O. Omikrine-Metalssi & V.-D. Le Université Paris-Est, Paris, France

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1 Fratur Mhani Conrt Conrt Strutur - Rnt Advan in Fratur Mhani Conrt - B. H. Oh, t al.(d) 2 Kora Conrt Intitut, Soul, ISBN Intgration ontat lmnt in RGIB-modul th finit lmnt tar CESAR-LCPC : Appliation to onrt trutur afftd by intrnal lling ration O. Omikrin-Mtali & V.-D. L Univrité Pari-Et, Pari, Fran J.-F. Signol Laboratoir Régional d Pont t Chaué (LRPC) d Clrmont-Frr, Fran S. Rigobrt, P. Humbrt & F. Toutlmond Univrité Pari-Et, Pari, Fran ABSTRACT: A ignifiant numbr xiting onrt dam ar at prnt dtrioratd by hmo-mhanial pro knon a intrnal lling ration (ISR). Th pathologi, hih inlud dlayd ttringit formation a ll a alkali-aggrgat ration, ar rponibl for raking xpanion th matrial hih lad to dgradation trutur bhaviour. Th hmo-mhanial modlling th phnomna i alrady laboratd by onidring th influn tmpratur humidity in th dvlopmnt xpanion. Thi modlling i valid only for ontinuou mdia. In ordr to idn th domain modlling appliability, thi papr drib a novl produr for modlling diplamnt diontinuiti in ISR-afftd onrt trutur, mainly to tak into aount to phnomna: firtly th modlling th iolatd rak opning that rult from intration btn pathology mhanial bhaviour th trutur, ondly, th valuation th potntial ffiiny tr rla (.g. aing dam), th produr hih onit in th ration utting lin in th trutur for rla xiv ompriv tr, rloing th ut bing a rult ubqunt lling. Th laboratd modl, aimd to poibly ombin ontat lmnt non-linar oupld modl durability mhani rlatd to ISR modlling, a applid to th 3D imulation ISR in an xampl impl gravity dam. Diplamnt tr at th ontat lmnt zon r omputd ompard ith rult th dam omputation ithout ontat lmnt. Th rult ho that th ration utting lin in th dam tudid lad to dra th tr in th trutur, hih onfirm a favourabl fft thi tr rla thniqu in ordr to dal ith ISR afftd trutur. INTRODUCTION Intrnal lling ration (ISR) an afft th longtrm durability onrt trutur mor partiularly hydrauli onrt trutur by auing raking xpanion th matrial. Th pathologi mainly onit in alkali-aggrgat ration (AAR) /or dlayd ttringit formation (DEF). Th Alkali-Silia Ration (ASR) i a ration hih our ovr tim in onrt btn th highly alkalin mnt pat rativ non-rytallin ilia, hih i found in many ommon aggrgat. Thi ration au th xpanion th altrd aggrgat by th formation a lling gl Calium Siliat Hydrat (CSH). Thi gl inra in volum ith atr xrt an xpaniv prur inid th matrial, auing lo trngth th onrt, finally lading to it failur. Manhil dlayd ttringit formation (DEF) onit in ttringit rytallization ithin onrt aftr hardning i ubtantially omplt, in hih no ulphat om from outid th mnt pat. It may our in matrial that hav bn ubjtd to tmpratur abov about 65 C to high humidity. Th ourrn th to phnomna in vral onrt trutur in th lat yar orldid ha pointd out th nd for improvmnt prvntiv maur to inhibit th ration in afftd trutur. Morovr, prdition onrt xpanion dgradation amnt ffiiny priodiity rhabilitation opration ar ruial point for xiting afftd trutur. Svral typ modl to imulat th bhaviour trutur afftd by (ISR) hav bn prntd in th litratur (Légr t al. 996, Ulm 999 for ASR Zhang t al. 22, Bruntaud 26, Hinz

2 & Ludig 987, Garbozi 997 for DEF). Hovr, in thi tudy, improvd modl propod by Li t al. (24) for ASR Baghdadi t al. (27), Baghdadi (28) for DEF. Th modlling th phnomna in CESAR- LCPC Finit Elmnt od a bad on a ub program alld ALKA, for ASR, nly dvlopd ub program alld RGIB that allo modling th bhaviour a trutur afftd by ASR or DEF. Thi program tak into aount vral oupld phnomna: fft th arly ag thrmal hitory, aniotropy th impod xpanion rlatd to th dirtion onrt ating, damag rulting from raking th matrial, drying hrinkag oupling btn th xpanion th tat tr. Many a-tudi hav hon that it i nary to modl diplamnt diontinuiti in th onrt, mainly to tak into aount to