STABILIZED FINITE ELEMENTS IN GEOMECHANICAL APPLICATIONS

Transcription

1 STABILIZED FINITE ELEMENTS IN GEOMECHANICAL APPLICATIONS S. Commnd GoMod consultng ngnrs, Lausann, Swtzrland Th. Zmmrmann Zac srvcs Ltd, Lausann, Swtzrland A. Truty Dpartmnt of Envronmntal Engnrng, Cracow Unvrsty of Tchnology, Poland ABSTRACT: In th framwork of lastoplastcty, th us of low ordr fnt lmnts s oftn dsrabl n ordr to smplfy mplmntaton, and also msh gnraton. Unfortunatly, standard low ordr lmnts bhav poorly n ncomprssbl and dlatant lastoplastc mda, xhbtng pathologs such as volumtrc lockng and oscllatons n th prssur fld. In Commnd (2001) & Commnd t al. (2004), w proposd a novl approach to ovrcom such pathologs wth th hlp of stablzd fnt lmnts, arlr dvlopd n th contxt of computatonal flud dynamcs. W rcall hr th ky faturs of th approach, and prsnt llustratons of th ffctvnss for smpl gomchancal applcatons n Z_Sol.PC (2009). In partcular, th stablzd fnt lmnts ar shown to allow smultanous us of low ordr quadrlatrals and trangls wthn th sam msh. Rlatd work n two-phas mda by th authors (s Truty and Zmmrmann (2006)) s also brfly addrssd. 1 INTRODUCTION Th us of low ordr fnt lmnts s oftn dsrabl n ordr to smplfy msh gnraton and also from an mplmntatonal pont of vw. Unfortunatly such lmnts bhav poorly n ncomprssbl and dlatant lastoplastc stuatons, xhbtng pathologs such as volumtrc lockng and oscllatons of th prssur fld. Dffrnt tchnqus hav bn proposd n th ltratur to ovrcom ths problm; n ths papr w dmonstrat th ffctvnss of stablzaton tchnqus, ntally dvlopd n th contxt of computatonal flud dynamcs and xtndd to a mxd dsplacmnt-prssur formulaton of lastoplastcty n Commnd (2001), on typcal gomchancal applcatons. In scton 2 w rcall th govrnng quatons, n scton 3 w prsnt th stablzaton tchnqus, valdaton tsts ar prsntd n scton 4, conclusons ar fnally drawn n scton 5. 2 GOVERNING EQUATIONS Eq. (1) dfns qulbrum for th statc cas: dv( σ) f = 0 (1) whr σ s th strss tnsor and f th body load.

2 Th ncrmntal consttutv rlaton rads: Δσ p = D Δε Δε (2) p wth D th lastc modulus matrx, Δε th total stran ncrmnt and Δε th plastc stran ncrmnt. Th soluton procdur s basd on a dscrtzd wak form n trms of dsplacmnt ncrmnts Δu. A mxd dsplacmnt-prssur form s obtand by ntroducng th followng volumtrc-dvatorc splt nto th consttutv quaton for th strss ncrmnt: p Δσ = D Δε Δε 1 (3) p p, th hydrostatc prssur ncrmnt n th sold phas, can b xprssd as: whr p = T p = K v K1 Δε Δε (4) v s th ncrmnt of th lastc volumtrc stran and K s th lastc bulk modulus. Th dvatorc projcton D s dfnd as: 1 T D = D I 11 (5) 3 whr I s th dntty matrx and 1 th vctoral rprsntaton of Kronckr's dlta j. A strong form of th problm can now b statd as follows: consdr a body, ts boundary = = g h wth g h =, wth = 1,...,nsd, whr n sd stands for th numbr of spatal dmnsons. Gvn p : R such that: f : R, g : R and h : R, fnd u : R and g h f = 0 (6) j, j p kk = 0 K (7) u = g (8) n = h (9) j j whr ( u, p) s computd ncrmntally usng Eq. (3). A corrspondng wak form s j constructd by multplyng Eq. (6) and Eq. (7) by approprat wghtng functons, ntgratng by parts and makng us of th natural boundary condton (s Commnd t al. (2004)). Dscrtzaton thn lads to a matrx form whch w can solv for Δu and Δp: K uu K up Δu Fu = (10) K pu K ppδp Fp whr F u and F ar gnralzd out-of-balanc forcs at currnt load stp and traton. p

