Fiber reinforced concrete characterization through round panel test - Part II: analytical and numerical study

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1 Fratur Mhani Conrt Conrt Strutur - High rforman Fibr Rinford Conrt Spial Loading Strutural Appliation- B. H. Oh t al. (d) Kora Conrt Intitut ISBN Fibr rinford onrt haratrization through round panl tt - art II: analytial numrial tudy F. Minlli & G.A. lizzari Univrity Bria Italy ABSTRACT: Th round dtrminat panl aording to ASTM a found to b a rliabl onitnt rpatabl tt mthod for th maurmnt th nrgy aborption in Fibr Rinford Conrt (FRC) ompoit. A mallr panl a propod xprimntally invtigatd by th author in th art I th prnt papr. Th rult vral tt on about round panl (ASTM mall panl) vidnd a lor attr a in th largr ASTM panl ith ignifiant advantag. In fat mall round panl ar air to pla hl thir ight i only 4 kg if ompard to th 9 kg th ASTM panl. Givn thi broad xprimntal databa an analytial approah i hrin rportd toard th dfinition a uitabl implifid ohiv tr-rak idth ontitutiv la for FRC dtrmind from round panl tt aording to th rquirmnt th main trutural od. To thi aim rak dvlopmnt a maurd during th tt. In addition a kinmati approah a propod to dtrmin th ontitutiv la from th load point diplamnt th lab. INTRODUCTION A tardizd Round Dtrminat anl (RD) tt i noaday availabl publihd by ASTM (4) for th maurmnt th nrgy aborption at 4 mm diplamnt for haratrizing Fibr Rinford Conrt (FRC) lmnt ith pial mphai on prayd onrt in tunnling appliation. It i a tatially dtrminat tt ith round hap lab having a diamtr (φ) 8 mm a thikn 75 mm upportd in thr point at dgr. Th tard tt i rathr traightforard only rquir load diplamnt maurmnt. Th pimn ight i 9 kg. In th firt part thi invtigation th author dmontratd that a mallr round panl ith a diamtr 6 mm a thikn 6 mm i haratrizd by a imilar attr in rult hih i muh lor ompard to th laial tt on mall nothd bam (Minlli & lizzari 7 9). Morovr th thr rak maurmnt ould b alo inludd in th tt a th rak pattrn i rpatabl prditabl; thrfor th pot-raking matrial proprti an b adquatly dtrmind. In addition hling plaing uh a pimn i air a ompard to th laial panl in it ight i only 4 kg. Finally tard rvoontrolld loading mahin gnrally fit ith th gomtry th mall panl alloing for a propr rak ontrolld tt ith a lo-loop ytm. In ordr to u thi tt produr for haratrizing FRC hih man dtrmining fratur proprti uh a toughn indx ridual pot raking trngth quivalnt pot-raking tr on hould b abl to find uffiint information from th xprimnt. Th produr adoptd in many tard for bam tt (UNI 3 RILEM 3 & CEN 5) ar bad on th dtrmination a tr-rak idth urv in hih: -th tr an b drivd from thory latiity i.. th flxural tr i jut th momnt ovr th ritan modulu. Thi i a rathr rough implifiation but it allo for th dfinition uitabl potraking trngth or toughn indx ithout any non-linar analyi hih i diffiult to prform; -th rak idth i maurd monitord (Crak Tip Opning Diplamnt CTOD) pially hil daling ith nothd bam. Str auing rak (i.. normal tr along th rak lin) rak idth rord ar thrfor ndd if th produr ll rognizd for bam i to b tranfrrd to round panl. Th virtual nrgy-bad yild lin mthod (Johann 97) in hih th uniaxial flxural apaity th matrial upon raking i ud togthr ith an aumd pattrn failur to prdit th point load apaity might b alo ud to dtrmin a impl rak idth a a funtion th point load diplamnt (Brnard 6). Hovr th yild lin mthod a originally dvlopd for nominally plati matrial; thrfor it appliation for th prdition load apaity in trutur mad lightly rinford onrt xhibiting brittl bhavior might b qutionabl.

