Fracture simulation of fiber reinforced concrete by visco-elasto-plastic suspension element method

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1 Fratur Mhani Conrt Conrt Strutur - High Prforman, Fibr Rinford Conrt, Spial Loading Strutural Appliation- B. H. Oh, t al. (d) 200 Kora Conrt Intitut, ISBN Fratur imulation fibr rinford onrt by vio-lato-plati upnion lmnt mthod H. Takahi & T. yauo Dpt. Arhittur, Faulty Sin Thnology, Mijo Univ., apan ABSTRACT: In thi papr, a imulation dirt tnion tt bnding tt fibr rinford onrt i arrid out by VEPSEM ha bn applid for th fratur imulation plain onrt undr variou ondition. Th fibr in fibr rinford onrt i imulatd by dynami lato-plati lmnt btn nodal point. A th rult, th analytial rult omparativly xpr th xprimntal rult in hih th maximum load bom high th dnding portion aftr maximum load bom dutil. INTRODUCTION Fibr rinford onrt i addd fibr to improv th brittl fratur bhavior plain onrt. It i tudid by many rarhr fibr rinford onrt that ha dutil fratur bhavior i idly ud in onrt ontrution ork. Th fibr ud in fibr rinford onrt ar tl fibr, gla fibr, arbon fibr o on, th othr kind fibr hav bn dvlopd. Hovr, in ordr to grap th fratur bhavior fibr rinford onrt ud a n fibr, many xprimntal xamination by tnil tt, bnding tt o on hould b arrid out atually. Bau th rlationhip btn th fratur bhavior fibr rinford onrt th proprty fibr i not narily grapd. Thrfor, th thortial tudy on thi rlationhip btn fratur bhavior fibr rinford onrt fibr on th mhanim rinforing by fibr ha bn arrid out. Gnrally, th mhanim boming dutil by fibr i alld bridg fft, bau fibr in onrt ontrut bridg intrnal rak, many modl on th bai thi mhanim ar propod. Hovr, th bridg fft fibr i inflund by mhanial proprti onrt fibr bond proprty btn onrt fibr. Th modl that an b applid for all fibr rinford onrt i not propod. Th author hav propod a vio-lato-plati upnion lmnt mthod (VEPSEM). Thi analytial mthod ha to haratriti. On i th u a impl non-ontinuum modl onitd aggrgat matrix. Th othr i th dynami analyi by th olution th quation motion. Th author hav arrid out th imulation onrt undr variou ondition, invtigatd th appliability thi analyi. In thi papr, in ordr to invtigat th fratur bhavior fibr rinford onrt, a imulation dirt tnion tt bnding tt fibr rinford onrt i arrid out by VEPSEM. Th fibr i imulatd by dynami lato-plati lmnt btn nodal point. In a fratur matrix btn nodal point, thi fibr lmnt ontrut bridg. Th ontitutiv la thi fibr lmnt onit th yild tr Young modulu. Th fft numbr, lngth, dirtion, yild tr Young modulu fibr lmnt on th fratur bhavior fibr rinford onrt i invtigatd. 2 ANALYTICAL METHOD Th vio-lato-plati upnion lmnt mthod (VEPSEM) ha bn propod to imulat th fratur bhavior onrt by th author. Th main haratriti thi analytial mthod ar to u a impl non-ontinuum modl onitd aggrgat matrix, to b abl to arry out th dynami analyi by uing th quation motion. Th dtail thi analytial mthod r rportd in th prviou papr. In thi haptr, th improvmnt fibr lmnt for analyi fibr rinford onrt i xplaind in dtail. Figur ho th onpt fibr lmnt. A hon in thi figur, th prviou analytial modl i truturd nodal point mortar lmnt. If th tr mortar lmnt rahd th trngth, th mortar lmnt i diappard. Thi diapparan mortar lmnt xpr th rak mortar lmnt. Th fibr lmnt ith dynami lato-plati proprty i addd btn nodal point for thi analytial modl. In a th xitn thi fibr lmnt, th tr i tranmittd btn nodal point by th fibr lmnt hn th mortar lmnt i diappard. Th bridg fft fibr i imulatd by adding thi fibr lmnt.

