Analysis of balanced double-lap joints with a bi-linear softening adhesive

Size: px

Start display at page:

Download "Analysis of balanced double-lap joints with a bi-linear softening adhesive"

Brianna Melton
5 years ago
Views:

1 Fratur Mhani Conrt and Conrt Strutur - Ant, Durability, Monitoring and Rtritting Conrt Strutur- B. H. Oh, t al. (d 00 Kora Conrt ntitut, Soul, SBN Analyi baland doubl-lap joint with a bi-linar tning adhiv C. S. Hann & H. Stang Dpartnt Civil Enginring, Thnial Univrity Dnark, Kg. Lyngby, Dnark J. W. Shidt Dpartnt Civil Enginring, Thnial Univrity Dnark, Kg. Lyngby, Dnark COW A/S, 800 Kg. Lyngby, Dnark ABSTRACT: Th rpon a bondd ytri baland doubl lap joint undr tnil loading with a bilinar tning adhiv i dribd with a lod olution. Sin bondd joint in onrt trutur undrgo tning, a vratil odl to drib th rpon a wid rang ontitutiv rlationhip i ndd. A ontitutiv rlationhip ontaining a bi-linar tning law ontain uh vratility. Th olution wa invtigatd odrat and tr tning paratr. Th olution tr tning paratr, hibitd a non-phyial bhavior, whr th iz th tr-fr rak drad inraing iz fratur pro zon. Thi uggt that in ordr to fully drib th loading and unloading rpon, an unloading law hould b iplntd in th ontitutiv odl. Apart fro adhivly bondd talli joint, th prnt olution ay b ud in analyi rakd onrt dik trngthnd with adhiv bondd fibr rind polyr (FRP, or in any othr trutur oparabl to a doubl lap joint with a tning intrfa. Th prnt ontitutiv odl an b hangd to fit any odl with th a hap ontitutiv rlationhip, Figur. NTRODUCTON. Gnral Sin bondd lap joint in onrt trutur tn hibit tning bhavior, th prdition th rpon rli on th ability to odl th tning in a ontitutiv law. n an analyi FRP or tl bondd to a onrt urfa, th typial tning law ud i linar. Suh a odl rul out odling or opl tning branh obtaind fro tt. Thi ould b ponntial tning, uddn drop or dutil tning bhavior. Th ability to analyz th bhavior bi-linar tning adhiv in joint i iportant, in a ipl bi-linar bhavior an b ud a approiation opliatd tning bhavior prind in tt. With a bi-linar tning rlationhip, it i poibl to approiat both onv and onav dnding tning branh. Thi papr analyz a ytri baland doubl lap joint with a bi-linar tning adhiv uing a ipl joint thory, firt dribd in Volkrn (938. Volkrn aud linar lati atrial, adhiv loadd and dd in har only and adhrnt allowing only aial dation. Etnion to Volkrn odl uh a tning and hardning bhavior hav bn propod in, and a vat aount work on tr analyi and bond and anhorag odl it bad upon it. S a apl, Hart-Sith (973, Ranih (98, Gutafon (987, Pihlr (993, Täljtn (995, Chn & Tng (00, Yuan t al. (004. Many th papr ak u a ohiv law or a bond-lip rlationhip a ontitutiv odl in uh data i ay to obtain fro tt. Howvr, in prinipl it i not iportant how th ontitutiv law th intrfa atrial i dfind in bond-lip or diffrnt tning har oduli ar jut diffrnt way dfining th a typ bhavior. Otton & Olon (988 invtigatd th rpon a ytri, baland doubl lap joint with linar hardning and tning uing th a auption a in Volkrn (938. Modling th adhiv wa don traditionally uing diffrnt har odulu th lati and hardning/tning tiffn. So iportant paratr dribing th failur a joint with tning bhavior wr idntifid, and dign tratgi wr uggtd. Thi papr tnd th olution by Volkrn and Otton & Olon (988 to inlud bi-linar tning. Fou i on alulation th diffrnt tag a joint undrgo fro firt applid load to failur. Th analyi rult in pliit ula poition tning initiation at ond tning branh, fratur initiation and load at diffrnt tning tag. Th ay a to uh iportant data and th ability to undrtand and invtigat all paratr i