2 J = Th D ( prnt h T ) h invtigation fou on th appliability round panl tt in dfining a implifid () ay-to-u Th proportionality ontitutiv fiint la for FRC D(hT) matrial. i alld Numrial moitur lati prmability analy it yild i a lin nonlinar mthod funtion ill b both th utilizd rlativ to humidity thi aim. h In partiular tmpratur th T lattr (Bažant approah & Najjar ill 97). b xamind Th moitur valuatd ma balan a far rquir a th prdition that th variation raking tim i onrnd th atr ma ompard pr unit againt volum th onrt xprimntal (atr rult. ontnt ) b qual to th divrgn th moitur flux J ANALYTICAL STUDY = J () t A diffuly rportd in art I thi papr mor than Th atr xprimnt ontnt on an round b xprd panl r a th arrid um out th in th vaporabl lat 4 yar atr at th Univrity Bria. (apillary atr atr Bad on thi broad xprimntation an analytial vapor adorbd atr) th non-vaporabl modl a thn dvlopd to th aim : (hmially bound) atr n (Mill 966 -dtrmining loal tr-rak opning ohiv la antazopoulo & Mill 995). It i raonabl to from round panl imilarly to th produr ll aum that th vaporabl atr i a funtion aknoldgd for bam; rlativ humidity h dgr hydration -dfining uitabl quivalnt pot-raking trngth dgr ilia fum ration dirtly from panl tt in th am i.. ay = a (h for bam ) = ag-dpndnt orption/dorption iothrm tt. (Norling Mjonll 997). Undr thi aumption Firt all appropriat gnral rlation aording to th thin plat thory r dtrmind for d- by ubtituting Equation into Equation on obtain ribing th maximum tangntial tr (a ll a th radial tangntial tr pattrn a hon in Fig. Fig. ) th diplamnt h at th load point. + ( D h) = & + & + & n (3) An lati h t finit hlmnt analyi through th ommrial tar STRAUS7 (4) ith to dimn- ional lmnt (plat) a prformd ith th folloing rult: hr /h i th lop th orption/dorption iothrm (alo alld moitur apaity). Th govrning quation (Equation 3) mut b ompltd D by σ appropriat t =.86 boundary initial ondition. () Th rlation tbtn th amount vaporabl atr rlativ humidity i alld adorption 3 iothrm if maurd D ith inraing rlativity humidity =.49 dorption () 3 E t iothrm in th oppoit a. Nglting thir diffrn (Xi t al. 994) in hr th folloing i th xtrnal orption load iothrm in th ntr ill b (a ud unit load ith rfrn kn a to both applid) orption D i th dorption diamtr ondition. th panl maurd By th ay from th if th upport hytri (55 mm in th th moitur a mall iothrm round ould panl) b takn t i th into panl aount thikn to (6 diffrnt mm) E rlation ν ar vaporabl Young atr oion v rlativ modulu humidity mut onrt b ud rptivly. aording to All th unit ign ar dfind th variation aording th to SI rlativity unit ytm. humidity. Th hap th orption iothrm Th lati for HC analyi inflund a alo by abl many to paramtr prdit th ditribution pially tho radial that influn tangntial xtnt tr rat along th lin hmial rak ration formation (i.. in th turn radial dtrmin bitor por btn trutur ah pair por iz pivot ditribution upport) (atr-to-mnt a dpitd in Figur ratio mnt 3. Not that hmial th lati ompoition olution i SF an ay-tou uring tool tim that allo mthod a implifid tmpratur alulation mix additiv th ontnt quivalnt t.). In th pot-raking litratur variou trngth formulation bing faibl an for b pratial found to drib dign th appliation orption iothrm in am ay normal a it onrt i don for (Xi bam t al. tt. 994). Hovr in th prnt papr Th th rult mi-mpirial ho that th xprion pak tnil propod tr our Norling nar Mjornll th loading (997) point i adoptd a broad bau ara by it xpliitly aount for th volution hydration ration SF ontnt. Thi orption iothrm rad ( h ) = G ( ) + ( g ) h Figur hr. th radial firt tr trm σ(gl r on a iothrm) mall round rprnt panl lati th analyi phyially ith FE bound program (adorbd) Strau7 (4). atr th ond trm (apillary iothrm) rprnt th apillary atr. Thi xprion i valid only for lo ontnt SF. Th fiint G rprnt th amount atr pr unit volum hld in th gl por at % rlativ humidity it an b xprd (Norling Mjornll 997) a G ( ) = k + k vg vg g Figur. tangntial tr σ t on a mall round panl G lati analyi ith FE program Strau7 (4). (6) K ( ) =. g Comparion ditribution σ r (r)/σ rmax -σ t (r)/σ tmax Small Round anl Str σ r /σ rmax σ t /σ tmax [Ma]. fr.8 (vaporabl) atr ontnt Sri t (r) in onrt at x t Sri r (r) r variou ag (Di Luzio & Cuati 9b)..6 ( g ) h K ( ) z Not.4 that at arly ag in th hmial ration y Support oint (4) (5) hr k vg k vg ar matrial paramtr. From th maximum amount atr pr unit volum that an fill all por (both apillary por gl por) on