2 ( Nodal h, T point h Mortar lmnt D Fibr lmnt () Th proportionality fiint D(h,T) i alld moitur prmability it i a nonlinar funtion th rlativ humidity h tmpratur T (Bažant & Najjar 972). Th moitur ma balan rquir (a) Strutur (b) Crak () Bridg fft by that analytial th variation modl in tim th atr ma fibr pr lmnt unit volum onrt (atr ontnt ) b qual to th Figur divrgn. Conpt th lmnt. moitur flux t Str σ σy (2) Th atr ontnt an b xprd a th um th vaporabl atr (apillary atr, atr vapor, adorbd Eatr) th non-vaporabl (hmially bound) atr n (Mill 966, εy Pantazopoulo & Mill 995). Strain It i ε raonabl to Figur aum 2. Contitutiv that th vaporabl la fibr lmnt. 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 2 on 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 Figur atr B rlativ modl (For humidity dirt tnion i alld tt). 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 HPC i inflund by many paramtr, (a) p 2% (b) p 3% () p 4% pially tho that influn xtnt rat th Figur hmial 4. Analytial ration modl, (Efft in turn, volum dtrmin fration por fibr p). trutur por iz ditribution (atr-to-mnt ratio, mnt hmial ompoition, SF ontnt, uring Th tim ontitutiv mthod, la tmpratur, fibr lmnt mix i additiv, hon in Figur t.). In 2. th Th litratur impl bi-linar variou tr-train formulation urv an b i ud found for to th drib ontitutiv th orption la iothrm fibr lmnt. normal Th ontitutiv onrt (Xi la t al. i onitd 994). Hovr, yild in tr th σy prnt Young' papr th modulu mi-mpirial E. In thi xprion analyi, th propod yild for by Norling tiffn Mjornll ar atually (997) nary. i adoptd Th bau input data it ar obtaind by multiplying yild tr σy 300N/mm 2 & xpliitly Young' aount modulu for E 20kN/mm th volution 2 gnral hydration tl by ration th tion ara SF ontnt. fibr rptivly. Thi orption Th iothrm volum fration rad fibr p i rprntd by th volum fibr dividd by th volum analytial modl. Th volum fibr i multiplid th tion ara fibr by th lngth fibr. ( h,, ) G (, ) 0( g ) h 3 SIMULATION OF DIRECT TENSION T EST (4) In thi haptr, th rult 0( imulation g ) h dirt tnion tt ar hon. K (, ) Figur 3 ho th analytial modl dirt tnion tt onitd 440 nodal point. Th loading board hr i th movd firt to trm dirtion (gl iothrm) arro ign rprnt to imulat th dirt phyially tnion bound tt. (adorbd) Tabl ho atr th input th data ond mortar. trm (apillary iothrm) rprnt th apillary atr. Thi xprion i valid only for lo ontnt Tabl SF.. Th Input fiint data mortar. G rprnt th amount tan atr pr Ft unit volum E hld η in v th gl d por H at 00% D rlativ humidity, it an b xprd (Norling 0. Mjornll ) 3 a φ : Angl intrnal frition, Ft : Pur tnil trngth (MPa), E : Elati modulu (GPa), η : Vioity (MPa. ), v : Shar loading rat (mm/), d :Diamtr G (, ) k k nodal point (mm), H : (5) Hight pimn vg (mm), vg D : Width pimn (mm) hr k vg k vg ar Lf matrial m paramtr. p 4% From th maximum amount atr pr unit volum p 3% that an fill all por (both apillary por gl p por), 2% on an alulat K a on Plain onrt Tnil tr σ (MPa) K (, ) g G 0 g Tnil train ε (%) Th matrial paramtr k vg k vg g an Figur b alibratd 5. Tnil by tr-train fitting xprimntal urv (Efft data volum rlvant fration to fibr p). fr (vaporabl) atr ontnt in onrt at variou ag (Di Luzio & Cuati 2009b). 