Size-scale effects on minimum flexural reinforcement in RC beams: application of the cohesive crack model

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "Size-scale effects on minimum flexural reinforcement in RC beams: application of the cohesive crack model"

Transcription

1 Fratur Mhani Conrt Conrt Strutur - Rnt Advan Fratur Mhani Conrt - B. H. Oh, t al.(d) 2010 Kora Conrt Intitut, Soul, ISBN Siz-al fft on mimum flxural rformnt RC am: appliation ohiv rak modl E. Cadamuro, M. Corrado & A. Carptri Politnio di Toro, Toro, Italy ABSTRACT: prription providd by od prati for amnt mimum rformnt amount rford onrt am uually drgard nonlar ontribution onrt tnion iz-al fft. In prnt papr, phnomna ar orrtly takn to aount dription flxural failur lightly rford onrt am by man a numrial algorithm bad on Nonlar Fratur Mhani. In th ontxt, appliation Dimnional Analy prmit a rdution numr govrng paramtr. In partiular, it analytially dmontratd only to nondimnional paramtr, N P, ar rponibl for brittl-to-dutil tranition mhanial rpon. Aordg to th approah, a n formula uitabl for valuation mimum rformnt pratial appliation propod. A omparon ith xprimntal rult dmontrat fftivn propod modl. 1 INTRODUCTION 1.1 Stard Cod Provion Limit Analy rford onrt (RC) am uually aum trthd onrt not arg load o rakg phnomnon not takn to aount valuation load arryg apaity. Th aumption not alay yild to a af dign ondition, a for tan a lightly rford onrt am, hr tnil onrt ontribution dtrm a hypr-trngth ith rpt to ultimat loadg ondition, ith a onqunt poibl tability ovrall mhanial rpon. In th a, fat, rtant ndg momnt aftr pak rakg momnt a monotoni drag funtion rak lngth, du to an untabl fratur propagation. For th raon, all national trnational tard od prati provid mpirial formula for dtrmation mimum rformnt amount hih nabl RC mmr to prvnt untabl rak propagation. Mot m onidr only to paramtr: onrt grad tl yild trngth, hra or important paramtr, uh a iz-al, ar ompltly ngltd. A an xampl, Modl Cod 90 (1993) Brith Stard BS (1997) giv valu mimum rformnt dpndg only on tl grad, hra Eurood 2 (2004) ACI (2005) provid valu mimum rformnt proportional to onrt ompriv trngth vr rlation to tl yild trngth. Only Norgian Stard NS 3473 E (1989) aount for fft mmr iz, by man iz-fft fator, k, qual to 1.5-h/h 1 1, hr h am dpth m, h 1 qual to 1 m. Fally, Autralian Stard AS 3600 (2001) onidr mimum rformnt a a funtion ratio ovrall to fftiv am dpth, D/d. 1.2 Modl for Computg Mimum Rformnt A mor aurat amnt mimum rformnt takg to aount iz-al fft hould aomplhd ordr to av tl rformnt mony a larg trutur. To th aim, mot modl availabl litratur propo dtaild analy takg to aount omplx havior du to fratur propagation tnion, on ba Lar Nonlar Fratur Mhani approah (Ghali t al. 1986, Baluh t al. 1992, Grtl t al. 1992, Fantilli t al. 1999, Ruiz t al. 1999, Appa Rao t al. 2007). In th ontxt, ignifiant ontribution driv from appliation Bridgd Crak Modl (Carptri 1981a, 1984, Boo & Carptri 1992), hih bad on Lar Elati Fratur Mhani, to tudy rak propagation prn a rformnt. Aordg to th modl, ovrall rpon RC am an drid by man rformnt brittln numr, N P, obtad through Dimnional Analy on ba mhanial gomtrial proprti (Carptri 1981a, 1984):

2 J D ( h, T ) 0.5 h σ h y N ρ P K (1) (1) proportionality offiint D(h,T) alld motur prmability it a nonlar funtion hr σ rlativ y humidity tl yild h trngth, tmpratur h T (Bažant am dpth, K onrt toughn, ρ & Najjar 1972). motur ma balan rquir rformnt variation prntag. tim A dutil atr havior ma pr prditd unit volum for onrt high valu (atr N ontnt P, hra ) a qual brittl to rpon divrgn xptd motur for flux lo valu N P. In J partiular, it poibl to df N PC, a ritial valu hih parat brittl rpon from dutil on, J orrpondg to mimum rformnt amount ondition. valu N PC 0.26 t a xprimntally obtad by Boo t al. (1990) atr ontnt an xprd a um for high trngth onrt am,, uquntly, vaporabl atr follog mpirial quation (apillary atr, atr ha n propod vapor, adord atr) non-vaporabl by Boo & Carptri (1992) to xpr dpndn N PC on onrt grad (hmially bound) atr n (Mill 1966, σ : Pantazopoulo & Mill 1995). It raonabl to aum vaporabl atr a funtion N σ rlativ PC humidity, h, dgr hydration,, dgr ilia fum ration,, i.. (h,, ) Equatg ag-dpndnt xprion orption/dorption N P (Eq. (1)) to orm N(Norlg PC (Eq. ) Mjonll 1997). olvg Undr ith rpt th aumption to ρ, follog by ubtitutg formula Equation for mimum 1 to Equation rformnt 2 on amount obta obtad: ρ m K ( σ ) D h) σ t h y h + (3) ( & + & + & n (3) hr In prnt papr, Dimnional Analy applid / lop orption/dorption orm to (alo algorithm alld propod motur by Carptri apaity). t al. (2007) govrng bad quation on (Equation Cohiv 3) mut Ovrlappg ompltd Crak by