an alulat K a on obtain Th matrial paramtr k vg k vg g an b alibratd by fitting xprimntal data rlvant to. Tmpratur volution aoiatd ith mnt hydration SF ration ar.xothrmi th tmpratur fild i not uniform for non-adiabati ytm vn if th nvironmntal tmpratur i ontant. Hat ondution an b. dribd in onrt at lat for tmpratur not xding C Radial (Bažant Coordinat & r Kaplan [mm] 996) by Figur Fourir 3. Elati la hih radial rad tangntial tr along th lin rak formation (radial oordinat). q = λ T (7) hightnd tnil tr aro along th rak lin. Whil th radial tangntial tr hav th am magnitud hr q i at th th ntr hat flux thy divrg T i th toard abolut th dg tmpratur th panl λ along i th ah hat radial ondutivity; bitor th in thi ra- roding FraMCoS-7 May 3-8

3 dial tr dropping to zro for mting th boundary ondition at th dg ( alo Brnard 6). By uing th abov quation for th dtrmination th loal tangntial maximum tnil tr along th yild lin it i poibl to om up ith a σ- ohiv la ompar it ith tho dtrmind from bam tt hrby dfin th xprimntal maurd rak idth. A an altrnativ ithout dirt maurmnt th rak idth a kinmati approah bad on th yild lin thory (mor proprly dfind a virtual nrgy-bad yild lin mthod Johann 97) ould b adoptd to alulat th rak idth a alo don by Brnard () Tran t al. () Lambrht (4). In a yild lin analyi a round panl th govrning mod failur i takn to ompri thr ymmtrial yild lin-rak manating from th ntr th fa oppoit th point load running radially to th dg hil biting ah tor btn adjant pivot upport. B Zro Diplamnt Q C Yld Lin Cntral Load oint ivot Lin Hing Lin Support ivot Figur 4. Yild Lin Approah for th dtrmination rak idth. Th rak idth a xprimntally maurd at a ditan from th point load mm (point Q in Fig. 4). Bad on gomtry onidration it an b rittn that: BC BQ = (3) δ Q hr CQ= mm (in th prnt xprimnt) hra B i th zro diplamnt point δ Q ar th vrtial diplamnt at point C Q rptivly. From trigonomtry on an driv that: r BQ = δq δ Q = BQ r (4) ith r =75 mm. J = D ( h T ) h Th rotation at th upport i qual to: δ Q = Q Th proportionality fiint (5) D(hT) moitur prmability it i a nonlina By ubtituting Q th rlativ δ q on humidity an obtain: h tmpratur & Najjar 97). Th moitur ma balan BQ BQ = that th variation in tim th (6) atr ma r volum 3 onrt (atr ontnt ) b q divrgn th moitur flux J Th rotation at th upport thrfor yild: 3 = J = t (7) r Th atr ontnt an b xprd a Th rotation in Q at th point xprimntal th vaporabl atr rak maurmnt an b drivd a: (apillary a vapor adorbd atr) th non- ϕ = = (hmially 3 bound) atr (8) n (Mil antazopoulo r & Mill 995). It i ra aum that th vaporabl atr i a fu Th orrponding rlativ bottom humidity rak opning h dgr (Fig. 5) hydration i thrfor givn dgr by: ilia fum ration i.. = ϕ = ag-dpndnt = t tg = t tg 3 orption/dorption m (Norling Mjonll 997). Undr (9) thi aum by ubtituting r Equation into Equati hr r i th radiu obtain (r = 75 mm) t i th panl thikn (t = 6 mm). It i orth notiing that th rak opning obtaind uing Equation 9 i + ontant ( D along h) = th & radiu + & + h h t h for a givn valu panl diplamnt. Hovr in th atual a th rak idth ill inra from th xtrnal urfa to hr th load point. /h i th lop th orption/ iothrm (alo alld moitur apa govrning quation (Equation 3) mut b by appropriat boundary / initial onditi Th rlation btn th amount atr rlativ humidity i alld iothrm if maurd ith t inraing humidity dorption iothrm in th a. Nglting thir diffrn (Xi t al. th folloing orption iothrm ill b t tg( /) rfrn to both orption dorption Figur 5. 3D rprntation By th ay th failur if th mod hytri a round th panl (a) (Brnard iothrm 6). Crak ould idth b bottom takn urfa into aount a to round panl (b). rlation vaporabl atr v rlativ humi b ud aording to th ign th varia Figur 6 ho th omparion btn xprimntal analytial rak idth: th xprimntal rlativity humidity. Th hap th iothrm for HC i inflund by many p urv a alulatd from a linar rgrion all pially tho that influn xtnt tt rult availabl. It i a bilinar urv ith a firt hmial ration in turn dtrm ontant lin ith zro rak rprnting th unrakd pha in hih th vrtial diplamnt in- trutur por iz ditribution (atrratio mnt hmial ompoition SF ra prior to raking (up to a valu around uring tim mthod tmpratur mix = mm in hih th lot valu i appropriat for normal trngth onrt hil th t.). In th litratur variou formulatio found to drib th orption iothrm hight i mor uitabl for high trngth onrt) onrt (Xi t al. 994). Hovr in th a ond linar branh hoing a quit good fitting th xprimnt (R mi-mpirial xprion pro papr th =.99). Th diffrn btn th to urv (ith th xprimntal Norling Mjornll (997) i adoptd b hoing roding FraMCoS-7 May 3-8

