Creep tests on GFRP pultruded specimens subjected to traction or shear

11 Pages·2013·4.092 MB·English

by  Marina Bottoni, Claudio Mazzotti, Marco Savoia

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Creep tests on GFRP pultruded specimens
subjected to traction or shear
ARTICLE in COMPOSITE STRUCTURES · FEBRUARY 2014
Impact Factor: 3.32 · DOI: 10.1016/j.compstruct.2013.09.057
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CompositeStructures108(2014)514–523
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Composite Structures
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Creep tests on GFRP pultruded specimens subjected to traction or shear
⇑
Marina Bottoni , Claudio Mazzotti, Marco Savoia
UniversityofBologna,FacultyofEngineering,DICAMDepartment,vialedelRisorgimento,2,40136Bologna,Italy
a r t i c l e i n f o a b s t r a c t
Articlehistory: Inthispaper,experimentallong-termtestsonpultrudedGFRPspecimensarepresented.Specimenshave
Availableonline7October2013 beensubjectedtoconstanttractionandshearforcesforabout760days.Thetestshavebeenperformed
under controlled temperature (20(cid:2)C) and humidity (60% RH) inside a climatic room. The specimens
Keywords: undertractionhavebeencutfromplatesandfromtheflangesofawideflangeGFRPpultrudedbeam;
Compositestructures the specimens undershearloadingshavebeen extracted fromthewebofthesame beam.Moreover,
Creep somespecimenshavebeenstiffenedbybondingCFRPsheetstobothsides,inordertostudyhowtheir
Experimentaltests
introductionmayreducethespecimendeformabilityunderlongtermloading.Resultsfromtheunstiff-
Rheologicalproperties
enedspecimenshavebeeninterpretedbymeansoftheFindleylaw,thereferencemodelforcreepin
FRP
polymersandFRP,intermsofstrainsandcreepcoefficient.Asimplemodelhasbeenfinallyproposed
topredictstrainsinstiffenedspecimensandbytakingstressredistributionintoaccount.
(cid:3)2013ElsevierLtd.Allrightsreserved.
1.Introduction Therelativelylowstiffnessoftheseprofilescanbeasignificant
drawbackinrealapplications:theprofilesmustbeoftendesigned
Modernstructuralengineeringhasrecentlyintroducedamong withreferencetoaservicelimitstate,i.e.,byprescribingamaxi-
thepossiblesolutionstheadoptionofcompositestructuralshapes mumvalueofdeformability(e.g.maximumdeflection)ratherthan
ascomponentsofframedsystemsortrusses.Theyareusuallypro- ofstrength.Verificationsagainstultimatelimitstatesmustbealso
ducedbyusingpultrusionprocess,whichallowtheirmanufactur- performed,inparticularwithrespecttolocalandglobalinstability
ing with different materials arranged with different proportions. failures[1–5].
Accordingly, their mechanical properties vary significantly with Moreover, as far as the service limit state verifications are
the choice and the relative quantities of the materials employed concerned, the pultruded shapes obtained with glass fibres may
forthematrixandthefibrereinforcement,sothatthecomposite undergo considerable long-term deformations under a load sus-
material can be designed with properties to meet specific design tainedintime[6–9].
needs. Themechanicalbehaviourintimeisstrictlyrelatedtotherhe-
In the production of pultruded shapes for civil engineering ologicalpropertiesofbothmatrixandfibres;forexample,glassfi-
applications,E-glassfibresaremostlyusedforthereinforcement. bresaresubjectedtocreepdeformationmuchgreaterthancarbon
Theyarecharacterizedbylowercostbutalsobyareducedstiffness fibreswhensubjectedtothesamesustainedstress,asstudiedfor
withrespecttootherfibres,e.g.carbonfibres.Thepultrusionpro- examplebyBarbero[10].Forpultrudedshapesmadewithglassfi-
cessallowscreatingprismaticprofileswherethefibresorientation bres,thecreepdeformationcanbecomparablewiththeinstanta-
isoptimized,beingorientedalongthelongitudinaldirection;nev- neous deformation of the beam. Moreover, in computing the
ertheless,otherslayersoffibreswithdifferentorientationmustbe deflectionofpultrudedbeams,thesheardeformabilitycanbesig-
mergedinthecompositeduringtheprocessinordertoassureto nificant even for slender elements (see for example [11–13]). In
the pultruded shape a sufficient strength also for local or impact fact,theratiobetweentheshearandtheYoungmodulusismuch
loadings.Usually,continuousstrandmatsarealternatedtounidi- smallerfororthotropicunidirectionalcompositesthanforisotropic
rectional layers with the purpose of improving the transversal materials. This circumstance may have significant consequences
strengthandstiffness;asadrawback,thelongitudinalstiffnessis also in the case of a sustained loadings: for a load kept constant
reducedwithrespecttoashapewithonlylongitudinalfibres. intime,thedeflectioncontributionduetothecreepshearstrains
is much greater than that of creep uniaxial strains (e.g., due to
bending). The different values of creep coefficients for shear and
⇑
Correspondingauthor.Presentaddress:EDFR&D,DepartmentAMA,1,avenue
bendingmayalsolead,forsomeconstraintandloadingcondition,
dugénéraldeGaulle,92141ClamartCedex,France.Tel.:[phone].
tosignificantstressredistributionwithtimeevenundersustained
E-mail addresses: [email protected] (M. Bottoni), claudio.mazzotti@
unibo.it(C.Mazzotti),[email protected](M.Savoia). serviceloading,asshownbyBottonietal.[14].