phnomna: In addition to multi-raking, obrvation iolatd rak ith an important opning, rulting from intration btn pathology mhanial tr applid to th trutur. Control th rak opning omtim rquir an xpliit modlling. Among thniqu availabl to dal ith ISR afftd trutur, th tr rla an b fftiv (.g. aing dam). To valuat th potntial ffiiny thi thniqu, ration utting lin in th trutur, rloing th ut a a rult ubqunt lling, nd to b modlld. In thi papr, drib th algorithm dvlopmnt aimd to poibly ombin ontat lmnt, hih th modlling a laboratd by Rihr (985), non-linar oupld modl durability mhani rlatd to ISR modlling, prnt validation illutrativ a impl gravity dam. 2 PRESENTATION OF SOFTWARE BACKGROUND AND DEVELOPMENTS CESAR-LCPC i a finit lmnt tar, dvlopd at th LCPC in 98 (Humbrt t al. 25). It i a finit lmnt gnral tar partiularly adaptd to th rolution Civil Enginring problm (trutural analyi, oil mhani, t). It inlud modul data managmnt, librari lmnt omputational modul. In thi tudy, ontat lmnt r intgratd in th ub-program alld RGIB aimd to th ralulation trutur afftd by intrnal lling ration, by uing an othr ub-program alld TCNL hih i ddiatd to olving ontat alulation btn to non-linar olid. Th finally improvd RGIBmodul ill pially allo modling th aing tratmnt ISR-afftd dam. 2. RGIB modul J Within th framork th mthodology laboratd for th xprti Th proportionality ISR-afftd fiint onrt D(h,T) trutur (LPC 23), moitur a mthod prmability trutur it i ralulation ha bn dvlopd. th rlativ Thi humidity mthod h onit, tmpratur a nonlina from th information & Najjar olltd 972). Th on moitur th damagd ma balan trutur from that th th poibl variation additional in tim tt th atr on ma rontitutd onrt, volum in alibrating onrt (atr th paramtr ontnt ) b q th la that ontitut divrgn th th modl. moitur An algorithm flux J rolution a dignd for th ralulation trutur afftd by DEF or by ASR (Baghdadi J 28). It i implmntd t in CESAR-LCPC a RGIB modul (Intrnal Slling Ration in Conrt) onit Th in an atr volution ontnt th an xiting b xprd a modul ALKA hih th vaporabl onrn th atr ralulation (apillary a onrt trutur vapor, afftd adorbd by ASR atr) (Li t al. th non- 24, Signol, Dubouht (hmially 26). bound) atr n (Mil To modl th kinti Pantazopoulo lling & Mill ration 995). in rlation ith it mhanial aum onqun that th vaporabl th urv atr i a fu It i ra dformation vru rlativ tim humidity, from th h, fr dgr xpanion hydration tt ralizd in laboratory dgr ilia undr fum vry ration, pifi nvironmntal ondition ag-dpndnt hav bn ud. orption/dorption Th xpan-, i.. ion urv hav (Norling th hap Mjonll a igmoid 997). hih Undr an thi aum b dribd mathmatially by ubtituting by uing Equation th folloing into Equati Equation (Bruntaud obtain 26, Baghdadi 28): ε ( t) τ + ( D h) & + & + ϕ h + h () t τ L + δ + t ε χ τ τ + t ) D ( h, T h h hr /h i th lop th orption/ iothrm (alo alld moitur apa hr: govrning quation (Equation 3) mut b ε : Potntial xpanion, by appropriat boundary initial onditi τ : Charatriti tim, Th rlation btn th amount τ L : Latny tim, atr rlativ humidity i alld ϕ δ : fiint haratrizing th lat iothrm if maurd ith inraing pudo-aymptoti pha xpanion. humidity dorption iothrm in th a. Nglting thir diffrn (Xi t al. Many rult ho that vral fator influn th folloing, orption iothrm ill b th amplitud th kinti th lling indud by DEF ASR (Bruntaud 26, Ptrov rfrn to both orption dorption By th ay, if th hytri th 23). Coupling la hav to drib th rlation iothrm ould b takn into aount, to btn th paramtr th xpanion la prntd in th quation () th nvironmntal rlation, vaporabl atr v rlativ humi b ud aording to th ign th varia ondition to hih th trutur i ubjtd, rlativity humidity. Th hap th hthr during it ontrution or during it liftim. iothrm for HPC i inflund by many p Th xprimnt mad by (Larn 995, Ptrov 23, Fu t al. 994) hod that a highr ur- pially tho that influn xtnt hmial ration, in