3 Th nonlnar problm s solvd tratvly usng Nwton-Raphson mthod. Convrgnc of th dscrbd mxd approach s known to b dpndnt on th choc of ntrpolaton functons adoptd for dsplacmnt and prssur flds. Applcaton of th sam ntrpolaton functons for both flds rqurs a crtan stablzaton trm to b addd to to th systm xprssd by Eq. (10) to avod spurous oscllatons n th prssur fld n th ncomprssblty lmt. 3 STABILIZATION TECHNIQUES Followng th approach appld by Hughs t al. (1986) for computatonal flud dynamcs - s also Franca (1987) and Hughs (1987) - w consdr addng to th systm of Eq. (10) stablzng trms of th form: n l =1 T h h T T h h L σw, q τl σu, p f d (11) whr ( h ) τ = 2 2 I (s Hughs t al. (1986)) s a stablzaton factor matrx. h s th dmnson of th lmnt, th matral's shar modulus and a scalar paramtr. L s a dffrntal oprator whch has th ffct of takng th dvrgnc of σ. Th h suprscrpt ndcats dscrtzd valus, whl w and q ar wghtng functons. In th most gnral cas, such a formulaton affcts both th uppr and th lowr part of th matrx systm of Eq. (10). An hurstc sarch of th most approprat wghtng trms n Commnd t al. (2004) lads, howvr, to a smplr formulaton n whch only th scond quaton n Eq. (10) s stablzd by th rsdual of th qulbrum quaton wghtd by a prssur trm. Ths formulaton s dfnd n th squl as Galrkn plus prssur stablzaton (GPS). Th corrspondng stablzaton trm s dfnd as (s Commnd t al. (2004)): n l =1 N T τl T σu h, p h f d (12) whr N ar prssur shap functons. Ths typ of stablzaton s usd n all th subsqunt analyss. Anothr possblty, frst proposd n th fld of computatonal sol mchancs by Pastor t al. (1997) s to consdr stablzaton trms of th form: n l =1 T N τnd (13) Ths lattr tchnqu shows vry smlar rsults whn compard to th GPS schm n th bnchmarks that w hav analyzd. Fnally, a drctonal charactr can b ntroducd n th stablzaton trms, followng th fnt ncrmnt calculus formulaton dscrbd by Oñat (2000) whch ylds: n l T 3 N hh T L T σu h, p h = 1 8 f d (14)

4 whr h s a untary drctonal vctor dpndng on th last convrgd ncrmnt of dsplacmnt. Dtals on how ths drctonal formulaton provds a tntatv physcal justfcaton to th GPS formulaton can b found n Commnd (2001). 4 VALIDATION TESTS 4.1 Barng Capacty of a Suprfcal Footng Ths frst xampl llustrats a classcal gomchancal applcaton. A suprfcal rgd foundaton on an ncomprssbl (ψ=0) mdum s subjctd to a ncrasng loadng (Fg. 1), untl falur s dtctd. Gravty loads ar nglctd (γ=0). Svral analytcal solutons to ths problm xst n th ltratur (s Trzagh (1951) or Matar and Salnçon (1979)). Fg. 1. Suprfcal footng gomtry, and FE msh composd of quadrlatrals (Q4) and trangls (T3) Followng Matar and Salnçon (1979), th ultmat load for a rough footng s gvn by: q u = c N (15) c ' c ' whr c and N c ar scalar coffcnts dpndng on th dmnsons of th footng, frcton angl and cohson c. In our cas, ths thortcal soluton gvs q u = =15.6 KN/ m. Whn standard blnar dsplacmnt quadrlatral lmnts (Q4) ar usd to smulat ncomprssbl mda, rsults show an ovrshoot of thortcal solutons, charactrstc of lockng, ths also holds for constant stran trangls, or whn a mxtur of th two typs of lmnts s bng usd; smlarly, dlatant mda show th sam lockng phnomna. Earlr solutons to ovrcom lockng, lk B stran-projcton (s Hughs (1987)), nhancd assumd strans (EAS) (n Smo and Rfa (1990)) or cross-dagonal trangls (n Nagtgaal t al. (1974)) solv th ncomprssbl cas for quads. EAS also ovrcoms lockng for dlatant mda whn msh s composd of quadrlatrals, but all fal whn