4 J = Drak ( h T ) hfor th am vrtial diplamnt) i largr () probably du to th fat that a littl lati dformationth i alay maurdfiint by th LVDT xproportionality D(hT)inithalld primnt (5 mm gaug lngth) mot impormoitur prmability it i a nonlinar funtion tantly thir loation mm from Tth load th rlativ humidityi hat tmpratur (Bažant point hil th yild lin thory ould math bttr & Najjar 97). Th moitur ma balan rquir ith a maurmnt at r/from th load point that th variation in tim th atr ma pr(i.. unit 37.5 mm). volum onrt (atr ontnt ) b qual to th divrgn th moitur flux J xpliitly aount ahydration tat (rak idth for fromth.6volution to 3 mm) follo ration (Fig. 9): SF ontnt. Thi orption iothrm rad (h ) = G ( ) (g )h (g K ( ) () t Th atr ontnt an b xprd a th um th vaporabl atr (apillary atr atr vapor adorbd atr) th non-vaporabl (hmially bound) atr n (Mill 966 antazopoulo & Mill 995). It i raonabl to aum that th vaporabl atr i a funtion rlativ humidity h dgr hydration dgr ilia fum ration i.. =(h) = ag-dpndnt orption/dorption iothrm (Norling Mjonll 997). Undr thi aumption by ubtituting Equation into Equation on obtain &+ & titutiv ohiv σ- la bad on round panl xprimnt. From a linar xprimntal i th loprgrion th orption/dorption hr /h rult an analytial rlation btn th rakth tip iothrm (alo alld moitur apaity). opning diplamnt (CTOD) th vrtial digovrning quation (Equation 3) mut b ompltd plamnt at midpan a bam 7) aording by appropriat boundary initial(fig. ondition. to UNI a drivd a follo: Th rlation btn th amount vaporabl atr rlativ humidity i alld adorption CTOD m = if.3maurd B () iothrm ith inraing rlativity humidity dorption iothrm in th oppoit a. Nglting thirmodl diffrn t al. 994) in From a kinmati hon(xi in Figur 8 it a th folloing obtaind that: orption iothrm ill b ud ith rfrn 4 tob both orption dorption ondition. ϕ ByB =th ay if th hytri th moitur () l iothrm ould b takn into aount to diffrnt rlation vaporabl ϕatr v rlativ humidity B mut m = (h a ) tg B = (h a ) tg () b ud aording to th ign th variation l th rlativity humidity. Th hap th orption iothrm for HC i inflund many paramtr Not that th am approahbyfor alulating th pially tho that influn xtnt th rotation rak idth an b follodrat for bam hmial ration in turn dtrmin por aording to CEN (5) providd that a noth trutur mm porrathr iz ditribution (atr-to-mnt than 45 mm i ud. Figur dpth a=5 hmial ontnt 8ratio ouldmnt alo apply for th ompoition CEN bam ttsf (5). uring tim mthod tmpratur mix additiv Th Italian Stard UNI 39 (3) dfin t.). In th quivalnt litratur variou formulation anfq(b to diffrnt pot-raking trngth found to drib th orption iothrm normal.6) f q(.6-3) hih rprnt th toughn givn onrt (Xi by t fibr al. 994). Hovr inrviability th prnt to th matrix in a onvntional papr th mi-mpirial xprion propod by (rak idth from to.6 mm) ultimat limit Norling Mjornll (997) i adoptd bau it roding FraMCoS-7 May 3-8 (4) hr th firt trm (gl iothrm) rprnt th phyially bound (adorbd) atr th ond trm (apillary iothrm) rprnt th apillary atr. Thi xprion i valid only for lo ontnt th amount SF.7.Th G rprnt Figur Crakfiint idth at bottom urfa a mall round panatr pr unit volum hld in th gl por at % l. rlativ humidity it an b xprd (Norling Load oint Mjornll 997) a + k G ( ) = k vg vg (5) hr kvg kvg ar matrial paramtr. From th maximum amount atr pr unit volum that an fill all por (both apillary por gl por) on an alulat K a on obtain B &+ + B Figur 6. Crak idth at bottom urfa a mall round panl. h + ( D h ) = (3) n h allo for th dfinition h tprodur Thi a on- = J )h Figur 8. Kinmati modl for 4 point bnding tt UNI bam. g K ( ) = +. G.88 g h (6) Th matrial paramtr kvg kvg g an b alibratd by fitting xprimntal data rlvant to fr (vaporabl) atr ontnt in onrt at variou ag (Di Luzio & Cuati 9b).. Tmpratur volution Not that at arlyag in th hmialtr ration Figur 9. Dfinition quivalnt pot-raking aording to UNI Stard (3). aoiatd ith mnt hydration SF ration ar xothrmi th tmpratur fild i not uniform l U for ytm if th nvironmntal (3) f non-adiabati = vn q (.6) ib(ontant. tmpratur Hat ondution an b. 6 h a ) dribd in onrt at lat for tmpratur not xding C l(bažantu & Kaplan 996) by (4) f = Fourir rad q (.6 la 3) hih b(h a ).4 q = λ T (7) By uing Equation it i poibl to orrlat rang vrtial B hr qrak i th hatithflux T diplamnt i th abolut ϕ bam. rotation B tmpratur λ i th hat ondutivity; in thi

