CALCULATION OF SHRINKAGE STRAIN IN EARLY-AGE CONCRETE STRUCTURES---AN EXAMPLE WITH CONCRETE PAVEMENTS

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1 Cmntitious Composits, April 212, Amstrdam, Th Nthrlands CALCULATION OF SHRINKAGE STRAIN IN EARLY-AGE CONCRETE STRUCTURES---AN EXAMPLE WITH CONCRETE PAVEMENTS Jun Zhang, Dongwi Hou and Yuan Gao Dpartmnt of Civil Enginring, Ky Laboratory of Strutural Safty and Durability, China Eduation Ministry, Tsinghua Univrsity, Bijing, 184, China Abstrat This papr fouss on th modling of th distribution of shrinkag strain in arly-ag onrt pavmnts. In th modling, an intgrativ modl for autognous and drying shrinkag prditions of onrt at arly-ag is introdud first. Sond, a modl taking both mnt hydration and moistur diffusion into aount synhronously is usd to alulat th distribution of intrior humidity in onrt. Th abov two modls ar xprimntally vrifid indpndntly by a sris of shrinkag and intrior humidity tsts on thr typs of onrts with diffrnt omprssiv strngth. Using th modls, th distribution of shrinkag strain in arly-ag onrt pavmnts undr th ondition that th pavmnt surfa suffrs to drying is alulatd. Th modl rsults show that th dvlopmnt of intrior humidity insid of pavmnt sin onrt ast obys two stag mod, that is a vapor saturatd stag with 1% rlativ humidity (stag I) and a stag with th rlativ humidity gradually drasing (stag II). Within th stag I, a uniform shrinkag strain is xptd throughout th slab. By ontrast, th shrinkag gradint along th slab dpth is obvious in th stag II. Th maximum and minimum shrinkag ours at slab top and bottom rsptivly. Th distribution of shrinkag strain along th pavmnt dpth is nonlinar and th nonlinarity is strong los to th drying fa and th rst is wak. Conrt strngth an signifiantly influn th magnitud of shrinkag strain within th slab. Ky words: Shrinkag strain; onrt pavmnt; rlativ humidity; modl 1. INTRODUCTION Shrinkag rsultd strss is on of th main origins of lading to arly-ag raking in onrt struturs, suh as pavmnt slabs. Conrt shrinks as moistur is lost to th nvironmnt or by slf-dsiation. As onrt shrinks, a rtain amount of tnsil strsss will b dvlopd in th strutur du to rstraints from adjunt matrials or onntd mmbrs. Th strsss may xd th tnsil strngth and aus onrt to rak. Craking in onrt mmbrs rdus th load apaity of th strutur. Morovr, raks allow watr and othr hmial agnts, suh as diing salt, to go through th ovr layr to om into ontat with th rinformnts, lading to rinformnt orrosion and ruptur in stl rinford onrt. Th magnitud of th shrinkag strain is normally proportional to th amount of moistur lost [1-3]. Gnrally, thr ar two mannrs lading th moistur loss in arly-ag onrt. As nvironmntal humidity is lowr than th humidity insid of onrt, watr in onrt vaporats and shrinkag of onrt ariss, whih is normally alld drying shrinkag. Anothr mannr of moistur loss is through mnt

2 Cmntitious Composits, April 212, Amstrdam, Th Nthrlands hydration, whih auss onrt to shrink also and normally is alld autognous shrinkag. In prati, mor watr loss may happn at th plas whr ar los to surfas of onrt lmnts. Thus, shrinkag gradint should xist in onrt struturs and orrsponding nonlinar shrinkag strsss should b rsultd. Howvr, th ffts of shrinkag gradint ourrd in onrt struturs hav not proprly bn takn into aount in th analyss of shrinkag strsss in th struturs du to th lak of appropriat modl to rlating th shrinkag strain and th amount of loal moistur loss. Thrfor, urrntly thr ar fw shrinkag strss modls whih ar abl to onsidr th influns of shrinkag gradint ompltly. This is an apparnt disadvantag for shrinkag strss modlling. This artil fouss on th numrial modlling of th distribution of shrinkag strain in arlyag onrt pavmnt in whih th shrinkag gradint is sriously takn into aount. In th modlling, an intgrativ modl for autognous and drying shrinkag prdition of onrt at arly-ag is introdud first. Sond, a modl taking both mnt hydration and moistur diffusion into aount synhronously is usd to alulat th distribution of intrior humidity in onrt. Th abov two modls ar xprimntally vrifid indpndntly by a sris of shrinkag and intrior humidity tsts on thr typs of onrts with diffrnt omprssiv strngth. Using th modls, th distribution of shrinkag strain in arly-ag onrt pavmnts undr th ondition of th pavmnt surfa suffring to dry is alulatd. 