appropriat modl boundary pifi itial a ondition. lightly rford onrt rlation am tn ndg. amount It ill vaporabl dmontratd atr rlativ to nondimnional humidity paramtr, alld adorption N P, orm ar rponibl if maurd for ith mhanial rag havior, rlativity not humidity N P only, a dorption onidrd o orm far. Fally, on oppoit ba a. Ngltg to ir brittln diffrn numr (Xi a t n al. 1994), formula ill follog, propod orption for valuation orm ill mimum ud ith rformnt rfrn to both amount. orption dorption ondition. By ay, if hytr motur orm ould takn to aount, to diffrnt 2 rlation, THE INTEGRATED vaporabl atr COHESIVE v rlativ AND humidity, mut OVERLAPPING ud aordg to CRACK ign MODEL variation rlativity humidity. hap orption Lt orm u onidr for HPC flund rford by onrt many paramtr, am lmnt pially Figur tho 1 ith flun a rtangular xtnt ro-tion rat thikn hmial b ration dpth, h, a tl turn, rformnt dtrm layr por dtant trutur from por lor iz dtribution dg a (atr-to-mnt rak lngth a. ratio, am mnt gmnt hmial ha a lngth ompoition, l qual to SF ontnt, dpth urg ubjtd tim mthod, to xtrnal tmpratur, ndg mix momnt additiv, M. W t.). aum In litratur middl variou ro-tion formulation an an onidrd found to a dri rprntativ orption mhanial orm havior normal onrt hol (Xi t lmnt, al. 1994). Hovr, all mhanial prnt nonlariti papr mi-mpirial du to rakg xprion tnion, propod tl yildg by Norlg ruhg Mjornll (1997) onrt adoptd omprion, au ar lo- it alizd xpliitly th aount tion, for hra volution outid hydration part xhibit ration an lati SF rpon. ontnt. Th orption orm rad 1 ( h,, ) G (, ) K (, ) Figur 1. Shm rford onrt lmnt. (4) hr firt trm (gl orm) rprnt 2.1 Contitutiv Modl phyially bound (adord) atr ond trm mhanial (apillary orm) rpon rprnt onrt tnion apillary drid atr. Th by xprion Cohiv valid Crak only Modl for lo (Hillrborg ontnt t al. SF. 1976, Carptri offiint 1985). G 1 rprnt In partiular, amount a larlati atr pr tr tra unit volum rlationhip hld gl aumd por at for 100% undamagd rlativ humidity, pha, hra it an a tng xprd tr rak (Norlg idth Mjornll rlationhip 1997) a dri pro zon up to t ritial opng,, rahd (Fig. 2). r tng funtion, σ f(), onidrd a a matrial G (, proprty, ) k a + k ll a ritial valu (5) 1 vg vg t rak opng,, fratur nrgy, G r F. hap hr k f() may vary from lar to bilar or vg k vg ar matrial paramtr. From vn maximum mor amount ompliatd atr rlationhip pr unit volum dpndg an on fill haratrti all por (both apillary onidrd por matrial gl por), on analyzd an alulat problm. K 1 a on Rntly, obta a gnral polynomial rlationhip ith frational or tgr por ha n propod to xpr all knon ohiv rak 10 la for onrt (Karihaloo & g Xiao 2008). K (, ) G g 1 1 Figur 2. Cohiv Crak Modl: (a) lar-lati σε la; (b) pot-pak tng σ rlationhip. Not, at arly ag, hmial ration aoiatd ith mnt hydration SF ration A far a modlg onrt ruhg failur ar xormi, tmpratur fild not uniform onrnd, Ovrlappg Crak Modl trodud for non-adiabati ytm vn if nvironmntal by Carptri t al. (2007, 2009) adoptd. Aordg to uh an approah, trongly onfirmd by x- tmpratur ontant. Hat ondution an drid onrt, at lat for tmpratur not primntal rult (van Mir 1984, Jann & Shah xdg 100 C (Bažant & Kaplan 1996), by 1997), drivd from pionrg ork by Fourir la, hih rad Hillrborg (1990), lati dformation pot-pak rgim drid by a fititiou trpntration matrial, hil rmag part (7) q λ T pimn undrgo an lati unloadg. A pair hr ontitutiv q la hat flux, trodud, T lo abolut analogy tmpratur, ith Cohiv λ Crak hat Modl: ondutivity; a tr-tra th rlationhip until ompriv trngth 1 (6) matrial paramtr k vg k vg g 1 an alibratd by fittg xprimntal data rlvant to fr (vaporabl) atr ontnt onrt at variou ag (Di Luzio & Cuat 2009b). 2.2 Tmpratur volution Prodg FraMCoS-7, May 23-28, 2010

3 ahivd (Fig. 3a), a tr-dplamnt (ovrlappg) rlationhip dribg phnomnon onrt ruhg (Fig. 3b). lattr la, uually aumd a a lar drag funtion, dri ho tr damagd matrial dra from it maximum valu don to zro a fititiou trpntration ra from zro to ritial valu,. It orth notg ruh- r g nrgy, G C, hih a dipatd urfa nrgy, dfd a ara lo pot-pak tng urv Figur 3b, an aumd a a tru matrial proprty, it almot dpndnt trutural iz. An mpirial quation for alulatg ruhg nrgy ha n rntly propod by Suzuki t al. (2006), takg to aount latral onfmnt xrtd by tirrup. (1994) Brkr J D ( t al. h, T ) (1999) h to modl a ohiv rak pla rford onrt am. mid-pan ro-tion proportionality offiint onidrd D(h,T) lmnt ubdividd motur to prmability n nod, hr ohiv it a nonla ovrlappg tr rlativ ar rplad humidity by h quivalnt tmpratur nodal for, F i, hih & Najjar dpnd 1972). on orrpondg motur ma balan rlativ nodal dplamnt variation aordg tim to ohiv or ovrlappg volum pot-pak onrt la (atr (Fig. 4a). ontnt ) q atr ma follog rlationhip divrgn hold tn motur horizontal flux J for, F i, horizontal dplamnt, i : J { F} [ K ]{ } + { K }M (4) M t hr [K ] matrix atr ontnt offiint an flun for nodal dplamnt, vaporabl atr {K M } (apillary a xprd a vtor offiint vapor, flun adord for atr) applid non- momnt M, omputd (hmially a priori bound) by a fit atr lmnt n (Mil analy. Pantazopoulo & Mill 1995). It ra aum vaporabl atr a fu rlativ humidity, h, dgr hydration dgr ilia fum ration,, i.. ag-dpndnt orption/dorption (Norlg Mjonll 1997). Undr th aum by ubtitutg Equation 1 to Equati obta Figur 3. Ovrlappg Crak Modl: (a) lar-lati σε la; (b) pot-pak tng σ rlationhip. tl rformnt ontribution modld by a tr v. rak opng rlationhip obtad by man prlimary tudi arrid out on tration tn rforg bar urroundg onrt. On ba bond-lip rlationhip providd by Modl Cod 90 (1993), by impog quilibrium ompatibility ondition, it poibl to orrlat rformnt ration to rlativ lip at rak dg, hih orrpond to half rak opng dplamnt. Typially, obtad rlationhip haratrizd by an andg branh up to tl yildg, to hih orrpond a ritial valu rak opng, y. Aftr, tl ration narly ontant. 2.2 Numrial Algorithm numrial modl hr propod to dri fraturg havior lightly rford onrt am drivd from mor gnral algorithm trodud by Carptri t al. (2007, 2009) for modlg mhanial rpon all poibl ituation rangg from pla to ovr-rford onrt am. propod modl, fat, prmit to orrtly dri rlvant nonlariti. Similar approah hav n propod by Carptri (1985), Plana & Eli (1992), Bažant & Bl + ( D h) h & + & + (a) hr / lop orption/ orm (alo alld motur apa govrng quation (Equation 3) mut by appropriat boundary itial onditi rlation tn amount atr rlativ humidity alld orm if maurd ith rag humidity dorption orm th a. Ngltg ir diffrn (Xi t al. follog, orption orm ill (b) rfrn to both orption dorption By ay, if hytr Figur 4. Fit lmnt nod (a); for dtribution ith ohiv rak tnion, orm ruhg ould omprion takn to aount, rformnt log for rlation, (b) aro vaporabl mid-pan atr ro-tion. v rlativ humi to ud aordg to ign varia Equation (4) ontitut rlativity a humidity. lar algbrai hap ytm (n) quation orm (2n+1) for HPC unknon, flund {F}, {} by many p M. With rfrn pially to tho gnri flun ituation xtnt rportd Figur 4b, hmial n additional ration quation, an turn, - dtrm trodud by onidrg trutur por ontitutiv iz dtribution la for (atrratio, mnt omprion hmial for ompoition, r- SF onrt tnion formnt urg nod tim r ( Carptri mthod, t tmpratur, al. 2007, mix 2009 for mor dtail). t.). In lat litratur additional variou quation formulatio driv from trngth found to ritrion dri adoptd orption to govrn orm propagation onrt pro. (Xi W t an al. 1994). t ir Hovr, th for fititiou papr rak tip, mi-mpirial m, qual to xprion ulti- pro Norlg Mjornll (1997) adoptd b mat tnil for, F u, or for fititiou Prodg FraMCoS-7, May 23-28, 2010

4 ruhg J D ( h, T ) tip, hp, qual to ultimat ompriv (1) for, F. It important to not rakg ruhg proportionality phnomna ar offiint phyially D(h,T) dpndnt alld ah motur or. prmability A a rult, it ituation a nonlar hih funtion lor to on rlativ humidity to poibl h ondition tmpratur hon T (Bažant to tablh & Najjar 1972). prvailg motur phnomnon. ma balan rquir drivg paramtr variation pro tim poition atr ma pr fititiou volum tip onrt (atr onidrd ontnt tp ) ha rahd qual to unit limit divrgn rtan. Fally, motur at flux ah J tp, an omput rotation, ϑ: T J ϑ t { D } { } + D M (5) M hr {D atr } ontnt vtor an xprd offiint a flun for vaporabl nodal dplamnt, atr (apillary Datr, M atr o- um ffiint vapor, adord flun atr) for applid non-vaporabl ndg momnt, (hmially M. bound) iz-al atr fft n ar (Mill takn 1966, to aount Pantazopoulo by man & Mill rlationhip 1995). It proportionality raonabl to aum afft offiint vaporabl flun atr a ntrg funtion Eq. (4) rlativ (5). humidity, h, dgr hydration,, dgr ilia fum ration,, i.. (h,, ) ag-dpndnt orption/dorption orm 3 (Norlg APPLATION Mjonll 1997). OF DIMENSIONAL Undr th aumption by ANALYSIS ubtitutg TO Equation LIGHTLY 1 to RC BEAMS Equation 2 on obta mot rlvant appliation Dimnional Analy Solid Mhani hav onrnd omplt + omplt ( D h) phyial & + imilarity & + & n trngth (3) h toughn dordrd matrial (Carptri 1980, 1981b, 1984, Phatak & Dhond 2003, Phatak & hr Dhp / 2005), lop a ll a orption/dorption tudy omplt orm lf-imilarity (alo alld motur fatigu apaity). rak groth (Barnblatt govrng quation & Botva (Equation 1980, Ciavarlla 3) mut t al. ompltd 2008). by Whn appropriat flxural boundary havior itial rford ondition. onrt am rlation tudid, tn aordg to amount numrial vaporabl modl propod atr rlativ prviou humidity tion, alld funtional adorption rlationhip orm among if maurd quantiti ith rag haratriz