2 otivation J = D ( h, T h analy lik thi, intad tudi ( prd with a finit lnt odl. Eapl ar Th givn proportionality in Stion 4. offiint D(h,T i alld oitur prability and it i a nonlinar funtion th rlativ huidity h and tpratur T (Bažant & Najjar GOVERNNG 97. Th DFFERENTAL oitur a EQUATONS balan rquir that th variation in ti th watr a pr unit. volu Matrial onrt (watr ontnt w b qual to th Th divrgn outr adhrnt th oitur ar arkd flu J with a ubript and th innr adhrnt with. S Figur a. Thy ar both aud linar lati with = J ( Young t odulu E and E. Thi rult i th rlation Th σwatr ( ontnt w an b prd a th u ε ( = ( th vaporabl E watr w (apillary watr, watr vapor, and adorbd watr and th non-vaporabl σ ( (hially ε ( = bound watr w n (Mill 966, ( E Pantazopoulo & Mill 995. t i raonabl to au that th vaporabl watr i a funtion whr rlativ ε, huidity, σ and ε, h, σ dgr dnot th hydration, train and tr in, and ah dgr adhrnt. ilia Th fu adhiv ration, layr i dribd uing, i.. w =w (h,, a = har ag-dpndnt odulu G and orption/dorption a thikn t a. Th ubript iothr (Norling th har Mjonll odulu 997. i dpndnt Undr thi on auption th adhiv and train by ubtituting and i dfind in th linar into lati tag a: on G 0 < γ γ, firt tning tag a: G γ < γ γ and ond tning tag a: G γ < γ γ f, whr γ and γ dnot th har train at tning initiation h tag on and two, and γ f dnot th + w& n (3 failur h train t th h adhiv. Th har tr at ont tning tag two i dfind a. S Fig- ur.. whr /h i th lop th orption/dorption iothr (alo alld oitur apaity. Th govrning quation ( 3 ut b opltd by appropriat boundary and initial ondition. Th rlation btwn th aount vaporabl G watr and rlativ huidity i alld adorption iothr if aurd G with inraing rlativity huidity and dorption iothr G in th oppoit a. Nglting thir diffrn (Xi t al. 994, in th following, orption iothr will b ud with γ γ γ f γ rfrn to both orption and dorption ondition. By th way, if th hytri th oitur Figur. Contitutiv rlationhip. iothr would b takn into aount, two diffrnt rlation, a vaporabl L/watr v L/ rlativ huidity, ut b ud aording to th ign th variation th rlativity P/huidity. Th hap th orption iothr HPC i inflund by any P paratr, pially P/tho that influn tnt and rat th hial ration and, in turn, dtrin por trutur b and por iz ditribution (watr-to-nt ratio, P/nt hial opoition, SF ontnt, uring ti and thod, tpratur, i additiv, P t.. n th litratur variou ulation an b P/ found to drib th orption iothr noral onrt (Xi t al Howvr, in th prnt Figur. a Gotry, dfinition and boundary ondition a papr doubl th lap joint. i-pirial b Dd hap prion joint. propod by Norling Mjornll (997 i adoptd bau it pliitly aount u ( th volution u ( + hydration σ σ + σ ration Eand, t SF ontnt. Thi orption iothr rad G γ(, t a ( γ( + σ E, t σ + σ w ( h,, = G (, + ( u ( 0 g h u ( + (4, u Figur 3. Equilibriu, dation 0( gand atrial h in a joint. K (, Th tning rlationhip i bad upon a trlip rlationhip and ulatd a a tr-train rlationhip whr th firt onvnin. tr (gl Thi iothr ipli rprnt that th tning phyially tr-train bound (adorbd rlationhip watr ut and b th hangd ond if th th tr adhiv (apillary thikn iothr hangd. rprnt th apillary watr. Thi prion i valid only low ontnt SF. Th offiint G rprnt th aount. Kinati ondition watr pr unit volu hld in th gl por at 00% Th rlativ rlationhip huidity, btwn and it an dation b prd and (Norling train ay Mjornll b dribd 997 a whn auing all train, by du ( G ( (, = k + k vg vg ε = (3 (5 du ( ( ε = (4 whr k vg and k vg ar atrial paratr. Fro th aiu ( aount watr pr unit volu that an γ = ( u( u( fill all por (both apillary por and gl por, (5 on ta an alulat K a on whr th diffrnt paratr ar dfind in Figur 3. Th adhiv layr i aud to b 0 gloadd and dd in pur 0 har. w G (6 K (, = 0.3 Equilibriu g Th horizontal quilibriu i obtaind by uation all Th atrial ating paratr in th infinitial k vg and k vg lnt, and g an Figur b alibratd 3. Not by that fitting only printal half th lnt data rlvant i hown. to Equilibriu fr (vaporabl provid: watr ontnt in onrt at variou ag (Di Luzio & Cuati 009b. dσ( t dσ ( t + ( = 0 (6. Tpratur volution ( = 0 (7 Not that, at arly ag, in th hial ration aoiatd with nt hydration and SF ration ar Sin othri, har th tr tpratur i rlatd fild i to not th uni har train non-adiabati γ by ( = γ(g yt, w vn obtain if th by nvironntal diffrntiation tpratur thi prion i ontant. with rpt Hat to ondution and inrtion an b dribd in (-(4. onrt, at lat tpratur not ding 00 C (Bažant & Kaplan 996, by Fourir d ( G law, σ( whih σ( rad = (8 t E a E q = λ T (7 Diffrntiation (8 again and inrtion th quilibriu whr q i th hat (6 flu, and (7 T i yild th th abolut diffrntial tpratur, quation and λ i th hat ondutivity; in thi Proding FraMCoS-7, May 3-8, 00