turn, dtrm ing tmpratur lad to inra th amplitud trutur por iz ditribution (atrratio, mnt hmial ompoition, SF final xpanion. Th duration xpour to high tmpratur alo amplifi th phnomnon (Bruntaud 26). uring tim mthod, tmpratur, mix t.). In th litratur variou formulatio Som ork on th fft th humidity on th found to drib th orption iothrm ISR hav bn mad. Tho Hinz & Luding onrt (Xi t al. 994). Hovr, in th (987) hod that no xpanion i obrvd blo papr th mi-mpirial xprion pro 9% rlativ humidity vn aftr 78 day. Th Norling Mjornll (997) i adoptd b final xpanion ar highr latny/haratriti Proding FraMCoS-7, May 23-28, 2

3 tim J D ( ar h, T ) hortr h hn th ampl ar kpt in () th atr rathr than % rlativ humidity. To Th drib proportionality th influn fiint th moitur D(h,T) i ontnt alld on moitur th kinti prmability th amplitud, i a th nonlinar la prntd funtion by th Poyt rlativ (23) humidity hav bn h implmntd tmpratur T aftr. (Bažant A nitivity & Najjar 972). analyi Th moitur dtaild ma in balan Signol rquir t al. (26). that th variation in tim th atr ma pr unit volum Th adoptd onrt approah (atr ontnt in th modlling ) b qual to th hmo-mhanial divrgn th moitur oupling flux i bad J on th aumption prribd hmial train uprimpod to mhanial dformation (Li t al. 24, Baghdadi t al. 28). J (2) t h + h ( D h) n (3) h & + & + & 2.2 Th TCNL atr modul ontnt an b xprd a th um th vaporabl atr (apillary atr, atr Th vapor, modul adorbd TCNL atr) i bad on th th non-vaporabl mthod pnalization (hmially hih bound) a dvlopd atr in CESAR for on- 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 obtain tat xpliitly dription aount pially for th uing volution ontat lmnt. hydration Thi ration mthod introdu SF ontnt. a pifi Thi orption variabl iothrm rigidity rad matrix at th lvl th intrfa bau it dpnd on th tat (opn/lod) thi intrfa at ah itration tp (Rihr 985). In prati, thi matrix i takn into aount by adding, in th mh ( h,, ) G (, ) + th trutur, th pial ontat lmnt along ( g ) h th urfa ontat. Th prinipl rolution (4) rquir to updat th rigidity matrix at ah itration tp, dpnding on th volution ( g ) h th ontat K (, ) intrfa, but avoid it ronditioning. Normally in a linar ytm hr dgr frdom ar ratd or upprd, th rigidity matrix mut b ronditiond. hr th firt With trm thi (gl mthod, iothrm) numbr rprnt dgr phyially frdom bound (adorbd) rmain ontant atr on th urfa ond th du trm to (apillary th prn iothrm) matrix rprnt pnalization. th apillary atr. Th Thi advantag xprion thi mthod valid only i that for lo th ontat 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 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 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 Figur. Rolution algorithm for RGIB-TCNL oupling problm. Proding FraMCoS-7, May 23-28, 2 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 29b). 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 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

4 lmnt on th potntial ontat urfa ar takn into aount in th am ay a th gnral lmnt th trutur, that i, through thir matrix rigidity, hih i ambld in th global matrix rigidity th trutur. Th ontat lmnt hav to nur th ontinuity normal tangntial diplamnt in th intrfa. For th modl ud hr th lmnt hav a ak thikn but not null; th intrfa i thn rprntd by a fititiou ontat matrial th intrfa lmnt hav mhanial proprti idntial to th rt th trutur. Th numbr nod mut b ompatibl ith that th lmnt hih ontitut th mdia on both id th intrfa. Th prinipl rolution onit in proding to a progriv loading in ordr to follo th volution th fftiv urfa ontat. For ah th load, vrify by an itrativ pro th quilibrium Equation th ritria ontat. Th hanging th fftiv urfa ontat rult in th modifiation th matrix rigidity th ontat lmnt thu th global matrix rigidity. For ah itration tp, prod to thr tt: a tt non-intrpntration, a tt on th normal tr a tt frition. Th firt vrifiation