An efficient response sensitivity analysis method for a bounding surface plasticity sandy soil model

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1 A fficit rsos ssitivity aalysis mthod for a boudig surfac lasticity sady soil modl Q. Gu Xiam Uivrsity, Xiam, Fujia, Chia G. Wag Hog Kog Uivrsity of Scic ad Tchology, Hog Kog S. Huag Xiam Uivrsity, Xiam, Fujia, Chia ABSTRACT: Fiit lmt (FE) rsos ssitivity aalysis is a imortat comot i gradit-basd algorithms, such as structural otimizatio, rliability aalysis, systm idtificatio, ad FE modl udatig. I this ar, th FE rsos ssitivity aalysis mthodology basd o th dirct diffrtiatio mthod (DDM) is alid to a boudig surfac lasticity matrial modl that has b widly usd to simulat oliar soil bhaviors udr static ad dyamic loadig coditios. Th DDM-basd algorithm is drivd ad imlmtd i a gral-uros oliar fiit lmt aalysis rogram OSs. Th algorithm is validatd through simulatio of th oliar cyclic rsos of a multilayrd soil groud i?? subjctd to liqufactio udr arthquak loadig. Th rsos ssitivity rsults ar validatd ad comard with thos from Forward Fiit Diffrc (FFD) aalysis. Furthrmor, th rsults ar usd to dtrmi th rlativ imortac of various soil costitutiv aramtrs to th dyamic rsoss of th systm. Th DDM-basd algorithm is dmostratd to b accurat ad fficit i comutig th FE rsos ssitivitis, ad has grat ottial i th ssitivity aalysis of highly oliar soil-structur systms. INTRODUCTION. Backgroud Fiit lmt (FE) rsos ssitivity aalysis is a sstial igrdit of gradit-basd otimizatio mthods ad is rquird i structural otimizatio, systm idtificatio, rliability, ad FE modl udatig (Klibr t al. 997, Cot t al. 23). Furthrmor, th ssitivity aalysis rsults may b usd to quatify th matrial ad loadig ucrtaity ad its roagatio from origial sourcs to th structural rsoss of itrst. I additio, FE rsos ssitivitis rovid ivaluabl isight ito th ffcts of systm aramtrs o, ad thir rlativ imortac to th systm rsos (Gu t al. 29). Svral mthods ar availabl for rsos ssitivityaalysis, icludig th Fiit Diffrc Mthod (FDM), th Adjoit Mthod (AM), th Prturbatio Mthod (PM), ad th Dirct Diffrtiatio Mthod (DDM) (Zhag & Dr Kiurghia 993, Gu t al. 29, Scott t al. 24, Haukaas t al. 26). Th FDM is th simlst mthod for rsos ssitivity comutatio, but is comutatioally xsiv ad ca b gativly affctd by umrical ois. Th AM is fficit for liar ad o-liar lastic systms, but is ot a comtitiv mthod for ath-ddt (i.., ilastic) roblms. Th PM is comutatioally fficit although grally ot vry accurat. Th DDM, o th othr had, is a gral, accurat ad fficit mthod that is alicabl to ay matrial costitutiv modl. Th DDMbasd rsos ssitivity aalysis mthodology shows grat romis i th aalysis of comlicatd structural or gotchical systms. Howvr th DDM mthod rquirs th aalytical drivatio ad umrical imlmtatio to diffrtiat th systm rsoss with rsct to ssitivity aramtrs. Ovr th ast dcad, th DDM-basd ssitivity aalysis mthod has b activly dvlod ad imlmtd i a o sourc FE aalysis framwork kow as OSs (Mcka & Fvs, 2). Th DDM has b dvlod for various costitutiv modls icludig uiaxial matrials, thrdimsioal J 2 lasticity modls ad rssuriddt multi-yild surfac J 2 lasticity modls (Fu l al. 2). Ths modls ca b usd to simulat truss ad bam comots i th structurs, ad oliar clay bhaviors. Dtaild dscritios of th DDM-basd ssitivity aalysis mthodology imlmtd i OSs ca b foud i th litratur (Dr Kiurghia & Haukaas 26, Gu 28). Rctly th mthod has b formulatd for sady soils, which usually xhibit diffrt bhaviors from clayy soils, such as rssur-ddt cyclic bhaviors, shar-iducd volumtric dilatio ad cotractio, as wll as flow liqufactio udr low ffctiv cofimt. Th objctiv of this ar is to summariz th DDM-basd ssitivity aalysis to a class of boudig surfac modls for sady soils. Th boudig