2. MODELING ON MOISTURE VARIATION INDUCED STRAIN In frsh onrt, all pors btwn mnt and othr solid partils ar initially filld with watr. Aftr stting of frsh onrt, a stiff sklton is formd and th hmial ontration produd by mnt hydration an not ompltly transfr to marosopi shrinkag of onrt. Thrfor, with ontinuation of mnt hydration, a numbr of apillary pors btwn mnt partils ar gradually formd and orrspondd mnisuss ar ratd to ompnsat th volum dras. Manwhil, th intrior humidity of onrt starts to dras from th initial saturatd stat of 1% du to th ontinuity of liquid watr gradually dstroyd with th formation of apillary pors. Thus, th dvlopmnt of intrior humidity of onrt (RH) at arly ags an b dsribd by a vapor saturatd stag with 1% rlativ humidity (stag I) followd by a stag that th rlativ humidity gradually rdud (stag II). Basd on th thory of apillary fors, th shrinkag strain dvlopd in stag I and stag II du to variation of watr ontnt rsultd ithr by mnt hydration or by nvironmntal drying may orrlatd with hmial shrinkag and intrior humidity rdution rsptivly as [4]: ( Vs Vs) for RH 1 S prt (1) 1 ( V ) s Vs ln( RH ) for RH 1 3M Ks K Whr is th influning fator of stiffnss, whih is normally a funtion of watr to mnt ratio. V s and V s ar th hmial shrinkag (in volum) at a givn mnt hydration dgr and at th point whr th intrior humidity starts to dras from 1% rsptivly. M is molar wight of watr (.182 kg/mol), is dnsity of watr and R is idal gas onstant (8.314 J/molK). K is bulk modulus of th whol porous body and K s is bulk modulus of th solid matrial. p is obtaind by introduing a paramtr k in th aumulat por volum as: xp( k r) (2) p 1 Whr is a paramtr rflting th influn of onrt ag (rfltd by mnt hydration dgr, ) on por volum and may b simulatd as =a, a and ar xprimntal dtrmind onstants. Paramtr k is obtaind by omparing modl and xprimntal rsults [5]. Paramtr S in (1) is alld saturation fration, S=V w /V p. V w is th vaporabl watr ontnt in th hardning

3 Cmntitious Composits, April 212, Amstrdam, Th Nthrlands mnt past, V p is th total por volum. S an b stimatd through Powrs volumtri modls [6-7]. Assuming th hydration dgr of mnt is, and th total volum of mnt partils and watr is 1, th phas omposition of a hardning Portland mnt past without silia fum addition, inluding hmial shrinkag V s, apillary por watr volum V w, gl watr volum V gw, gl solid volum V gs and unhydratd mnt volum V an b alulatd through th following quation: Vs.2(1 p) ; Vw p 1.3(1 p) ; Vgw.6(1 p) ; Vgs 1.5(1 p) ; V (1 p)(1 ) (3) w/ Whr Vi=1, p. w and ar th wight of watr and mnt rsptivly in w/ / w onrt mixtur. w and ar dnsity of watr and mnt rsptivly. So th saturation fration S an b alulatd by: p.7(1 p) S (4) p.5(1 p) Th abov quations an b usd only in th shrinkag alulation for onrt without silia fum appliation. For th onrt with silia fum addition, th phas omposition, inluding hmial shrinkag V s, apillary por watr volum V w, gl watr volum V gw, gl solid volum V gs and unhydratd mnt partils volum V as wll as silia fum volum V s an b stimatd by Jnsn and Hansn [7]: Vs k(.2.7s / )(1 p) ; Vw p k( s / )(1 p) ; Vgw k(.6 1.6s / )(1 p) ; (5) Vgs k(1.6.7s / )(1 p) ; V k(1 p)(1 ); Vs k(1.4s / )(1 p)(1 ) Whr V i =1, s, ar th wight of silia fum and mnt rsptivly. w/ 1 p, k. Thus, th saturation fration S for onrt w/ w / ( w / )( s / ) 11.4( s / ) with silia fum addition an b alulatd by: p.8k(1 p) S (6) p k.6.7( s / ) (1 p) Th mnt hydration dgr, an b alulatd from isothrmal tsts. By masuring th adiabati tmpratur ris of onrt at diffrnt tim, th mnt hydration dgr is stimatd by: Tad ( t) u (7) Tad ( ) Whr T ad (t) is th adiabati tmpratur rising at tim t, T ad ( ) is th ultimat adiabati tmpratur rising. u is ultimat dgr of hydration and is a funtion of watr to mnt ratio (w/) as [8]: 1. 