0263-8223/$-seefrontmatter(cid:3)2013ElsevierLtd.Allrightsreserved.
http://dx.doi.org/10.1016/j.compstruct.2013.09.057
M.Bottonietal./CompositeStructures108(2014)514–523 515
Somestudiescanbefoundconcerningtherheologicalproper- pultrudedwide-flangebeamsandplates(Fig.5).Evenifthethick-
tiesofpultrudedcomposites.Mottram[15]testedunderthethree ness of both beam flange and plate is the same (9.53mm), they
point bending scheme two assembled beams, composed of two have different mechanical properties, because the percentage of
I-beamsbondedtogetherbyplatesgluedontheirtopandbottom strandmatsisgreaterfortheplatesinordertoassureasufficient
faces. The tests have been performed under controlled environ- strengthevenwhensubjectedtobiaxialstressstate.Theaverage
mentalconditions,buttheirdurationwasveryshort(24h).MacC- mechanicalpropertiesofthe pultrudedelementsdeclared by the
lureand Mohammadi[16]haveperformedaccuratecompression producerarelistedinTable1.
testsoncouponsandanglestubformorethan2500h,undercon- The specimens extracted from the plates and from the beam
stantenvironmentalconditions.Duringthecreeptests,bothstubs flanges have been subjected to long-term traction tests, whereas
and coupons have been subjected to a constant load 45% of the thoseobtainedfromthebeamwebshavebeenusedfortestsunder
instantaneous bucklingload, with straingages placedin orderto long-termshearloadings.Anumberofspecimenshavebeenexter-
verifytheonsetofapossiblebucklingfailure.ShaoandShanmu- nallystrengthenedbybondingcarbonfibressheets,inordertoin-
gam [17] tested coupons cut from the flanges and the web of a crease their stiffness. Unidirectional plies have been added to
wide flange beam under compression. Finally, Scott and Zureick specimensfortractiontests(a),whilesheetswith±45(cid:2)havebeen
[18]testedacompositesheetpilingunderthree-pointbendingtest bondedtospecimensforsheartests(b).
during9000h,withloadsequalto50%and25%ofthefailureload, Adetailedschemeoftestedspecimens,withtheirsizeandcar-
respectively.Predictionofthemid-spandeflectionisobtainedby bonfibresdirection,isgiveninFig.1.Fourplain(unstiffened)spec-
meansoftheTimoshenkobeammodelandusingaveragedtensile imensextractedfromtheplateshavebeensubjectedtolong-term
andshearmoduli.Thesearecalculatedfromstrainmeasurements tractiontests.Twodifferentwidthshavebeenadopted,inorderto
onwebandflangeofthesamebeam.Resultsareinteresting,how- study the effect of the stress level on the creep strain evolution
evertheyareobtainedwithoutcontroloftheambientconditions with time: two specimens are 46mm wide, while the remaining
andstrainmeasurementsaretakenwithonlytwogages(onegage twoare65mmwide;thespecimenshavesimilarwidth/lengthra-
intractionandarosetteforshear). tios,sothattheirlengthsare245mmand335mm,respectively.A
In the literature there are still a limited number of long-term largewidth/lengthratioisrequiredfororthotropicmaterialsinor-
experimentaltestsofpultrudedcomposites,notsufficienttoassess der to have the central portion of the specimens free of distur-
thecreepbehaviourofpultrudedmaterials.Thetestsareoftenof bances due to the end restraints, as explained by Tullini and
shortdurationandperformedonlyonstructuralelementsandal- Savoia[19,20].Fouradditionalspecimenshavebeenstrengthened
mostneveroncompositecoupons.Intestsonstructuralelements, byexternallybondingthreeunidirectionalFRPcarbonplieseach;
thestressstatemayvaryduringtimeduetostressredistributions twoofthemhavebeenobtainedfromplatesandare31mmwide
withinthecross-section;moreover,itisnotpossibletodistinguish and 245mm long, while the remaining two have been cut from
the two deflection contributions due to flexural and shear defor- beamflangesandare44mmwideand335mmlong.
mation,respectively.Thislackofexperimentaldataiscomplicated As far as the long-term shear tests are concerned, four speci-
by the fact that very often the manufacturers do not provide the mens with dimensions 120mm(cid:2)220mm have been extracted
compositionoftheirpultrudedmaterials,inparticulartheamount from the beam web; two of them have been strengthened with
anddirectionoffibres.Asaconsequence,thereisaneedofknowl- twocarbonfibresheetswith±45(cid:2)fibresorientation.Asexplained
edge for the rheological properties of commercial pultruded in Section 2.2, each specimen for shear test contains two loaded
elements. portions, which are given two different names (SP5 and SP6 for
The experimental tests described in the present paper, per- example).
formed at the Laboratory of Structural Testing of the University The strengthening of the pultruded specimens subjected to
ofBologna,enlargetheknowledgeonthistopic.Creeptestshave tractionhasbeendonewithtwounidirectionalcarbonfibreplies
been performed on composite pultruded specimens subjected to on one surface and one ply on the opposite free surface. Hence,
twokindoflongtermforces,i.e.,uniaxialtractionandshearforce. thecross-sectionisnotperfectlysymmetric,sothata smallflex-
During tests, temperature and humidity have been kept constant uralmomentarises.However,theaverageofthetwostrainsmea-
andequalto20(cid:2)Cand60%RH,respectively.Thespecimenshave suredonoppositesurfaceswillbeequaltothestrainmeasuredon
beenkeptunderconstantloadingformorethan2years,soprovid-
ing information on the material long-term behaviour out of the
usualtimerange. Sometestshavebeen performedon specimens Table1
strengthened with additional layers of carbon fibres, glued on Average properties of pultruded shapes used to obtain specimens. For plates:
LW=lengthwise,CW=crosswise.
external faces of pultruded coupons. The idea is to propose this
kindofstrengtheningasameantoreducetheshort-andlong-term Property Value Unit
deformabilityoftheglassfibrepultrudedspecimens. Wideflangebeam,152.4(cid:2)152.4
The long-term behaviour of plain specimens has been inter- Youngmodulusintension 28.6 GPa
preted by the Findley creep model, usually employed for plastics Youngmodulusincompression 26.5 GPa
and often also applied to FRP. A simple model is then proposed Shearmodulus 3.4 GPa
Tensilestrength 275 MPa
for prediction of long-term deformation of CFRP stiffened speci-
Compressivestrength 315.7 MPa
mens,whosebehaviourismorecomplexduetostressredistribu- Poissonratio 0.35 –
tion, related to the presence of different materials on the cross-
Plate
section(pultrudedcouponandCFRPlayer). Youngmodulusintension(LW) 12.4 GPa
Youngmodulusintension(CW) 6.9 GPa
2.Testingmethodandmaterials Youngmodulusincompression(LW) 12.4 GPa
Youngmodulusincompression(CW) 6.9 GPa
Tensilestrength(LW) 137 MPa
2.1.Thepultrudedspecimens
Tensilestrength(CW) 69 MPa
Compressivestrength(LW) 165 MPa
Thespecimensforthelong-termtestshavebeenextractedfrom Compressivestrength(CW) 110 MPa
commercialpultrudedshapesmadeofE-glassfibresembeddedina Poissonratio(LW) 0.32 –
Poissonratio(CW) 0.25 –
polyester matrix. Specimens have been extracted from both
516 M.Bottonietal./CompositeStructures108(2014)514–523
Table3
MaterialpropertiesofepoxyresinfortheCFRPapplication.
