ADAPTIVE LIMIT ANALYSIS USING DEVIATORIC FIELDS

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1 VI Intrnational Confrnc on Adaptiv Modling and Simulation ADMOS 2013 J. P. Moitinho d Almida, P. Díz, C. Tiago and N. Parés (Eds) ADAPTIVE LIMIT ANALYSIS USING DEVIATORIC FIELDS ANDREI V. LYAMIN *, KRISTIAN KRABBENHOFT * AND SCOTT W. SLOAN * * Cntr of Excllnc for Gotchnical Scinc and Enginring (CGSE) Th Univrsity of Nwcastl -mail: andri.lyamin@nwcastl.du.au, Ky words: Limit Analysis, Finit Elmnts, Error Estimation, Adaptiv Rmshing Abstract. Accurat stimats of limit loads for difficult stability problms in gotchnical nginring can rarly b obtaind from a singl finit lmnt limit analysis without using an xcssiv numbr of lmnts. Thrfor, fficint adaptiv stratgis which maximiz th solution accuracy using minimum numbr of lmnts in th msh ar of grat intrst. This study xplors th possibility of using th intrnal dissipation calculatd from dviatoric strsss and strain rats as suitabl control fild for purly frictional matrials. Th prformanc obsrvd for considrd st of problmatic for othr adaptiv schms gotchnical xampls is vry promising. Morovr, th proposd approach works vry wll also for cohsiv and cohsiv frictional matrials, suggsting its us as gnral ngin for adaptiv msh rfinmnt. 1 INTRODUCTION For complx, practical stability problms in gotchnical nginring, accurat stimats of th collaps load or factor of safty can rarly b obtaind from a singl analysis and a trial and rror procss is usually rquird. Th ky to obtaining accurat solutions lis in accuratly capturing th aras of plasticity within th problm domain, as thir pattrn and intnsity govrn th solution. Th dvlopmnt of an fficint msh adaptivity stratgy, which is abl to pinpoint th fin dtail of a structur s collaps mchanism, is thus of th highst priority in modrn limit and shakdown analysis. A critical aspct of any adaptiv mshing procss is th stimation of th discrtisation rror prsnt in a givn finit lmnt solution. Sinc a priori rror stimats play only an indicativ rol (Borgs t al. [1] ), usful rror stimats must mploy a postriori tchniqus to prdict th ovrall discrtisation rror in on or mor solution norms (or control variabls). Gnrally spaking, two major approachs hav bn practicd so far. Th first is th Hssian basd rror stimation, whr th spatial distribution of th rror in solution is obtaind on th basis of information gathrd from th matrix of scond drivativs of som control variabl (Zinkiwicz t al. [2], Almida t al. [3], Lyamin t al. [4] ). And th scond is a so-calld gap adaptivity schm, which is basd on th fact that for limit analysis applications th global rror in th solution can b radily obtaind as th sum of lmntal diffrncs btwn uppr and lowr bound stimats (Ciria t al. [5], Muñoz t al. [6] ).

2 Th major advantag of Hssian basd rror stimation whn combind with optimalmsh-adaptiv schm is that it usually provids th lmnt siz distribution which convrgs (kping th numbr of lmnts in th msh constant) vry quickly to a stady, smoothly gradd msh pattrn, which can b ithr isotropic or anisotropic. It is vry gnral and basd on th fact that, at som point x in th vicinity of a point x 0, th diffrnc btwn th variabl of intrst u and its discrt approximation u h can b stimatd using th following xprssion u u C ( x x ) H ( u ( x ))( x x ) (1) T h 0 R h 0 0 whr C is a positiv constant and HR( uh( x0)) dnots a rcovrd Hssian matrix. An anisotropic rror stimator for lmnt of a partition h of th domain can thn b introducd as 2 12 T n ( x ) h ( x x ) H ( u ( x ))( x x ) d ; (2) n 0 n 0 R h 0 0 whr n is th problm dimnsionality, h n is th minimum dimnsion of lmnt, and n is th largst ignvalu of th lmnt Hssian matrix. It is assumd also that th stimatd rror yilds th sam valu in any dirction, i.. h h h n n Th choic of a suitabl control variabl is not obvious for plasticity problms. Svral approachs hav bn practicd so far including thos basd on powr dissipation or its gap (Ciria t al. [5], Muñoz t al. [6] ), plastic multiplirs (Lyamin t al. [4] ) and strain rat (Christiansn & Pdrsn [7] ) filds mployd as control variabls. All ths schms work quit wll for cohsiv or cohsiv-frictional matrials, but for purly frictional soils thir prformanc stalls as.g. plastic multiplirs hav substantially high valus for all zro strss points on th surfac of soil domain, thrfor cannot indicat rliably plastic aras. Similar conclusion can b mad about prformanc of schms basd on powr dissipation or strain rats. This study xplors th possibility of using th intrnal dissipation calculatd from dviatoric strsss and strain rats (calld also shar powr in th rst) 1 1 s : ε d, ij ij 3 1ij, ij ij 3 1 (3) ij u s I I as suitabl control fild for purly frictional matrials. In abov ij, sij and ij, ij ar th Cartsian and dviatoric strsss and strain rats, rspctivly, and I 1, I 1 ar th first invariants for strsss and strain rats. Th prformanc obsrvd for considrd st of problmatic for othr adaptiv schms gotchnical xampls is vry promising. Morovr, th proposd approach works vry wll also for cohsiv and cohsiv frictional matrials, suggsting its mploymnt as gnral ngin for adaptiv msh rfinmnt. 2 THE OPTIMAL MESH ADAPTIVE SCHEME Usually msh rfinmnt procds with gradual adjustmnt of th lmnt siz aiming to distribut local rror uniformly ovr th problm domain. Th othr altrnativ is to obtain th