3. Efft volum fration fibr Th 2.2 analytial Tmpratur modl volution hih hav ah volum fration fibr ar hon in Figur 4. Th fft Not that, at arly ag, in th hmial ration volum fration fibr i invtigatd by uing th aoiatd ith mnt hydration SF ration modl. Figur 5 ho th fft volum fration ar xothrmi, th tmpratur fild i not uniform fibr on tnil tr-train urv. A hon in for non-adiabati ytm vn if th nvironmntal thi figur, th tr plain onrt uddnly dra aftr th tr rah th maximum tr. tmpratur i ontant. Hat ondution an b dribd in onrt, at lat for tmpratur not It i hon that th fratur bhavior plain onrt i brittl. On th othr h, though th maxi- xding 00 C (Bažant & Kaplan 996), by Fourir la, hih rad mum tr fibr rinford onrt i not hangd ompard ith that plain onrt, th q λ T dnding portion in tnil tr-train urv bom gntl ith inra volum fration fi- (7) br. hr It i q hon i th that hat th fratur flux, T bhavior i th bom abolut dutil tmpratur, ith adding λ th i fibr th hat lmnt. ondutivity; in thi (6) Proding FraMCoS-7, May 23-28, 200

3 Pur tnil failur D ( h, T h iothrm ould b takn into aount, to rlation, vaporabl atr v rlativ humi hmial ration, in turn, dtrm (a) Plain onrt (b) p 2% () p 3% Th atr ontnt an b xprd a th () vaporabl p 4% atr (apillary a Figur 6. Intrnal rak (Efft volum fration fibr p). vapor, adorbd atr) th non- (hmially bound) atr n (Mil Figur 6 ho th lat tat intrnal rak. Th intrnal rak plain onrt our in th am by adjuting Pantazopoulo numbr fibr & lmnt. Mill 995). Figur It 0 i ra ho th analytial aum rult. that Th th fft vaporabl lngth atr i a fu lor onntratd part pimn. On th othr fibr on tr-train rlativ urv humidity, i mall, h, dgr th inraing tnil trngth dgr dutil ilia fum fratur ration, bhavior, i i.. hydration h, th intrnal rak fibr rinford onrt our in th hol part pimn. Th rult ho th bridg fft fibr lmnt, that i, th obrvd at all lngth ag-dpndnt fibr lmnt in orption/dorption thi figur. (Norling Mjonll 997). Undr thi aum tr i tranmittd by bridg fft fibr lmnt by ubtituting Equation into Equati 3.3 Efft dirtion hn th intrnal rak our in mortar lmnt. fibr Thi bridg fft inra ith inraing volum It i aid that th dirtion fibr afft th fratur fration fibr. bhavior fibr rinford onrt. Thrfor, th h Figur 8 ho th rult train maurd in fft dirtion fibr i invtigatd. ( D h) & & h t h th rang hon in Figur 7. A hon in thi figur, Figur ho th analytial modl hih ha th train plain onrt inra only in No. 2. Bau th intrnal rak our only in th only horizontal fibr lmnt anothr on hih ha only vrtial hr fibr lmnt. /h i Figur th lop 2 ho th th orption/ lor part pimn a hon in Figur 6. Hovr, analytial rult obtaind iothrm by (alo th modl. alld moitur Th rult apa in th a fibr rinford onrt, all train inra. It i hon that th train inra in hol pimn by th bridg fft fibr lmnt. by only vrtial govrning fibr lmnt quation i almot (Equation am 3) a mut b that by all fibr lmnt. by appropriat On th othr boundary h, th initial rult onditi by only horizontal fibr Th rlation lmnt btn i almot th am amount a that by plain onrt. atr Th rlativ rult ho humidity th fft i alld dirtion fibr iothrm on dirt if tnion maurd tt almot ith inraing dpnd 3.2 Efft lngth fibr on only vrtial humidity lmnt. dorption iothrm in th a. Nglting thir diffrn (Xi t al. Figur 9 ho th analytial modl in hih th th folloing, orption iothrm ill b 3.4 Efft proprti fibr lngth fibr lmnt ar varid in th thr lvl rfrn to both orption dorption, 2 3m. Th volum fration fibr i th Th fibr that hav