Property Value Unit
Percentageofhardener/resin 30% –
Lifeat25(cid:2)C 60 min
Lifeat18(cid:2)C 90 min
Timeforcompletehardeningat25(cid:2)C 5 days
Tensilestrength 43 MPa
Elasticmodulus 2520 MPa
Compressivestrength 68 MPa
Elasticcompressivemodulus 1900 MPa
tional carbon fibre sheets are reported in Table 2. Carbon fibre
sheetshavebeenmanuallygluedtotheglass-fibrepultrudedcom-
positebymeansofanepoxyresin:itsmechanicalcharacteristics
aregiveninTable3.Thestrengthenedspecimenshavebeenvac-
uumcuredforfivedaysandtestsstartedatleastthreeweekafter
thespecimenpreparation.
2.2.Experimentalset-up
Twoframeshavebeenspecificallydesignedfortheexperiments
andusedforbothtractionandsheartests.Thetwoexperimental
setups are based on the leverage system depicted in Fig. 2, so
assuring that the load applied to the specimens is perfectly con-
stantintime.Theloadisprovidedbyanumberofsteelplatessus-
pendedtothelongerleverarmandistransmittedtothespecimens
attached at the shorter lever arm. The amplificationfactor of the
appliedforcehasbeenobtainedfromacalibrationtestwhichled
tothevalue8.12fortheframeadoptedforthetractiontestsand
7.71for the framefor thesheartests. At theshorterarmof each
frame,asecondaryleveragesystemisusedtoloadtwospecimen
chainsatatime,eachchaincarrying50%ofthetotalappliedload.
Thespecimensusedforthetractiontestshavetheirextremities
glued to two steel plates, which are then bolted to other steel
platesconnectingthedifferentspecimens(seeFig.3a);eachspec-
imenbelongingtooneofthetwochainsoftheframeissubjected
toanaxialloadingQ=8.89kN,constantintimeduringthetest.
The specimens for the shear tests have steel plates glued to
theirextremitiesandtotheirmiddlesection,whereathirdseries
Fig.1. Notationanddimensionsforthespecimenssubjectedtothelong-termtests.
ofsteel elements are bolted (Fig. 3b). As better shownin Fig. 4b,
the central bolts separate the specimen in two portions loaded
asymmetricspecimen.Theplyfibrecontentis320g/m2withlon- withthesameshearforceF(thegreypartinthepicture).Dueto
gitudinalelasticmodulusequalto240GPa.Inspecimensforshear thesmall width, shearstressesare approximatelyuniform in the
tests, stiffening consisted in one bidirectional ±45(cid:2) sheet applied
on each free surface; the sheet fibre content is 200g/m2 in each
directionandtheelasticmodulusis240GPainthefibredirection.
The mechanical properties of the unidirectional and two-direc-
Table2
Materialpropertiesofuniaxialand±45(cid:2)CFRPsheets.
Property Value Unit
UniaxialCFRPsheet
Weight 320 g/m2
Effectivethickness 0.177 mm
Tractionloadatfailureforunitwidth 640 N/mm
Fibredensity 1.78 g/m2
Elasticmodulus(infibresdirection) 240 GPa
Elongationatfailure 1.5% –
±45(cid:2)CFRPsheet
Totalweight 400 g/m2
Weightforeachdirection 200 g/m2
Effectivethickness 0.45 Mm
Elasticmodulus(infibresdirection) 240 GPa
Elongationatfailure 1.5% –
Fig.2. Leveragesystemsforthecreeptestsonpultrudedcompositespecimens.
M.Bottonietal./CompositeStructures108(2014)514–523 517
(a)
(a)
F F
(b)
Fig. 3. Mechanical systems for the load transmission for specimens under (a)
tractionand(b)shear.
(b)
twoportions.Eachspecimenchainisloadedwithaconstantforce
2F=9.39kN.Theadoptedmechanicalsystemforshearissimilarto Fig.4. Theinstrumentationadoptedforthespecimensunder(a)tractionand(b)
thatrecommendedinASTMD4255standard[26]forinstantaneous shear.
sheartestsoncomposites.
Exacttestdurationhasbeenof760days. about 15% of the plates tensile strength (see Table 1); the larger
specimens have been designed in order to achieve a stress level
2.3.Instrumentation about 70% of that of the narrow specimens (i.e., 14.7MPa). The
width of the stiffened specimens is smaller with respect to that
Inthetractiontests,thedeformationismeasuredbymeansof of the corresponding unstiffened specimens, in order to attain a
twolongitudinalstraingagesplacedonthetwooppositefacesof stresslevelinthe pultrudedcouponcomparablewiththatofthe
the specimens; in Fig. 4a the upper gauge is visible. For the unstiffened ones, taking into account the contribution of the
shear-loaded specimens, two strain gages are placed on the two carbonfibresheetsreinforcement,characterizedbythicknessand
sides with ±45(cid:2) orientation withrespect to the load direction;in E-modulusreportedinTable2.
