Principle of virtual work and the discontinuous Galerkin method

Transcription

1 Rakntidn mkaniikka (Jornal of Strctral Mchanics) Vol. 43, No 4, 010, pp. -57 Principl of virtal work and th discontinos Galrkin mthod Joni Frnd and Ero-Matti Salonn Smmary. Th articl dscribs th application of th so-calld discontinos Galrkin mthod to bar and bam trsss. Th formlation is basd on th principl of virtal work modifid by additional intrfac trms. This xtndd principl of virtal work and discontinos approximation giv th sal discontinos Galrkin mthod as a particlar cas and allows on to procd in th sam mannr as with th standard principl of virtal work and continos approximation. Th ky fatrs ar a two-fild formlation sitabl for strctrs combining diffrnt nginring modls, and a comptationally fficint on fild implmntation. Particlar mphasis is placd on xplaining th logic of th trms of th xtndd principl withot too many tchnical dtails that may shadow th ndrlying ida. Ky words: principl of virtal work, discontinos Galrkin mthod, bam, trss Introdction Main contnts Th so-calld discontinos Galrkin finit lmnt mthod has obtaind mor and mor attntion in rcnt yars. In it th finit lmnt rprsntations for crtain qantitis ar allowd on prpos to hav th possibility of discontinitis. This givs flxibility to th approximation. For xampl, th dgr of approximation can vary from lmnt to lmnt as may b locally ndd withot problms sinc strict continity at th lmnt intrfacs is not rqird. On th othr hand, for a fixd msh and dgr of polynomials in th approximation, th dgrs of frdom of a discontinos Galrkin mthod oftn xcd considrably that of th standard Galrkin mthod, althogh th two mthods convrg roghly at th sam rat. Implmntation on an xisting softwar dsignd for continos approximations may not also b straightforward. Th prpos of this articl is to try to xplain to a radr not familiar with th discontinos Galrkin mthod how th mayb initially mystriosly looking trms in th varios qations can b motivatd withot going dp into analysis of th soltion mthod proprtis and mathmatics ndd thrin. W frthr concntrat on strctral applications and spcially on trss applications, as th writrs ar not awar of attmpts to s th mthod in complx trss gomtry. Th two-fild formlation prsntd

2 allows on to trat strctrs combining diffrnt nginring modls straightforwardly mch in th sam mannr as whn sing th standard Galrkin mthod. Th othr prpos is to discss th ways to mak a discontinos Galrkin mthod comptitiv with th standard Galrkin mthod in comptational simplicity. Althogh introdction of an additional nknown may sm initially to b a bad ida in this rspct, it trns ot to b th ky fatr. Th first option is basd on an additional algbraic rlationship btwn th filds and lads to what is calld hr as th standard discontinos Galrkin mthod. An altrnativ option, basd on static condnsation on th lmnt lvl, is a novlty in connction with th discontinos Galrkin mthod, as far as th writrs ar awar of. As a first stp, th mthod is xplaind starting from a simpl stting and dtaild xplanations. Scond, th otcom is gnralizd to an xtndd principl of virtal work. Th principl corrsponds to local forms of th balanc laws of mchanics writtn ndr lss svr continity assmptions than sally. To b mor spcific, diffrntial qations ar rplacd by physically corrct jmp conditions on lins or srfacs of discontinitis. In th third and final stp, th gnric xtndd principl of virtal work is applid to bar- and bam-trsss. Ths applications indicat that th xtndd principl of virtal work and discontinos approximation can b sd mch in th sam mannr as th standard principl of virtal work and continos approximation. Nitsch s formlation Som rfrncs considr an articl by Nitsch [1] as containing th main ingrdints on which th discontinos Galrkin mthod can b basd. W thrfor start hr with an analogos stting consisting of th qation st s + b = 0 in Ω, (1) t t = 0 on t, () = 0 on, (3) in which t = n s, n is th nit otward normal vctor to and th constittiv qation is takn as s = S. (4) Th qations may b considrd to rprsnt for instanc th small transvrs dflction = xy (, ) of a strtchd mmbran in a plan domain Ω with a givn niform tnsion forc S pr nit lngth. Th traction t is givn on th traction bondary t. Similarly, th dflction is givn on th displacmnt bondary. Th disjoint bondary parts t and form togthr th whol bondary. Th gnric symbols Ω and ar commonly in s in th finit lmnt litratr althogh if wantd thy cold b rplacd hr say with A and s. Th maning of th rst of th notations in (1) to (4) is obvios. In what to follow, w shold rmmbr that s= s ( ) and t= t ( ), that is, ths qantitis dpnd on th dflction. Howvr, to shortn th formlas, ths dpndncis ar sally not shown. 3

