectionsanddoubleTbeams)forbuildingsfrom
loadtablesprovidedbythevendor. Pretensionedbridgegirders(e.g.AASHTOandbulbTgirders)can
bedesignedbythevendorusingspecialcomputersoftware.
Studentscangainanunderstandingofthebehaviorofpretensionedbeamsbyanalyzingtheresponseof
atypical
pretensioned
beam
at
each
stage
of
its
life.
Example: SelectapretensionedDoubleTbeamfromthePCImanualandcheckitagainstcriteriainACI
31808. Span=52ft,SDL=0,LL=60psf.
1. Selectashapeandprestressinglayout. FromthePCIloadtableshowninFigure1,selecta10DT26with10diameter270ksilowrelaxationstrandswithoneharppoint(atmidspan).
Figure1. SpanloadtablefromPCIManual(5th
Edition)for10DT26.
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CE437/537,Spring2011 PretensionedBeamExample 2/15
Usethefollowingsectionandmaterialproperties. Assumethatthetendonsarejackedto70%oftheir
tensilestrength. Alsoassumethatattransfer,10%ofthestressislostduetoseatingoftendon
anchorages. Calculatethetendoneccentricitiesatthecriticalsections(transferpoint=strand
developmentlengthfromendofbeam,0.4span,andmidspan).
sp a nlength L 52 ft SDL 0 psf
LL 60 psf Concrete:
SectionProperties Mat'lProperties
Shape 10DT26 f'c 5,000 psi strength@28days
A 689 in2
1 0.80
bf 120 in UW 150 pcf unitweight
tf 4 in SW 718 plf selfweight
I 30,716 in4
yt 5.71 in f'ci/f'c 80% strengthlevel at transfer
yb 20.29 in f'ci 4,000 psi strengthattransfer
sectionmodulus St 5,379 in3
=I/yt Ec 4287 ksi mod.ofelasticity
Sb 1,514 in3
=I/yb Eci 3834 ksi
=0.033*UW^1.5*SQRT(f'ci)
SteelStrands:SectionProperties Mat'lProperties
numberofstrands Nstrands 10 fpu 270 ksi tensilestrength
stranddiameter ps 0.5 in Eps 28,500 ksi modulus
area ofstrand Astrand 0.153 in2
pu 0.045 max.rec'dstrain
area ofal l strands Aps 1.53 in2
frombeambottom ys_end 10 in
ys_mid 3 in
Jacking&Release
ja cki ng stress level fjacking/fpu 75%
seatingloss ftrans/fjacking 90%
fpo 182 ksi =f pu*fjacking/fpu*ftrans/fjacking
tress force a ttransfer Po 279 k =fpo*Aps
developmentlength Ldevel 30.38 in =fpo*ps/3
TendonProfile
Beam
e n d ra n sf e r
Pt 0.4
S pa n Mi d
Spaneccentricity e 10.29 10.97 15.89 17.29 in =yb ys_mid
Moments
TransferPt 0.4Span MidSpan
x/L 0.0487 0.400 0.500
MCoef 0.0232 0.120 0.125 =0.5*(x/L x/L^2)
omentdu etose l f wt MSW
44.9 233 243 kft =SW/1000*L^2*MCoef
mentdu etolive load ML 37.6 194.7 203 kft =LL/1000*bf/12*L^2*MCoef
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CE437/537,Spring2011 PretensionedBeamExample 3/15
2. Calculatethelossofprestressduetoelasticshortening,creepandshrinkageoftheconcreteandrelaxationoftheprestressingstrands. SeePrestressLossesontheclasswebsiteforan
explanationoftheconcretestressesduetoprestressing.