2 surfac modl has b widly usd ad rov to b a ffctiv ad robust modl to simulat th bhaviors of sady matrials udr cyclic ad sismic loadig coditios (Dafalias 986, Li 22). Th DDM-basd ssitivity algorithm is articularly fficit for strogly oliar, larg-scal roblms with a larg umbr of ssitivity aramtrs. Th gotchical roblms modld by th boudig surfac modl ar a xaml. Thus dvloig a DDM ssitivity algorithm for th boudig surfac modl will allow us to solv a larg umbr of challgig gotchical roblms, such as th arthquak-iducd liqufactio homo i sady soils. Wh combid with th xistig ssitivity aalysis framwork for clayy soils ad soil-structur itractio systms, th DDM-basd ssitivity aalysis may b radily alid to ral soil-foudatio-structur itractio systms (Gu 28). This ar rovids a brif summary of th boudig surfac modl ad DDM formulatio, followd by a xaml to validat th DDM-basd rsos ssitivity algorithm. Th algorithm is alid to study th ssitivity of liqufid groud rsoss at Port Islad i Jaa udr a ral arthquak scario. Th rsults ar furthr usd to idtify th rlativ imortac of th soil aramtrs to th surfac rsoss..2 Numrical imlmtatio of a boudig surfac modl Th boudig surfac modl mloys a strss ratio ivariat, dfid as R = r : r, whr r is th strss ratio of th dviatoric strss s ovr rssur, i.., r = s /, ad th otatio : is th doubl cotractio btw two scod-ordr tsors, i.., A:B = A ij B ij. Accordigly, a ultimat failur surfac, or a failurboudig surfac, is dfid as: f ˆ = R ˆ R f =, whr th hats ^ dot strss quatitis o failur surfac, th aramtr R f is th strss ratio ivariat at th failur surfac, which is rlatd to th corrsodig classical critical stat triaxial aramtr M by R f = M / 3, ad th aramtr itrolats btw comrssio ad xtsio. Similarly, th maximum rstrss mmory boudig surfac is dfid as: f = R R m =, whr R m is a history aramtr rovidig th maximum rstrss lvl. Th two boudig surfacs ar combid to comut th lastic modulus, as show i Fig.. Figur. Modl mchaism i dviatoric strss ratio sac Isid th failur-boudig surfac, th hyolastic rsos, i.., th lastic strai ratε&, is dfid as th summatio of dviatoric strai & ad volumtric strai trε& as: ε& = & + ( trε& ) I = s& + & I 3 2G 3K = r& + r + I & 2G 2G 3K () Whr G ad K ar th rssur-ddt lastic shar ad bulk moduli, rsctivly. Similarly, th hyolastic rsos, i.., th lastic strai rat ε&, ca b writt as: ε& = D + I r H r 3Kr ( & : ) + r + I h( m ) & H 3K N (2) whr H r ad K r ar, rsctivly, th lastic shar ad bulk moduli associatd with th dviatoric strss ratio r& ; aramtrs H ad K ar, rsctivly, th lastic shar ad bulk moduli associatd with th rssur rat &. Th vctors D ad N ar uit vctors i strss sac alog th dviatoric art of ε& ad th associatd dviatoric loadig dirctio, rsctivly. I this ar both D ad N ar tak to b th sam as th uit vctor ormal to th maximum rstrss mmory boudig surfac f = (i.., vctor i Figur ). Th m is th maximum valu of ma rssur xricd i ast loadig. Th Havisid st fuctio h( m ) ad th Macaulay brackts aroud & idicat that th lastic mchaism du to & orats oly wh = m ad & >. As show i Figur, th rvious uloadig strss oit (i.., α i Figur ), th currt dviatoric strss ratio r ad a rorly dfid imag strss r o th maximum rstrss mmory boudig surfac f ( σ ) = ar combid to dtrmi variabl lastic moduli H r ad K r, which ar cotiuous fuctios of th distac ρ from α to r ( ρ = r α ) ad th distac ρ from α to r ( ρ = r α ) (Dafalias 986). It is worth mtioig that for ractical alicatios, th shar-iducd lastic strais usually domiat. Thrfor th lastic strai rat ε& ca b simlifid as: ε& = + I ( r& : ) (3) Hr 3Kr

3 2 THE RESPONSE SENSITIVITY ALGORITHM BASED ON THE DIRECT DIFFERENTIATION METHOD Rsos ssitivity is dfid as th first drivativ of a rsos quatity r (.g., dislacmt, strai, strss) with rsct to a ssitivity aramtr θ, i.., dr dθ. Th ssitivity aramtr could b a gomtric, matrial or loadig aramtr. I gral, th scalar rsos quatity r( θ ) = r ( f ( θ ), θ ) dds o th aramtr θ both xlicitly ad imlicitly through th vctor fuctio f ( θ ). Th DDM-basd rsos ssitivitis ar comutd aftr covrgc of ach tim or loadig st i oliar FE rsos aalysis. This rquirs cosistt diffrtiatio of th FE algorithm for th rsos-oly comutatio with rsct to ach ssitivity aramtrθ. Cosqutly, th rsos ssitivity comutatio algorithm ivolvs th various hirarchical lvls of FE rsos aalysis: () th structur/systm lvl, (2) th lmt lvl or sctio lvl, ad (3) th matrial lvl. Dtails about th DDM-basd ssitivity formulatios i classical dislacmt-basd, forc-basd ad mixd fiit lmt mthods ca b foud i th litratur (Gu 29, Scott t al. 24, Haukaas & Dr Kiurghia 26). 2. Dislacmt-basd FE rsos ssitivity aalysis usig DDM Aftr satial discrtizatio usig th fiit lmt mthod, th quatios of motio of a structural systm ca b rrstd by th followig oliar diffrtial quatio: M( θ) u&& ( t, θ) + C( θ) u& ( t, θ) + R( u( t, θ), θ) = F( t, θ) (4) whr t is tim, θ is a scalar ssitivity aramtr, u(t) is a vctor of odal dislacmts, M is th mass matrix, C is th damig matrix, R(u, t) is a history-ddt itral rsistig forc vctor, F(t) is th alid dyamic load vctor, ad u& ad u&& dot, rsctivly, th first ad scod drivativs of u with rsct to tim. Without loss of grality, Eq. (4) ca b itgratd umrically usig tim-stig mthods such as th wll-kow Nwmark-β mthod. Th systm of quatios ca b solvd usig th Nwto-Rahso itratio rocdur, which cosists of solvig a liarizd systm of quatios at ach itratio. I th followig discrtizd format, a subscrit + is usd to dot th variabls at th tim st +. Assumig that u + is th covrgd solutio for th currt tim st t +, ad rcogizig that R( u+) = R( u +( θ ), θ ) dds o θ xlicitly ad imlicitly through u +, w obtai th followig rsos ssitivity quatio at th structural lvl usig th chai rul of diffrtiatio(cot t al. 23, Gu t al. 29): α M + C + ( K ) stat + 2 T + β( t) β( t) dθ d F + R( u+(θ), θ) = dθ θ whr u+ dm α dc + 2 u β( t) dθ β( t) dθ du % (5) df% + df + dm = + u 2 + u& dθ dθ dθ β( t) β( t) 2β u&& du du& d&& u + M + 2 β( t) dθ β( t) dθ 2β dθ dc α α α + u ( t) dθ β( t) β u& 2β && u + α d u α d u& ( t) α d && u + C β( t) dθ β dθ 2β dθ I Eq. (5), α ad β ar Nwmark itgratio stat aramtrs, ad ( K T ) + dots th static algorithmic (cosistt) tagt stiffss matrix of th structur/systm, which is dfid as th assmbly of th cosistt tagt stiffss matrics of th lmts as T alg ( d ) R( u ) ( K ) = = A B C B Ω l stat + T + + u = Ω + l whr ( ) (6) A dots th dirct stiffss assmbly = orator, l rrsts th umbr of lmts i th FE