3 On wll-known convntional starting point towards th finit lmnt tratmnt of th problm dscribd by qations (1) to (4) is to writ down th fnctional (potntial nrgy of th systm corrsponding to th standard Galrkin mthod whn minimizd in a finit dimnsional spac) V = s dω b dω t d Ω Ω. (5) t G 1 Normally in (5) is dmandd to satisfy in advanc (3). Th variation of (5) givs first G δv = s δdω bδdω t δd Ω Ω. (6) Intgration by parts in th first trm of th xprssion givs frthr th idntity s δdω = tδd ( s) δdω. (7) Ω Ω t It is to b notd that th bondary intgral is hr only ovr t sinc d to (3), δ = 0 on. Collcting th trms abov, th variation is fond to bcom δ G V = ( s + b) d ( t t ) d Ω δ Ω + δ. (8) Stting th rqirmnt δ V G = 0 for arbitrary δ givs th fild qation (1) and th traction bondary condition (th so-calld natral condition) (). On th basis of ths rslts, th standard stps applid in th convntional finit lmnt mthod with continos approximation making s of (5) can b ndrstood. Nitsch changd th stting dscribd abov by not dmanding to satisfy in advanc qation (3) in th variational principl. Th chang can b prformd as follows. Lt s altr th sitation by first sing th wll-known Lagrang mltiplir mthod. Ths w considr (3) as a constraint on fnctional (5) and s a modifid fnctional L G V = V + λ ( )d, (9) whr λ is th Lagrang mltiplir. Whn w now tak th variation, w obtain with rspct of th right-hand sid of (6) th xtra trms λδ d + δλ ( )d. (10) Frthr, in th intgration by parts th bondary trm in th maniplation (7) is now ovr th whol as constraint (3) is not pt in advanc on and δ is ths arbitrary also on. W obtain L δv = ( s + b ) δdω t Ω t + ( t t) δd + [( t+ λ) δ+ δλ( )]d. (11) t 4

4 Hr δ and δλ and ar arbitrary. W can ddc now qations (1), () and (3) and frthr, th intrprtation λ = t. (1) This givs th possibility to sbstitt rslt (1) back into (9) to arriv at th fnctional V N G 1 = V t ( )d + τ( ) d. (13) Th additional last trm which has mrgd on th right-hand sid is xplaind as follows. Dpr mathmatical analysis shows that for th formlation to work proprly in finit dimnsional cass (in th finit lmnt mthod) som additional wighting shold b pt on th satisfaction of constraint (3). This is achivd by adding on th right-hand sid of (9) th last-sqars (bondary) fnctional (th so-calld pnalty trm) LSB 1 V = ( ) d τ, (14) whr τ is a non-ngativ paramtr which may vary on. It is sn that dmanding th variation of (14) to vanish givs immdiatly qation (3). It may b notd that th factor 1/ in (14) is sd hr jst for asthtic rasons and is not ssntial as th magnitd of th trm dpnds ltimatly on th slction of τ. Exprssion (14) is in dtail LSB 1 1 V = τd τ d + τ( ) d. (15) As th last trm is a constant, it will disappar whn th variation is takn and, if wantd, it can ths b nglctd. In [1], th corrsponding trm has bn droppd. W prfr th mor compact form (14). Th formlation by Nitsch in [1] has now prodcd th two last xtra intgrals in (13). Factors lik τ ar calld in th litratr sally stabilization paramtrs or tning paramtrs or wightings and mch rsarch has bn dvotd on th slction of appropriat vals for thm. Th slction of th paramtr shold mak th xprssion dimnsionally homognos and nsr that th bondary trms imply that th ssntial bondary condition is satisfid also in a finit dimnsional spac. In fact, from th strctral point of viw, xprssion (14) can b intrprtd hr as th strain nrgy contribtion to th total potntial nrgy from an imaginary spring with a spring constant τ attaching th strctr to th srrondings. Discontinitis in Ω A lin loading in th mmbran problm or a point forc in a string problm may indc a discontinity in intrnal forc. Th discontinity in forc or vn in transvrs dflction may also b d to fnction st whrin th soltion is soght and thrfor th cas can b calld as nmrical. In th discontinos Galrkin mthod, 5