3 , d ( G ω 0, ω + t a Et Et λ ω = = = (9 a < b b* * d f g < f f f L/ L/ L/ L/ L/ L/ L/ L/ L/ Figur 4. Shati ovrviw th diffrnt tag a-g during bi-linar tning a baland joint: a Elati, b Elati-tning(, b* Fully tning(, Elati-tning(+, * Fully tning(+, d Ont failur at Elati-tning (+, Elati-tning(+-failur, f Stning(+ failur, g Unloading. At = -L/ it i Jknown = D ( h, that T σ h = P/(t w and σ = 0, whr t i th thikn, w i th width th joint and P i th applid Th. proportionality nrtion in offiint (8 D(h,T and applying th a oitur thod prability = -L/ and whr it i a σnonlina = P/(t w and σ = 0 th yild rlativ huidity h and tpratur & Najjar 97. Th oitur a balan d PG L = that = th variation in ti th (0 watr a t we a t volu onrt (watr ontnt w b q d PG divrgn L th oitur flu J = = ( t we a t w = J Sin w ar analyzing t a baland ytri joint, E t = E t, ( hold and w dfin th uful ontant QTh. watr ontnt w an b prd a th vaporabl watr w (apillary wa vapor, and adorbd watr and th non- d PG PG L = Q = (hially = bound = watr ( w n (Mil tawe t tawet Pantazopoulo & Mill 995. t i ra au that th vaporabl watr i a fu n th thoing rlativ olution huidity, w will h, dgr only invtigat poitiv -valu dgr in ilia th joint fu i ytri. ration,, i.. w =w hydration = ag-dpndnt orption/dorption (Norling Mjonll 997. Undr thi au 3 JONT ANALYSS by ubtituting into Equati Th rpon th joint i alulatd by uing appropriat boundary ondition and opatibility ondition. Th olution h h will loly follow th + w tag a-g hown in hfigur t 4, and h at th a ti plain th tag in qution. whr /h i th lop th orption/ 3. a Elati tag( iothr (alo alld oitur apa govrning quation ( 3 ut b n tag a th har tr in th joint ha not yt by appropriat boundary and initial onditi rahd th aiu har tr,. Th olution Th rlation btwn th aount th govrning (9 i thn givn in th watr and rlativ huidity i alld intrval 0 L/ a iothr if aurd with inraing huidity and dorption iothr in th ( = Cinh ( λ + C oh( λ a. Nglting thir diffrn (3 (Xi t al. th following, orption iothr will b Du to ytry rfrn th to hyprboli both orption in and funtion, dorption C=0 and th following By th boundary way, ondition if th hytri applid: th iothr would b takn into aount, two d L rlation, vaporabl watr v rlativ hui = Q = (4 b ud aording to th ign th varia rlativity huidity. Th hap th Th olution yild iothr HPC i inflund by any p pially tho that influn tnt and Q oh ( λ hial ration and, in turn, dtr ( = (5 trutur L and por iz ditribution (watrratio, nt hial opoition, SF λ inh λ uring ti and thod, tpratur, i t.. n th litratur variou ulatio Th aiu thi tag i obtaind by found to drib th orption iothr tting (L/ = and applying th prion onrt (Xi t al Howvr, in th Q, (5. W obtain: papr th i-pirial prion pro Norling Mjornll (997 i adoptd b Proding FraMCoS-7, May 3-8, 00

4 J = D ( h, T h L P λtawe t tanh λ G = (6 ( w Th i th proportionality width th joint. offiint D(h,T i alld oitur prability and it i a nonlinar funtion th rlativ huidity h and tpratur T (Bažant 3. & Najjar b Elati-tning( 97. Th oitur a balan rquir n that tag th variation b th har in ti tr in th th watr joint a ha rahd pr unit th volu aiu onrt har (watr tr, ontnt, at a w point b qual dfind to th a divrgn. Th olution th oitur th govrning flu J (9 i th a typ a (3. Uing th a ytry ondition a in a and applying th boundary ondition = J ( t ( = (7 Th watr ontnt w an b prd a th u th th rpon vaporabl th watr trutur w (apillary i givn in th watr, intrval watr vapor, and adorbd watr and th non-vaporabl 0 (hially < a bound watr w n (Mill 966, Pantazopoulo & Mill 995. t i raonabl to au that th vaporabl watr i a funtion oh( λ rlativ ( = huidity, h, dgr hydration, (8 oh( λ, and dgr ilia fu ration,, i.. w =w (h,, = ag-dpndnt orption/dorption iothr (Norling Th rpon Mjonll in 997. th intrval Undr thi auption L/ ut and b olvd by ubtituting uing diffrntial into (9. Th root on th haratriti quation will b iaginary in G and G ar ngativ. Th olution i givn a ( C h 3in( λ C4o( λ + D h = h t h = + (9 ( & + & + w& n (3 By applying boundary