onit in nuring that thr i no intrpntration at th lvl th inativ ontat lmnt th potntial urfa ontat. Th ond tt onit in vrifying that th normal tr in a point th fftiv urfa ontat i lor or qual to th tnil trngth (dtahmnt tt). Th lat tt onit in vrifying that Coulomb frition la (gnrally ud) i atifid at vry point th ativ trutur ara. 2.3 Coupling algorithm Th oupling algorithm RGIB ontat lmnt i hon in Figur. Firtly, From th fild tmpratur humidity, hmial paramtr th matrial, RGIB-modul alulat th advanmnt dgr th ration for ah tim tp (in th a intrnal ulfat attak, hav to mak th arly ag thrmal alulation firt in ordr to hav lling potntial in th trutur). Thn, thi advanmnt dgr i introdud into th hmo-mhanial modl hih lad to alulat, from a plati or lati la, th fft ISR (train, diplamnt, tr). For oupling RGIB ith ontat lmnt, it i matory to inlud in th mh th loation ontat lmnt. Th ar non-ativ for prliminary thrmal hydri alulation bom ativ in th mh during mhanial alulation. Th intgration th ontat lmnt in th alulation i mad by an amblag thir rigidity matrix by alulating th tr train at tim t t+ t, by hking for ah itration tp, th ontitutiv la that govrn Jontat D ( h, T ) ritria h (intrpntration, frition, dtahmnt). Whn th alulation onvrg, thi oupling Th RGIB- proportionality ontat lmnt fiint allo D(h,T) to hav th volution moitur train prmability tr throughout th tudid trutur th rlativ inluding humidity at th h ontat tmpratur it i a nonlina lmnt zon. & Najjar 972). Th moitur ma balan that th variation in tim th atr ma volum onrt (atr ontnt ) b q 3 APPLICATION divrgn TO A SIMPLE th DAM moitur STRUCTURE flux J 3. Matrial, mh haratriti paramtr J An idalizd part t an ASR-afftd gravity dam ha bn ud for illutrating th apaiti th dvlopd modul. Th atr numrial ontnt imulation an b hav xprd a bn arrid out, auming th vaporabl onrt Young atr modulu (apillary a Poion ratio vapor, qual to Eadorbd 3 atr) MPa υ th non-.2 rptivly. (hmially Ma dnity bound) onrt atr i takn n (Mil a ρ 24 kg/mpantazopoulo 3. Th duration & Mill th analyi 995). i It 5 i ra yar, auming aum on yar-long that th tim vaporabl tp. Total atr i a fu 65 thr dimnional rlativ lmnt humidity, h, 3392 dgr nod in hydration th ntir modl ar dgr ud for ilia finit fum lmnt ration, alulation. Th main gomtri ag-dpndnt haratriti orption/dorption th fi-, i.. nit lmnt mh (Norling ud Mjonll for th 997). imulation Undr ar thi aum hon in Figur by 2. Th ubtituting valu Equation th thrmal into Equati humidity fild ar obtain onidrd a ontant. Th ambint tmpratur th rlativ humidity takn into aount ar 5 C % rptivly. h Tmpratur humidity + fild ( Dr h) omputd & + uing th DTNL modul in CESAR-LCPC a hon & + h h in Figur th rult ar ud to alulat th hmial train hr in th RGIB /h i modul th lop (Baghdadi th orption/ 28). iothrm (alo alld moitur apa For th dtrmination govrning th quation ration (Equation kinti 3) (Eq. mut b ) ud th folloing by appropriat valu: boundary τ 6 yar; initial τ L onditi 6 yar; ε.5, Th φ rlation δ. btn Th paramtr th amount ar attahd to a rfrn atr tmpratur rlativ humidity 5 C. i alld A for th boundary iothrm ondition, if maurd th tranvr ith inraing diplamnt r humidity rtraind dorption on both id, iothrm hil in th th foundation a. th dam Nglting a onidrd thir diffrn a dampd (Xi t al. (Fig. 3). Th noth th folloing, th dam orption i rprntd iothrm by a ill b ontat lmnt rfrn zon (P to both P6). orption Variou ontitutiv la an b By nviagd th ay, for if th typ hytri l- th dorption mnt: prft liding, iothrm adhion, ould b Coulomb takn into frition. aount, to Eah th la rlation, i aoiatd vaporabl to a ontitutiv atr v rlativ matrix that allo to b nur ud th aording ontinuity to th on ign th orr- th varia humi ponding diplamnt. rlativity For humidity. prft liding, Th th hap on-otat lmnt mut iothrm only for nur HPC th i inflund ontinuity by many p th normal diplamnt, pially hil tho