5 discontinitis appar insid th comptational domain along lmnt intrfacs as th finit dimnsional st of fnction dos not hav any bilt-in continity rstrictions at th lmnt intrfacs. Hr, w do not mak diffrnc btwn a physical or nmrical sorc bt rathr try to trat both possibl sorcs on th sam footing. I y Ω n + + n Ω + Figr 1 Domain consisting of two sb-domains. To xplain th notation to follow, w considr nxt th domain of Figr 1. Jst on lin of discontinity crosss th domain dividing it into two sb-domains Ω and Ω + or with a mor compact notation Ω E = { +., } With th finit lmnt mthod in mind, th sb-domains cold b considrd as two larg lmnts and I th common intrfac of th two lmnts. Th nknown in th mmbran or string problm is th transvrs dflction, which is considrd to b continos by its natr. This is rprsntd mathmatically ± ± x + = 0 on I, (16) whr = Ω (with th rstriction notation) dnot dflctions insid sb-domains Ω ±. To b prcis, ± in (16) man limit vals at I takn from Ω ± bt w omit th dtail as tchnical on. Th continity condition (16) can altrnativly b writtn in th form + = 0 on I, (17) = 0 and th displacmnt condition = 0 on can b xprssd = 0 = 0 on. (18) Finally, w may xprss (17) and th first qation of (18) in a concis form = 0 on, (19) 6

6 in which th sprscript taks as its vals all th indics of th sb-domains E = { +, } having a point p in common. In th xampl cas of Figr 1, E p = { +, } p I, E p = {} + on of Ω +, and E p = {} on of Ω. Now w considr as th ntir bondary consisting of th intrior and xtrior bondaris. This is consistnt with th sag of in th prvios chaptr as thr th intrior bondary did not xist. Abov w hav introdcd an intrfac displacmnt dfind vrywhr on th sb-domain bondaris. Th s of an additional nknown of this typ sms to b a novlty in connction with a discontinos Galrkin mthod althogh th ida of sing a fram approximation has bn discssd arlir.g. in [] and [3]. W try now to xtnd th Nitsch s ida of trating th displacmnt bondary condition in [1] to discontinitis insid th domain with th intrfac displacmnt concpt. Instad of qations (1) to (3), w assm th following st s + b = 0 in Ω \ I, (0) + t = 0 on \, (1) t = 0 on, () = 0 on, (3) in which = t I. Th rprsntation is implid by th gnric balanc laws of mchanics: diffrntial qation (0) is obtaind only ndr crtain continity assmptions on s if th matrial lmnt is chosn insid th mmbran and thrfor it is not valid on. If a highly localizd forc b is idalizd as lin loading t acting on I or t, th otcom is (1) whr th sm is ovr st Ep E. As discssd in connction to th simplifid stting of Figr 1, th st contains th indics of th sbdomains having point p on thir bondaris. In (), th indx taks all th vals in E p. In string and mmbran applications, E p contains on or two lmnts bt latr in trss applications any nmbr of sb-domains may hav point p on thir bondaris. W start with th fnctional L 1 V = s dω b dω t d Ω \ I Ω \ I \ + λ ( )d + λ( )d. (4) Extrnal potntial nrgy xprssion has now a contribtion from t also on I. Th constraints () and (3) hav bn takn into accont with th Lagrang mltiplirs λ E and λ. Abov w mak diffrnc btwn a sm insid intgral and otsid of it. In th lattr cas, th intrfac points ar xcldd. This will b indicatd by notation I fdω f d Ω \ Ω E. (5) Ω 7