ondition whr /h i th lop th orption/dorption iothr (alo alld d oitur apaity. Th ( = = and = Q, = L (0 govrning quation ( 3 ut b opltd by appropriat boundary and initial ondition. w Th obtain rlation th olution btwn th aount vaporabl watr and rlativ huidity i alld adorption iothr if aurd λ o ( λ ( L + with inraing Q in ( λ ( rlativity huidity ( = and dorption iothr in th oppoit ( a. Nglting λ o ( λ ( L thir diffrn (Xi t al. 994, in th following, orption iothr will b ud with rfrn Thi olution to both i orption valid until and th dorption har tr ondition. rah at By a th ditan way, if = th, whr hytri th tning th oitur odulu hang. iothr Thi would ditan b takn an into b found aount, by two itration, diffrnt danding rlation, vaporabl th prion watr in v ( rlativ qual huidity, to ut and olving b ud aording to th ign th variation th rlativity huidity. Th hap th orption iothr HPC i inflund by any paratr, L pially = λ o λ + tho that influn tnt and rat th hial ration and, in turn, dtrin ( por trutur and por iz L Q in ( λ ( ditribution (watr-to-nt λ o λ ratio, nt hial opoition, SF ontnt, uring ti and thod, tpratur, i additiv, t.. Q n i dtrind th litratur by variou applying ulation a ontinuity an ondition found at to σ b drib and σ th both orption th lati iothr and tning noral id onrt th (Xi olution t al. in 994. Howvr, (8. W in obtain th prnt papr th i-pirial prion propod by Norling Mjornll (997 i adoptd bau it pliitly aount th volution hydration d d, ration = and SF ontnt. = Thi orption iothr (3 G G rad Fro thi rlationhip, it i now poibl to olv Q. w ( h,, = G (, + 0( g h λ Q = λ o ( λ( L tanh ( λ (4 λ (4 0( g h + in ( λ( L K (, whr th rlationhip G /G = -λ /λ ha bn ud. whr Th th firt i tr dtrind (gl iothr by rprnt (. th phyially bound (adorbd watr and th ond 3.3 tr b* (apillary Stning( iothr rprnt th apillary watr. Thi prion i valid only low ontnt n tag SF. Th b* tning offiint ha Grahd rprnt = 0 th b aount tag tning watr pr unit ha volu bgun. Thi hld in phnonon th gl por i at uually 00% n rlativ in vry huidity, hort joint, and it or an joint b prd with a t (Norling adhiv. Mjornll Whn 997 th ntir a olution i dfind to b in th tning rgion, th th olution i givn in (9. Uing th a ytry ondition a G ( in tag, = a kand applying + k th boundary ondition (5 vg d = Q, = L (5 whr k vg and k vg ar atrial paratr. Fro th aiu aount watr pr unit volu that an fill all por (both apillary por and gl por, on an th alulat rpon K th trutur in th intrval 0 a on L/ yild Q o ( λ 0 g ( = w G 0 L λ in λ (, K = 0 g 3.4 Elati-tning(+ vg (6 (6 Th atrial paratr k vg and k vg and g an n b tag alibratd th by intrval fitting 0 printal < i data till lati rlvant and to th fr olution (vaporabl in watr (8 ontnt an b in ud. onrt at variou Th intrval ag (Di Luzio & < Cuati, whr 009b. wa dtrind fro (, tning i in tag. Boundary th following boundary ondition ar applid. Tpratur to th gnral volution olution, (9 Not that, at arly ag, in th hial ration ( = = and ( = = aoiatd with nt hydration and SF ration (7 ar othri, th tpratur fild i not uni Th non-adiabati olution yild yt vn if th nvironntal tpratur i ontant. Hat ondution an b dribd inin ( λonrt, ( at lat in ( λ ( tpratur not ( ding = 00 C (Bažant & Kaplan 996, (8 by in ( λ( Fourir law, whih rad q = Th λ T lat intrval L/ i in tag. Thi (7 part th olution i obtaind by uing th following boundary ondition in (9 whr q i th hat flu, T i th abolut tpratur, and λ i th hat ondutivity; in thi Proding FraMCoS-7, May 3-8, 00

5 d ( and Q, L ( = = = = (9 Th rpon yild = + L Q λ o ( λ ( in( λ ( L λ o ( λ ( (30 To obtain th orrt valu Q and ooth tranition btwn th two tning tag, th ontinuity btwn σ and σ in (8 rquir d d =, = G G (3 Diffrntiating (8 and (30, inrting into (3 w obtain Q λ λ o ( λ ( L λ = + in ( λ( ot ( λ( tan ( λ ( L (3 Solving Q and uing th rlationhip in ( to olv th P, giv P a t we t Q = (33 G 3.5 * tning(+ n tag * = 0 and tag tning ha bgun, but failur nar = L/ i not initiatd. Th whol joint i thrby in th tning tat. Th olution in th intrval 0 < ay b alulatd by applying ytry lading to C3 = 0. Th othr boundary ondition in (9 i = = (34 ( ( Th olution yild o ( λ = (35 o ( λ For L/ th olution in (30 i till uabl. Th aiu in th joint an b obtaind uing (3, tting = d Elati-tning(+, ont failur. Th olution Stag d i idntial to th on in Stag. Thi tag i howvr J = D ( h, intrting T h in failur and growth a aro (tr-fr rak bgin at L/. Growth th aro Th rak proportionality i dribd in Stion offiint 3.7. D(h,T Th aial load oitur arrid prability by th joint and in thi it i tat a nonlina i alulatd by danding th rlativ (L/ huidity = 0 in h and tpratur (30. Rduing th prion & Najjar and 97. olving Th Qoitur giv u: a balan that th variation in ti th watr a λ Q = volu onrt (watr ontnt (36 w b q in ( λ ( L divrgn th oitur flu J Applying (, th in tag d an = J b writtn a t tawe t Th λ watr ontnt w an b prd a Pd = (37 G in ( λ th ( L vaporabl watr w (apillary wa vapor, and adorbd watr and th non- (hially bound watr w To dtrin th aial load at ont failur, n (Mil Pantazopoulo & Mill 995. t i ra th ditan orrponding to (L/ = 0, ha to b au that th vaporabl watr i a fu known. Th olution thi a i not dirtly availabl in lod, and i thr obtaind by it- rlativ huidity, h, dgr hydration dgr ilia fu ration, ration. A ipl itration h an b tup, i.. w a =w = ag-dpndnt orption/dorption follow. f w quat (30 to 0 at L/, w (Norling Mjonll 997. Undr thi au obtain th following prion: by ubtituting into Equati L λ = + in (38 Q λ h + w h t h Thi quation i ud along with (36 to alulat prion, P d and Q. an b obtaind fro whr (8 /h i to th plot lop th full th rpon. iothr (alo alld oitur apa orption/ govrning quation ( 3 ut b by appropriat boundary and initial onditi 3.7 Elati-tning-failur Th rlation btwn th aount Th ont th watr aro and rak rlativ will tart huidity at = i L/ alld whn = 0. A faild, iothr tr fr if aurd rgion will with now tart inraing to grow fro th huidity nd. Th and ditan dorption to thi nd iothr dfind a f. Thi ditan a. Nglting an b dtrind thir diffrn by firt (Xi t al. in th olving th diffrntial th following, quation, orption and ondly iothr ap-wilplying ontinuity ondition rfrn to to both part orption th olution. and dorption b On part th By joint th i till way, lati if th in th hytri intrval 0 th < and th iothr olution would in b takn (8 into i aount, till two uabl. rlation, vaporabl watr v rlativ hui n th intrval b ud < aording, whr to th wa ign dtrind fro rlativity (, huidity. th rpon Th an hap till b th th varia dtrind by iothr (8. HPC i inflund by any p Solution th pially lat intrval tho that influn f progr tnt and a prviouly, whr hial boundary ration ondition and, in turn, olution trutur (9 i and por iz ditribution (watr- dtr ratio, nt hial opoition, SF uring d ti and thod, tpratur, ( = = and = Qf, = f (39 i t.. n th litratur variou ulatio found to drib th orption iothr Hr Q f i th onrt firt drivativ (Xi t al. har 994. tr Howvr, at f. in th Th olution yild papr th i-pirial prion pro Norling Mjornll (997 i adoptd b Proding FraMCoS-7, May 3-8, 00

6 J = D ( h, T h λ o ( λ ( f + Qf in ( λ ( ( = λ o ( λ ( ( (40 Th proportionality offiint f D(h,T i alld oitur prability and it i a nonlinar funtion To th find rlativ th ditan huidity f h w and ut tpratur dand har T (Bažant tr to & b Najjar zro 97. at f. nrting Th oitur ( f = a 0 in balan rquir (40 w obtain that th th variation prion in ti Q f th to b watr a pr unit volu onrt (watr ontnt w b qual to th divrgn th λ oitur flu J Q f = (4 in ( λ ( f = J ( t To obtain ooth tranition btwn th two tning Th tag, watr ontnt th ontinuity w an b btwn prd σ a and th σu in th vaporabl (8 rquir watr w (apillary watr, watr vapor, and adorbd watr and th non-vaporabl (hially d d = bound watr, = w n (Mill 966, (4 Pantazopoulo G G & Mill 995. t i raonabl to au that th vaporabl watr i a funtion rlativ Diffrntiating huidity, h, dgr (8 hydration, and (40, inrting, and Qdgr f fro ilia fu (4 ration, and rarranging,, i.. w =w yild (h,, = ag-dpndnt orption/dorption iothr (Norling Mjonll 997. Undr thi auption and by ubtituting o ( λ ( into on λ ot f = + (43 λ in ( λ( λ h + w& n (3 h t h whr G /G = λ /λ ha bn ud. Th rpon in (40 i thn alulatd at a valu by alulating whr /h f fro i th lop th (43 orption/dorption followd by dtrination iothr (alo Q f fro alld oitur (4. apaity. t i thn poibl govrning to alulat quation th ( rpon. Th 3 ut b an opltd b ob- Th taind by appropriat by uing boundary rlationhip and initial ondition. Th rlation btwn th aount vaporabl watr Pand f G rlativ huidity i alld adorption Q, f = = f (44 iothr ta we tif aurd with inraing rlativity huidity or and dorption iothr in th oppoit a. Nglting thir diffrn (Xi t al. 994, in tawe t λ th Pf = following, orption iothr will b ud (45 with G in ( λ ( f rfrn to both orption and dorption ondition. By th way, if th hytri th oitur iothr would b takn into aount, two diffrnt 3.8 rlation, f tning(+-failur vaporabl watr v rlativ huidity, ut n b tag ud f, aording th ditan to th ign ha jut th rahd variation zro and th (0 rlativity =. n huidity. th intrval Th 0 hap < th th rpon orption i dtrind iothr fro HPC th i inflund a ula by any a in paratr, tag *, pially (35. tho that influn tnt and rat th hial n th intrval ration and, < in f, th turn, olution dtrin fro trutur (40 and i till por uabl. iz ditribution (watr-to-nt por ratio, Th nt at hial full tning opoition, i obtaind SF by ontnt, tting uring = 0 in ti and thod, (43 and tpratur, uing (45. i additiv, t.. n th litratur variou ulation an b found to drib th orption iothr noral 3.9 g Unloading onrt (Xi t al Howvr, in th prnt n papr tag th g i-pirial th tning rgion prion will rain propod ontant by and Norling th load Mjornll will dra (997 linarly. i adoptd Whn th bau baring it apaity pliitly i aount opltly hautd, th volution th tnd hydration rgion will ration fail, Otton and SF & ontnt. Olon (998. Thi orption Thi ha alo iothr bn hown rad in Yuan t al. (004. Th prinipl i hown in Figur 4g with a dottd lin. A fully orrt analyi th unloading rpon rquir a dfinition th unloading rpon in th ontitutiv rlationhip. w ( h,, = G (, + 0( g h (4 4 EXAMPLES 0( g h K (, Two apl ar givn to illutrat o th fatur and apabiliti thi odl. So th fa- tur ar diud in Stion 5. whr n th th apl firt tr th lip (gl i iothr alulatd rprnt at L/ a th phyially bound (adorbd watr and th ond tr ( L (apillary = γ ( L t a iothr rprnt th apillary (46 watr. Thi prion i valid only low ontnt Whn SF. Th a dbondd, offiint tr-fr G rprnt zon th i prnt, aount th lati watr pr longation unit volu in adhrnt hld in th ut gl b por addd at a 00% rlativ huidity, and it an b prd (Norling Mjornll 997 a P L ( L = γ ( f ta + f (47 Etw G (, = k + k (5 vg vg 4. Modrat tning bhavior whr k vg and k vg ar atrial paratr. Fro th Modrat tning bhavior drib a joint whr aiu aount watr pr unit volu that an th abolut valu G do not diffr uh fro fill all por (both apillary por and gl por, on G. G ut not vary ignifiantly fro G. Th an alulat K a on following ontant hav bn hon: L = 60; ta = ; t = ; t = t; w = ; E = 00000; E = E; G = 30000; G = -0000; G = -5000; 0 = g5; =.5. w G Rult ar 0 found in Figur 5 - Figur 6 (6 K (, = 0 g Th atrial paratr k vg and k vg and g an b alibratd by fitting printal data rlvant to fr (vaporabl watr ontnt in onrt at variou ag (Di Luzio & Cuati 009b.. Tpratur volution Not that, at arly ag, in th hial ration aoiatd with nt hydration and SF ration ar othri, th tpratur fild i not uni non-adiabati yt vn if th nvironntal tpratur i ontant. Hat ondution an b dribd in onrt, at lat tpratur not ding 00 C (Bažant & Kaplan 996, by Fourir law, whih rad q = λ T whr q i th hat flu, T i th abolut tpratur, and λ i th hat ondutivity; in thi Figur 5. Shar tr ditribution at diffrnt lngth. (7 Proding FraMCoS-7, May 3-8, 00