By th bn ay, ud if in th fibr hytri rinford th b ud aording to th ign th varia Mauring portion Mauring rlativity portion humidity. Th hap th iothrm for 4HPC i inflund by many p pially 3 tho that influn xtnt trutur por iz ditribution (atrratio, mnt 2hmial ompoition, SF 3 4 4,2 uring tim mthod, tmpratur, mix ,4,5,6 t.). In th litratur variou formulatio found to drib th orption iothrm Tnil train pimn (%) onrt Tnil train (Xi t pimn al. 994). (%) Hovr, in th papr th mi-mpirial xprion pro (a) Plain (b) p 4% Figur 7 Maurring rang train Norling Mjornll (997) i adoptd b Figur 7. Maurring rang train. Figur.8 Strain urfa pimn. Tnil train (%) Tnil train (%) Th proportionality fiint D(h,T) moitur prmability it i a nonlina th rlativ humidity h tmpratur & Najjar 972). Th moitur ma balan that th variation in tim th atr ma volum onrt (atr ontnt ) b q divrgn th moitur flux t Proding FraMCoS-7, May 23-28, 200

4 D ( h, T h () Th proportionality fiint D(h,T) i alld moitur prmability it i a nonlinar funtion th rlativ humidity h tmpratur T (Bažant & Najjar 972). Th moitur ma balan rquir that th variation in tim th atr ma pr unit volum onrt (atr ontnt ) b qual to th divrgn th moitur flux t (2) 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, Pantazopoulo (a) Lf & 2m Mill 995). It (b) i Lf raonabl 3m to Figur aum 9. Analytial that th modl vaporabl (Efft atr lngth i a fibr funtion Lf). rlativ humidity, h, dgr hydration,, onrt dgr ar ilia gla fum fibr, ration, arbon fibr,, i.. vinylon (h,fibr, ) aramid ag-dpndnt fibr o on orption/dorption bid tl fibr. iothrm In thi haptr, (Norling th Mjonll fft 997). proprti Undr thi fibr aumption th fratur by ubtituting bhavior fibr Equation rinford into onrt Equation i invti- 2 on gatd by hanging yild tr σy Young' modulu E hih ar input data for ontitutiv la fibr lmnt. h ( D h) Figur 3 ho th fft & yild & tr & n σy. Th (3) h t h rult 3000 MPa i am a that 300 MPa. It i hon that th fft yild tr i nothing, if yild hr tr /h i ovr i th 300 lop MPa. Figur th orption/dorption 4 ho th fft iothrm Young' (alo modulu alld E. moitur Th tnil apaity). trngth Th inra govrning quation th fratur (Equation bhavior 3) mut bom ompltd dutil ith by appropriat inraing boundary Young' modulu. initial ondition. It i hon that th Th fft rlation Young' btn modulu th amount i larg. Th vaporabl rult man atr that th rlativ inraing humidity rinformnt i alld adorption by fibr at arly iothrm loading if improv maurd th ith fratur inraing bhavior rlativity fibr rinford humidity onrt dorption but that iothrm at latr loading th oppoit don't afft a. Nglting th fratur thir bhavior. 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 HPC 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 Figur onrt 0. Tnil (Xi t tr-train al. 994). urv Hovr, (Efft in lngth th prnt fibr Lf). 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 (, ) 0( g ) h hr th firt trm (gl iothrm) rprnt th phyially bound (adorbd) atr th ond trm (apillary (a) Only horizontal iothrm) rprnt (b) Only vrtial th apillary atr. fibr Thi lmnt xprion i valid fibr only lmnt for lo ontnt SF. Th fiint G rprnt th amount Figur atr. pr Analytial unit volum modl hld (Efft in th dirtion gl por fibr). at 00% rlativ humidity, it an b xprd (Norling Mjornll 997) a All fibr lmnt Only vrtial fibr lmnt Only horizontal fibr lmnt G (, ) k k Plain onrt (5) vg vg Tnil tr σ (MPa) Tnil train ε (%) g G 0 Figur 2. Tnil tr-train urv (Efft dirtion fibr). K (, ) (6) 0 g h Tnil tr σ (MPa) ( g ) h K (, ) (4) 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 Lf m σy 3000 MPa p 4% σy 300 MPa Th matrial paramtr k vg σy k vg 30 MPa g an b alibratd by fitting xprimntal Plain data onrt rlvant to fr (vaporabl) atr ontnt in onrt at variou ag (Di Luzio & Cuati 2009b). 