3 lmnt siz distribution which minimizs th global rror givn by quation (2). This approach is known as optimal-msh-adaptiv tchniqu and is dscribd in dtail.g. by Almida t al. [3] In brif, th optimal-msh-adaptiv procdur can b cast as constraind optimization problm, which for two-dimnsional cas bcoms p p 2 minimis ( h ) ( h ) 2 h p 2T 2T 2T 2T T k 2 T 2T 2T k T k subjct to N (4 / 3) ( s h ) to find h, T whr h2t and s T ar th nw siz and th strtching of lmnt T, k is th finit lmnt discrtization at th adaptation stp k and N is th dsird numbr of lmnts at th stp k+1. For p = 2 and th cas of quilatral lmnts (no strtching) th solution to problm (4) is givn by h2 T 4 3N (5) Th advancing front algorithm (Prair t al. [8] ) has bn mployd for gnrating th msh. As th mshing tim is only a small fraction of th total CPU tim in adaptiv limit analysis, this algorithm was chosn in ordr to giv full control of th msh quality, including th shap of th lmnts and th rat of chang of th lmnt siz throughout th msh from on itration to th nxt. Both rfinmnt and coarsning of th msh hav bn allowd. 3 LIMIT ANALYSIS Th lowr (LB) and uppr (UB) bound limit analysis formulations usd in this invstigation stm from th mthods originally dvlopd by Sloan [9][10], but hav volvd significantly ovr th past two dcads to incorporat th major improvmnts dscribd in Lyamin and Sloan [11][12] and Krabbnhoft t al. [13][14]. Ky faturs of th mthods includ th us of linar finit lmnts to modl th strss/vlocity filds, and collapsd solid lmnts at all intr-lmnt boundaris to simulat strss/vlocity discontinuitis. Th solutions from th lowr bound formulation yild statically admissibl strss filds, whil thos from th uppr bound formulation furnish kinmatically admissibl vlocity filds. This nsurs that th solutions prsrv th important bounding proprtis of th limit thorms. Both formulations rsult in convx mathmatical programs, which (considring th dual form of uppr bound problm) can b cast in th following form: maximiz subjct to 0 f ( σ) 0, i {1,, N} i Aσ p p whr λ is a load multiplir, σ is a vctor of strss variabls, A is a matrix of quality constraint cofficints, p 0 and p ar vctors of prscribd and optimizabl forcs, rspctivly, f i is th yild function for strss st i and N is th numbr of strss nods. Th (4) (6)

4 solutions to problm (6) can b found fficintly by using gnral Intrior-Point mthods (IPM) or spcialisd conic optimization solvrs (SOCP). 4 NUMERICAL EXAMPLES Two rprsntativ xampls from th soil mchanics ar considrd in this sction to illustrat th fficincy of proposd adaptiv approach. First xampl, so-calld N problm, is about stimating th baring capacity of rigid footing rsting on cohsionlss soil (sand). Scond xampl is known as passiv arth prssur cas. Hr th maximum latral prssur which can b xrtd to th soil cut, bfor it collapss upwards, nds to b found. Both xampls ar tratd as two-dimnsional problms and considrd undr plain strain conditions. Adaptiv rfinmnt procds by spcifying th initial and targt numbr of lmnts in th msh, and th numbr of adaptiv itrations. If this targt numbr of lmnts is rachd bfor th maximum numbr of itrations has xcdd, no additional lmnts ar injctd. Howvr, som improvmnt can still b achivd by rdistributing th lmnt sizs in th rmaining itrations if a bttr pattrn of th control variabl can b found. In xampls considrd th thrsholds on msh rfinmnt and coarsning factors btwn 2 itrats wr st to 0.25 and 1.5, rspctivly. 4.1 Rigid rough strip footing on cohsionlss soil (N problm) For a rigid strip footing rsting on a pondrabl purly frictional soil with no surcharg th baring capacity is usually stimatd by using rducd Trzaghi [15] quation of th form q = 0.5 BN (7) whr is soil unit wight, B is th width of th footing and N is th baring capacity factor, which dpnds on soil friction angl,. Thr is no xact solution availabl for N and ovr th yars svral mpirical xprssions wr suggstd and usd in practic (Brinch Hansn [16], Caquot & Krisl [17] ). Rcntly vry accurat stimats don by numrical limit analysis wr rportd (Hjiaj t al. [18] ) and vntually quasi-xact valus of N wr obtaind by th mthod of charactristics (Martin [19] ). Thrfor, bsids th standard for limit analysis UB-LB gap rror stimation, this allows dirct chck of th accuracy of adaptivly obtaind q a) b) c) rough Prandtl failur mchanism =30, =1 SharPowr 1.6E E E E E E E E E E E E E E E E+01 Figur 1. Gomtry (a), initial msh (b) and shar powr dissipation plot (c) for strip footing.