3. Calculatethestressesintheconcreteattransferandatservice. TheallowablestressesfromACI31808areshownonFigureA2.
LossofPrestress (@0.4L)PSforcea ttransfer P
o 279 k
e 15.89 in
fci_CGS 1.251 ksi =Po/A + Po*e^2/I MSW
*12*e/I
elas ticshortening ES 9.30 ksi =fci_CGS/Eci*Eps
creep CR 16.64 ksi =2*ES*Eci/Ec
l tosurfacearea ratio V/S 2.05
relativehumidity RH 75 %
shrinkage SH 5.12 ksi =0.0000082*Eps*(1 0.06*V/S)*(100 RH)
C 1.0
relaxation RE 3.76 ksi =f'c/1000 0.04*(ES+CR+SH)*C
Total_Loss 34.8 ksi
fpe 168 ksi =fpu*fs_jacking/fpu Total_Loss
ectiveprestressforce Pe 257 k =fpe*Aps
StressesatTransfer(allstressesinpsi)Transfer
Pt 0.4
Span Mid
Span
Allowable
concrete stressesat: compression tension
top ofbeam ft 64 2800 379
bottomofbeam fb 2069 =0.7*f'ci =6*SQRT(f'ci)
ft 101 50 2400 190
fb 1486 1666 =0.6*f'ci =3*SQRT(f'ci)
where:
ft =(Po/A+Po*e/St MSW
*12/St)*1000
fb =(Po/A
Po*e/Sb
+
M
SW
*12/Sb)*
1000
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CE437/537,Spring2011 PretensionedBeamExample 4/15
4. Checktheflexurestrengthunderoverload(atultimateconditions).Thestressstraingraphforprestressingstrands(seeFigureA3)isnotbilinear,asassumedfor
rebar. Thestressinthestrandscanbecalculatedasafunctionofthetotalstrainintheprestressing
usingtheequationsatthebottomofthefigure.
The
total
strain
in
the
tendons
is
the
sum
of
the
strain
due
to
the
effective
prestress
force
(Pe)
plus
thestrainintheconcreteattheCGSduetothefailureloads(seeFigureA4). ThestrainattheCGS
duetothefailureloadsismosteasilycalculatedbyfirstcalculatingthestrainrequiredto
decompresstheconcrete,thencalculatingthetensilestrainintheconcreteatultimate(similarto
anormallyreinforcedconcretebeam).
Theinternalforcesmustbecalculatediteratively,sincetheforceintheprestressingisafunctionof
thestrainintheprestressing,andthestraindistributionisafunctionoftheinternalforces. When
checkingtheflexurestrengthusinghandcalculations,itsconvenienttostartwithanassumedvalue
ofthestressintheprestressingthatisclosetotheultimatetendonstrength,saywithin5ksi.
Theavailableflexurestrength,Mn,mustbegreaterthanthemomentduetofactoredloads,Mu,andmustbegreaterthan1.2xthecrackingmoment,Mcr. Thislaststipulationistoensureaductile
failure: iftheflexurestrength(Mn whichisbasedontheassumptionthattheconcreteinthetensilezonehascracked)islessthantheuncrackedstrengthofthebeam,thenwhenthe
overloadedbeamdoescrackitwillfailsuddenly.
StressesatService(allstressesinpsi) Allowable
TransferPt 0.4Span MidSpan compression tension
cretestresses du e to:
sustainedloads ftsustained
51 134 89 2250 849
=0.45*f'c =12*SQRT(f'c)
a l l service loads ftall
33 568 541 3000 849
=0.6*f'c =12*SQRT(f'c)
a l l service loads fball
1578 324 228 3000 849
=0.6*f'c =12*SQRT(f'c)
where:
ftsustained
=(Pe/A+Pe*e/St MSW
*12/St)*1000
ftall
=(Pe/A
+
Pe*e/St
M
SW
*12/St
ML
*12/St)*
1000
fball
=(Pe/A Pe*e/Sb+MSW
*12/Sb+ML*12/Sb)*1000
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CE437/537,Spring2011 PretensionedBeamExample 5/15
5. CheckDeflectionsTheengineermustcalculatethecamberofthebeamwhenitiserectedandthelongtermcamber
ofthebeamsothatthefinishedstructureperformsasintended. Deflectionsduetoliveloadsare
alsochecked
against
the
ACI
allowable
deflections
listed
at
the
bottom
of
Figure
A
2.