7 J = D ( h, T h Th proportionality offiint D(h,T oitur prability and it i a nonlina th rlativ huidity h and tpratur & Najjar 97. Th oitur a balan that th variation in ti th watr a volu onrt (watr ontnt w b q divrgn th oitur flu J t w = J Figur 6. Rlativ load a a funtion rlativ lip at L/. 4. Etr tning bhavior Etr tning bhavior i dfind by tning branh whr at lat on ha a nar vrtial or horizontal inlination. Thi apl ha th following ontant: L = 60; t a = ; t = ; t = t ; w = ; E = 00000; E = E ; G = 30000; G = ; G = - 50; = 5; = Rult ar found in Figur 7 - Figur 8. Th watr ontnt w an b prd a th vaporabl watr w (apillary wa Figur 8. Rlativ load vapor, a a funtion and adorbd rlativ watr lip L/. and th non- (hially bound watr w n (Mil Whn valu Pantazopoulo approah & 0 Mill th valu 995. t f, i ra ay, in tr a, au b that n th to hibit vaporabl non-phyial watr i a fu bhavior, a it i rlativ th a huidity, with th urrnt h, dgr apl. hydration To obtain quilibriu dgr in th ilia olution, fu ration, th obination, and = f i ag-dpndnt in Figur 9 n orption/dorption to au th, i.. w =w ont th tr-fr (Norling rak Mjonll to dra Thi Undr i diud furthr in Stion by ubtituting 4.3 and 5. into Equati thi au h h = t h & + & + w Figur 7. Shar tr ditribution at diffrnt lngth. whr /h i th lop th orption/ iothr (alo alld oitur apa govrning quation ( 3 ut b by appropriat boundary and initial onditi Th rlation btwn th aount watr and rlativ huidity i alld iothr if aurd with inraing huidity and dorption iothr in th a. Nglting thir diffrn (Xi t al. Figur 9. f and a th a funtion following, orption. Noti th iothr pial b-wilhavior f all rfrn valu to. both orption and dorption b By th way, if th hytri th iothr would b takn into aount, two 4.3 Paratri tudy, variation G rlation, vaporabl watr v rlativ hui A pial obrvation b ud wa aording ad whn to th Gign diffrd th varia fro G. At o rlativity valu huidity. G, th Th lngth hap f th would inra iothr draing HPC valu i inflund. A phnonon that i pially larly non-phyial, tho that influn but nvrth- tnt and by any p l rquird by th hial olution. ration Th bhavior and, i in hown turn, dtr in Figur 0. t i trutur n that and f will por bhav iz ditribution a ptd (watrratio, valu nt at = 0 hial fro G opoition, SF and obtain lowt and lowr. G uring ti and and highr, thod, a pak tpratur, rror i around 4% on f i t.. prnt. n th litratur variou ulatio So variation found on othr to drib paratr th orption hav hown iothr to produ th a onrt typ (Xi bhavior. t al Th ubjt Howvr, i in th furthr diud papr in Stion th i-pirial 5. Matrial paratr prion pro thi a ar: Norling L = 60; t a Mjornll = ; t = ; (997 t = t ; i w = adoptd ; b Proding FraMCoS-7, May 3-8, 00

8 EJ = D ( 00000; h, T h E = E ; G = 30000; G = ; ( G = ; = 5; =.5. Th proportionality offiint D(h,T i alld oitur prability and it i a nonlinar funtion th rlativ huidity h and tpratur T (Bažant & Najjar 97. Th oitur a balan rquir that th variation in ti th watr a pr unit volu onrt (watr ontnt w b qual to th divrgn th oitur flu J t w = J ( Th watr ontnt w an b prd a th u th vaporabl watr w (apillary watr, watr vapor, and adorbd watr and th non-vaporabl (hially bound watr w n (Mill 966, Pantazopoulo & Mill 995. t i raonabl to Figur au 0. that Rlativ th diplant vaporabl a watr a funtion i a funtion th ond branh tning odulu, G. rlativ huidity, h, dgr hydration,, and dgr ilia fu ration,, i.. w =w (h,, = ag-dpndnt orption/dorption iothr 5 DSCUSSON AND CONCLUSON (Norling Mjonll 997. Undr thi auption and by ubtituting into on A bi-linar tning urv to drib th bhavior th adhiv ha bn iplntd in th olution a tniond adhiv joint in pur har uing wll known auption. h Th joint wa odld aftr th + w& n lai Volkrn (938 thory, whih in othr a (3 h t h ha provn to work wll with tt. S intan Yuan t al. (004, and Täljtn (008 whr a thory bad /h i th lop th orption/dorption whr on Volkrn (938 wa ud to valuat iothr (alo alld oitur apaity. Th th har tr ditribution in a trngthnd tl govrning quation ( 3 ut b opltd plat. by appropriat boundary and initial ondition. Th lod olution th propod probl Th rlation btwn th aount vaporabl ha provn apabl prditing th rpon watr and rlativ huidity i alld adorption both odrat and tr valu tning paratr. Th odrat tning paratr r- iothr if aurd with inraing rlativity huidity and dorption iothr in th oppoit ultd in a tady tat dlaination pro with a a. Nglting thir diffrn (Xi t al. 994, in alot ontant fraturing load whih nd up in a th following, orption iothr will b ud with hort unloading pha. Etr tning paratr rultd rfrn in to a both pak-typ orption load-lip and dorption rpon, ondition. followd by By a th hort way, unloading