Fig.4b,theyaredepictedasblackandwhiterectangles,withthe Forspecimenssubjectedtocreepsheartest,theaverageshear
black one being glued on the hidden side. From the strain gage stressis4.1MPa.Forthestiffenedspecimens,thesamedimensions
measurements, assuming a state of pure shear in the two panels of the pultruded coupons have been adopted, due to the dimen-
ofeachspecimens(confirmedbyverysimilarabsolutevalueregis- sionsofthemechanicaldeviceusedtotransfertheshearload.
teredbyeachcoupleofstraingauges),theshearstrainchasbeen
computedastwicetheaveragevalueofstrainse ande ,measured 3.Simplifiedmodelsforpredictionoflong-termbehaviourof
1 2
bythetwogauges. strengthenedpultrudedelements
2.4.Loadlevelforcreeptests 3.1.TheFindleymodel
Theloadlevelprescribedonspecimenssubjectedtocreeptests FollowingFindley[22–24],strainevolutionintimeofpolymers
has beendeterminedonthe basisofthedesignrequirements for under constant stress can be described by the following power
wideflangebeamsunderserviceloadings,i.e.,byprescribingalim- law:
itvalueofthedeflectionofthebeamsubjectedtouniformlydis- eðtÞ¼e þmðt(cid:3)t Þa ð1Þ
tributed transverse loading (see also [21]). In particular, uniaxial 0 0
creep traction tests are characterized by an applied force of wheret istheinstantofapplicationofthesustainedload,e(t)isthe
0
9.1kNforeachspecimenschain.Thecorrespondinguniformuni- strain evolution, e ¼e is the instantaneous strain and m, a are
0 t0
axial stress of the narrow unstiffened specimens is 20.7MPa, i.e. the law parameters. Eq. (1) is applied also to fibre reinforced
518 M.Bottonietal./CompositeStructures108(2014)514–523
composites(FRP).ItisworthnotingthatEq.(1)becomesastraight Under the linear viscoelastic hypothesis, the strain evolution
line in bi-logarithmic chart, which enables a simple calibration of under variable stress is given by a convolution integral (see for
thelawparameters,throughalinearregressionanalysisofexperi- example [32]). This integral cumulates at time t the effect of dr
mentaldata.Eq.(1)canbealsousedforshearstrainsevolutionwith attimet0 givenbyEq.(6a),accordingtotheexpression:
time,withshearstraincreplacingtractionstraine.Parametersm,a
Z rðtÞ
fortractionorsheartestsoncompositesareusuallydifferent,since eðtÞ¼r Jðt(cid:3)t Þþ Jðt(cid:3)t0Þdrðt0Þ ð7Þ
0 0
thematerialisorthotropicalsowithrespecttolong-termloadings, r0
sogivingdifferentstrainevolutionlaws. wherer istheinitialinstantaneousstress,t istheinstantofappli-
0 0
TheFindleylawisnon-linearwithrespecttothestresslevel,so
cationoftheinstantaneousstress,andJisthecreepfunction.Rigor-
that,ifstressdependenceisintroducedexplicitly,Eq.(1)becomes: ously,Eq.(7)isapplicableonlyforincreasingstresslevels;itsuse
(cid:2)r (cid:3) (cid:2)r (cid:3) underdecreasingstresseswouldrequire,infact,theknowledgeof
eðtÞ¼e0 sinh 0 þm0sinh 0 ðt¼t Þa ð2Þ
0 r0 r0 0 the material recovery function. In the present case, the stress on
the pultruded material is decreasing in time. In fact, its effective
wherer istheapplied(constant)stressandr0isareferencestress
0 stiffnessdecreasesintimeduetocreepdeformation,whereascar-
level; moreover, e0 and m0 are the instantaneous strain and the
0 bon fibres exhibit usually negligible creep effects (as shown in
material parameter m at the reference stress level; a is the same
[25,28]).Then,thestiffeningcarbonlayerscarryintimeanincreas-
parameter introduced in Eq. (1), independent from stress level.
ingquotaoftheappliedload,andconsequentlytheirstresslevelin-
Eq.(2)canbesimplifiedbyexpandingthehyperbolicsineinTaylor
creases;stressonthepultrudedmaterial,onthecontrary,decreases
series and considering only the first order approximation, so
tosatisfyequilibriumonthecrosssection.
obtaining:
Forthisreason,inthepresentpaperaformulationmoreappro-
eðtÞ¼e0 r0þm0r0ðt(cid:3)t Þa ð3Þ priatetothecaseathandhasbeenconsideredmakinguseofthe
0r0 r0 0 dualformulationofcreep,i.e.stressevolutionintimeunderanim-
posedstrainhistory:thisiscalledarelaxationproblem.Inthiscase,
Eq.(3)representsthelinearviscoelasticbehaviourofthematerial.
thestrainlevelmustbegrowingintime.Theformulacorrespond-
Thebehaviourofthespecimenssubjectedtoconstanttraction
ingtoEq.(7)isbasedupontherelaxationfunctionR(t(cid:3)t )accord-
intimecanbedescribedbythecreepcoefficient/(t(cid:3)t ),givenby: 0
0 ingtotheexpression:
e e
/ðt(cid:3)t Þ¼ v ¼E v ð4Þ Z eðtÞ
0 e0 r0 rðtÞ¼e0Rðt(cid:3)t0Þþ Rðt(cid:3)t0Þdeðt0Þ ð8Þ
e
whereev=e(t)(cid:3)r0/Eistheviscousstrain.Eq.(4)givestheratioof 0
the strain increment due to creep over the instantaneous strain In the same way as the creep function, the relaxation function
which,inlinearviscoelasticity,isindependentfromthestresslevel. R(t(cid:3)t0) depends on the time interval t(cid:3)t0 elapsed from load
For shear loadings, Eq. (4) can be written in a similar way as / application.