5 Tabl. Midpan diplamnt rotation orrponding to th CTOD m dfind by th UNI tard 39 (3). CTOD m B ϕ [mm] [mm] B For round panl it a xprimntally found that (Fig. 6):.67.3 (5) m = By impoing th rotation round panl to b qual to th on bam aording to th yild lin thory uing Equation 8 on an firt dtrmin th vrtial diplamnt at th point load thn th rak idth orrponding to th rotation givn (Equation 5). Th rak idth valu thn dfin to rang that allo th dfinition to quivalnt pot-raking trngth rlvant for rviability ultimat limit tat rptivly. Th rult thi produr ar rportd in Tabl. Tabl. Vrtial diplamnt rak idth valu orrponding to th am yild lin rotation btn panl bam. ϕ =ϕ B m [mm] [mm] Thrfor on an dfin to rak idth rang dtrmin th orrponding quivalnt potraking trngth hih ar: f q from to + ; f q from + to +. bing th rak idth at firt raking =.55 mm =.75 mm (Tabl ). Th to quivalnt pot-raking trngth an thrfor b dfind a: k D U f q = (6) t k D U f q = (7) t hr: f q i th quivalnt pot-raking trngth rlvant for rviability limit tat orrponding to f q(-.6) UNI bam; f q i th quivalnt pot-raking trngth rlvant for rviability limit tat orrponding to f q(.6-3) UNI bam; U i th ara undr th - xprimntal graph alulatd in th rak rang from to + (Fig. ); U i th ara Jundr = D ( th h T ) - h xprimntal graph alulatd in th rak rang from + to + (Fig. ); Th proportionality fiint D(hT) k =.86 moitur mm - th prmability ontant alrady it alulatd throughout th rlativ lati humidity analyi h prviouly tmpratur i a nonlina hon. & Najjar 97). Th moitur ma balan mcari that th variation in tim th atr ma volum onrt (atr ontnt ) b q divrgn th moitur flux J = J t Th atr ontnt an b xprd a th vaporabl atr (apillary a U U vapor adorbd atr) th non- (hmially bound) atr n (Mil antazopoulo & Mill 995). It i ra + + aum that th vaporabl atr i a fu Figur. Load-rak rlativ idth urv humidity for mall h round dgr panl dpiting ara U dgr U. ilia fum ration i.. = hydration = ag-dpndnt orption/dorption Thi approah (Norling an b ud Mjonll to dtrmin 997). Undr avrag thi aum valu tard dviation by ubtituting haratriti Equation valu into Equati th toughn paramtr obtain dfind taking a ignifiant advantag from th lor attr panl tt ovr bam ( Conti Fllli 9). h + ( D h) = & + & + h t h Load hr /h i th lop th orption/ iothrm (alo alld moitur apa govrning quation (Equation 3) mut b by appropriat boundary initial onditi Th rlation btn th amount atr rlativ humidity i alld iothrm if maurd ith inraing humidity dorption iothrm in th a. Nglting thir diffrn (Xi t al. th folloing orption iothrm ill b rfrn to both orption dorption Figur. Typial Load- urv for a mall round panl dfining ara E EBy. th ay if th hytri th iothrm ould b takn into aount to A imilar approah rlation an vaporabl b drivd atr tarting v rlativ from humi th load-diplamnt b ud urv aording panl to th (Fig. ign ) nglting th rak rlativity idth maurmnt humidity. Th (tn hap not th th varia providd). In uh iothrm irumtan for HC i on inflund an dfin by many p uitabl diplamnt pially rang tho for th that dfinition influn xtnt f q f q. Th diplamnt hmial ration rang alulatd in turn from dtrm Equation 4 ar trutur inorporatd por into iz th dfinition ditribution (atrratio pot-raking mnt hmial trngth a: ompoition SF th to quivalnt uring tim mthod tmpratur mix k D E t.). In th litratur variou formulatio f q = (8) found to drib th orption iothrm t onrt (Xi t al. 994). Hovr in th k D Epapr th mi-mpirial xprion pro f q = (9) t Norling Mjornll (997) i adoptd b roding FraMCoS-7 May 3-8

6 bing J = D ( h =.8 T ) h mm =4. mm (Tabl ). () Aording to th Italian Guidlin for th Dign Contrution Th proportionality rodution fiint Control D(hT) Fibr i alld Rinford moitur Conrt prmability Strutur it (CNR i a nonlinar 6) on funtion uitabl th bam rlativ tt humidity ar prformd h tmpratur th orrponding T (Bažant valu & Najjar 97). f q Th f q moitur (alo indiatd ma balan a f q(-.6) rquir fthat q(.6-3) th if variation th aformntiond in tim UNI th atr tt i ma onidrd) pr unit it volum i poibl onrt to dfin (atr a implifid ontnt tr-rak ) b qual idth to th ontitutiv divrgn ohiv th moitur la a flux follo: J f Ft =.45 fq () = J () t u f FtuTh = k atr f Ft ontnt ( f Ft an.5 b f qxprd +. f q ) a th () um i th vaporabl atr (apillary atr atr vapor adorbd atr) th non-vaporabl hr: (hmially bound) atr n (Mill 966 antazopoulo k i a fiint & Mill qual 