7 If a sm is placd insid intgral, only th sb-domains having a point in common ar accontd for. This will b indicatd by notation d ( f f E )d. (6) Th somwhat nconvntional dfinition (6) bcoms ndrstandabl latr, whn mor than two sb-domains may shar an intrfac point. Th variation of (4) is L δv = s δdω bδ dω t δ d Ω \ Ω \ \ I I [ ( ) ( + δλ + λ δ δ )]d + [ δλ( )+ λδ ]d. (7) Th stps ndd ar rathr tdios bt to gain confidnc in th mrging formlas w obviosly shold go throgh thm. Intgration by parts ovr th sb-domains givs th idntity s δdω = t δ d ( s) δdω Ω \ (8) Ω \ I and th variation is fond to bcom δ L \ ( ) d \ ( V s b δ Ω t λ ) δ = + + Ω d I + ( t + λ ) δ d + δλ ( )d p ( )d ( ) + δλ + λ λ δ d. (9) W obtain ths from th rqirmnt δ V = 0 th corrct fild qations and th displacmnt continity qations () and (3). Frthr, w hav th intrprtations t L I λ = on, (30) λ = λ on (31) and whn ths ar applid in (9) w finally obtain th satisfaction of (1). Th nxt stp is to sbstitt (30) and (31) back to (4) to obtain th fnctional L 1 V = s dω b dω t d Ω \ I Ω \ I \ ( )d t t ( )d. (3) Smmarizing, th rlvant fnctional is ths 8

8 G JI JB LSI LSB V = V + V + V + V + V, (33) whr th standard contribtion, taking into accont qation (0) and partly qation (1), is G 1 V = s dω b dω t d Ω \ I Ω \. (34) I \ Points whr th intgrand may b discontinos hav xplicitly bn xcldd as indicatd by notation Ω \ I. Th jmp contribtions taking into accont partly qation (1) and qations () and (3) ar JI V = t ( )d, (35) JB V = t ( )d. (36) W hav not yt prformd a similar stp as in qation (13) of ptting xtra wight on th constraints. Ths w can spclat on sing additionally th last-sqars intrfac and bondary fnctionals LSI 1 V = τ ( )d, (37) LSB 1 V ( ) d τ =, (38) whr paramtrs τ and τ ar dimnsionally corrct bt othrwis arbitrary. Now, xprssions (37) cold also b intrprtd as th strain as th strain nrgy contribtions from imaginary springs with spring constants τ attaching th sb-domains to th fram. W notic that th xact soltion to qations (0) to (3) maks all th trms in G xcss to V vanish. To simplify th xprssions, on may impos additional assmptions on th fnction st whrin th stationary val of th fnctional is soght in th sam mannr as with th standard formlation. For xampl, th displacmnt bondary condition = 0 on can b satisfid in th strong sns so that trms (36) and (38) vanish. Th slction is particlarly sfl in th on-dimnsional cas whn consists of points as no intrpolations ar involvd. Principl of virtal work for th mmbran Extndd principl of virtal work For comparison with th prsntation to follow, w may rprsnt th formlation (33) also sing th principl of virtal work. Thn th goal is to find, sch that δ W = 0 δ,. (39) δ 9