modl, B is th strai-dislacmt alg trasformatio matrix, C + dots th algorithmic (cosistt) tagt moduli obtaid through cosistt liarizatio of th costitutiv law itgratio schm (Simo & Hughs 998), i.., C σ ( σ, ε, ε...) = alg ε+ (7) whr σ + is th strss at currt tim st t +. Th scod trm o th right-had sid of Eq. (5) rrsts th artial drivativ of th itral rsistig forc vctor, R(u + ), with rsct to th ssitivity aramtr θ udr th coditio that th odal dislacmt vctor u + rmais fixd. It is comutd through th dirct stiffss assmbly of th lmt rsistig forc drivativs as: R( u ) l + T σε ( (θ),θ) + = A ( ) dω θ = B x θ u+ Ω ε+

4 I this ar, th dtaild drivatio of th DDM basd ssitivity quatios ad th cosistt tagt orator of th boudig surfac modl is ot show, but ca b foud lswhr ( Gu & Wag 23). 3 NUMERICAL EXAMPLE I this sctio, a xaml is rstd to vrify th abov DDM algorithm ad illustrat its alicatio i modlig liqufiabl soils. Th soils ar cosidrd fully saturatd, ad a simlifid mthod is usd to simulat th fluid-soil itractio: Aftr th iitial rssur is alid, th volum of ach lmt is kt costat by fixig th vrtical dislacmt of all ods ad imosig th sam horizotal dislacmts to ach air of ods at th sam dth. This mthod is basd o th followig assumtio: (a) Th rocss of watr sag is much slowr tha that of th arthquak loadig, thus ca b igord; (b) Volumtric modulus of watr is much largr tha that of soil, thus watr is cosidrd as icomrssibl. Basd o ths assumtios, th volum of watr isid soil ks costat durig arthquak, ad th volum of soil lmt is also costat. For a horizotal layrd soil subjct to horizotal arthquak loadig, th total rssur at ay oit ks costat ad is qual to th iitial rssur. Thus, th or watr rssur ca b obtaid as th diffrc btw th iitial rssur ad th soil ffctiv rssur, ad it is ot modld as a iddt variabl. Th soil ffctiv rssur is comutd by usig th boudig surfac soil modl rstd hri. Comard to usig a fully could fluid-soil lmt, i.., u- formulatio (ZiKiwicz & Cha 999), th limitatio of this aroximatio mthod is that it ca ot rorly modl th ost-liqufactio rocss, which ivolvs th watr draiag ad or watr rssur dissiatio. 3. Modl dscritio This xaml studis th rsos ssitivitis of a multi-layrd soil sit locatd at Port Islad, Kob, Jaa udr arthquak loadig. Th soil thr is comosd of a layr of 8-m-thick rclaimd sad o to of silty clay, sad ad silt layrs. Th soil rofil is illustratd i Fig. 2. Th to layr of rclaimd sad udrwt xtsiv liqufactio, latral sradig, ad liqufactio-iducd sttlmt durig a arthquak o Jauary 7, 995. Groud motio acclratio tim historis hav b rcordd usig a dowhol array with statios at th groud surfac, at 6 m, 32 m, ad 82 m blow groud surfac, rovidig valuabl iformatio for studyig th liqufactio homo (Elgamal t al. 996) mtr z Rclaimd sad (3) Silty clay (4) Sad ad silt (5) Gravl sad ad silt (Diluvium) Silty clay (Diluvium) Groud surfac Figur 2. Soil rofil at Port Islad (Toki, 995) Dis. [m] Acclratio (g) -2. () (2) (6) Figur 3 Dislacmt history at groud surfac ad acclratio historis at various dths I this study, th soil colum is discrtizd ito a two-dimsioal la-strai fiit lmt modl cosistig of 82 quadrilatral lmts. Th soil colum dforms udr th la-strai siml-shar coditio. Th sad ad clay matrials ar both modld usig th boudig surfac modl rstd i this ar, ad th matrial aramtrs ar listd i (Gu & Wag 23). Gravity is first alid statically, which grats th iitial cofiig rssur withi th