(t(cid:3)t )=Gc /s , being s the applied constant shear stress and G TwosimplifiedalgebraicsolutionsofEq.(8)aregivenbythefol-
0 v 0 0
theshearelasticmodulus. lowingapproximations:
Ifthematerialislinearviscoelastic(so thatEq.(3)holds)and rðtÞffie Rðt(cid:3)t ÞþðeðtÞ(cid:3)e ÞRðt(cid:3)t Þ¼eðtÞRðt(cid:3)t Þ; ð9aÞ
theFindleylawisadopted,thecreepcoefficientcanthenbewritten 0 0 0 0 0
intheform: rðtÞffie Rðt(cid:3)t ÞþðeðtÞ(cid:3)e ÞR ð9bÞ
0 0 0 m
/ðt(cid:3)t0Þ¼m0rE0ðt(cid:3)t0Þa ð5Þ InthecaseofEq.(9a),knownasEffectiveModulus(EM)approxima-
tion,thewholestraine(t)isconsideredasappliedattheinitialtime
with the shear modulus G replacing the Young modulus E for t ,andthelargestrelaxationmodulusR(t(cid:3)t )isappliedtoit.Con-
0 0
shear loadings. For sake of simplicity, formulas with notation for sequently, material viscosity is usually overestimated, and this
traction loadings will be employed also for shear loadings in the approximationconstitutesanupperlimitforlong-termstrainpre-
following. In that case, symbols e, r, E should be understood as diction. The approximation given by Eq. (9b) is known as Mean
c, s and G. Stress (MS) method. In this case, the convolution integral of Eq.
(8) is solved by multiplying the strain increment e(t)(cid:3)e by
0
3.2.Asimplifiedmodelforthecreepbehaviourofstrengthened the algebraic average of the relaxation function with time
specimens evaluated at the beginning and end of the loading period:
R (t)=0.5[R(t(cid:3)t )+R(t(cid:3)t)]=0.5[R(t(cid:3)t )+E]. These methods
m 0 0
Strainevolutionunderconstantstressr attimetcanbewrit- areproposedbytheFIB-CEBModelCode1990topredictconcrete
0
teninageneralformbymeansofthecreepfunctionJ(t,t ).Inthe creep[27].
0
caseofnon-agingmaterials,i.e.,ifrheologicalpropertiesaretime- OnceEq.(8)issolved,equilibriumofforcesonthecross-section
independent,thecreepfunctioncanbemoresimplywrittenasa ofthestrengthenedspecimencanbeimposed,forthegeneraltime
function of the time from load application, i.e., t(cid:3)t . It is also instant t. In the case of traction loading, equilibrium is given by
0
straightforwardtofindtherelationshipbetweenthecreepfunction N=A r (t(cid:3)t )+Ar(t(cid:3)t ), being A, r the cross-section of the
p p 0 c c 0
andthecreepcoefficient: materialcomponentandthestressactingonit,andwithc,pindi-
catingthesinglecomponents(c=carbonfibres,p=pultrudedFRP).
Uðt(cid:3)t Þþ1
eðtÞ¼r Jðt(cid:3)t Þ withJðt(cid:3)t Þ¼ 0 ð6a;bÞ Forshearloading,equilibriumcanbewrittenbymakinguseofthe
0 0 0 E
average stresses s ,s on the cross section of each layer: F=A s
p c p p
The creep behaviour of specimens stiffened with FRP is more (t(cid:3)t0)+Acsc(t(cid:3)t0).Accordingtoliteraturedata[25],CFRPsheets
complicated than of the plain pultruded material, since the dif- constitutivelawhasbeenconsideredaslinearelastic,i.e.itscreep
ferent rheological properties of pultruded element and carbon fi- is neglected. Perfect bond between carbon fibres and FRP is also
bres lead to stress redistributions. Hence, Eq. (1) cannot be considered. Few mathematical manipulations on equilibrium
applied in its simple form, being valid under constant stress equation and Eq. (9) yield the following expressions for strain
condition only. evolution:
M.Bottonietal./CompositeStructures108(2014)514–523 519
eðtÞ¼ N ð10aÞ By operating a linear regression for the data of four types of
A Rðt(cid:3)t ÞþA E specimens, the values of the regression index R2 are very high
p 0 c c
(about0.995forallregressionlines)meaningthatlinearbehaviour
eðtÞ¼N(cid:3)Ape0DR=2 ð10bÞ isagoodassumption.Thisresultwillbeexploitedinthemodelling
A R þA E stage.Theloadingphasehasbeenusedtoassessmateriallinearity,
p m c c
but corresponding linear coefficients should not be employed as
whereDR(t)=R(t(cid:3)t )(cid:3)R(t(cid:3)t)=R(t(cid:3)t )(cid:3)E.Eqs.(10a)and(10b)
0 0 the elastic moduli. In fact, during the loading phase (of duration
correspondtotheEffectiveModulusandMeanStressapproximations
of about 1600s) specimens could have developed some creep
ofEqs.(8),respectively.
strains,andmeasuredstrainsduringtheloadingphasecouldhave
Finally, the relaxation function R(t(cid:3)t ) of the pultruded ele-
0 beenoverestimatedwithrespecttoapureelasticloading.Hence,
ments has to be obtained from the creep function J(t(cid:3)t ) given
0 in further computations the elastic moduli provided by the pro-
by Eq. (6b) and calibrated making use of unstiffened specimens
ducerandexperimentallyverifiedhavebeenemployed,inparticu-
creep data. This conversion is possiblebecause creep and relaxa-
larforpredictingstrainsofstiffenedspecimens.
tionformulasreflectthesamerheologicalpropertiesofthemate-
rial under different loading conditions. In particular, the
4.2.Viscoelasticphasefortheunstiffenedspecimens
followingrelationshipholds(see[29]):
1 Z t dJðt0Þ In this section, the experimental data from unstiffened speci-
Rðt(cid:3)t Þþ Rðt(cid:3)t0Þ ¼1 ð11Þ
E 0 dt0 mens are described, together with their interpretation according
0
totheFindleylaw.