995). to.7 It for i unrakd raonabl tion aum undr that tnion th vaporabl qual atr to in i th a funtion othr a to k= rlativ in thi humidity a; h dgr hydration dgr i i th ilia avrag fum ration th rak i.. idth =valu (h for ) hih = ag-dpndnt f q i alulatd. orption/dorption In thi a i =.8 mm iothrm (avrag (Norling valu Mjonll btn 997)..6 Undr 3 mm thi aording aumption to UNI 39); by ubtituting Equation into Equation on obtain u i th gratt valu th rak rang for hih f q i alulatd. In thi a u =3 mm (aording to UNI 39); Figur h dpit a typial linar + ontitutiv la ( D h) = & + & + & n (3) obtaind h t from th h quivalnt pot-raking tr alulatd from bam tt. hr /h i th lop th orption/dorption iothrm σ t (alo alld moitur apaity). Th govrning quation (Equation 3) mut b ompltd by appropriat boundary initial ondition. Th rlation btn th amount vaporabl atr rlativ humidity i alld adorption iothrm if maurd ith inraing rlativity humidity f Ft dorption iothrm in th oppoit a. Nglting thir diffrn (Xi t al. 994) in.5f q -.f q th folloing orption iothrm ill b ud ith rfrn to both orption dorption f Ftu ondition. By th ay if th hytri th moitur iothrm ould b takn into aount i to u diffrnt rlation vaporabl atr v rlativ humidity mut Figur. Typial σ- implifid ohiv ontitutiv la aording b ud aording to CNR DT 4/6. to th ign th variation th rlativity humidity. Th hap th orption iothrm On an for noti HC that i inflund to th aim by many oming paramtr up ith a pially implifid tho that ay-to-u influn ontitutiv xtnt rat la th tard hmial nglt ration th firt in branh turn from dtrmin th tnil por trngth trutur to th por valu iz f Ft ditribution. Morovr (atr-to-mnt th σ- indiatd ratio in mnt th pitur hmial rfr ompoition to a train-tning SF ontnt matrial uring undr tim tnion. mthod tmpratur mix additiv t.). Figur In th 3 litratur Figur variou 4 rport formulation a an xampl an th b haratriti found to drib valu th orption quivalnt iothrm pot-raking normal tr onrt alulatd (Xi t al. by 994). uing Hovr th produr in th propod prnt hrin papr th for mi-mpirial mall round panl xprion th propod laial by Norling Mjornll (997) i adoptd bau it mthod inludd in UNI 39 (3) for bam. Th xpliitly xmplifiation aount rfr for th to volution a normal trngth hydration onrt ration ith to SF diffrnt ontnt. ontnt Thi orption tl fibr iothrm 3 rad 6 kg/m 3 aording to Conti Fllli (9). Th advantag onrn th ignifiantly mallr diprion xprimntal rult that man highr haratriti valu thrfor an improvd ( h ) = G ( ) + tr-rak idth ohiv la a dpitd in ( g ) h Figur 5. Th fratur nrgy a th ara undr (4) th σ- diagram i around 5% largr in th to round panl diagram plottd hih ( grfr to th ) h haratriti valu hon in Figur 3 Figur 4. Th K ( ) ontitutiv la rlatd to Bam in th plot ha th only advantag to ho th bt prforman for high hr rak th idth firt trm xding (gl iothrm) all la rprnt dpitd. th Thi phyially vidn bound om (adorbd) from th atr fat that th th ond to quivalnt trm (apillary haratriti iothrm) pot-raking rprnt th tr apillary ar rathr atr. imilar Thi xprion thn i in valid th alulation only for lo th ontnt ubtrahnd SF. ould Th b fiint lor (Equation G rprnt ). Thi th ould amount alo uggt atr pr that unit thr volum i a hld nd in to th furthr gl por orroborat % rlativ validat humidity th propoal it an inludd b xprd in CNR (Norling (6) againt Mjornll a broadr 997) a t xprimntal data. 4 G ( ) = k + kstl Fibr; NSC vg vg roprti dtrmind at SLS 3.3 f q / f qb f q / f hr 3 k qb vg =. k vg.9 ar matrial paramtr. =.58 From th.38 fill all por (both apillary por gl por) on.9 an alulat K a on obtain [Ma] Comparion f q(-.6)k -f qk Figur Th 3. matrial Analytial paramtr alulation k quivalnt pot-raking vg k vg g tr from xprimnt. NSC bam panl from idntial an b onrt alibratd bath by (Conti fitting xprimntal Fllli 9). data rlvant to fr (vaporabl) atr ontnt in onrt at variou 4 ag (Di Luzio & Cuati 9b). Comparion f q(.6-3)k -f qk Stl Fibr; NSC roprti dtrmind at ULS. Tmpratur volution 3 f q / f qb =.56 aoiatd f q ith / f qb mnt = hydration.3 SF ration ar xothrmi.4 th tmpratur fild i not uniform for non-adiabati ytm vn if th nvironmntal.6 [Ma] xding C (Bažant & Kaplan 996) by Fourir la UNI hih rad UNI q = λ T 3 kg/m 3 6 kg/m 3 (5) maximum amount atr pr unit volum that an g G UNI UNI K ( ) = g 3 kg/m 3 6 kg/m 3 Figur 4. Analytial alulation quivalnt pot-raking tr hr from q xprimnt. i th hat NSC flux bam T i panl th from abolut idntial tmpratur onrt bath (Conti λ i th Fllli hat 9). ondutivity; in thi (6) Not that at arly ag in th hmial ration tmpratur i ontant. Hat ondution an b dribd in onrt at lat for tmpratur not (7) roding FraMCoS-7 May 3-8