for adhion, that influn th xtnt lmnt mut alo hmial nur th ration quality, tangntial in turn, dtrm diplamnt. For trutur frition, th por rlation iz ditribution btn (atrratio, tr mnt i ontrolld hmial by Coulomb' ompoition, SF normal har la. uring tim mthod, tmpratur, mix In th a t.). th tudid In th litratur trutur, variou th aing formulatio opration i aumd found to to hav drib bn th prformd orption aftr iothrm yar (aing onrt thikn (Xi i t m). al. Th 994). mhanial Hovr, in th bhaviour th papr trutur th mi-mpirial an thu b ompard xprion pro ith th rfrn Norling tat (ithout Mjornll aing). (997) Furthr- i adoptd b Proding FraMCoS-7, May 23-28, 2

5 mor, J D ( th h, T ) rloing h th to id th noth folloing th lling th dam a imulatd. Th () rult Th r proportionality ompard fiint ith rfrn D(h,T) tat i ithr alld ithout moitur aing, prmability or ith a it noth i a nonlinar ratd from funtion th ontrution th rlativ tim. 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 J t J (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 Figur 2. Gomtri haratriti th dam finit lmnt by ubtituting Equation into Equation 2 on mh. obtain h + h ( D h) n (3) h & + & + & xpliitly aount for th volution hydration ration SF ontnt. Thi orption iothrm rad ( h,, ) G (, ) + ( g ) h (4) ( g ) h K (, ) Figur 4. Vrtial diplamnt v. tim along th aing lin (ith ithout aing). hr Th th variation firt trm tranvr (gl iothrm) diplamnt rprnt vru th tim phyially i prntd bound in (adorbd) Figur 5. atr Th noth lo th ond progrivly trm (apillary from point iothrm) P to P5 rprnt in l than th apillary yar. Only atr. point Thi (P6) xprion i opn i a valid it only tranvr for lo diplamnt ontnt SF. tay Th lor fiint than th G valu rprnt aing th amount thikn. Th atr rult pr unit for volum th train hld at in thi th point gl por (P6) hav at % no phyial rlativ humidity, maning in it thi an on b xprd i onidrd (Norling a a onntion Mjornll 997) point a btn th nd th noth th olid blok hih rprnt th ba th dam. G (, ) k + k (5) vg vg 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 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 Figur 3. Contat lmnt boundary ondition in th dam tudid. 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 Figur hmial 4 ho ration th, variation in turn, vrtial dtrmin diplamnt trutur vru tim por iz point ditribution (P to (atr-to-mnt P6) for th dam por tudid, ratio, mnt ith hmial ithout ompoition, aing on SF th ontnt, graph, CE uring t tim for mthod, ontat tmpratur, lmnt. Thi mix additiv, diplamnt t.). inra In th litratur ith tim variou bom formulation mor important found hn to drib on approah th orption th uppr iothrm urfa for normal both an b a. onrt It tabiliz (Xi t al. aftr 994). about Hovr, 25 yar. By in th omparing prnt th papr rult, th thi mi-mpirial vrtial diplamnt xprion tay propod mor important Norling in Mjornll th a ithout (997) aing i adoptd (by about bau 5 %). by it 3.2 Rult 3.2. Saing aftr ontrution g G (6) K (, ) g Figur 5. Tranvr diplamnt v. tim along th aing lin aftr aing. Th matrial paramtr k vg k vg g For th variation tranvr tr vru an b alibratd by fitting xprimntal data rlvant to tim, Figur 6 ho th urv for both a: ith fr (vaporabl) atr ontnt in onrt at ithout aing. For th rfrn a, th variou ag (Di Luzio & Cuati 29b). tranvr tr inra ith tim until it tabilization at a valu 45 MPa. Thi valu i roughly onitnt 2.2 Tmpratur ith E. ε volution. Whil in th a aing, th tranvr tr tay qual to zro until th progriv rloing th noth. Thn it inra Not that, at arly ag, in th hmial ration aoiatd ith mnt hydration SF ration tabiliz at valu 3 to 35 MPa. Th tr ar xothrmi, th tmpratur fild i not uniform valu ar in agrmnt ith th thortial valu for non-adiabati ytm vn if th nvironmntal alulatd at th nd th lling. Indd, ith tmpratur i ontant. Hat ondution an b th aing lin thikn m mad on th part dribd in onrt, at lat for tmpratur not dam ho idth i L 2 m, th maximum xding C (Bažant & Kaplan 996), by tranvr tr i givn by (Eq. 2): Fourir la, hih rad q λ T σ E. ε E. (2) (7) L hr q i th hat flux, T i th abolut tmpratur, Th tnil valu λ i th tranvr hat ondutivity; tr at point in (P6) thi highlight that th noth at thi point tay opn. A Proding FraMCoS-7, May 23-28, 2