EvenifsomeanalyticalsolutionsexistforspecificJfunctions(Gross
Consideringagroupofnspecimens,wheremsamplingsofthe
[30]),Eq.(11)isusuallysolvednumerically.
viscousstrainevattimestkareavailable,asimplemeasuredev of
datadispersion(aspercentage)canbedefinedas:
44..1R.eInssutlatnstaannedouinstleorapdrientgaptihoanseofuniaxialtests dev ¼m1Xkm¼1"De(cid:3)evvððttkkÞÞ# ð12Þ
with DevðtÞ¼maxi;j¼1...njeviðtÞ(cid:3)evjðtÞj and e(cid:3)v the average value of
Theloadhasbeenincreasedbystepsthroughapplicationofsix viscousstrainsattimet.
steelplatestothelongerleverarmofthesystem(Fig.2).Knowing ForTNPorTWPspecimens,thedatadispersionwithinthesame
theleverageratioequalto8.12andtheplateweight,theforceon groupisverylowandsmallerthen1%inbothcases(seeFig.7).
eachspecimenchainisalsoknown.Longitudinalstrainshavebeen CreepstrainsforTNPandTWPspecimensaregiveninFig.7ina
measuredbothduringtheloadingphaseandintheconstantload bi-logarithmicforminordertoverifyiftheFindleylawofEq.(1)is
phase.Fig.6showstheloadingpathforfourgroupsofspecimens: applicable.Eachlinereferstotheaverageofthetwogaugesonthe
plain narrow specimens from plates, plain wide specimens from specimen opposite surfaces. It is worth noting that data do not
plates, stiffened specimens from plates and stiffened specimens standonastraightline,asexpectediftheywouldperfectlyobey
from beam flanges. All points represent the load per unit width the Findley law. Nevertheless, deviation from linearity is particu-
vsstrain,inordertocomparespecimenswithdifferentwidths. larly strong only in the very first two hours after loading
(a)
(b)
Fig.5. (a)ExtractionofthespecimensfrombeamsfortractiontestsandstrengtheningwithCFRPsheetsand(b)extractionofthespecimensfrombeamwebfortheshear
testsandstrengtheningwithCFRPsheets.
520 M.Bottonietal./CompositeStructures108(2014)514–523
0.6
TNP5 Experimental values
−] 0.5 TTNWPP67 EExxppeerriimmeennttaall vvaalluueess
nt [ 0.4 TWP8 Experimental values
e TWP − Findley
effici 0.3 TNP − Findley
o
c
p 0.2
e
e
Cr 0.1
0
10−3 10−2 10−1 100 101 102 103
Time [days]
Fig.8. CreepcoefficientfortheunstiffenedspecimensTNPandTWPundertraction:
experimentaldataandmodelpredictionsbyFindleylaw.
103
Fig.6. Loadingphaseforthespecimensundertractionandcorrespondinglinear
regressionlines.
E6] 102
1
x
103 ains [ 101
ains [x 1E6] 110012 TNP5 Experimental values Creep str 1100−01 TTTTNNWWSSSS1234 EEEExxxxpppp.... ddddaaaattttaaaa TTTNNWSSS::: EMEffeffeaecnct tiSivvtere e MMssoodduulluuss
p str TTNWPP67 EExxppeerriimmeennttaall vvaalluueess 10−3 10−2 10−1 100 101 102 103
ee 100 TWP8 Experimental values Time [days]
Cr TWP − Findley
TNP − Findley Fig.9. CreepstrainsofstiffenedspecimensTNSandTWS:experimentaldataand
10−1 modelpredictions.
10−3 10−2 10−1 100 101 102 103
Time [days]
close during the whole test duration. An indicator similar to Eq.
Fig.7. CreepstrainsofunstiffenedspecimensTNPandTWPandregressionlines (12)canbedefined:
accordingtotheFindleylaw.
" #
dU¼m1Xm UððtktÞÞ ð13Þ
completion.Byneglectingthefirsthourafterloadapplication(con- k¼1 k
sidering the reduced technical relevance of this time range), the being DUðt Þ¼j/ ðt Þ(cid:3)/ ðt Þj and Uðt Þ¼0:5½/ ðt Þþ
k TNP k TWP k k TNP k
Findley law can be properly calibrated. With this hypothesis, the / ðt Þ(cid:5). From creep data, d is smaller than 10%, so confirming
TWP k
R2 value is higher than 0.99, showing good agreement between thevalidityoflinearviscoelasticityhypothesis,atleastforthepre-
the linear model and the experimental data. The two dashed scribedstresslevels(10–15%offailureload),typicaloftheservice
regression lines in Fig. 7 represent the Findley laws for TNP and condition,sothatEqs.(3)and(4)apply.Thenormalizedcoefficients
TWPspecimens,calibratedfromtheaveragecurvesofspecimens m0aregiveninTable4;thelowerstresslevelr0=14.7MPahasbeen
withthe samewidth.Theparametersm, aoftheFindleylaw are chosen as reference stress. The linearization allows to define an
given in Table 4 for each specimen or family of specimens (TNP average Findley curve for both TNP and TWP specimens, whose
andTWP);R2coefficientsarealsoreported. parameters a, m0 are also given in Table 4. Fig. 8 also shows that,
For the four unstiffened specimens, the experimental creep after 760days from loading, strains have increased of more than
coefficientisshowninFig.8.Dataarereportedwithlogaxisonly 50%oftheinstantaneousstrains.
fortimeontheabscissas,inordertoperceivecorrectlytheorderof
magnitudeofthecreepcoefficient;ontheotherside,thelogarith- 4.3.Viscoelasticphaseforthestiffenedspecimens
micscalefortimeisusuallypreferredforcreepstrainsinorderto
better capture the initial strain evolution. It is worth noting that LongitudinalcreepstrainsforthestiffenedspecimensTNSand
curvesobtainedfromthewideandthenarrowspecimensarequite TWSareshowninFig.9.Dispersionofexperimentaldataislarger
thanforunstiffenedspecimens,beingparameterdev (seeEq.(12))
equalto7%forTNSand4%forTWSspecimens.
Table4 Asignificantdifferencebetweentheabsolutestrainsmeasured
SummaryofthecalibratedFindleyparametersintraction(r0¼14:69MPa). on the wide (TWS) and the narrow (TNS) specimens can be ob-
106m 106m0 a(–) R2 served, which is not proportional to the ratio between the two
widths.This is due todifferent mechanicalpropertiesofthe pul-
TWP7 128.73 128.73 0.249 0.997
truded specimens, obtained from a beam flange in one case and
TWP8 134.06 134.06 0.243 0.996
TNP5 245.74 173.89 0.211 0.993 fromaplateintheother,soconfirmingthedifferencesremarked
TNP6 241.67 171.75 0.212 0.993 duringtheinstantaneousloadingphaseofthespecimens.