31w w u (8). 194 To alulat th hydration dgr undr diffrnt tmpratur history, th quivalnt ag is usd. Th quivalnt ag onpt assums that sampls of a onrt mixtur of th sam quivalnt ag will hav th sam mhanial proprtis or mnt hydration dgr, rgardlss of th ombination of tim and tmpratur yilding th quivalnt ag. Basd on abov dfinition, th quivalnt ag t an b xprssd as t t 1 U ar U at R T Whr t is th quivalnt ag at th rfrn tmpratur (hr th rfrn tmpratur is qual to 2 o C is assumd). U ar and U at ar th apparnt ativation nrgy (J/mol) at rfrn and atual dt (9)

4 Cmntitious Composits, April 212, Amstrdam, Th Nthrlands tmpratur rsptivly. R is th univrsal gas onstant, 8.314J/molk. T is tmpratur in Clsius ( o C). Rgarding apparnt ativation nrgy, a numbr of rsarhrs hav onludd that it ould not b onsidrd as a onstant indpndnt of tim xpt during th bginning of mnt hydration [9-1]. Basd on ths findings, th apparnt ativation nrgy of onrt is xprssd as a funtion of tmpratur and uring tim as [11]:.17 U T t a T (1) Whr T is uring tmpratur ( o C) and t is uring tim in days. Du to th atual tmpratur T insid of onrt is varid with tim, it is onvnint to solv t q in matrix form instad of intgrating. If th uring tim is dividd into n stions and th tmpratur in ah tim intrval is assumd to b a onstant, thn w hav t n i1 1 U U ar at i R Ti Th stion numbr n may dpnd on th rquird auray and normally an b qual to th tim intrvals for tmpratur masurmnt. Basd on th quivalnt ag, th hydration dgr of mnt dfind in (8) an b simulatd by [11-12]: A u xp t (12) Whr A and B ar two mpirial onstants whih an b dtrmind by fitting isothrmal xprimntal rsults and quation (13). Undr drying ondition, th moistur ontnt in onrt will b lss than that undr sald stat and this moistur rdution will rdu th mnt hydration dgr. Th fft of moistur ontnt on mnt hydration should b takn into onsidration in th modl by [5]: B1 d B B u n P RH P t ln A ( ) (13) d Constants n and P an b dtrmind from intrior humidity masurmnts and isothrmal tsts. Thus for diffrnt drying pross, th fft of intrior humidity variation on mnt hydration an b stimatd by (13). Th mnt hydration dgr at givn tim t an b obtaind by intgrating (13) from to t. As shown in (1), lasti modulus of onrt is also an important paramtr for shrinkag alulation. Aftr stting th lasti modulus of onrt starts to grow from zro. Basd on th quivalnt ag, th dvlopmnt of lasti modulus of onrt with ag undr varid tmpratur and drying status an b stimatd by [13]: E E (14) u Whr is th hydration dgr at onrt stting. b is a onstant that an b dtrmind by fitting (14) with xprimntal data. Th shrinkag modl prsnt abov is basd on th formation of apillary pors and rsulting apillary strsss during th formation of mnt matrix sklton. Thrfor, th modl may b usd for moistur loss rsultd shrinkag prdition rgardlss if th moistur loss is ausd by mnt hydration or by nvironmntal drying. Furthr, by applying prsnt modl, th alulation of shrinkag distribution in onrt struturs in th as of humidity gradint xistd boms possibl. Th influns of uring onditions, inluding nvironmntal humidity and tmpratur on onrt shrinkag should b rfltd in th modl by mnt hydration dgr and intrior rlativ humidity. Th shrinkag modl is vrifid by xprimnts [5]. Th rlatd matrial paramtrs usd in th modl ar listd in Tabl 1. Fig.1 displays th mnt hydration dgr and quilibrium ag diagrams of th thr kinds of onrts with omprssiv strngth at 28 days of 34.1 MPa, 5. t t B i i1 b (11)