6 for th train imulation, thi loal tr valu ha no phyial maning in it orrpond to ingular onntion ontat olid lmnt. J ) 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 J Figur 8. Tranvr diplamnt v. tim along th aing J lin bfor aftr aing. t Figur 6. Tranvr tr v. tim along th aing lin ith ithout aing Saing aftr yar Figur 7 ho th variation vrtial diplamnt vru tim point (P to P6) blonging to th aing lin. Bfor aing, th vrtial diplamnt orrpond to Figur 4. Aftr aing (aftr yar th trutur lif), th diplamnt dra intantanouly inra again but tay about 5 % lor than th vrtial diplamnt rfrn a (ithout aing). Figur 7. Vrtial diplamnt v. tim along th aing lin bfor aftr aing. Tranvr diplamnt ar diplayd in Figur 8. Thy ar zro for point (P to P6) until yar for raon ymmtry. Aftr aing ( m), th noth lo intantanouly diplamnt on both id thi noth tak xatly th valu th aing thikn. Th noth rmain opn only at point P6 in th maximum diplamnt at thi point (.2 mm) tay lor than th valu th aing thikn (onntion point btn th nd th noth th olid blok). Figur 9 ho th horizontal tranvr diplamnt th trutur tudid aftr 5 yar from th ontrution. At thi ag, th tranvr diplamnt on both id th noth hav ompnatd th valu th aing thikn. 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 obtain Figur 9. Tranvr diplamnt th dam aftr 5 yar. h + ( D h) & + & + h h Morovr, th variation tranvr tr vru tim ith ithout aing ar prntd in Figur. In hr fat, bfor /h aing, i th lop th tranvr th orption/ tr inra ith iothrm tim until (alo it alld tabilization moitur at a apa valu E. ε 45 govrning MPa. Th quation tr (Equation dra 3) intantanouly jut by aftr appropriat aing for boundary all th point initial b-onditlonging to th aing Th lin, rlation thn btn inra th again amount mut b ith tim until atr tabilization rlativ valu humidity 3 i alld to 35 MPa (Eq. 2). Sin iothrm tr if maurd rla i ith only partial, a ompard humidity to th total xpanion dorption potntial, iothrm fi- in th inraing nal valu ar imilar a. for Nglting th a thir hon diffrn in Figur (Xi t al. Figur 6. th Th folloing, noth at orption point P6 tay iothrm opn, ill b hih i highlightd rfrn by a tnil to both tr orption in thi zon: dorption thi i a loal artfat By th in ay, thi ingular if th zon hytri th th mh. iothrm ould b takn 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 trutur por iz ditribution (atrratio, mnt hmial ompoition, SF uring tim mthod, tmpratur, mix t.). In th litratur variou formulatio found to drib th orption iothrm onrt (Xi t al. 994). Hovr, in th papr th mi-mpirial xprion pro Figur. Tranvr tr v. tim along th aing lin bfor aftr aing. Norling Mjornll (997) i adoptd b Proding FraMCoS-7, May 23-28, 2

7 4 J SIMULATION D ( h, T ) h OF PROGRESSIVE () CRACKING Th proportionality fiint D(h,T) i alld Tt moitur hav prmability bn alo prformd it i for a nonlinar dmontrating funtion an othr th apaity rlativ humidity th dvlopd h tmpratur modul hih T (Bažant i aounting & Najjar for 972). progriv Th moitur raking ma du balan to hmial rquir lling that th variation aud by in th tim ISR. th Th atr numrial ma pr imulation volum hav onrt bn prformd (atr ontnt ) a b impl qual to to- th unit dimnional divrgn a th moitur (Fig. ) flux taking J into aount th am input paramtr a for th dam alulation. Thi a-tt, ith a quar hap ith dampd ba, ontain J ontat lmnt in th middl hih (2) t ar initially providd in th mh. Th ambint tmpratur Th atr rlativ ontnt humidity an b takn xprd into aount a th um ar 5 C th vaporabl % atr rptivly. (apillary atr, atr vapor, Vrtial