if th pha. hytri Th failur th har oitur train wa iothr not rahd would b in thi takn a. into A aount, poibl two planation diffrnt rlation, th load vaporabl lip bhavior watr ight v rlativ b found huidity, in Figur ut 9, whr b ud th aording ont to th th tr-fr ign th rak variation i plottd. th Thi rlativity point huidity. ov bak Th through hap an alrady th orption dlainatd iothr part HPC th rak i inflund at all by valu any paratr,, whih i pially not phyially tho poibl. that influn Thi ay tnt diturb and rat th olution th hial owhat. ration and, in turn, dtrin por trutur and por iz ditribution (watr-to-nt ratio, nt hial opoition, SF ontnt, uring ti and thod, tpratur, i additiv, t.. n th litratur variou ulation an b found to drib th orption iothr noral onrt (Xi t al Howvr, in th prnt papr th i-pirial prion propod by Norling Mjornll (997 i adoptd bau it pliitly A paratri aount tudy howd th volution a puliar bhavior hydration ration f whn o and paratr SF ontnt. wr Thi varid. orption Thi i iothr in partiular rad th a with variation th inlination th ond tning branh, G. Rult howd that th alulatd ditan to th dbondd zon did not dra all valu, but intad inra w ( h,, = G (, + nar = 0. Thi olution i not phyially allowabl, 0( g h but it i ditatd by th quilibriu ondition and (4 olution produr. No final onluion an b drawn fro th initial tudy 0 ( gthi probl; h howvr K (, it i lar that th probl annot b olvd without a oplt ontitutiv rlationhip th adhiv dribing th tning a wll a th unloading bhavior. whr th A firt poibility tr (gl bypaing iothr rprnt th probl th aud phyially by tr bound (adorbd tning paratr, watr and i th to ond driv th tr rpon (apillary iothr joint a rprnt a funtion th diplant watr. in Thi an adhrnt; prion.g. i dirt valid odling only low a ontnt d- apillary ation SF. ontrolld Th offiint tt. G rprnt th aount watr pr unit volu hld in th gl por at 00% rlativ huidity, and it an b prd (Norling REFERENCES Mjornll 997 a Chn, J.F., Tng, J.G. 00. Anhorag Strngth Modl G ( FRP, and = Stl k Plat + k Bondd to Conrt. Journal (5 vg vg Strutural Enginring 7(7: Gutafon, P.J Analyi gnralizd Volkrnjoint k in vg tr and k vg non-linar ar atrial fratur paratr. hani. Fro Mhani- th whr aiu al bhavior aount adhiv watr joint. pr Europan unit volu hani that an olloquiu all por 7. (both (Vrhry, apillary G. and por Cardon, and A.H.. gl por, on fill Hart-Sith, L.J Adhiv-bondd doubl-lap joint. an alulat K a on NASA-CR-35. Langly Rarh Cntr. Otton, N.S. & Olon, K.G., N. 988., Hardning/Stning Plati Analyi Adhiv Joint. Journal Enginring 0 g Mhani 4(:97-6. w G 0 Pihlr, D Di Virkung von Anprungdrükn auf (6 K ( di, Vrankrung = von Klblalln(in Gran. Dirtation. Univrität nnbruk. 0 g Ranih, E-H. 98. Zur Tragfähigkit von Vrklbrungn Zwihn Bautahl und Bton Gklbt Bwhrung(in Gran. Th atrial Dirtation. paratr TU Braunhwig. k vg and k vg and g an Täljtn, B Strngthning onrt pri uing th b alibratd by fitting printal data rlvant to plat-bonding thniqu. ntrnational Journal Fratur. (vaporabl watr ontnt in onrt at fr Täljtn, variou B., ag Hann, (Di Luzio C.S., Shidt, & Cuati J.W. 009b Strngthning old talli trutur in fatigu with prtrd and non-prtrd CFRP lainat. Contrution and Building. Matrial. Tpratur volution Volkrn, O Di Nitkraftvrtilung in zugbantpruhtn Not that, at arly ag, in th hial ration Nitvrbindungn it kontantn Lahnqur-hnittn. aoiatd Luftfahrthung. with nt 5:4-47. hydration and SF ration Yuan, ar othri, H., Tng, J.G., th Sraino, tpratur R., Wu, fild Z.S., i Yao, not uni J Full non-adiabati rang Bhavior yt FRP-to-onrt vn if th bondd nvironntal joint. Enginring Strutur i ontant. 6: Hat ondution an b tpratur dribd in onrt, at lat tpratur not ding 00 C (Bažant & Kaplan 996, by Fourir law, whih rad q = λ T (7 whr q i th hat flu, T i th abolut tpratur, and λ i th hat ondutivity; in thi Proding FraMCoS-7, May 3-8, 00

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

Fracture simulation of fiber reinforced concrete by visco-elasto-plastic suspension element method 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 978-89-5708-82-2 Fratur imulation fibr rinford