2.2 Tmpratur volution Not that, at arly ag, in th hmial ration aoiatd 0 ith 0.0 mnt 0.02hydration SF ration 0.05 ar xothrmi, th Tnil tmpratur train ε fild (%) i not uniform Figur for non-adiabati 3. Tnil tr-train ytm vn urv (Efft if th nvironmntal yild tr fibr tmpratur σy). i ontant. Hat ondution an b dribd in onrt, at lat for tmpratur not 4 xding SIMULATION 00 C OF (Bažant BENDING & Kaplan TEST 996), by Fourir la, hih rad In thi haptr, th rult imulation bnding q tt λ T ar hon. (7) Figur 5 ho th analytial modl onitd 650 nodal point to imulat ntr-point loading hr q i th hat flux, T i th abolut bnding tt. Tabl 2 ho th input data mortar. tmpratur, λ i th hat ondutivity; in thi Proding FraMCoS-7, May 23-28, 200

5 Tnil tr σ (MPa) Lf m p 4% E 20 GPa E 2 GPa E 2. GPa Plain onrt Tnil train ε (%) Figur 4. Tnil tr-train urv (Efft Young' modulu fibr E). Tabl 2. Input data mortar. tan Ft E η v d H D φ : Angl intrnal frition, Ft : Pur tnil trngth (MPa), E : Elati modulu (GPa), η : Vioity (MPa ), v : Shar loading rat (mm/), d :Diamtr nodal point (mm), H : Hight pimn (mm), D : Width pimn (mm) 4. Efft volum fration fibr Figur 6 ho th analytial modl in hih th volum fration fibr i varid. Figur 7 ho th load-dfltion urv by uing th modl. In a plain onrt, th load aftr rahing maximum load dra uddnly. Hovr, in a fibr rinford onrt, th maximum load inra th load aftr rahing maximum load hardly dra. Th tndni bom obviouly ith inraing th volum fration fibr. Th rult ar xplaind by Figur 8 hih ho th intrnal rak. Th intrnal rak plain onrt our in on traight lin only at ntr pimn. Thi i th raon uddn dra Figur B modl (For ntr-point loading bnding tt). load aftr maximum load. On th othr h, th intrnal rak fibr rinford onrt our in larg ara pimn pially at high volum fration fibr. It i thought that th rult xpr th bridg fft fibr. Th xtnion rak idth i rtraind by D ( th h, T ) bridg h fft fibr, th ourrn rak prad in hol pimn. Th proportionality fiint D(h,T) 4.2 Efft lngth moitur fibr prmability it i a nonlina th rlativ humidity h tmpratur Figur 9 ho & th Najjar analytial 972). modl Th moitur in hih ma th balan lngth fibr lmnt that th ar variation varid in in th tim thr th lvl atr ma,2 3m volum th volum onrt fration (atr ontnt fibr i ) b q am. Figur 20 divrgn ho th analytial th moitur rult flux th load-dfltion urv. Th fft lngth fibr Fibr lmnt t 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 () p.0% if maurd ith inraing humidity dorption iothrm in th Figur 6. Analytial a. modl (Efft Nglting volum thir fration diffrn fibr p). (Xi t al. th folloing, orption iothrm ill b rfrn Lf m to both p.0% p orption 0.5% dorption By th ay, if p th 0.25% hytri th iothrm ould Plain b takn onrt into aount, to rlation, vaporabl atr v rlativ humi b ud aording to th ign th varia rlativity humidity. Th hap th iothrm for HPC i inflund by many p pially tho that influn xtnt hmial ration, in turn, dtrm 0 0.trutur por iz 