7 Str [Mpa] Cohiv σ- ontitutiv La Bam v. Round anl BEAM ROUND ANEL BEAM ROUND ANEL Crak Width [mm] Figur 5. σ- la aording to CNR DT 4/6. Bam tt v. anl tt. NSC bam panl from idntial onrt bath (Conti & Fllli 9). 3 CONCLUDING REMARKS In th prnt papr numrial lati th yild lin approah r adoptd to om up ith implifid ohiv ontitutiv la from round panl mall tt. anl tt i quit appropriat for rprnting th atual bhaviour FRC matrial. It an b onidrd a a omplt tt for th haratrization FRC: uitabl rang rak idth r dfind th orrponding quivalnt (or ridual) potraking trngth r drivd (from σ- plot) folloing th am produr a don for bam tt. Th kinmati approah for alulating rak idth turnd out to b impl fftiv rathr aurat if ompard againt xprimnt. anl tt an bom a vry promiing tt mthod for th haratrization FRC ompoit a it i rlativly ay to prform it i not xpniv it might rquir only on vrtial diplamnt trur lat but not lat it i haratrizd by a vry lo attr. Furthr tudi ar atually going on at th Univrity Bria for upplmntary rfin validat th propod alulation. Among th opn iu for furthr tudi th folloing ky-point hould b kpt in mind: - th poition th rak maurmnt: th rak idth vari along th rak lin hang in a diffrnt ay bfor aftr raking. Morovr hould alo rall that tangntial tr hav to b onidrd ith radial tr hih ar non zro for a lngth /3 th rak lin. On hould think at mauring rak lo to th pivot upport hr radial tr ar ngligibl but thn th rak idth ould b lor mor notably th tr diturban J = D ( th h pivot T ) h upport might afft th rord. - Th 6 mm thikn Th proportionality th mall round fiint panl D(hT) hould b ll valuatd moitur pially prmability ith longr it i fibr a nonlina (longr than 5 mm). th Suitabl rlativ limitation humidity h hould tmpratur b dfind onrning & Najjar maximum 97). alloabl Th moitur fibr lngth ma balan for hih th mall that round th variation panl tt in an tim b onidrd ignifiant. Similar volum limitation onrt hovr (atr ontnt hould ) b q th atr ma b dfind alo for divrgn laial round th moitur panl hr flux J th thikn i only lightly highr (75 mm). - Th thin plat gomtry might au a D orintation hih hould b = ll Jvaluatd ompard t ith that bam tt ampl takn dirtly from th fild trutural Th appliation. atr ontnt an b xprd a th vaporabl atr (apillary a vapor adorbd atr) th non- 4 ACKNOWLEDGMENTS (hmially bound) atr n (Mil antazopoulo & Mill 995). It i ra Th author ould lik to inrly thank Enginr aum that th vaporabl atr i a fu Moira Conti Giovanni Fllli & Niola Vzzoli for rlativ humidity h dgr hydration thir aitan in data proing in prforming dgr ilia fum ration numrial analytial tudi. i.. = = ag-dpndnt orption/dorption (Norling Mjonll 997). Undr thi aum by ubtituting Equation into Equati REFERENCES obtain ASTM C55-4 Stard tt mthod for flxural toughn fibr rinford onrt (uing ntrally loadd h round panl) pp ( D h) = & + & + h t h Brnard E.S. Bhaviour round tl fibr rinford onrt panl undr point load Matrial Strutur/Matériaux t Contrution Vol. 33 April pp hr 8-88 /h i th lop th orption/ Brnard E.S. Corrlation iothrm in (alo th bhaviour alld moitur fibr rinford hotrt bam govrning panl quation pimn (Equation Matrial 3) mut b apa Strutur/Matériaux by appropriat t Contrution boundary Vol. 35 initial April onditi pp Th rlation btn th amount Brnard E.S. 6 atr Influn Toughn rlativ humidity on th Apparnt i alld Craking Load Fibr-Rinford Conrt Slab Journal Strutural Enginring iothrm Vol. if 3 maurd No. Dmbr ith inraing 6. humidity dorption iothrm in th CEN 5 rat a. Conrt Nglting rodut thir Tt mthod diffrn for mtalli fibr onrt th folloing Mauring orption th flxural iothrm tnil ill b (Xi t al. trngth Europ. rfrn Stard EN to 465 both orption pp.8. dorption CNR DT 4/6. By 6. th Guidlin ay if for th th hytri Dign Contrution rodution Control Fibr Rinford Con- th rt Strutur" iothrm National Rarh ould b Counil takn into Italy. aount to Conti M. & Fllli rlation G. vaporabl 9 Carattrizzazion atr v rlativ dl humi altruzzo fibrorinforzato b ud aording attravro to prov th ign u piatr th varia ottili M.S. Thi rlativity Dpartmnt humidity. DICATA; Th Univrity hap th Bria Marh 9 iothrm only in for Italian. HC i inflund by many p Johann K.W. 97. Yild lin thory Cmnt Conrt pially tho that influn xtnt Aoiation U.K. Lambrht A.N. Th hmial variation