FlexureStrength0.4Span MidSpan
e 15.89 17.29 in
du e toprestressing sPe
0.00588 0.0058836 =Pe/(Aps*Eps)
decompression
i nconcrete a tCGS CGSPe
0.00058 0.00067 =Pe/(A *Ec) Pe*e^2/(I*Ec)
effective depthtoPS dp 21.60 23.00 in =yt+e
depthto neutral axis c 1.01 1.01 in mustbe
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CE437/537,Spring2011 PretensionedBeamExample 6/15
DeflectionduetoPrestressing. Equationstocalculatethedeflectionduetoprestressingcanbe
derivedfromthemomentdistributionscausedbytheprestressingforces.
= dxEIM
PS
Forabeam
with
asingle
depression
point,
the
moment
diagram
due
to
prestressing
is
as
shown
below:
Doubleintegrationoftheequationabove:
+= dxLx
ePdxePIE oendoci2/
')(
12
'
8
22L
IE
ePL
IE
eP
ci
o
ci
endomidspan +=
eend
emide
Poeend
Poe
TendonProfile
MomentsduetoPS
x
L/2 L/2
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CE437/537,Spring2011 PretensionedBeamExample 7/15
CrackedMomentofInertia. Ifthebeamwillcrackunderserviceloads,thenthecrackedmomentof
intertia(Icr)mustbecalculated. Thetransformedsectionshownbelowisconstructedinwhichthe
areaoftheprestressingismultipliedbythemodularratioofsteeltoconcrete,n=Eps/Ec. the
concretebelowtheneutralaxis(NA)isneglectedbecauseitisintensionandcracked. Inthefigure
below,K=curvature
=strain
/(distance
from
NA).
Thefirsttaskistocalculatethelocationoftheneutralaxis,xinchesbelowthetopoftheflange,
forwhichthecompressionforcesarebalancedbythetensionforces.
02
)(2
1
)(2
1
)(
,2
1
2
2
__
_
=+
=
=
==
==
=
=
ppspsf
ppsf
cppsfc
cppspsps
cctopctopc
pspsftopc
dAnxAnx
b
xdAnxb
EnxdKAxbExK
EnxdKEf
ExKEf
fAxbf
TC
Usingthequadraticequationtosolveforxgives(andlettingnAps=A,bf=b,anddp=d)
bdAbAAxx '2'',
2
21 +=
Thecrackedmomentofinertiaisthen
23
)('3
xdAxb
Icr +=
bf
x
nAps
dp
flange
NA
strains concrete
stress
K
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CE437/537,Spring2011 PretensionedBeamExample 8/15
EffectiveMomentofInteria. Thebeammaycracknearmidspan(at0.4Lforbeamswithasingle
depressionpoint)butitwillnotcrackunderserviceloadsoverthewholelength. Equation98inACI
31808canbeusedtocalculateaweightedaverageofthecrackedmomentofinertiaandthe
uncracked(grossmomentofinertia,Ig)dependingontherelativemagnitudeofthecracking
momentMcr andthemomentduetoappliedloads,Ma. Theeffectivemomentofinertiaforthe
entire
span,
Ie
is
cra
crg
a
cre I
M
MI
M
MI
+
=
33
1
Thesituationiscomplicatedbythepresenceofprestressingforces. Forpretensionedbeams,
Ma=theliveloadmoment
Mcr=theportionoftheliveloadmomentnecessarytocausecracking
Crackingis
imminent
when
at
the
bottom
of
the
beam,
the
stresses
due
to
dead
load
plus
apercent
oftheliveloadequalthedecompressionstressplusthetensilestrengthoftheconcrete
riondecompress
bcrackingcausetoL
bDb ffff +=+
%
Dbr
Pb
crackingcausetoLb ffff
e +=% , wherethedecompressionstrain= +vestrainduetoPe
Writingtheequationsaboveintermsofmoments
bLb
bDbr
Pb
a
cr
Sf
Sfff
M
M e )( +
=
Definethetotalstressatthebottomofthebeamas
Lb
Db
Pb
Tb ffff
e ++=
then
Db
Pb
Tb
Lb ffff
e =
and
Lb
rT
b
a
cr
Lb
rT
bLb
a
cr
f
ff
M
M
f
fff
M
M =
+= 1,
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CE437/537,Spring2011 PretensionedBeamExample 9/15
Toillustratethecalculationprocedureforabeamthatcracksunderliveload,increasetheliveload
to80psfforthisexample.