5 quadrlatrals and trangls ar usd smultanously wthn th sam msh, a stuaton whch s oftn unavodabl. In th prsnt study, thr dffrnt mshs ar usd: on composd of quads, anothr on composd of trangls and th thrd on contanng a mxtur of both typs of lmnts (Fgur 1). Th proposd stablzd formulaton slghtly ovrshoots th lmt load ndcatd by Q4B (BBAR) lmnts, whch s qual 15.6 kn/m for th ncomprssbl cas (Fgur 2). It yld rsults vry clos to th EAS on (16.3 kn/m). Mxtur of T3/Q4 stablzd lmnts ylds slghtly stffr rspons. Th lmt load n th dlatant cas (Fgur 4) ndcatd by EAS lmnts and stablzd as wll s qual to 16.5 kn/m. Th falur mchansm s rproducd n Fgur 4 (dlatant cas, ψ = ) for th Q4 + T3 msh. It s dntfd by dsplacmnt ncrmnts color maps btwn q = 16.2 kn/ m and 16.4 kn/ m Q4B Q4U Q4EAS Q4UP T3UP Q4UP+T3UP Matar t al Fg. 2. Forc-sttlmnt dagram (ncomprssbl cas) Q4U Q4EAS Q4UP T3UP Q4UP+T3UP Matar t al Fg. 3. Forc-sttlmnt dagram (dlatant cas ψ = )

6 Fg. 4. Falur mchansm dntfd by dsplacmnt ncrmnts ntnsts (Q4+T3 msh, dlatant cas) All tsts hav bn prformd wth a stablzaton paramtr = 1.0, on a msh of mtrs wth fxd boundary condtons at th bottom and sldng boundary condtons on both sds. Th wdth of th footng s 2a = 2 mtrs. Th msh s composd of 1296 quads, or 2592 trangls, or a mxtur of both (only half of th footng has bn modlld du to symmtry). 4.2 Rockng Foundaton Th cas of a rockng foundaton s consdrd nxt. A momnt s appld to a suprfcal foundaton of wdth 2a = 10 mtrs undr plan stran condtons, and w ar lookng for a lmt momnt and th assocatd falur mchansm. Gomtry and proprts of th sol ar gvn n Fgur 5. Fg. 5. Rockng foundaton: gomtry and proprts An stmat of th ultmat momnt n th rgd-prfctly plastc cas s dfnd n Yodr (1981): M u = (1 ) (16) 2 a 2 M whr M s th shar strngth of th mdum, qual to cohson c f w us a Mohr- Coulomb modl. Two typs of mshs hav bn usd n ths study. Frst, a msh composd of trangls and quads dsposd n an llptcal pattrn around th foundaton. Scond, a rctangular msh pattrn ( 6344 quads) of 80 mtrs dpth and 100 mtrs wdth usd to analys th nflunc of msh algnmnt wth th falur mchansm. Rsults show that a falur mchansm can b found n both cass (s Fgur 6). Rsults obtand wth th stablzd approach ar found to b btwn th thortcal lmt momnt and th rsults found by Yodr (1981), whl standard Q4 lmnts ovrstmat th barng capacty of th foundaton (s Fgur 7).

7 Fg. 6. Rockng foundaton: falur mchansms (lft: llptcal Q4+T3 pattrn, rght: rctangular Q4 pattrn) Fg. 7. Rcaptulaton of rsults Th sam rockng foundaton tst s rproducd nxt wth boundary condtons closr to th foundaton n an ncomprssbl lastc mdum n ordr to show th ffct of stablzaton on prssur oscllatons. Fgur 8 shows solns of th frst strss nvarant I 1( σ ) for th mxd unstablzd cas (lft), and for th stablzd soluton (rght). Instablts notcd n th formr cas ar clarly ovrcom n th lattr on. Fg. 8. Frst strss nvarant solns (lft: standard mxd formulaton, rght: stablzd soluton)

8 4.3 Convrgnc Study on th Thck Cylndr Tst In Commnd t al. (2004) w prsnt a convrgnc study on a narly ncomprssbl lastoplastc thck cylndr loadd by an ntrnal prssur (Fgur 9). Fg. 9. Elasto-plastc thck cylndr loadd by an ntrnal prssur Th xstnc of an analytcal soluton maks t possbl to prov convrgnc whn msh * sz h s rfnd, on th normalzd L 2 prssur and dsplacmnt rror norms. In partcular, Fgur 10 compars th voluton of th rror wth rspct to N - th numbr of lmnts n th radal drcton of th cylndr - for two ntrnal prssurs, and for two dffrnt fnt lmnt mshs: B nhancd Q4 lmnts vs. a mxtur of GPS stablzd Q4+T3 lmnts. Fg. 10. Convrgnc study on th thck cylndr tst