5 Cmntitious Composits, April 212, Amstrdam, Th Nthrlands Tabl 1. Paramtrs usd in shrinkag alulation Conrt C3 C5 C8 Hydration dgr paramtr Elasti modulus paramtr Por strutur paramtr η u A B E 28 (GPa) E s (GPa) b k a Hydration dgr of mnt Equivalnt ag (hours) Tst-C3 Tst-C5 Tst-C8 Fig.1 Rlationship of mnt hydration dgr and quivalnt ag Shrinkag(1-6 ) C5 Tst data-dry Tst data-sal Shrinkag(1-6 ) Shrinkag(1-6 ) C3 Tst data-dry Tst data-sal C8 Tst data-dry Tst data-sal Tim (days) (a) Tim (days) Tim (days) (b) () Fig.2 Comparison btwn modl and tst rsults on shrinkag of C3 (a), C5 (b) and C8 () onrts MPa and 88.7 MPa. Fig.2 prsnts th omparisons btwn modl prditions and xprimntal rsults for th thr kinds of onrts in trms of shrinkag-ag diagrams starting from onrt st to 28 days undr both saling and drying uring onditions. From th figur, w an obsrv that th modl an wll ath th haratristis of th dvlopmnt of shrinkag of onrt starting from st. Undr drying ondition, a high shrinkag is obtaind in th xprimnts and in modl prdition as wll. Th modl an prdit th shrinkag whatvr it is rsultd by drying or by

6 Cmntitious Composits, April 212, Amstrdam, Th Nthrlands mnt hydration. In addition, baus th modl ombins th fft of ag and position into a singl physial paramtr, RH, th modl an prdit shrinkag strain in onrt struturs not only for diffrnt tim, but also for diffrnt positions. Crtainly, in ordr to do so, th moistur distribution, rprsntd by rlativ humidity insid of onrt is rquird prior to using th modl. 3. MODELING ON THE MOISTURE DISTRIBUTION IN EARLY AGE CONCRETE Th loss of watr in arly-ag onrt is normally ausd by both of mnt hydration and watr diffusion. At th initial priod aftr onrt ast, most of th pors in onrt ar filld by liquid watr. Th rlativ humidity in onrt is almost qual to 1%. Du to th pross of watr onsuming is so slow that th priod with 1% humidity, whih is dfind as stag I in prsnt papr, an list quit long tim. Whn th watr ontnt in onrt pors drass to a ritial valu, at whih th vapor prssur boms lowr than th saturatd valu, th rlativ humidity starts to dras. Starting from this momnt, th dvlopmnt of intrnal humidity gos into th stag II. Hr w may dfin th lngth of stag I as th ritial tim t, whih is a funtion of both watr to mnt ratio and loation from asting surfa and an b dtrmind by xprimnts. In th stag II, th variation of watr ontnt (C) is rsultd from both mnt hydration (C s ) and watr diffusion to nvironmnt (C d ). Th driving for of watr diffusion should b th variation of (CC s ) with loation and tim. If on-dimnsional watr diffusion is onsidrd (along th oordinat dirtion x), aording to th sond Fik s law, th moistur ontnt balan rquirs ( C C ) ( C C D t x x s s) Whr paramtr D is th moistur diffusion offiint dpnding on th por humidity and on th omposition of onrt [14-17]. Both mnt hydration and diffusion through onrt is so slow that various phass of watr in ah por (vapor, apillary watr and absorbd watr) rmain almost in thrmodynami quilibrium at any tim [15]. Thrfor, th rlationship btwn humidity H and watr ontnt C an b rlatd by th wll-know dsorption or sorption isothrms [6, 15]. That mans th humidity (H) is a linar funtion of watr ontnt (C). Thus, quation (15) an b rwrittn as ( H H s ) ( H H s ) T D k (16) t x x t Hr, H s is th humidity rdution du to mnt hydration. Th third itm rflts th fft of tmpratur variation on th humidity. k is th hang in H du to on dgr hang in tmpratur T at onstant C and a fixd dgr of hydration. Exprimntal rsults hav indiatd that th hang in H du to tmpratur variation is rathr small (i.. k) that an b ngltd [13]. Thn, quation (16) boms ( H H s ) ( H H s) D (17) t x x Lt H =HH s, th abov partial diffrntial quation boms H H D (18) t x x For alulating th moistur distribution in onrt xposd to a givn atmosphr with initial ondition of 1% rlativ humidity, quation (18) must b solvd taking adquat boundary onditions and initial onditions into onsidration. Howvr, th rlativ humidity H is a funtion of tim (t) and loation (x) in onrt and th moistur diffusivity D is also a funtion of por humidity. Thus, th distribution of rlativ humidity along x dirtion an not b solvd from quation (18) dirtly. In ordr to ovrom this diffiulty, w dfin paramtr F as (15)