soil colum. Th actual acclratio rcordd usig th dowhol array at a dth of 82 m is alid to th bas of th FE modl, s Fig. 3. Th Nwmark-bta itgratio mthod is usd with aramtrs α = 5 ad β =.2756 ad a costat tim st t =. sc. Good agrmt is obtaid btw th rcordd ad comutd horizotal dislacmt at th groud surfac ad btw th rcordd ad comutd acclratios at diffrt soil dths, as ca b s i Fig.3. A tyical shar strss vs. shar strai rsos ad a tyical shar strss vs. ffctiv u u 2 u 3 u 4 u 5 x rcordd comutd Surfac Dth 6 m Dth 32 m Dth 82 m Tim [sc]

5 cofiig rssur rsos i th to soil layr ar show i Fig.4. ad Fig.5. rsctivly. Durig shakig, xcssiv or-rssur builds u rogrssivly i th rclaimd sads, rsultig i rducd ffctiv cofiig rssur. Liqufactio of th to layr occurs at about scods, as is vidcd by th sigificat loss of strgth ad stiffss of th soil matrial. Ths figurs dmostrat that th umrical simulatio agrs wll with th ral rcordd data, ad th rstd boudig surfac modl is abl to catur th ky faturs of th sad bhaviors icludig arthquakiducd liqufactio. Shar strss [Pa] Shar strss [Pa].5 x Shar strai x 3 Figur 4. Shar strss vs. strai rsoss.5 x rssur [Pa] x 4 Figur 5 Shar strss vs. ma ffctiv rssur rsos i to soil layr at a dth of 3.2 m For ractical itrsts, th ssitivity of th groud surfac rsos to various matrial aramtrs of th to soil layr (i.., layr #) is ivstigatd. Th DDM-basd rsos ssitivity rsults ar vrifid usig th FFD mthod with diffrt lvls of aramtr rturbatio ad ar show i Figurs 6 ad 7. Th FFD rsults ar show to covrg asymtotically to th DDM rsults as th FFD rturbatio rducs from - ad -3 th -5. Thus th DDM-basd ssitivity algorithm ad its imlmtatio ar vrifid to b corrct for this multilayr soil systm. 3.2 Obsrvatios ad fidigs Th advatag of th DDM mthod ovr th FFD mthod is vidt from th followig rror aalysis. If th roud-off rror of u from FE aalysis is δ, th th rror of aramtr ssitivity ( u θ ) θ from th DDM mthod is also i th ordr of δ. Howvr, th rror from th FFD mthod cosists of two arts: th roud-off rrors i th ordr of O( ( θ θ ) δ ) whr O is th Ladau symbol; ad th trucatio rror du to fiit diffrc aroximatio is i th ordr of O( θ θ ) M, whr M = θ 2 2 u θ 2 ( ξ ) is a fiit umbr, ξ [ θ, θ + θ ]. Wh θ/θ is larg, th trucatio rror trm domiats, ad so th total rror from th FFD mthod dcrass as θ/θ is rducd. O th othr had, wh θ/θ is small ough, th roud-off rror O( θ θ δ ) bcoms domiat. I this cas, rducig θ/θ cotiuously will iduc larg roud-off rrors i th FFD rsults. Th sigificat limitatio of th FFD mthod ca b obsrvd but ot show i this ar. Figur 6. Ssitivity of groud surfac dislacmt to aramtr a obtaid usig DDM vs FFD with diffrt lvls of aramtr rturbatio ( u surfac / θ)θ [m] u surfac / a [m] x Tim[sc] 9 x Figur 7. Zoomd viw of Fig. 6 DDM Tim[sc] Figur 8. Rlativ imortac of soil matrial aramtrs i rgards to th horizotal dislacmt of th groud surfac i G R f h r γ

6 ( Accl surfac / θ)θ [m/s 2 ] Figur 9. Rlativ imortac of soil matrial aramtrs i rgards to th horizotal acclratio of th groud surfac Th rlativ imortac of systm aramtrs to th systm rsos ca b quatifid accordig to th ak absolut valu of th ormalizd rsos ssitivity tim history u θ θ. Fig. 8 shows th ormalizd ssitivitis of th horizotal dislacmt rsos of th groud surfac to th fiv most ssitiv matrial aramtrs of th to soil layr. Th ordr of imortac of ths aramtrs (i dscdig ordr) is as follows: () th iitial void ratio