rlativity phnomnon humidity dorption follog: orm oppoit a. Ngltg ir diffrn (Xi t al. 1994), M follog, Φ (σ u, G F, orption σ, G C, E orm, σ y, ρ t, h; ill b/h, l/h, ud ϑ) ith (6) rfrn to both orption dorption ondition. hr By M ay, rtant if hytr ndg momnt, σ u motur, G F, σ, Gorm C, E ar, ould rptivly, takn to tnil aount, trngth, to diffrnt fratur rlation, nrgy, vaporabl ompriv atr v rlativ trngth, humidity, ruhg mut nrgy, ud aordg lati to modulu ign onrt, variation σ y ρrlativity t rprnt humidity. yildg trngth hap prntag orption orm tnil for HPC rformnt, flund h by many haratrti paramtr, iz pially tho body, b/h flun l/h df xtnt gomtry rat hmial ampl ration aordg, to Figur turn, 1, dtrm ϑ loal por rotation trutur por lmnt. iz dtribution S (atr-to-mnt ar trtd ratio, mhanial mnt hmial rpon ompoition, lightly rford SF ontnt, onrt urg am, tim mthod, t variabl tmpratur, an mix rdud additiv, a follo: t.). In litratur variou formulation an found to dri orption orm normal onrt M (Xi Φ (σ u, G t F, al. E, 1994). σ y, ρ t, h; Hovr, ϑ) prnt (7) papr mi-mpirial xprion propod by Norlg Mjornll (1997) adoptd au it hr xpliitly aount paramtr for dribg volution havior hydration onrt ration omprion, SF ontnt. σ Th Gorption C, ar not orm xpliitly rad onidrd, ruhg failur not volvd failur mhanm. On or h, only am dpth, h, onidrd if gomt- rial ratio ampl, b/h 1L/h, ar aumd ( h,, ) G (, ) 1 + to ontant. 1 appliation Bukgham Π- orm (Bukgham 1915) for phyial imilitud 1 (4) al modlg prmit to mimiz dimnion pa primary 10variabl ( g by ) h ombg K (, ) 1 1 m to dimnionl 1 group, a follo: M σ h σ h u y E h hr firt Φ trm (gl, ρ orm), ϑ rprnt (8) t h E E E E phyially GF bound G(adord) G G F Fatr F ond trm (apillary orm) rprnt apillary if atr. h Th G xprion E ar aumd valid a only for dimnionally lo ontnt F dpndnt SF. offiint variabl. GIt 1 rprnt orth notg amount formr atr pr paramtr unit volum rprntativ hld gl por iz-al at 100% rlativ pimn, humidity, hra it an lattr xprd a (Norlg matrial proprty. Mjornll 1997) In partiular, a trm G E orrpond to onrt fratur toughn, K, a- F ordg to fundamntal rlationhip propod by G (, ) k + k (5) Ir 1 (1957). vga a onqun, vg dimnionl funtional rlation for propod modl om: hr k vg k vg ar matrial paramtr. From maximum M % Φ ( amount, N, ϑ ) atr pr unit volum (9) an 2 P n fill all por (both apillary por gl por), on an alulat K hr: 1 a on obta K g G σ h u K (, ) 1 g 1 1 (10) (6) 1 2 matrial σ h paramtr k vg k vg y N ρ (11) Palibratd t by fittg xprimntal data rlvant to K fr (vaporabl) atr ontnt onrt at variou ag (Di Luzio & Cuat 2009b). ar govrng nondimnional numr, M % nondimnional ndg momnt, ϑ n normalizd 2.2 Tmpratur loal rotation. volution It orth notg Eq. (10), Not hih, at rprnt arly ag, tr brittln hmial ration numr trodud aoiatd ith by Carptri mnt hydration (1981b, 1982), SF lud ration only ar xormi, mhanial tmpratur proprti fild matrix not uniform iz for non-adiabati al ytm problm. vn On if or nvironmntal h, Eq. (11) tmpratur rprnt ontant. rformnt Hat ondution brittln numr an trodud drid aga onrt, by Carptri at lat (1984) for tmpratur ontxtually not ith xdg Bridgd 100 C Crak (Bažant Modl, & Kaplan ontag 1996), by proprti Fourir la, hih rformnt. rad A a rult Dimnional Analy, aordg to Eq. (9), trutural q λ T rpon, trm M % vru ϑ n, a (7) funtion N P. g 1 an hr q hat flux, T abolut tmpratur, λ hat ondutivity; th Prodg FraMCoS-7, May 23-28, 2010

5 4 COMPARISON OF NUMERAL PREDTIONS AND EXPERIMENTAL RESULTS xprimntal vtigation hr onidrd a arrid out Matrial Strutur Laboratory Dpartmnt Strutural Gothnial Engrg Politnio di Toro, by Boo t al. (1990) ith ma purpo vrifyg xtn iz fft trutural havior. Thirty thr-pot-ndg tt r prformd on rford high-trngth onrt am. Thr diffrnt iz-al hav n onidrd, haratrizd by a dpth, h, qual to 0.1, m, a ontant thikn, b, qual to 0.15 m. pan to dpth ratio a t qual to 6. onrt proprti r follog: yldrial ompriv trngth σ 76 MPa, ant lati modulu E MPa, fratur nrgy G F N/mm. A rgard rformnt, nomal diamtr adoptd tl bar r 4, 5, 8 10 mm. orrpondg J D ( h, T ) hyild trngth r: 637, 569, MPa. It orth notg prntag r not proportionality aumd ontant offiint by varyg pimn motur iz. On prmability ontrary, y it r a nonla D(h,T) varid kpg ontant rlativ brittln humidity numr h tmpratur N P, dfd Eq. (1). & Najjar 1972). motur ma balan Som numrial variation imulation tim ompard atr to ma orrpondg volum xprimntal onrt rult, (atr ontnt trm ) q applid load v. divrgn mid-pan dfltion motur urv, flux J ar hon Figur 5 to 7. In numrial imulation RC lmnt Figur 1 aumd to rprntativ mid-pan J portion am t ubjtd to thr-pot-ndg tt. Hn, mid-pan dfltion obtad atr ontnt a um an xprd loalizd rotation