7 Cmntitious Composits, April 212, Amstrdam, Th Nthrlands F H H m DdH (19) Hr H m is a rlativ humidity that an b sltd arbitrarily, normally is qual to th minimum humidity that may ours in onrt. From (19), w hav F H Using abov quations in (18), w obtain F H D, D (2) t t F t 2 F D 2 x Thus, th problm for solving H undr givn tim and loation boms solving F from (21). Chmial ration btwn mnt and watr an lads rdution of watr ontnt (rprsntd by RH) also. And th magnitud of th humidity rdution rsultd from mnt hydration must b a funtion of hydration dgr. In th prsnt work, a modifid mnt hydration dgr basd modl, whih taks th initial liquid-watr saturatd stag (stag I) into aount, is usd to dsrib th humidity rdution du to mnt hydration as indiatd in quation (22). for H s (22) H s, u 1 1 for u Whr H s,u is th rlativ humidity onsidring slf-dsiation at ultimat dgr hydration, whih is a funtion of w/ and an b dtrmind from xprimnts. is a hydration paramtr alld ritial hydration dgr at whih th humidity insid of onrt starts to dras from 1% lvl, whih an b alulat by applying th xprimntal dtrmind ritial tim t in quation (12). Th paramtr is a onstant. Using th dvlopd modl, w ar abl to obtain th omplt humidity distribution fild in arly-ag onrt. To vrify th modl, th dvlopmnt of th humidity insid of onrt is xprimntally dtrmind. In th xprimnts, on dimnsional hat and moistur transportation in onrt is ratd. Watrproof plywood mold with innr dimnsions of 228 mm was usd. To allow hat and moistur movmnt only along th spimn thiknss dirtion, th innr surfas of th mold wr ovrd with plasti sht to prvnt moistur lost and th fiv outr surfas was ovrd with polystyrn board to prvnt hat lost. Only th asting fa was kpt to ontat with air dirtly. In th tsts, a digital humidity and tmpratur ombind snsor was usd to masur th humidity and tmpratur. Dtaild spimn prparation and tst produrs an b found in [13]. Manwhil, th humidity distributions of th onrt slabs ar alulatd using th dvlopd modl. Th humidity dpndnt diffusivity of C3, C5 and C8 onrts usd in th modl is shown in Fig.3, whih is dtrmind from xprimnts [17]. Th othr rlatd paramtrs usd in th modl ar list in Tabl 2. Fig.4 displays th omparisons btwn modl prditd humidity profils and xprimntal rsults at som typial ags. Th graph shows that th modl and xprimntal rsults agr wll with ah othr. Tabl 2. Input paramtrs usd for humidity fild alulation Conrt H s,u β a m (m/day) C C C (21)

8 Cmntitious Composits, April 212, Amstrdam, Th Nthrlands Diffusion offiint (1-9 m 2 /s) D-C3 D-C5 D-C RH(%) Fig.3 Watr diffusion offiint usd in th modl RH(%) (a) C3 Tst-72h Tst-168h Tst-336h Tst-672h RH(%) C5 Tst-72h Tst-168h Tst-336h Tst-672h RH(%) 6 C8 4 Tst-72h Tst-168h Tst-336h 2 Tst-672h (b) () Fig.4 Comparisons btwn prditd humidity profil and xprimntal rsults of C3 (a), C5 (b) and C8 () onrt slabs 4. SHRINKAGE STRAIN IN CONCRETE PAVEMENT As an xampl of appliation of abov modls, th distribution of shrinkag strain in onrt pavmnts mad of C3, C5 and C8 onrt rsptivly is alulatd. A simpl diagram illustrating th strutur of onrt pavmnt is shown in Fig.5. As alulating sampl, assum th slab was ast in spring morning that should influn th dvlopmnt of tmpratur insid of th slabs. Baus th tmpratur within th slab in arly-ag is ritially ndd to alulat th dgr of mnt hydration, th dvlopmnt of tmpratur insid th pavmnt slab was alulatd first in th modling. Aftr th dvlopmnt of tmpratur insid of onrt pavmnt Thrmal onvtion Wind spd Solar radiation Pavmnt lb Poor onrt bas 25 3m Soil bas Fig.5 Conrt pavmnt modl for humidity fild alulation