i, (2) th modl costat G, (3) th failur dviatoric strss ratio R f, (4) th costat aramtr h r for th lastic shar modulus, ad (5) th costat aramtr γ for th critical stat li. Th void ratio i is idtifid as th most imortat aramtr affctig th groud surfac dislacmt rsos. From Fig. 8, o ca s that most arts of th ssitivity historis ( usurfac i ) i ar ositiv. Thus, rducig th void ratio will rduc th groud surfac dislacmt. Fig. 9 shows th ormalizd ssitivitis of th horizotal acclratio rsos of th groud surfac to th fiv most ssitiv aramtrs of th to soil. Ths rsults idicat that th groud surfac acclratio is most ssitiv to th sam st of aramtrs as th groud surfac dislacmt, xct that th ordr of imortac is slightly chagd (i dscdig ordr) to () i, (2) R f, (3) G, (4) h r, ad (5) γ. From ths obsrvatios, it is clar that th iitial void ratio i is th cotrollig aramtr affctig sigificatly both groud surfac dislacmt ad acclratio durig arthquak xcitatio. 4 CONCLUSION Tim[sc] Th DDM mthod is a gral, accurat ad fficit mthod for comutig FE rsos ssitivitis to modl aramtrs, scially i th cas of oliar matrials. This ar alis th DDM-basd rsos ssitivity aalysis mthodology to a boudig surfac lasticity matrial modl that has b widly usd to simulat i R f G h r γ oliar sady soil bhaviors udr static ad dyamic loadig coditios. Th algorithm is imlmtd i th gral-uros oliar FE aalysis softwar OSs. Th w algorithm ad its softwar imlmtatio ar validatd through two alicatio xamls, i which th DDM-basd rsos ssitivitis ar comard with thir coutrarts obtaid usig FFD aalysis. Th advatag of th DDM mthod ovr th FFD mthod is also highlightd through covrgc tsts. I th alicatio xaml, th ormalizd rsos ssitivity aalysis rsults ar also usd to masur th rlativ imortac of th soil costitutiv aramtrs i rgards to th groud surfac dislacmt ad acclratio i th cas of groud liqufactio. Th xaml illustrats th us of fiit lmt rsos ssitivity aalysis to dtrmi th rlativ imortac of matrial aramtrs for scifid systm rsos aramtrs. Th work rstd i this ar sigificatly broads th alicatio of DDM-basd rsos ssitivity aalysis, sic it abls umrous alicatios ivolvig th us of th boudig surfac lasticity matrial modl. Work is udrway to us th work rstd hr i rsos ssitivity aalysis of larg-scal oliar soilstructur itractio systms. 5 REFERENCES Cot, J.P. 23. Vijalaura PK, Mghlla M. Cosistt fiit lmt rsos ssitivity aalysis, Joural of Egirig Mchaics: 29(2): Dafalias, Y.F Boudig surfac lasticity. I: Mathmatical foudatio ad hyolasticity. Joural of Egirig Mchaics, 2(9): Dr Kiurghia A, Haukaas T, Fujimura K. 26. Structural rliability softwar at th Uivrsity of Califoria, Brkly. Structural Safty. 28: Elgamal A, Zghal M, Parra E Liqufactio of rclaimd islad i Kob, Jaa. Joural of Gotchical Egirig. 22 (): Gu Q, Cot JP, Elgamal A, Yag Z. 29. Fiit lmt rsos ssitivity aalysis of multi-yild-surfac J2 lasticity modl by dirct diffrtiatio mthod. Comutr Mthods i Alid Mchaics ad Egirig. 98 (3-32): Gu Q, Barbato M, Cot JP. 29. Hadlig of costraits i fiit lmt rsos ssitivity aalysis. Joural of Egirig Mchaics. 35(2): Gu Q, Barbato M, Cot JP, Li Y. 2. OSs Commad Laguag Maual --- Rsos Ssitivity Aalysis basd o th Dirct Diffrtiatio Mthod (DDM), htt://oss.brkly.du/wiki/idx.h/ssitivity_a alysis. Gu Q. 28. Fiit lmt rsos ssitivity ad rliability aalysis of soil-foudatio-structur-itractio systms,

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