0.4ditribution 0.5 (atrratio, mnt hmial ompoition, SF Dfltion δ (mm) Figur 7. Load-dfltion uring urv tim (Efft mthod, volum tmpratur, fration mix fibr p). t.). In th litratur variou formulatio found to drib th orption iothrm on load-dfltion urv i mall, th dutil onrt (Xi t al. 994). Hovr, in th fratur bhavior i hon at all lngth fibr lmnt in thi figur. papr th mi-mpirial xprion pro Norling Mjornll (997) i adoptd b Load P (kn) Th atr ontnt an b xprd a th vaporabl atr (apillary a vapor, adorbd atr) th non- (hmially bound) atr n (Mil Pantazopoulo & Mill 995). It i ra aum (a) p that 0.25% th vaporabl atr i a fu rlativ humidity, h, dgr hydration dgr ilia fum ration,, i.. ag-dpndnt orption/dorption (Norling Mjonll 997). Undr thi aum by ubtituting Equation into Equati h ( D h) (b) h p t0.5% h & & Proding FraMCoS-7, May 23-28, 200

6 D ( h, T h () Th proportionality fiint D(h,T) i alld moitur prmability it i a nonlinar funtion th rlativ humidity h tmpratur T (Bažant & Najjar 972). Th moitur ma balan rquir that th variation in tim th atr ma pr unit volum onrt (atr ontnt ) b qual to th divrgn th moitur flux t (2) 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, Pantazopoulo & 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 2 on 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 Figur 8. Intrnal rak (Efft volum fration fibr p). iothrm if maurd ith inraing rlativity humidity dorption iothrm in th oppoit 4.3 a. Efft Nglting lngth thir diffrn fibr (Xi t al. 994), in Figur th folloing, 9 ho orption th analytial iothrm modl ill in b hih ud ith th lngth rfrn fibr to both lmnt orption ar varid dorption in th thr ondition. lvl By,2 th ay, 3m if th th hytri volum fration th moitur fibr i am. iothrm Figur ould 20 b ho takn th into analytial aount, rult to diffrnt th load-dfltion rlation, vaporabl urv. atr Th fft v rlativ lngth humidity, fibr mut on load-dfltion b ud aording urv to th i mall, ign th th variation dutil fratur th rlativity bhavior humidity. i hon Th at all hap lngth fibr th orption lmnt in iothrm thi figur. for HPC i inflund by many paramtr, pially tho that influn xtnt rat th hmial ration, in turn, dtrmin por 4.4 trutur Efft dirtion por iz ditribution fibr (atr-to-mnt Figur ratio, mnt 2 ho hmial th analytial ompoition, modl hih SF ha ontnt, only horizontal uring tim fibr mthod, lmnt tmpratur, th analytial mix additiv, modl hih t.). In ha th only litratur vrtial fibr variou lmnt. formulation Figur 22 an b found ho to th drib analytial th rult. orption Th rult iothrm by only normal vrtial fibr onrt lmnt (Xi t i al. almot 994). Hovr, am a that in th by prnt plain onrt. papr th Th mi-mpirial rult by only horizontal xprion fibr propod lmnt i by almot Norling am Mjornll a that by all (997) fibr lmnt. i adoptd Th rult bau ho it & that xpliitly th fft aount horizontal for fibr th lmnt volution i high hydration that vrtial ration fibr lmnt SF i ontnt. lo in bnding Thi tt. orption iothrm rad 4.5 Efft proprti fibr In ordr to invtigat th fft proprti fibr ( h,, ) G (, ) on th fratur bnding bhavior, th imulation in 0( g ) h hih yild valu σ y Young' modulu E i varid i arrid out. (4) 0( g ) h K (, ) hr k (b) Lf 3m vg k vg ar matrial paramtr. From th Figur maximum 9. Analytial amount modl atr (Efft pr unit lngth volum fibr Lf). that an fill all por (both apillary por gl por), on