ontribution a vaporabl givn atr by Eq. (5) (apillary a lati ontribution, vapor, aordg adord to atr) follog non- xprion: (hmially bound) atr n (Mil Pantazopoulo & Mill 1995). It ra 3 ϑlaum 1 Pl vaporabl atr a fu δ δ + δ + (12) lo l 4rlativ 48 E Jhumidity, h, dgr hydration dgr ilia fum ration,, i.. ag-dpndnt orption/dorption (Norlg Mjonll 1997). Undr th aum by ubtitutg Equation 1 to Equati obta + ( D h) h & + & + hr / lop orption/ orm (alo alld motur apa govrng quation (Equation 3) mut by appropriat boundary itial onditi rlation tn amount atr rlativ humidity alld orm if maurd ith rag humidity dorption orm th a. Ngltg ir diffrn (Xi t al. follog, orption orm ill rfrn to both orption dorption By ay, if hytr orm ould takn to aount, to rlation, vaporabl atr v rlativ humi ud aordg to ign varia rlativity humidity. hap orm for HPC flund by many p pially tho flun xtnt hmial ration, turn, dtrm trutur por iz dtribution (atrratio, mnt hmial ompoition, SF urg tim mthod, tmpratur, mix t.). In litratur variou formulatio found to dri orption orm onrt (Xi t al. 1994). Hovr, th papr mi-mpirial xprion pro Norlg Mjornll (1997) adoptd b (a) ρ % (N P 0.10) (b) ρ % (N P 0.26) () ρ % (N P 0.53) Figur 5. Comparon tn numrial xprimntal applid load v. mid-pan dfltion urv for am dpth h 0.2 m. (a) ρ % (N P 0.10) (b) ρ % (N P 0.26) () ρ % (N P 0.53) Figur 6. Comparon tn numrial xprimntal applid load v. mid-pan dfltion urv for am dpth h 0.4 m. (a) ρ % (N P 0.10) (b) ρ % (N P 0.26) () ρ % (N P 0.53) Figur 7. Comparon tn numrial xprimntal applid load v. mid-pan dfltion urv for am dpth h 0.6 m. Prodg FraMCoS-7, May 23-28, 2010

6 hr J D ( l h, T ) ham pan, J momnt (1) rtia ro-tion. urv Figur 5 to 7 vidn proportionality a gnral good offiint agrmnt D(h,T) tn alld numrial motur prmability xprimntal rult, it a nonlar although funtion a lar tng rlativ la humidity ha n h trodud tmpratur T ohiv (Bažant modl & Najjar tad 1972). a mor motur omplx ma rlationhip. balan rquir variation tim atr ma pr unit volum onrt (atr ontnt ) qual to 5 divrgn NUMERAL motur RESULTS flux AND J DISCUSSION 5.1 Paramtri Analy J In h t tion, aordg to rult dimnional analy, dutil-to-brittl tranition trutural atr rpon ontnt analyzd an xprd on ba um to nondimnional vaporabl atr numr (apillary dfd by atr, Eq. atr (10) vapor, (11). adord yildg atr) trngth non-vaporabl lati modulu (hmially bound) tl rformnt, atr n (Mill a ll a 1966, it rlativ Pantazopoulo dtan & from Mill 1995). tnil It dg, raonabl /h, ar aumd, aum rptivly, vaporabl qual to atr 600 MPa, a funtion to N/mm rlativ 2, humidity, 0.10, for h, all dgr numrial hydration, imulation., Alo dgr am ilia idth, fum ration, b, aumd, i.. to (h, ontant, ) qual ag-dpndnt to 0.15 m. In orption/dorption th a, gomtrial (Norlg ratio b/h Mjonll no 1997). longr Undr a ontant, th aumption xprion orm by ubtitutg nondimnional Equation momnt 1 to om: Equation 2 on obta M M% 3 2 bh G E + h (13) F ( D h) & + & + & n (3) dimnionl momnt vru normalizd rotation hr urv / hon lop Figur 8 orption/dorption rprnt mhanial orm havior (alo alld RC motur am haratrizd apaity). by valu govrng Nquation P rangg (Equation from 0.003) up mut to 1.85, ompltd ith by appropriat In gnral, boundary obtad itial urv ondition. ar haratrizd by rlation a firt andg tn branh, amount du to vaporabl lati matrial atr havior rlativ humidity to itial alld tabl adorption rak propagation, orm if up maurd to pak ith rakg rag momnt, rlativity n humidity by a tng dorption vn orm nap-bak branh, oppoit du to an a. untabl Ngltg rak ir propagation. diffrn n, (Xi t dpndg al. 1994), on follog, rformnt orption amount, orm loadg ill pro ud ith an rturn rfrn to to both tabl orption up to tl dorption yildg. Fally, ondition. mhanial By ay, rpon if tnd hytr aymptotially to motur ultimat orm momnt, ould hih takn orrpond to aount, to to tl diffrnt plati flo rlation, vaporabl omplt atr donntion v rlativ humidity, onrt mut ro-tion. ud aordg diagram to ign Figur variation 8 vidn an rmnt rlativity humidity. load-arryg hap apaity orption a mor dutil orm for tabl HPC mhanial flund havior by many paramtr, lmnt pially by rag tho Nflun P. Mor prly, xtnt for rat onidrd hmial valu ration,, global rpon turn, dtrm om por ta- bl, trutur i.., por aymptoti iz dtribution valu (atr-to-mnt ultimat momnt ratio, mnt om hmial highr ompoition, than pak SF rakg ontnt, momnt, urg tim hn mthod, N P gratr tmpratur, than 0.23 mix additiv, approximatly. t.). In litratur variou formulation an found to diagram dri Figur orption 9 ho orm dimnionl normal momnt onrt v. (Xi normalizd t al. 1994). rotation Hovr, rlationhip for prnt diffrnt papr valu mi-mpirial tr xprion brittln propod numr, by, rangg Norlg