9 4.4 Stablzaton n two-phas mda: Consoldaton Analyss of a Suprfcal Footng It s also possbl to apply such stablzaton tchnqus to two-phas mda consoldaton problms n ordr to crcumvnt volaton of th LBB condton, ladng to spatal prssur oscllatons whn th sam ntrpolaton flds ar usd for both dsplacmnt and por prssur flds. Dffrnt classs of stablzd mthods ar dscrbd n Truty and Zmmrmann (2006). To llustrat th ffcncy of th approach, Fgur 11 shows th por prssur dstrbuton n a consoldaton analyss of a suprfcal footng, and t s shown that whl standard lmnts xhbt strong oscllatons n th prssur fld, stablzd lmnts ovrcom ths problm and yld a smooth soluton. Th formulaton adoptd hr combns th nhancd assumd strans (EAS) for th sold wth stablzaton for th flud prssur quaton. Fg. 11. Por prssur contours at t = 0.1 yars 5 CONCLUSIONS Th prformanc of stablzd fnt lmnt formulatons s xamnd n ths papr. A novl stablzaton schm dvloppd n Commnd (2001) and Commnd t al. (2004) and appld hr to a mxd dsplacmnt-prssur formulaton of lastoplastcty s usd. Th proposd formulaton s shown to provd an approprat rmdy to problms of lockng n ncomprssbl and dlatant mda and allows th us of low ordr quadrlatral and trangular lmnts, ncludng n assocaton wth a mxtur of trangular and quadrlatral lmnts wthn th sam msh. Illustratons on classcal gomchancal applcatons ar prsntd, whch dmonstrat th ffctvnss of th approach. ACKNOWLEDGEMENT Th fnancal support of th Fund of th Swss Natonal Commtt on Larg Dams and of th Swss Natonal Scnc Foundaton undr grant for th frst author s gratfully acknowldgd.

10 REFERENCES Commnd, S. (2001), Stablzd Fnt Elmnts n Gomchancs, PhD Dssrtaton 2391, Swss Fdral Insttut of Tchnology (EPFL), Lausann Commnd, S., Truty, A. and Zmmrmann, Th. (2004), Stablzd Fnt Elmnts Appld to Elastoplastcty: I. Mxd Dsplacmnt-Prssur Formulaton, Comp. Mth. n Appl. Mch. and Eng., Vol. 193, Franca, L.P. (1987), Nw Mxd Fnt Elmnt Mthods, PhD Dssrtaton, Stanford Unvrsty, Palo Alto (CA) Hughs, T.J.R., Franca, L.P. and Balstra, M. (1986), A Nw Fnt Elmnt Formulaton for Computatonal Flud Dynamcs: V. Crcumvntng th Babuska-Brzz Condton: A Stabl Ptrov-Galrkn Formulaton of th Stoks Problm Accomodatng Equal-Ordr Intrpolatons, Comp. Mth. n Appl. Mch. and Eng., Vol. 59, Hughs, T.J.R. (1987), Th Fnt Elmnt Mthod: Lnar Statc and Dynamc Fnt Elmnt Analyss, Prntc-Hall. Hughs, T.J.R., Franca, L.P. and Balstra, M. (1989), A Nw Fnt Elmnt Formulaton for Computatonal Flud Dynamcs: VIII. Th Galrkn/Last-Squars Mthod for Advctv-Dffusv Equatons, Comp. Mth. n Appl. Mch. and Eng., Vol. 73, Matar, M. and Salnçon, J. (1979), Capacté portant ds smlls flants, Rvu Fr. d Géotchnqu, Vol. 9, Nagtgaal, J.C., Parks, D.M. and Rc, J.R. (1974), On Numrcally Accurat Fnt Elmnt Solutons n th Fully Plastc Rang, Comp. Mth. n Appl. Mch. and Eng., Vol. 4, Oñat, E. (2000), A Stablzd Fnt Elmnt Mthod for Incomprssbl Vscous Flows Usng a Fnt Incrmnt Calculus Formulaton, Comp. Mth. n Appl. Mch. and Eng., Vol. 182, Pastor, M., Qucdo, M. and Znkwcz, O.C. (1997), A Mxd Dsplacmnt-Prssur Formulaton for Numrcal Analyss of Plastc Falur, Computrs and Structurs, Vol. 62(1), Smo, J.C. and Rfa, M.S. (1990), A Class of Mxd Assumd Stran Mthods and th Mthod of Incompatbl Mods, Int. J. Num. Mth. Eng., Vol. 29, Trzagh, K. (1951), Mécanqu théorqu ds sols, Dunod, Pars Truty, A. and Zmmrmann, Th. (2006), Stablzd mxd fnt lmnt formulatons for matrally nonlnar partally saturatd two-phas mda, Comp. Mth. n Appl. Mch. and Eng., Vol 195, Yodr, P.J. (1981), A Stran-Spac Plastcty Thory and Numrcal Implmntaton, PhD Dssrtaton, Calfornan Insttut of Tchnology Z_Sol.PC (2009), Usr manual, Zac srvcs Ltd, Lausann