5 Limit prssur Andri V. Lyamin, Kristian Krabbnhoft and Scott W. Sloan solutions for this problm. Th problm dscription (including Prandtl [20] failur mchanism) togthr with th initial msh usd for analysis and corrsponding shar powr dissipation is givn in Figur 1. Nxt, in Figur 2 th distributions of svral traditionally usd for adaptiv limit analysis control variabls ar plottd. Du to th absnc of cohsion in soil mass it is vidnt that powr dissipation (all zros) is not an altrnativ to govrn rfinmnt procdur in this cas. And a) powr dissipation (UB) b) rat of work by gravity (UB) c) strain rat (LB) d) plastic multiplir (LB) Figur 2. Distributions of commonly usd control variabls in th cas of N problm. vn if powr loss du to soil unit wight is takn into account (Figur 2b) th rsultant distribution dos not rsmbl th actual collaps mchanism (slip lin) to b considrd as a good choic. Nithr it will work whn UB-LB gap of lmntal powr dissipation would b usd. Similar commnts ar applid to anothr pair of control variabls, strain rat and plastic multiplir filds. It is clar that all zro-strss points (soil surfac boundary, LB cas) ar at plastic stat, thrfor will hav som non-zro plastic multiplirs as shown in (Figur 2d). This nois prvnts plastic multiplirs to b mployd as adaptivity guid ithr. On th othr hand, th dissipation computd using dviatoric trms of strsss and strains (shar powr) has vry distinctiv distribution rsmbling classical Prandtl [20] collaps mchanism for strip footing. And, as can b judgd from rsults prsntd in Figur 3c, it works fficintly for both lowr and uppr bound discrtizations. Th final msh and corrsponding shar powr dissipation ar illustratd in Figur 3a,b. a) b) SharPowr Figur 3. Final msh (a), shar powr dissipation (b) and convrgnc diagram for strip footing E E E E E E E E E E E E E E E E c) Uniform UB Shar UB Shar LB Uniform LB XY (Scattr) Itrations

6 Limit prssur Andri V. Lyamin, Kristian Krabbnhoft and Scott W. Sloan 4.2 Passiv arth prssur This is anothr classical problm in soil mchanics, whr th latral prssur, p, is applid to th soil mass to caus its collaps, as shown in Figur 4. Thr ar svral thoris for this problm (th most famous ar du to Coulomb [21] and Rankin [22] ) with diffrnt analytical solutions accounting for various soil slop angls, soil/wall intrfac conditions, mod of a) b) c) p =40, =1 rough Figur 4. Gomtry (a), initial msh (b) and shar powr dissipation plot (c) for passiv arth prssur. failur (no rotation or rotation allowd), tc. But our main focus hr is not actually to compar rsults obtaind to xisting solutions, rathr dmonstrat that proposd msh a) powr dissipation (UB) b) rat of work by gravity (UB) c) strain rat (LB) d) plastic multiplir (LB) Figur 5. Distributions of commonly usd control variabls in th cas of passiv arth prssur problm. rfinmnt approach prforms rliably whn applid to sands. For this purpos, in th sam way as for N cas, th distributions of most popular control variabls traditionally usd within th limit analysis adaptiv schms ar givn in Figur 5. It appars that th sam commnts as thos givn in prvious sction ar applicabl hr as wll - non of th distributions in Figur 5 sms to b suitabl to assist with ffctiv msh rfinmnt. On th othr hand, using proposd adaptiv schm basd on shar powr dissipation rsults in robust rfinmnt procdur as prsntd in Figur 6. a) b) c) Uniform UB Shar UB Shar LB Uniform LB XY (Scattr) Itrations Figur 6. Final msh (a), shar powr dissipation (b) and convrgnc plot (c) for passiv arth prssur problm.