AverageTWP 131.40 131.40 0.246 – PredictionofcreepstrainsofTNSspecimenscanbegivenbythe
AverageTNP 243.71 172.82 0.212 –
modeldescribedin Section3.2. The Findleycreep coefficientand
M.Bottonietal./CompositeStructures108(2014)514–523 521
0.6 averagevaluesobtainedfromstiffenedandun-stiffenedspecimens
TNP−TWP, experimental values arereportedinFig.11.Similarlytospecimensundertraction,alin-
nt [−] 00..45 TTTNWWSSP, e−exx Fppieenrrdiimmleeeynnttaall vvaalluueess esiaornreingdreesxsiRo2nhhaavsebbeeeennofpoeurnadtefdo.rVaelrlythheigshpevcailmueesnsof(Rth2e=r0e.9gr9e9s)-,
cie TNS − Effective modulus thus proving that the instantaneous shear loading phase can be
oeffi 0.3 TBWeaSm −fl Eanffgeecst i−v eF minoddleuylus alsoconsideredlinear.
c
p 0.2
ee 5.2.Viscoelasticphaseforunstiffenedspecimens
Cr 0.1
As in thecase of specimensundertraction,in thissection the
0
10−3 10−2 10−1 100 101 102 103 experimentaldatafromunstiffenedspecimenssubjectedtoshear
Time [days] are given, together with the numerical predictive curves by the
Findleylaw.Creepshearstrainsareshowninabi-logarithmicform
Fig. 10. Creep coefficient for the unstiffened and stiffened specimens under in Fig. 12. Each curve reportedin thegraph is the averageof the
traction:experimentaldataandnumericalpredictions. absolute value of shear strains recorded by the two strain-gages
placedonoppositefacesofeachsingleloadedareaofthepultruded
element(seeFig.3b).Asremarkedinthecaseofspecimensintrac-
creep function have been obtained from the master curve for
tion, the Findley law is not able to properly describe the creep
unstiffened plates, whose parameters are given in Table 4. The
behaviour immediately after loading. Again, excluding strains
adopted Young modulus is given in Table 1 for plates, i.e.,
measured in the first hour of loading, an accurate matching can
E=12.4MPa. The relaxation function R(t(cid:3)t ) has first been ob-
0
be found with the analytical law and the calibration process can
tained from the numerical conversion of Eq. (11). However, it
besuccessfullycarriedout.Theparametersobtainedforallspeci-
wasfoundtobeveryclosetothefunction1/J(t(cid:3)t ),whichauto-
0 mensand R2 indexes are providedin Table 5. Average valuesare
matically satisfies the initial condition R(0)=1/J(0)=E. Predicted
strains are shownwiththickblue1 linesin Fig.9. Continuous and alsogiven,sodefiningtheFindleymastercurveforshearreported
dashed lines represent the Mean Stress and the Effective Modulus inFig.12.Dispersionofdataisverylow,beingdev ¼4%basedon
Eq.(12).
methods, respectively. Both methods show a very good accuracy,
Results for unstiffened specimens are also given in Fig. 13 in
eveniftheglobaltrendseemstobebettercapturedbytheEffective
termsofcreepcoefficient.Thefinalvalueoftheufunctionisabout
Modulusmethod.
0.55,verysimilartothatofspecimensintraction.Moreover,Fig.13
AsfarasTWSspecimensareconcerned,strainscannotbepre-
showsclearlythatcreepisstillinitssteadystate,sinceattheend
ciselypredictedbyEq.(10),sincecreeppropertiesofbeamflanges
of loading the strain rate can be still approximated with a linear
were not available. Parameters of the Findley model have been
lawinthelog–logscale.
found a posteriori by least squared optimization on logarithmic
scale and by using the Effective Modulus method, i.e., a, m0 equal
to0.301and12.7(cid:2)10(cid:3)6,respectively.InFig.9,theredthickline 5.3.Viscoelasticphaseforstiffenedspecimens
indicatesthislaw.
ThecreepshearstrainsforthestiffenedspecimensSSaregiven
Again,resultsarealsogivenintermsofcreepcoefficient,sopro-
inFig. 14by a bi-logarithmic representation. Inthiscase, disper-
vidingacomparisonbetweenallspecimengroups.InFig.10,the
sionofexperimentaldataismoresignificantthanthatforunstiff-
creep coefficients for the four different types of specimens are
ened specimens. This is due to the manual manufacturing
compared. For stiffened specimens, the instantaneous strain at
procedureforbondingCFRPsheets,whichdoesnotallowforaper-
thedenominatorofEq.(4)iscomputedasN/(AE +A E ).Numer-
c c p p
fectcontroloffibreinclination.DispersionaccordingtoEq.(12)is
icalcreepcoefficientsarealsoprovided.Sincetestsonunstiffened
specimensfrombeamflangesarenotavailable,onlythenumerical thenquitehigh,beingdev ¼30%.
PredictionofcreepstrainsfromSSspecimenshasbeendoneby
calibrated Findley law has been depicted. It is worth noting that
themodeldescribedinSection3.2,i.e.byassumingperfectbond
material from flanges is much stiffer than material from plates,
betweencarbonfibresandthepultrudedmaterial,aswellasline-
notonlywithrespecttoinstantaneousloadings(Table4)butalso
aritywithrespecttothestresslevel.TheFindleyshearcreepcoef-
iflong-termstrainsareconsidered;thefinalcreepcoefficientisin
ficientandshearcreepfunctionforthepultrudedspecimenshave
factabout70%smaller.Fig.10showsthatthestrengtheningwith
been obtained from the master curve obtained for unstiffened
CFRPhasreducedthefinalcreepcoefficientofabout60%forpul-
trudedspecimensobtainedfromplates(TNSwithrespecttoTNP
and TWP). The reduction for pultruded specimens obtained from
beamflangesisabout35%.