9 Cmntitious Composits, April 212, Amstrdam, Th Nthrlands is known, th shrinkag strain indud by mnt hydration and nvironmntal drying an thn b alulatd. Fig.6 prsnts th modl rsults of th dvlopmnt of intrior humidity and orrsponding shrinkag strain at diffrnt plas from top to bottom of th pavmnts mad of C3, C5 and C8 onrts rsptivly. From th rsults, first w an obsrv that th dvlopmnt of intrior humidity insid of onrt with ag obys th two stag mod, that is a vapor saturatd stag with 1% rlativ humidity (stag I) and a stag with th rlativ humidity gradually drasing (stag II). Th humidity gradint along th slab dpth is signifiant and is varid with ag. Undr th ondition that th slab surfa undrgos drying, th lngth of stag I inrass with th loation from th slab top. Sond, th shrinkag strain is wll rlatd with intrior humidity. Within th stag I, a uniform shrinkag strain is xptd throughout th slab. By ontrast, th shrinkag gradint along th slab dpth is quit obvious in th stag II and th maximum and minimum shrinkags our at slab top and bottom rsptivly. That is baus th humidity gradint starts to our in this stag and th maximum and minimum humidity rdution appars at slab top and bottom rsptivly at th momnt. Th rat of shrinkag dvlopmnt is gradually rdud from slab top to bottom in this stag, maning that th fft of surfa drying is onfind within a rtain rang. Th distribution of shrinkag strain along th C3, C5 and C8 onrt slab at som typial ags is displayd in Fig.7. Clarly, th shrinkag distribution along th pavmnt dpth is apparntly nonlinar. With dvlopmnt of ag, th shrinkag gradint is vn pronound. Conrt strngth an signifiantly influn th magnitud of shrinkag strain as wll as its distribution in th slab. For a givn ag and loation, th high th onrt strngth, th largr th shrinkag strain and th gratr th shrinkag gradint. Hr w should not that th shrinkag at slab bottom is los to th magnitud of autognous shrinkag of onrt and at th slab top surfa is a rsult of a ombination of autognous and drying shrinkag. Apparntly, high strngth onrt will rsult in high shrinkag strain as wll as high shrinkag gradint in onrt mmbrs. Shrinkag(1-6 ) C3-spring-8: Shrinkag RH,1, 2, 5, 25m Shrinkag(1-6 ) C5-spring-8: Shrinkag RH,1, 2, 5, 25m Tim (hours) (a) Tim (hours) 1 (b) Shrinkag(1-6 ) C8-spring-8: Shrinkag RH,1, 2, 5, 25m Tim (hours) () Fig. 6 Dvlopmnt of shrinkag strain at diffrnt loations in onrt pavmnt, (a) C3, (b) C5 and () C8

10 Cmntitious Composits, April 212, Amstrdam, Th Nthrlands Spring-C3-8: 24h 48h 72h 12h 168h Spring-C5-8: 24h 48h 72h 12h 168h Shrinkag (X1-6 ) (a) Shrinkag (X1-6 ) (b) Shrinkag (X1-6 ) Spring-C8-8: 24h 48h 72h 12h 168h () Fig.7 Distribution of shrinkag strain along th slab dpth at som typial ags, (a) C3, (b) C5 and () C8 In addition, w may b notd also from Fig.7 that th influning sop of surfa drying is a funtion of onrt strngth and ag. For givn ags, highr strngth onrt has a shortr influning dpth of drying. Th influning dpth inrass with ag for all thr kind of onrt in arly-ag. In th viw of durability dsign of onrt struturs, all abov haratristis rlatd to shrinkag ourrd in onrt mmbrs should onsidrably b takn into aount. 5. SUMMARY AND CONCLUSIONS In this papr, an intgrativ modl for autognous and drying shrinkag prditions of onrt at arly-ag is first introdud. Sond, a modl taking both mnt hydration and moistur diffusion into aount synhronously is usd to alulat th distribution of intrior humidity in onrt pavmnt. Using th abov two modls, th distribution of shrinkag strain in onrt pavmnts at arly-ag from onrt ast is alulatd and analyzd. Thr kinds of onrts wr usd in th alulation to invstigat th ffts of onrt strngth. Th modl rsults show that th dvlopmnt of intrior humidity insid of pavmnt sin onrt ast obys two stag mod, that is a vapor saturatd stag with 1% rlativ humidity (stag I) and a stag with th rlativ humidity gradually drasing (stag II). Undr th ondition that th pavmnt undrgos surfa drying, th lngth of stag I inrass with loation from th slab top. Within stag I, a uniform shrinkag strain is xptd throughout th slab. By ontrast, shrinkag gradint along th slab dpth is quit signifiant in th stag II and th maximum and minimum shrinkag our at slab top and bottom rsptivly. Th distribution of shrinkag strain along th pavmnt dpth is nonlinar and th nonlinarity is strong los to th drying ara. Conrt strngth an signifiantly influn th magnitud of shrinkag strain as wll as its distribution in slab. For givn ag and loation, th high th onrt strngth, th largr th shrinkag strain and th gratr th shrinkag gradint.