from Mjornll 0.43 up (1997) to 2.63, ith adoptd N P au W it xpliitly aount for volution hydration ration SF ontnt. Th orption orm rad 1 ( h,, ) G (, ) K (, ) (4) hr firt trm (gl orm) rprnt phyially bound (adord) atr ond Figur trm 8. (apillary Numrial orm) dimnionl rprnt momnt v. normalizd apillary rotation atr. urv Th by xprion varyg N P valid for only for lo ontnt SF. offiint G 1 rprnt amount an atr orv pr unit volum valu hld pak gl por rakg at 100% momnt, rlativ dirtly humidity, orrlatd it an to xprd onrt (Norlg tnil trngth, Mjornll 1997) a drag funtion. On or h, pot-pak branh ollap to am aymptoti valu ultimat ndg momnt, G ( hih, ) k a funtion + k tl ontnt yild (5) 1 vg vg trngth. From a global pot vi, a tranition mhanial havior from a vry dutil tabl rpon to a brittl untabl on, ith hr k vg k vg ar matrial paramtr. From maximum amount atr pr unit volum an apparan a nap-bak tability, vidnd fill all por (both apillary por gl por), on by drag from 2.63 to an alulat K 1 a on obta K (, ) 1 g G g 1 1 (6) matrial paramtr k vg k vg g 1 an alibratd by fittg xprimntal data rlvant to fr (vaporabl) atr ontnt onrt at variou ag (Di Luzio & Cuat 2009b). 2.2 Tmpratur volution Not, at arly ag, hmial ration Figur aoiatd 9. Numrial ith mnt dimnionl hydration momnt v. normalizd SF ration rotation ar xormi, urv by varyg tmpratur for N P fild not uniform for non-adiabati ytm vn if nvironmntal 5.2 tmpratur Mimum Rformnt ontant. Hat ondution an drid onrt, at lat for tmpratur not In xdg th tion, 100 C a (Bažant n rlationhip & Kaplan tn 1996), by mimum Fourir la, rformnt hih rad mhanial gomtrial paramtr propod. To th aim, four q diffrnt λ T valu am dpth, h 0.1, 0.2, (7) m, fiv diffrnt valu onrt grad, σ 16, 30, 40, 65, 76 MPa, hav n hr q onidrd. All hat or flux, mhanial T proprti abolut tmpratur, onrt, a, for λ tan, hat ondutivity; tnil trngth th Prodg FraMCoS-7, May 23-28, 2010

7 fratur nrgy, hav n valuatd aordg to rlationhip providd by Modl Cod 90. A rgard tl rformnt, a yild trngth σy 600 MPa, an lati modulu E MPa hav n aumd. For ah onidrd am, vral imulation hav n arrid out by varyg tl prntag, ordr to fd mimum rformnt amount. In partiular, uh a valu dtrmd hn pak rakg load, P r, qual to ultimat load, P u, a hon Figur 10. valu N P th a N PC, it onta ritial valu rformnt amount orrpondg to ritial ondition ar hon Figur 11. obtad trnd an drid ith a vry good approximation (goodn fit r ) by follog hyprboli urv: Figur 10. Dfition mimum rformnt. Figur 11. Bt-fit rlationhip numrial rult (not filld ymbol) tn N PC. Filld- ymbol rfr to xprimntal rult from Boo t al. (1990). N PC N PC r (14) By ubtitutg Eq. (10) (11) to (14), follog rlationhip tn mimum rformnt mhanial gomtrial proprti am obtad: J D ( h, T ) h σ h K σ K u u m K σ h σ h y y ρ. (15) proportionality offiint D(h,T) motur prmability it a nonla mimum rformnt prntag v. am rlativ humidity h tmpratur dpth urv aordg to diffrnt modl availabl litratur ar ompard to Eq. (15) & Najjar 1972). motur ma balan variation tim atr ma Figur 12. All urv rfr to f k MPa volum onrt (atr ontnt ) q σy 450 MPa. rlativ dtan tn divrgn motur flux J rformnt layr tnil ro-tion dg ha n kpt ontant qual to 0.10 at any al. Mot urv larly J highlight a dra mimum tl t amount a am dpth ra. propod urv vry lo to by Ruiz t al. (1999), but Eq. atr (15) ontnt air an pratial xprd a appliability. n vaporabl propod formula atr (apillary alo a ompard to prription vapor, adord dign atr) od non- Figur 13. (hmially bound) atr n (Mil Pantazopoulo & Mill 1995). It ra aum vaporabl atr a fu 6 CONCLUSIONS rlativ humidity, h, dgr hydration dgr ilia fum ration,, i.. appliation Dimnional ag-dpndnt Analy orption/dorption to flxural havior (Norlg lightly rford Mjonll 1997). onrt Undr am th aum prmit ovrall by mhanial ubtitutg rpon Equation to 1 to govrnd only by to Equati obta nondimnional numr, N P. In partiular, phyial imilitud nondimnional momnt vru normalizd rotation dia- + ( D h) & + & + gram prditd a brittln h numr ar kpt ontant, pritably ud to lt mimum rformnt amount. A n formula, fat, obtad hr a t-fittg / numrial lop rult orption/ trm N orm PC. In (alo Eq. (15), alld ma motur mhanial gomtrial apa govrng paramtr quation (Equation afftg 3) mut phnomnon g by appropriat tudid ar boundary orrtly takn itial to onditi aount by man rlation Nonlar tn Fratur amount Mhani bad modl atr adoptd rlativ for humidity numrial alld imulation. A a orm rult, mimum if maurd rformnt ith rag amount an rag humidity funtion dorption onrt orm tnil trngth th a. toughn, Ngltg hra ir it diffrn dra a (Xi t al. tl yildg trngth follog, orption am orm dpth ra. fft ill rfrn bond-lip to both orption tn onrt dorption rforg bar By potd ay, out