diplamnt adorbd atr) vru th tim non-vaporabl point (P to (hmially P4) ar prntd bound) in Figur atr 2. Th urv all n (Mill 966, th Pantazopoulo point ho & an Mill inra 995). It thi i raonabl vrtial diplamnt to aum that until th vaporabl it tabiliation, atr i a it funtion bom mor rlativ important humidity, hn h, on dgr approah hydration, th uppr urfa, dgr th ilia trutur fum tudid. ration,, i.. (h,, ) Figur ag-dpndnt 3 ho th orption/dorption variation tranvr iothrm diplamnt (Norling Mjonll for thr 997). diffrnt Undr thi tim aumption (8, 9 5 by yar). ubtituting At 8 yar, Equation th tranvr into Equation diplamnt 2 on rah obtain th maximum valu about.86 mm, thr i till no rak. Aftr on mor yar, i.. at th 9 th yar, a rak appar at th ontat lmnt h + ( D h) & + & + & n (3) h 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 Figur. Contat lmnt layr boundary ondition in th trutur tudid. 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 Figur 2. Vrtial diplamnt v. tim at th ontat lmnt aftr 5 yar. Norling Mjornll (997) i adoptd bau it ho xpliitly opning aount i mor for important th volution at th uppr hydration urfa. ration Thu, maximum SF ontnt. valu Thi tranvr orption diplamnt rad xd mm. Th valu ar doubld aftr iothrm 5 yar th opning th rak bom mor important. ( h,, ) G (, ) + ( g ) h ( g ) h K (, ) atr pr unit volum (a): hld At 8 yar in th gl por at % (b): At 9 yar g G g (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 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 (, ) (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 29b). 2.2 Tmpratur volution (): At 5 yar Not that, at arly ag, in th hmial ration Figur 3. Tranvr diplamnt th trutur tudid aoiatd ith mnt hydration SF ration aftr 8, 9 5 yar. ar xothrmi, th tmpratur fild i not uniform for Figur non-adiabati 4, hih ytm prnt vn th if variation th nvironmntal tranvr tmpratur tr i vru ontant. tim, Hat highlight ondution thi an rak b opning. dribd In in fat, onrt, th boundary at lat ondition for tmpratur allo obrving xding a tnion C zon (Bažant on th & Kaplan uppr fr 996), urfa by not (point Fourir la, 2). hih Whn rad th tnil tr rah th tnil trngth (2 MPa) aftr 8 yar, th rak our ngndring th tr rla. Thi tr tnd (7) q λ T to zro for th point (P, P2 P3) (Fig. 4). Only th hr tr q at i point th (P4) hat ontinu flux, T to i inra th abolut until tabilization. tmpratur, λ i th hat ondutivity; in thi Proding FraMCoS-7, May 23-28, 2

8 Figur 4. Tranvr tr v. tim along th ontat lmnt. Th rult th variation tranvr tr ar in prft agrmnt ith tho th variation tranvr diplamnt. In fat, th rak appar aftr th 8th yar, hih i rfltd by th tr rla hon in Figur 4. 5 CONCLUSION Thi papr prnt n funtion a Finit Elmnt Star for modlling diplamnt diontinuiti in onrt trutur afftd by Intrnal Slling Ration. It mak it poibl to valuat th potntial ffiiny tr rla (.g. by aing dam). It an hlp th nginr in diding on th hoi th numbr utting lin thir loation in th trutur in ordr to inra it liftim. Furthrmor, thi Finit Elmnt Star allod to modl th lling th trutur vn if th on ar rakd. Morovr, alulation prmit to follo rak volution. It hould b furthr applid to varid ral-a in upport xprt nginring. ACKNOWLEDGMENTS Th author ould lik to thank STUCKY. SA (pially Mr L. Bolda) for it finanial thnial upport. REFERENCES Baghdadi, N., Toutlmond, F., Signol, J.F. 27. Modéliation d l fft d ontraint ur l aniotropi d l xpanion dan l béton attint d réation d gonflmnt intrn, 25 èm rnontr d l AUGC, Bordaux. Baghdadi, N., Toutlmond, F., Signol, J.F. 28. Chmomhanial modl dribing th xpanion du to intrnal ulphat attak: numrial imulation, Conrt modlling CONMOD 8, Dlft, (Th Nthrl), May 26-28, RILEM PRO 58, pp Baghdadi, N. 28. J Modéliation D ( h, T ) h du ouplag himioméaniqu d un béton attint d un réation ulfatiqu intrn, PhD thi, Eol National d Pont t Chaué, Pari, Fran. Th proportionality fiint D(h,T) Bruntaud, X. 