7 5 CONCLUSIONS Basd on dviatoric strss and strain filds lmntal powr dissipation was mployd to control msh rfinmnt procss in limit analysis computations for purly frictional matrials. Both lowr and uppr bound countrparts of limit analysis wr tstd. Th obtaind rsults show that th proposd approach works rliably for dmanding applications, whr traditionally usd control variabls fail to prform. REFERENCES [1] Borgs, L.A., Zouain, N., Costa, C. and Fijóo, R. An adaptiv approach to limit analysis, Intrnational Journal of Solids and Structurs, (2001) 38: [2] Zinkiwicz, O.C., Huang, M. and Pastor, M. Localization problms in plasticity using finit lmnts with adaptiv rmshing, Intrnational Journal for Numrical and Analytical Mthods in Gomchanics, (1995) 19: [3] Almida, R.C., Fijóo, R., Glão, A.C., Padra, C. and Silva, R.S. Adaptiv finit lmnt computational fluid dynamics using an anisotropic rror stimator, Computr Mthods in Applid Mchanics and Enginring, (2000) 182: [4] Lyamin AV, Sloan SW, Krabbnhoft K and Hjiaj M Lowr bound limit analysis with adaptiv r-mshing, Intrnational Journal for Numrical Mthods in Enginring (2005) 63(14): [5] Ciria H, Prair J and Bont J. Msh adaptiv computation of uppr and lowr bounds in limit analysis. Int. J. Numr. Mth. Engng. (2008) 75: [6] Muñoz JJ, Bont J, Hurta A, Prair J. Uppr and lowr bounds in limit analysis: Adaptiv mshing stratgis and discontinuous loading. Int. J. Numr. Mth. Engng. (2009) 77: [7] Christiansn, E. and Pdrsn, O.S. Adaptiv msh rfinmnt in limit analysis, Intrnational Journal for Numrical Mthods in Enginring, (2001) 50: [8] Prair, J., Vahdati, M., Morgan, K. and Zinkiwicz, O.C. Adaptiv rmshing for comprssibl flow computations, Journal of Computational Physics, (1987) 72: , [9] Sloan, S.W. Lowr bound limit analysis using finit lmnts and linar programming. Intrnational Journal for Numrical and Analytical Mthods in Gomchanics, (1988) 12(1): [10] Sloan, S.W. Uppr bound limit analysis using finit lmnts and linar programming. Intrnational Journal for Numrical and Analytical Mthods in Gomchanics, ( 1989) 13(3): [11] Lyamin, A.V., and Sloan, S.W. Lowr bound limit analysis using nonlinar programming. Intrnational Journal for Numrical Mthods in Enginring, (2002) 55(5): [12] Lyamin, A.V., and Sloan, S.W. Uppr bound limit analysis using linar finit lmnts and nonlinar programming. Intrnational Journal for Numrical and Analytical Mthods in Gomchanics, (2002) 26(2): [13] Krabbnhoft, K., Lyamin, A.V., Hjiaj, M., and Sloan, S.W. A nw discontinuous uppr

8 bound limit analysis formulation. Intrnational Journal for Numrical Mthods in Enginring, (2005) 63(7): [14] Krabbnhoft, K., Lyamin, A.V., and Sloan, S.W. Formulation and solution of som plasticity problms as conic programs. Intrnational Journal of Solids and Structurs, (2007) 44(5): [15] Trzaghi, K. Thortical Soil Mchanics. John Wily & Sons, NwYork (1943). [16] Brinch Hansn, J. A Rvisd and Extndd Formula for Baring Capacity. Th Danish Gotchnical Institut, (1970) Bulltin No. 28. [17] Caquot, A. and Krisl, J. Sur l trm d surfac dans l calcul ds fondations n miliu pulvrnt. Proc. 3 rd ICSMFE, Zurich, (1953) 1: [18] Hjiaj, M., Lyamin, A.V., Sloan, S.W. Numrical limit analysis solutions for th baring capacity factor N γ. Intrnational Journal of Solids and Structurs, (2005) 42(5-6) : [19] Martin, C.M. Exact baring capacity calculations using th mthod of charactristics. Proc. 11th Int. Conf. of IACMAG, Turin, (2005) Vol. 4, pp [20] Prandtl, L. Übr di Härt plastischr Körpr. Göttingn Nachr, Math. Phys. K1, 12, 74-85, (1920). [21] Coulomb C.A. Essai sur un application ds rgls ds maximis t minimis a qulqus problms d statiqu rlatifs a l'architctur. Mmoirs d l'acadmi Royal prs Divrs Savants, (1776) 7. [22] Rankin, W. On th stability of loos arth. Philosophical Transactions of th Royal Socity of London, (1857) 147.

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