ration tl fibr onrt in turn haratriti. Study trutur on toughn por rult iz -3 ditribution ro-(atrding th Intrnational ratio mnt Workhop hmial Fibr ompoition Rinford SF dtrm Conrt. From uring thory to tim prati mthod Brgamo tmpratur Sptmbr mix pp. t.) In th litratur variou formulatio Marinoni Y. t al.: rov di frattura u altruzzi fibrorinforzati in found aordo to drib on l prinipali th orption normativ iothrm intrnazionali Thnial onrt Rport (Xi t Univrity al. 994). Hovr Bria in th Dpartmnt DICATA papr 7. th mi-mpirial xprion pro Minlli F. & lizzari Norling G.A. (7) Mjornll Fibr(997) Rinford i Conrt haratrization: Round anl v. bam tt toard adoptd b a roding FraMCoS-7 May 3-8

8 J = harmonization D ( h T ) h roding th 3rd Cntral Europan () Congr on Conrt Enginring CCC 7 Vigrád Hungary 7-8 Sptmbr 7. G.L. Baláz & S.G. Nhm Th proportionality (d) ublihing fiint Company Budapt D(hT) Univrity i alld moitur Thnology prmability Eonomi it pp. i a 3-. nonlinar funtion Minlli th rlativ F. & lizzari humidity G. A. 9 h Round tmpratur panl v. T bam (Bažant tt & Najjar toard a 97). omprhniv Th moitur harmoni ma balan haratrization rquir FRC matrial roding th BMC-9 Confrn that th variation in tim th atr ma pr unit Wara ol 5-8 Otobr 9 A.M. Brt t al. volum (d.) pp. onrt 3-3. (atr ontnt ) b qual to th RILEM divrgn Final Rommndation th moitur flux TC-6-TDF J Tt Dign Mthod for Stl Fibr Rinford Conrt; σε D- Mthod Matrial Strutur Vol 36 Otobr ign 3 = pp. J () Strau7. t 4. Finit Elmnt Analyi Sytm.tr7.om.trau7.om. Tran Th V.G. atr t al. ontnt. Th applia-tion an b xprd Yild a Lin th Thory um to th Round vaporabl Dtrminat atr anl Shotrt: (apillary Enginring atr atr Dvlopmnt adorbd Brnard (d.) atr) pp th non-vaporabl St & Zitlin- vapor (hmially gr Li. bound) atr UNI 39 Stl Fibr Rinford Conrt n (Mill art I: Dfinition Claifiation antazopoulo & Mill Spifiation 995). It Conformity i raonabl - art II: to aum Tt Mthod that th for Mauring vaporabl Firt atr Crak i Strngth a funtion Dutility Indx humidity Italian h Board dgr for Stardization hydration 3. rlativ dgr ilia fum ration i.. = (h ) = ag-dpndnt orption/dorption iothrm (Norling Mjonll 997). Undr thi aumption by ubtituting Equation into Equation on obtain h + h t ( D h) = n (3) h & + & + hr /h i th lop th orption/dorption iothrm (alo alld moitur apaity). Th govrning quation (Equation 3) mut b ompltd by appropriat boundary initial ondition. Th rlation btn th amount vaporabl atr rlativ humidity i alld adorption iothrm if maurd ith inraing rlativity humidity dorption iothrm in th oppoit a. Nglting thir diffrn (Xi t al. 994) in th folloing orption iothrm ill b ud ith rfrn to both orption dorption ondition. By th ay if th hytri th moitur iothrm ould b takn into aount to diffrnt rlation vaporabl atr v rlativ humidity mut b ud aording to th ign th variation th rlativity humidity. Th hap th orption iothrm for HC i inflund by many paramtr pially tho that influn xtnt rat th hmial ration in turn dtrmin por trutur por iz ditribution (atr-to-mnt ratio mnt hmial ompoition SF ontnt uring tim mthod tmpratur mix additiv t.). In th litratur variou formulation an b found to drib th orption iothrm normal onrt (Xi t al. 994). Hovr in th prnt papr th mi-mpirial xprion propod by Norling Mjornll (997) i adoptd bau it & xpliitly aount for th volution hydration ration SF ontnt. Thi orption iothrm rad ( h ) = G ( ) + ( g ) h ( g ) h K ( ) (4) hr th firt trm (gl iothrm) rprnt th phyially bound (adorbd) atr th ond trm (apillary iothrm) rprnt th apillary atr. Thi xprion i valid only for lo ontnt SF. Th fiint G rprnt th amount atr pr unit volum hld in th gl por at % rlativ humidity it an b xprd (Norling Mjornll 997) a G ( ) = k + k vg vg (5) hr k vg k vg ar matrial paramtr. From th maximum amount atr pr unit volum that an fill all por (both apillary por gl por) on an alulat K a on obtain K ( ) = g G g (6) Th matrial paramtr k vg k vg g an b alibratd by fitting xprimntal data rlvant to fr (vaporabl) atr ontnt in onrt at variou ag (Di Luzio & Cuati 9b).. Tmpratur volution Not that at arly ag in th hmial ration aoiatd ith mnt hydration SF ration ar xothrmi th tmpratur fild i not uniform for non-adiabati ytm vn if th nvironmntal tmpratur i ontant. Hat ondution an b dribd in onrt at lat for tmpratur not xding C (Bažant & Kaplan 996) by Fourir la hih rad q = λ T (7) hr q i th hat flux T i th abolut tmpratur λ i th hat ondutivity; in thi roding FraMCoS-7 May 3-8