an alulat K a on p.0% Lf 3m Lf 2m Lf m g Plain onrt G 0 (6) K (, ) 0 g h Load P (kn) 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 (a) GLf 2m rprnt th amount atr pr unit volum hld in th gl por at 00% rlativ humidity, it an b xprd (Norling Mjornll 997) a G (, ) k k vg vg Th matrial 0 0.paramtr 0.2 k0.3 vg 0.4 k vg 0.5 g an b alibratd by fitting Dfltion xprimntal δ (mm) data rlvant to Figur fr 20. (vaporabl) Load-dfltion atr urv ontnt (Efft lngth in onrt fibr Lf). at variou ag (Di Luzio & Cuati 2009b). 2.2 Tmpratur volution hr q i th hat flux, T i th abolut (b) Only vrtial fibr lmnt tmpratur, λ i th hat ondutivity; in thi Figur 2. Analytial modl (Efft dirtion fibr). (5) 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 (a) Only ontant. horizontal Hat fibr ondution lmnt an b dribd in onrt, at lat for tmpratur not xding 00 C (Bažant & Kaplan 996), by Fourir la, hih rad q λ T (7) Proding FraMCoS-7, May 23-28, 200

7 Load P (kn) p.0% Lf m All fibr lmnt Only horizontal fibr lmnt Only vrtial fibr lmnt Plain onrt Dfltion δ (mm) Figur 22. Load-dfltion urv (Efft dirtion fibr). Figur 23. Load-dfltion urv (Efft yild tr fibr σy). D ( h, T h Th proportionality fiint D(h,T) moitur prmability it i a nonlina th rlativ humidity h tmpratur & Najjar 972). Th moitur ma balan that th variation in tim th atr ma volum onrt (atr ontnt ) b q divrgn th moitur flux Figur 25. Load-dfltion urv (Efft Young' modulu t fibr E). Th atr ontnt an b xprd a th vaporabl atr (apillary a vapor, adorbd atr) th non- (hmially bound) atr n (Mil Pantazopoulo & Mill 995). It i ra aum that th vaporabl atr i a fu rlativ humidity, h, dgr hydration dgr ilia fum ration,, i.. ag-dpndnt orption/dorption (Norling Mjonll 997). Undr thi aum by ubtituting Equation into Equati h ( D h) h t h & & Figur 24. Intrnal rak (Efft yild tr fibr σy). Figur 23 ho th fft yild tr σy on load-dfltion urv. Th fratur bhavior in a 30MPa i brittl almot am a that plain onrt. Hovr, th maximum load bom high th fratur bhavior bom dutil ith inraing yild tr σy. Figur 24 ho th lat tat intrnal rak. Th rak our at only ntr pimn in a 30MPa, th fibr lmnt yild. In a 3000MPa, th rak our in larg ara pimn, th fibr lmnt don t yild. hr /h i th lop th orption/ Figur 26. Intrnal rak iothrm (Efft (alo Young' alld modulu moitur fibr apa E). govrning quation (Equation 3) mut b by appropriat boundary initial onditi Figur 25 ho Th th rlation fft btn Young' th modulu amount E. Th maximum atr load bom rlativ high humidity ith inraing i alld Young' modulu. iothrm Hovr, if th maurd fratur ith bhavior inraing don't imply humidity bom dutil dorption ith inraing iothrm in th Young' modulu. a. Th Nglting load aftr maximum thir diffrn load inra gradually th in folloing, a orption 2.GPa. In iothrm a ill b (Xi t al. 20GPa, th load rfrn aftr maximum to both load orption dra dorption ttl don. Figur By 26 th ho ay, intrnal if th rak. hytri Th th rak in a 2.GPa iothrm our ould on b a f takn lin into at aount, ntr pimn. On rlation, th othr vaporabl h, th atr onntratd v rlativ humi to rak at ntr b ud pimn aording our to th in ign a th varia 20GPa. It i thought rlativity that humidity. thi diffrn Th in hap our-orn intrnal iothrm rak happn for HPC th i inflund diffrn by many p th fratur bhavior. pially tho that influn xtnt Th fft hmial proprti ration fibr on, th in bnding turn, dtrm tt i diffrnt from trutur that on th por dirt iz tnil ditribution tt. (atrratio, mnt hmial ompoition, SF uring tim mthod, tmpratur, mix 5 CONCLUSIONS t.). In th litratur variou formulatio found to drib th orption iothrm In thi papr, th onrt fratur imulation (Xi t al. 994). dirt Hovr, tnion in th tt bnding papr tt th fibr mi-mpirial rinford onrt xprion by pro uing th vio-lato-plati Norling Mjornll upnion (997) lmnt i adoptd b Proding FraMCoS-7, May 23-28, 200