if by prviou hytr tudi 15,16 drgardd, orm, ould ith rfrn takn to to Figur aount, to 10, it flun rlation, valu vaporabl applid atr load v rlativ humi mid-pan dfltion ud orrpondg aordg to to tl ign yildg, P y, hra varia rlativity P r P humidity. u do not onidrably hap hang. orm for HPC flund by many p A far a pially iz-al tho fft ar flun onrnd, xtnt it ha to notd hmial prn ration ohiv, log turn, dtrm tr dtrm trutur a variation por iz ρ m ith dtribution am (atrratio, iz l pronound than mnt hmial prditd ompoition, by SF Bridgd Crak Modl. urg An tim xponnt mthod, qual tmpratur, to 0.15, mix fat, obtad t.). tad In 0.50, litratur typial variou Lar formulatio Elati Fratur found Mhani. to dri Suh a diffrn orption orm larly hon onrt Figur 12, (Xi hr t al. 1994). urv Hovr, from th Boo & Carptri papr (1992) mi-mpirial ompard to xprion n pro propoal. Norlg Mjornll (1997) adoptd b Prodg FraMCoS-7, May 23-28, 2010

8 J ) D ( h, T h (1) proportionality offiint D(h,T) alld motur prmability it a nonlar funtion rlativ humidity h tmpratur T (Bažant & Najjar 1972). motur ma balan rquir variation tim atr ma pr unit volum onrt (atr ontnt ) qual to divrgn motur flux J t J atr ontnt an xprd a um vaporabl atr (apillary atr, atr vapor, adord atr) non-vaporabl Figur 12. Mimum rformnt v. am dpth aordg to (hmially variou modl. bound) atr n (Mill 1966, Pantazopoulo & Mill 1995). It raonabl to aum vaporabl atr a funtion rlativ humidity, h, dgr hydration,, dgr ilia fum ration,, i.. (h,, ) ag-dpndnt orption/dorption orm (Norlg Mjonll 1997). Undr th aumption by ubtitutg Equation 1 to Equation 2 on obta + ( D h) n (3) h & + & + & hr / lop orption/dorption orm (alo alld motur apaity). govrng quation (Equation 3) mut ompltd by appropriat boundary itial ondition. Figur 13. rlation Mimum tn rformnt amount v. am dpth vaporabl aordg to variou dign od. atr rlativ humidity alld adorption orm if maurd ith rag rlativity humidity dorption orm oppoit REFERENCES a. Ngltg ir diffrn (Xi t al. 1994), ACI-318 follog, orption orm ill ud ith Buildg Cod Rquirmnt for Rford rfrn Conrt. to Dtroit. both orption dorption ondition. Appa By Rao, G., ay, Aravd, if J. & hytr Elighaun, R Evaluation motur orm mimum ould flx. rformnt takn to aount, RC am to ug diffrnt fititiou rak vaporabl approah. atr Journal v rlativ Strutural humidity, Engrg mut rlation, (Madra) 34(4): ud aordg to ign variation Baluh, M., Azad, A. & Ahmai, W Fratur mhani appliation humidity. to rford onrt hap mmr orption flxur. rlativity orm In A. Carptri for HPC (d), flund Appliation by many Fratur paramtr, Mhani pially to Rford tho Conrt: flun xtnt London: Elvir rat Applid Sin. ration, turn, dtrm por hmial Barnblatt, trutur G.I. por & Botva, iz dtribution L.R (atr-to-mnt Inomplt lfimilarity fatigu lar rang fatigu rak ratio, groth. mnt Fatigu hmial Engrg ompoition, Matrial SF Strutur ontnt, urg 3(3): tim mthod, tmpratur, mix additiv, Bažant, t.). In Z.P. & litratur Bl, S. variou Smard-tip formulation uprpoition an found mthod to for dri ohiv fratur orption ith orm rat fft normal rp, onrt Intrnational (Xi t Journal al. 1994). Fratur Hovr, 65(3): prnt Brkr, R., Hnrikn, M.S., Chrtnn, F.A. & Hh, G. papr mi-mpirial xprion propod by Siz fft on ndg haviour rford Norlg onrt Mjornll am. In (1997) A. Carptri adoptd (d), Mimum au R- it xpliitly formnt aount Conrt for Mmr: volution Oxford hydration (UK): ration Elvir Sin SF Ltd. ontnt. Th orption orm BSI, rad BS Cod prati for trutural u onrt. London. Bukgham, E Modl xprimnt form mpirial quation. ASME Tranation 37: Boo, C. & Carptri, A Fratur 1 mhani valuation,, mimum ) G ( ( h rformnt, ) onrt trutur. In A. Carptri (d), Appliation 10 Fratur ( g Mhani ) h to Rford Conrt: London: Elvir Applid Si- (4) 1 n. 10( g Boo, C., Carptri, A. & Drnardi, P.G. ) h Mimum K (, ) 1 1 rformnt 1high-trngth onrt. Journal Strutural Engrg 116: Carptri, A Siz fft fratur toughn ttg: A hr dimnional firt analy trm approah. (gl orm) In G.C. Sih, rprnt & M. Mirabil (d), Prodg an Intrnational Confrn on Analytial bound Exprimntal (adord) Fratur atr Mhani:_ ond phyially trm Alphn (apillary an dn Rijn. orm) Sijthf & rprnt Noordhf. apillary Carptri, atr. Th A. xprion 1981a. A fratur valid mhani only for modl lo for ontnt rford SF. onrt offiint ollap. GIn 1 Prodg rprnt a IABSE amount Colloquium: pr unit volum Dlft: Dlft hld Univrity Pr. gl por at 100% atr Carptri, A. 1981b. Stati nrgti fratur paramtr rlativ humidity, it an xprd (Norlg for rok onrt. Matrial & Strutur 14(81): Mjornll ) a Carptri, A Noth nitivity fratur ttg aggrgativ matrial. Engrg Fratur Mhani 16: G ( , ) k + k (5) 1 vg vg Carptri, A Stability fraturg pro RC am. Journal Strutural Engrg 110(3): Carptri, hr