5
5.Resultsfromsheartests
4
N]
5.1.Instantaneousloadingphase k
e [ 3
c
tainFeodr adlulrtinhge sthpeecliomaedninsgsupbhjaesceteadretoprsehseeanrtetedsti,nthteermresoufltsshoebar- hear for 2 Data for SP specimens
force–straindiagrams(F(cid:3)c),beingFtheshearforceoneachspec- S Data for SS specimens
1
imen,i.e.,halfoftheloadappliedoneachspecimenchain(seeFig Linear regression for SP specimens
3bfortheloadingscheme).Asinthecaseofthetractiontests,the Linear regression for SS specimens
loadingphaseisrepresentedthroughpointsintheF–cplane.The 00 200 400 600 800 1000 1200 1400
Shear strains [x 1E6]
1 Forinterpretationofcolorin‘Fig.9’,thereaderisreferredtothewebversionof Fig.11. Loadingphaseforthespecimensundershearandcorrespondingregression
thisarticle. lines.
522 M.Bottonietal./CompositeStructures108(2014)514–523
103 0.6
Effective Modulus
Creep strains [x 1E6] 111000012 SSSSPPPP5678 EEEExxxxppppeeeerrrriiiimmmmeeeennnnttttaaaallll vvvvaaaalllluuuueeeessss Creep coefficient [−] 00000.....12345 MSSSSSSSSe1234a−−−−nEEEE Sxxxxtppppreeeeesrrrrsiiiimmmmeeeennnnttttaaaallll vvvvaaaalllluuuueeeessss
SP − Findley
10−1 0
10−3 10−2 10−1 100 101 102 103 10−3 10−2 10−1 100 101 102 103
Time [days] Time [days]
Fig.12. CreepshearstrainsofunstiffenedspecimensSP:experimentaldataand Fig.15. CreepcoefficientforthestiffenedspecimensSS:experimentaldataand
regressionlinesaccordingtotheFindleylaw. modelpredictionsbytheEMandMSmethods.
Table5 AG =F/c (cid:3)A G , where the last term is the contribution of the
SummaryofthecalibratedFindleyparametersforshearðs0¼s0¼4:11MPaÞ. pucltcruded0specpimpenalone.
106m=106m0 a(–) R2 TheshearrelaxationfunctionR(t)hasbeenfoundfromEq.(11).
SP5 179.97 0.206 0.997 Asin thecase ofspecimensintraction,theobtainedR(t)function
SP6 170.26 0.208 0.997 was found very close to the function 1/J(t), the difference being
SP7 177.16 0.211 0.997 about1%.
SP8 194.65 0.196 0.994
Predictedshearcreepcoefficientsareshownasthickblacklines
inFig.15.Thetwoanalyticalcurves(byEMandMSmethods)are
veryclosetothelowestexperimentalcurve,representingstrainsof
specimenSS1. Taking the unavoidable scattering in experimental
SP5 Experimental values
ent [−] 00..56 SSSPPP678 EEExxxpppeeerrriiimmmeeennntttaaalll vvvaaallluuueeesss wdboahtniac,fihtbhrceeasrneonsrouiteltnbitsaetsipaotrniesdcfyaicinntegcda.uInbseyfaacthts,iegenvmeifinocdaaenslt.mlMoaslolsrdoeefovvsiteairtf,ifontnehseisn[gc3la1ur]e-,
oeffici 00..34 SP − Findley (mepayoxeyx)huibseitdsthoebaroncdretehpebCeFhRaPvisohuere:tsittsoctrheeeppuplrtorupdeertdieeslesmhoeunltds
c
p then affect the long-term behaviour of these specimens far more
e 0.2
Cre than for specimens under traction, since relative slip between fi-
0.1 bresisonlylimitedbyepoxyinthepresentcase.
0 ComparisonbetweenFigs.15and13alsoshowsthatcarbonfi-
10−3 10−2 10−1 100 101 102 103 bresreducecreepcoefficienttolessthanonehalfwithrespectto
Time [days] thecasewithoutreinforcement.
In this case, due to data dispersion the accuracy of Effective
Fig.13. CreepcoefficientfortheunstiffenedspecimensSP:experimentaldataand
ModulusandMeanStressmethodsarecomparable,evenifthetrend
meancreepcoefficientevolutionaccordingtotheFindleylaw.
seemstobebettercapturedbytheEMmethod.
103 6.Conclusions
6]
E In this paper, the results from creep tests on specimens ex-
x 1 102 tractedfromcommercialpultrudedprofilesareshown.Bothtrac-
ns [ tion and shear tests have been carried out, with specifically
strai 101 Effective Modulus designedsteelframes.Somepultrudedspecimenshavebeenstiff-
ar Mean Stress enedwithadditionallayersofcarbonfibres,alongtheloaddirec-
he SS1−Experimental values tion for specimens in traction and with direction ±45(cid:2) for
eep s 100 SSSS23−−EExxppeerriimmeennttaall vvaalluueess specimenssubjectedtoshear.Thespecimenshavebeenkeptunder
Cr SS4−Experimental values constant loading for more than 2years, so providing information
10−1 onthematerialbehaviouroutoftheusualtestrange.Ithasbeen
10−3 10−2 10−1 100 101 102 103 verifiedthatadditionofcarbonfibresallowsreducingsignificantly
Time [days] theshort-andlong-termdeformabilityofthespecimens.
Shearcreeparelargerthancreepundertraction,soconfirming
Fig.14. CreepshearstrainsofstiffenedspecimensSS:experimentaldataandmodel
thattherheologicalpropertiesofthepultrudedcompositesrelated
predictionsbytheEMandMSmethods.
tonormalorshearloadingsareindependent,duetothedirectional
arrangement of fibres. Experimental data for stiffened specimens
specimens(seeTable5).Theadoptedshearmodulusistheonegi- under shear show a significant dispersion. It can be due to the
veninTable1forplates,i.e.G=3.4MPa. manualprocedureadoptedtogluecarbonfibresheetstothepul-
SincetheshearmodulusoftheCFRPsheetsisquitedifficultto trudedspecimens.
be measured experimentally, it has been obtained indirectly by Creepofunstiffenedspecimenshasbeenmodelledbymeansof
computing the fibres stiffness AG from the average measured Findleypowerlawoftime,thereferencelawforpolymersandFRP.
c c
shearstrainc attheendoftheloadingphase.Equilibriumequa- Availabledataatearlyagesafterloading(firsthour)showedthat
0
tionunderpureshearstatehasbeenalsoconsidered,soobtaining: the model is not able to properly catch the growing creep rate,