8 mthod D ( h, a T ) arrid h out. Th fft fibr lmnt () on th fratur bhavior fibr rinford onrt a Th timatd proportionality by thi analyi. fiint D(h,T) i alld moitur Aording prmability to th rult, it i it a i nonlinar obtaind funtion by onidring th rlativ fibr lmnt humidity that h th tmpratur maximum T load (Bažant bom & Najjar high 972). Th th moitur dnding ma portion balan aftr rquir th maximum that th variation load bom in tim dutil th both atr in ma th rult pr unit dirt volum tnion onrt tt (atr bnding ontnt tt. ) Th b qual analytial to th rult divrgn an omparativly th moitur xpr flux th xprimntal on. Th rult fratur bhavior onidring fibr lmnt ho that th intrnal rak train pimn ar diprd ompard ith that (2) t pimn plain onrt. It i thought that th au Th atr th ontnt rult an an b b xplaind xprd by a bridging th um fft th vaporabl fibr lmnt. atr It man that thi analytial (apillary atr, atr mthod ha th poibility that an b applid for th vapor, adorbd atr) th non-vaporabl invtigation fibr rinford onrt. (hmially bound) atr n (Mill 966, Pantazopoulo & Mill 995). It i raonabl to aum that th vaporabl atr i a funtion REFERENCES rlativ humidity, h, dgr hydration,, dgr ilia fum ration, A.M.Nvill 995. Proprti Conrt,, i.. Paron Eduation (h,, ) F.H.Wittman ag-dpndnt 993. Numrial orption/dorption modl in fratur mhani iothrm (Norling onrt, Mjonll Bakma, 997). Sitzrl Undr thi aumption M.Arai, by ubtituting T.Funami, Equation Y.Kurokaa, into H.Mori Equation & Y.Tanigaa 2 on 995. Failur Analyi Conrt bad on Non- Continuum Modl, Tranation AI, Vol.47, pp.-9 (in apan) S..Park, hy.tanigaa, H.Mori, T.Hiraia & M.Sugiura 2004 Study on Efft ( D Coar h) Aggrgat & on Fratur & & n Bhavior (3) h t h Conrt, Proding th Fourth Intrnational Confrn on Conrt undr Svr Condition in Soul, hr pp /h i th lop th orption/dorption T.Hiraia, Y.Tanigaa, H.Mori & S..Park 200. Fratur iothrm (alo alld moitur apaity). Th Simulation Confind Conrt by Vio-Elato-Plati govrning Supnion quation Elmnt (Equation Mthod, 26th 3) mut Confrn b ompltd on Our by World appropriat in Conrt boundary & Strutur, initial pp ondition. T.Hiraia, Th rlation Y.Tanigaa, btn H.Mori, th amount Y.Kurokaa, vaporabl M.Arai & atr Y.Nanbu 998. rlativ Fratur humidity Simulation i alld Conrt adorption by Vioiothrm Elato-Plati if maurd Supnion Elmnt ith inraing Mthod, Fratur rlativity Mhani Conrt Strutur, AEDIFICTIO, Vol. 2, humidity pp 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 HPC 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 (, ) 0( g ) h 0( 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 00% 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 K (, ) g G 0 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 2009b). 2.2 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 00 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 Proding FraMCoS-7, May 23-28, 200