k vg A k vg Intrprtation ar matrial paramtr. Griffith tability From a maximum a bifuration amount global atr quilibrium. pr unit volum In S.P. Shah (d), an fill Appliation all por (both Fratur apillary Mhani por to Cmntitiou gl por), Compoit: Dordrht: Martu Nijhf Publhr. on an alulat K Carptri, A., Corrado, 1 a on obta M., Paggi, M. & Mani, G Cohiv vru ovrlappg rak modl for a iz fft analy RC lmnt ndg. In A. Carptri, P. g Gambarova, G. Frro, & G. Plizzari (d), 1 Fratur G 1 Mhani Conrt Strutur: London: Taylor 0 1 (6) K (& Fran., ) 1 Carptri, A., Corrado, M., Paggi, M. g & Mani, G N modl for analy 1 iz-al 1 fft on dutility rford onrt lmnt ndg. Journal Engrg matrial Mhani paramtr 135(3): k vg k vg g 1 an CEN TC/ Eurood 2: Dign Conrt Strutur, Part 1-1: Gnral Rul Rul for Buildg. alibratd by fittg xprimntal data rlvant to fr Brul. (vaporabl) atr ontnt onrt at Ciavarlla, variou ag M., Paggi, (Di Luzio M. && Carptri, Cuat A. 2009b) On, no on, on hundrd thou rak propagation la: a gnralizd Barnblatt Botva dimnional analy approah Tmpratur to fatigu rak volution groth. Journal Mhani 2.2 Phyi Solid 56(12): Comité Not, Euro-Intrnational at arly ag, du Béton hmial CEB-FIP ration Modl aoiatd Cod ith In Bullt mnt No. hydration 213/214. Thoma SF Tlford ration Ltd, ar Lauann. xormi, tmpratur fild not uniform Fantilli, for non-adiabati A.P., Frrtti, ytm D., Iori, I. vn & Valli, if P. nvironmntal Bhaviour tmpratur R/C lmnt ontant. ndg Hat tnion: ondution problm an mimum rformnt ratio. In A. Carptri (d), Mimum Rformnt Conrt Mmr: Oxford drid onrt, at lat for tmpratur not xdg (UK): Elvir 100 C Sin (Bažant Ltd. & Kaplan 1996), by Grtl, Fourir W.H., la, Dy, hih P.P., rad Praad, N.N.V., Rahulkumar, P. & Xi, M Crak groth flxural mmr-a fratur λ T mhani approah. ACI Strutural Journal: 89(6) q (7) Ghali, A., Favr, R. & Elbadry, M Conrt trutur: hr tr q dformation, hat J. flux, W. Arromith T Ltd., abolut Brtol, tmpratur, UK, λ hat ondutivity; th Prodg FraMCoS-7, May 23-28, 2010

9 Hillrborg, A Fratur mhani onpt applid to momnt apaity rotational apaity rford onrt am. Engrg Fratur Mhani 35: Hillrborg, A., Modr, M. & Ptron, P.E Analy rak formation rak groth onrt by man fratur mhani fit lmnt. Cmnt Conrt Rarh 6: Jann, D.C. & Shah, S.P Efft lngth on ompriv tra tng onrt. Journal Engrg Mhani 123(1): Karihaloo, B.L. & Xiao, Q.Z Aymptoti fild at tip a ohiv rak. Intrnational Journal Fratur 150(1-2): Norgian Stard, NS 3473 E Conrt Strutur, Dign Rul. (Englh Tranlation). Norgian Counil for Buildg Stardization, Olo, Noray. Plana, J. & Eli, M Aymptoti analy a ohiv rak: 1. ortial bakground. Intrnational Journal Fratur 55: Phatak, D.R. & Dhp, N.P Prdition 28 day ompriv trngth 53-grad mnt ug dimnional analy. Journal J D ( h, Matrial T ) h Civil Engrg 17(6): Phatak, D.R. & Dhond, H.B Dimnional analy rford onrt am proportionality ubjtd to pur torion. offiint Journal Strutural Engrg motur prmability 129(11): it a nonla D(h,T) Ruiz, G., Eli, M. & Plana, rlativ J humidity Siz fft h tmpratur bondlip dpndn & lightly Najjar rford 1972). onrt motur am. ma In A. balan Carptri (d), Mimum Rformnt Conrt Mmr: Oxford (UK): Elvir Sin Ltd. variation tim atr ma Stard Autralia, volum AS onrt Autralian (atr Stard ontnt for ) q Conrt Strutur. divrgn Sydny. motur flux J Suzuki, M., Akiyama, M., Matuzaki, H. & Dang, T.H Conntri loadg tt RC olumn ith normal- high-trngth matrial avragd J tr-tra modl for onfd onrt onidrg t ompriv fratur nrgy. In Prodg 2nd fib Congr, Napl, Italy, Jun 5-8, 2006, (CD-ROM). atr ontnt an xprd a van Mir, J.G.M Stra-tng vaporabl atr Conrt (apillary undr a Multiaxial Loadg vapor, Condition. adord PhD, atr) Edhovn non- Univrity Thnology. (hmially bound) atr n (Mil Pantazopoulo & Mill 1995). It ra aum vaporabl atr a fu rlativ humidity, h, dgr hydration dgr ilia fum ration,, i.. ag-dpndnt orption/dorption (Norlg Mjonll 1997). Undr th aum by ubtitutg Equation 1 to Equati obta + ( D h) h & + & + hr / lop orption/ orm (alo alld motur apa govrng quation (Equation 3) mut by appropriat boundary itial onditi rlation tn amount atr rlativ humidity alld orm if maurd ith rag humidity dorption orm th a. Ngltg ir diffrn (Xi t al. follog, orption orm ill rfrn to both orption dorption By ay, if hytr orm ould takn to aount, to rlation, vaporabl atr v rlativ humi ud aordg to ign varia rlativity humidity. hap orm for HPC flund by many p pially tho flun xtnt hmial ration, turn, dtrm trutur por iz dtribution (atrratio, mnt hmial ompoition, SF urg tim mthod, tmpratur, mix t.). In litratur variou formulatio found to dri orption orm onrt (Xi t al. 1994). Hovr, th papr mi-mpirial xprion pro Norlg Mjornll (1997) adoptd b Prodg FraMCoS-7, May 23-28, 2010