11 Cmntitious Composits, April 212, Amstrdam, Th Nthrlands ACKNOWLEDGEMENTS This work has bn supportd by a grant from th National Sin Foundation of China (No ) and a grant from National Basi Rsarh Program of China (No. 29CB6232) to Tsinghua Univrsity. REFERENCES [1] Ayano, T., Wittmann, F.H., Drying, moistur distribution, and shrinkag of mnt-basd matrials, Matrials and Struturs, 35(247) (22) [2] Bissonntt B, Pirr P, Pigon M., Influn of ky paramtrs on drying shrinkag of mntitious matrials, Cmnt and Conrt Rsarh, 29(1) (1999) [3] Zhang, J., Hou, D. W. and Sun, W., Exprimntal study on th rlationship btwn shrinkag and intrior humidity of onrt at arly ag, Magazin of Conrt Rsarh, 62(3) (21) [4] Zhang, J., Hou, D. and Chn, H., Exprimntal and thortial studis on shrinkag of onrt at arly-ags, ASCE Journal of Matrials in Civil Enginring, 23(3) (211) [5] Zhang, J., Hou, D. and Han Y., Miromhanial modl of autognous and drying shrinkags of onrt, In prss, Building and Constrution Matrials, 211. [6] Powrs, T.C. Brownyard, T.L., Studis of th physial proprtis of hardnd Portland mnt past (nin parts), J. Am. Conr. Inst. 43, Bulltin 22, Rsarh Laboratoris of th Portland Cmnt Assoiation, Chiago, 1946, [7] Jnsn, O.M. and Hansn, P.F., Watr-ntraind mnt-basd matrials: I. Prinipls and thortial bakground, Cmnt and Conrt Rsarh, 31(5) (21) [8] Rastrup, E., Hat of Hydration, Magazin of Conrt Rsarh, 6(17) (1954) [9] Kjllsn, K. and Dtwilr, R.J., Latr-ag strngth prdition by a modifid maturity modl, ACI Matrials Journal, May-Jun, (1993) [1] Chanvillard, G. and Daloia, L., Conrt strngth stimation at arly ag: Modifiation of th mthod of quivalnt ag, ACI Matrials Journal, 94(6) (1997) [11] Kim, J.K., Estimation of omprssiv strngth by a nw apparnt ativation nrgy funtion, Cmnt and Conrt Rsarh, 31(2)(21) [12] Pan, I. and Hansn, W., Conrt hydration and mhanial proprtis undr nonisothrmal onditions, ACI Matrials Journal, 99(6) (22) [13] Zhang, J., Qi, K. and Huang, Y., Calulation of moistur distribution in arly-ag onrt, ASCE Journal of Enginring Mhanis, 135(8) (29) [14] Akita, H., Fujiwara, T. and Ozaka, Y., A pratial produr for th analysis of moistur transfr within onrt du to drying, Magazin of Conrt Rsarh, 49(179) (1997) [15] Bazant, Z.P., and Najjar, L.J., Nonlinar watr diffusion in nonsaturatd onrt, Matrials and Struturs, 5(25) (1972) 3-2. [16] Nilsson, L.O., Long-trm moistur transport in high prforman onrt, Matrials and Struturs, 35 (22) [17] Zhang, J., Hou, D. Gao Y. and Sun, W., Dtrmination of moistur diffusion offiint of onrt at arly-ag from intrior humidity masurmnts, Drying Thnology, 29(6) (211)