26. moitur Etud d prmability l influn d it différnt i a nonlina paramètr t d lur th rlativ intration humidity ur la h inétiqu tmpratur t l amplitud d la & réation Najjar ulfatiqu 972). Th intrn moitur au béton, ma PhD balan thi, Eol Cntral d Pari, Fran. that th variation in tim th atr ma Fu, Y., Xi, P. & Baudoin, J.J Efft th tmpratur on ulfat adorption volum dorption onrt by (atr trialium ontnt iliat ) b q hydrat, Cmnt divrgn Conrt th Rarh, moitur 24 (8), flux 428- J 432. Garbozi, E.J Str, diplamnt, xpaniv raking around a ingl phrial J aggrgat undr diffrnt xpaniv ondition, t Cmnt Conrt Rarh, Vol 27, No 4, pp Hinz, D. & Ludig, U. Th 987. atr Mhanim ontnt ondary an b xprd ttringit formation in th mortar vaporabl onrt atr ubjtd (apillary to a a hat tratmnt, Conrt vapor, durability, adorbd Amrian atr) Conrt Intitut, K. & B. Mathr, ACI SP-, th non- (hmially bound) atr Humbrt, P., Dubouht, A., Fzan, G. & Rmaud, D. 25. n (Mil CESAR-LCPC, un Pantazopoulo progiil d alul & Mill dédié au 995). géni ivil, It i ra Bull. d laboratoir aum d that pont th t haué, vaporabl , atr 7- i a fu 37, LCPC, Pari. rlativ humidity, h, dgr hydration Larn., C.D dgr Dlayd ilia ttringit fum formation: ration, an iu?, i.. Matrial Sin Conrt. (d. J. Skalny & S. Mind), ag-dpndnt orption/dorption pp Amrian Crami Soity, Wtrvill, OH, USA. (Norling Mjonll 997). Undr thi aum Légr, P., Cot, P., Tinair, by ubtituting R Finit Equation lmnt analyi into Equati onrt lling obtain du to Alkali-Aggrgat Ration in Dam, Computr Strutur, Vol. 6, pp. 6-6, 996. Li, K., Couy, O. & Lariv, hc. 24. Modéliation himio-méaniqu du omportmnt h d béton h affté par la ré- + ( D h) & + & + ation d alali-ili. Exprti numériqu d ouvrag d art dégradé, Etud t rhrh d LPC, OA43, Laboratoir Cntral d hr Pont t Chaué, /h i th Pari. lop th orption/ LPC 23. Aid à la iothrm gtion d (alo ouvrag alld attint moitur d réation apa d gonflmnt intrn, Guid thniqu, Laboratoir Cntral d Pont t Chaué, govrning Pari. quation (Equation 3) mut b Ptrov, N. 23. Combind by appropriat Efft boundary Diffrnt Fator initial on onditi Conrt Expanion Th du to rlation DEF, PhD btn thi, Univrité th amount d Shrbrook, Shrbrook, atr Canada. rlativ humidity i alld Poyt, S. 23. Etud iothrm d la dégradation if maurd d ouvrag ith n béton inraing attint par la réation alali-ili : Approh xpérimntal t modéliation numériqu multi-éhll d dégrada- humidity dorption iothrm in th tion dan un nvironnmnt a. Nglting hydro-hmo-méaniqu thir diffrn variabl, PhD thi, th Univrité folloing, d Marn-La-Vallé. orption iothrm ill b (Xi t al. Rihr, S Réolution rfrn d problèm to both orption d ontat ntr dorption olid élatiqu, PhD By thi, ay, Eol National if th hytri d Pont t th Chaué, Pari, iothrm Fran. ould b takn into aount, to Signol, J.F. & Dubouht, A. 26. Cahir d dévloppmnt du od d alul rlation, ALKA, vaporabl CESAR-LCPC atr vrion v rlativ rhrh, LCPC. b ud aording to th ign th varia humi Signol J.F., Ngo, T.T., rlativity Toutlmond humidity. F. 26. Th Modling hap th th oupling btn iothrm moitur for HPC alkali-ilia i inflund ration by in many p onrt. - "Computational pially Modling tho that influn Conrt Strutur", proding Euro-C 26 onfrn, Mayrhn xtnt hmial ration, in turn, dtrm (Autria), 27-3 Marh, 26, Mhk t al.d, Taylor & Frani Group, London, trutur pp por iz ditribution (atrratio, thrmohémoméaniqu mnt hmial ompoition, dan l SF Ulm F. J Couplag béton : un prmir uring bilan, tim Laboratoir mthod, ntral d tmpratur, pont t mix haué, Etud t.). rhrh In th d litratur LPC, OA3. variou formulatio Zhang, Z. Olk, J. Diamond S. 22. Studi dlayd ttringit found to drib th orption iothrm formation in arly-ag hat urd mortar: 2-Charatriti mnt that may onrt b uptibl (Xi t al. to DEF. 994). Hovr, in th Cmnt Conrt Rarh papr 32, 22, th pp mi-mpirial xprion pro Norling Mjornll (997) i adoptd b Proding FraMCoS-7, May 23-28, 2