Deflection
ImmediateDeflectionsduePSandSW:
eend 10.29 in
emid 17.29 in
e' 7.00 in =emid eend
deflection a ttransfer transf 1.72 i n
=Po*eend*(L*12)^2/(8*Eci*I)+ Po*e'*(L*12)^2/(12*Eci*I)
lection du e tos e l f wt SW
1.00 in =5*SW/1000*L^4*1728/(384*Eci*I)
LL
Icr:
n 6.65 =Eps/Ec
A 1.53 in2
=Aps
nA 10.17 in2
=n*A
d 21.60 in =dp
b 120.00 in =bf
sqrt 229.9 in2
=SQRT(nA^2+2*b*nA*d)
x1 1.83 in =
(
nA
+
sqrt
)
/
bx2 2.00 in =(nA sqrt)/b
fromtop fibertoNA x 1.83 in
i a fo rcrackedsection Icr 4,221 in4
=b*x^3/3 + nA *(d x)^2
Ieff:
s du e a ll l oads a t0.4L fbT
838 psi
fbL
2058 psi =ML*12000/Sb
Mcr/Ma 0.850 =1 (fbT fr)/fb
L
Ieff 20,508 in4
=(Mcr/Ma)^3
*
I
+(1
(Mcr/Ma)^3)
*
Icr
deflection du e toLL LL
1.50 in =5*LL/1000*bf/12*L^4*1728/(384*Ec*Ieff)
owable deflectdu e LL LL
max 1.73 in =L*12/360
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CE437/537,Spring2011 PretensionedBeamExample 11/15
FigureA1. Stressesinconcreteduetopretensioning.
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CE437/537,Spring2011 PretensionedBeamExample 12/15
PrestressingSteel: ThefollowingcriteriaarespecifiedbyACIfortheprestressingsteel(Section18.5.1,
pg287):
Maxstressduetojackingforce=min(0.94fpy,0.80fpu)
Maxstressattransfer=min(0.82fpy ,0.74fpu)
Stage DesignCriteria
1. Concrete
stressesat
transferofPT
forceto
concrete
(ACI18.4.1,
pg284)
atends elsewhere
maxtension '6ic
f '3ic
f
maxcomp.
'7.0ic
f
'6.0ic
f
2. Concrete
stressesunder
serviceloads
(ACITable
R18.3.3,
pg284)
Sustained
loadsAllloads
maxtension '12ic
f
maxcomp. '45.0ic
f
'60.0ic
f
3. Flexure
strength) 290)pg18.8.2,(ACIcrn
un
MM
MM
2.1
4. Deflections
(ACITable
9.5b,pg124)
240max
360max
L
L
erectionafter
LL
=
=
FigureA2. RelevantdesigncriteriainACI31808
PP
wSW+SDL+LL
PP
wSW
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CE437/537,Spring2011 PretensionedBeamExample 13/15
FigureA3. Stressvsstrainforprestressingtendon.
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CE437/537,Spring2011 PretensionedBeamExample 14/15
FigureA4. Straininprestressingsteelatultimateflexurestrengthofbeam
straininsteelduetodecompressing
theconcreteattheCGS:
straininsteelduePe:
+
PePe
psps
ePs
EA
Pe =
1. Aftertransfer&losses
(butnogravityloads)2. Gravityoverloadsareapplied
tobeamuntilfailure
straindistribution
in
concreteduePe
dp
straindistribution
in
concreteduetooverloads
=
cc
e
cc
eP
EI
eP
EA
Pe
CGS
2
+=
strainin
theCGSg
Pe
CG
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CE437/537,Spring2011 PretensionedBeamExample 15/15
FigureA5. Deflectionmultipliersforestimatinglongtermdeflections.