9 Th dtaild spcification of th fnction st whrin th soltion is soght is a tchnical dtail in this connction. Roghly, th fnction st is assmd to b consistnt with th assmptions sd in th strong form. Th virtal work xprssion consists of th standard, jmp, and last-sqars contribtions ach giving two trms to INT EXT JI JB LSI LSB δw = δw + δw + δw + δw + δw + δw. (40) This will b calld hr as xtndd principl of virtal work to discrn it from th standard form. Th standard contribtions of intrnal and xtrnal forcs, taking into accont qation (0) to and partly qation (1) ar INT δw = s δdω, (41) Ω \ δ EXT W b δ d Ω t δ = + d. (4) Ω \ \ Th jmp contribtions taking into accont partly qation (1) and qations () and (3) ar and th last-sqars contribtions ar JI δw = δ[ t ( )]d, (43) JB δw = δ[ t ( )]d, (44) LSI 1 δw = δτ [ ( ) ]d, (45) LSB 1 δw = δτ [ ( ) ]d. (46) Th principl of virtal work can b takn as th stationarity condition of fnctional (33), with a slightly modifid notation and, thrfor, th two formlations ar qivalnt in th prsnt cas. Howvr, th principl of virtal work maks sns also whn a stationary principl of a fnctional dos not xist and thrfor it can b takn as mor gnral. Whn xpandd with rspct to th δ oprator, th jmp contribtions bcom JI δw [ βδt ( = ) + t ( δ δ )]d, (47) JB δw [ βδt ( = ) + t( δ δ )]d, (48) in which β = 1 and δ = 0. If paramtr β is givn som othr val, consistncy with (0) to (3) is not affctd, bt th principl of virtal work dos not rprsnt a stationarity condition of any fnctional. Th sign chang modification β = 1 discssd.g. in [4], [5] has a favorabl ffct on stability proprtis on finit dimnsional spacs, 30

10 and on may spclat that paramtr β can b sd to balanc th accracy and stability of th discrt mthod. Th virtal work qation (39) is satisfid by th soltion to qations (0) to (3) and thrfor th formlation is consistnt. Standard maniplation sing intgration by parts and fndamntal lmma of th calcls of variations show that th principl implis qations (0) to (3) and thrfor, th two rprsntations ar qivalnt. Bsids that, on wold lik to show that th xtndd principl of virtal work givs an niq soltion with, say, picwis continos polynomial approximations to th nknown and. This thm is, howvr, ot of th scop of th prsnt articl. On-fild formlation Th xtndd principl of virtal work discssd in th prvios sction dos not rqir intr-lmnt continity of th approximation in a finit lmnt application, bt as a drawback involvs sparat approximations for lmnts and thir intrfacs. From th practical viwpoint, this maks th mthod lss attractiv than a on-fild formlation. Th sal Galrkin mthod wold b obtaind by slcting th intrfac displacmnt to b th rstriction of lmnt displacmnt to i.. by sing condition = on (49) in th virtal work xprssion. Thn also th approximation shold b continos to b consistnt with th assmptions of th formlation. A slightly lss rstrictiv algbraic rlationship, along th sam lins, follows from ( ) = 0 on. (50) Th slction can b motivatd.g. as follows. As τ is a fr paramtr of th formlation, on may considr trm (37) as constraint or pnalty trm and lt τ so that th minimizr of (33) ssntially satisfis (50). If, as in most cass, th nmbr of trms in th sm is two on I and on on \ I, th qation givs th man val = on I and = on \ I. With ths rlationships th virtal work xprssions (43) and (44) bcom th sal trms of th standard discontinos Galrkin mthod JI 1 δw = δ [ t ]d, (51) JB δw = δ[ t ( )]d. (5) Th last-sqars contribtions associatd with jmp contribtions tak also th familiar forms LSI 1 δw = δ [ τ ]d, (53) 4 31

11 LSB 1 δw = δ [( ) τ ( )]d, (54) assming that τ = τ. A discontinos Galrkin mthod of this typ is known to shar th convrgnc and stability proprtis of th standard mthod basd on a continos approximation [5], [6]. Althogh an algbraic rlationship lik (50) is simpl in th lastic mmbran or string problm, th otcom can b qit complx whn a strctral modl combins bams, plats, tc. Also th improvmnt in comptational simplicity is not nogh to mak th mthod comptitiv with th standard Galrkin mthod. W sggst hr a qit diffrnt approach basd on th limination of lmnt approximation withot additional rstrictions lik (50). Thn maniplations ar not possibl on th virtal work xprssion lvl, bt thy ar rathr prformd in discrt form of lmnt contribtions T δ a K K a F δw = (55) δ a K K a F in which a and a ar th paramtrs of th lmnt and intrfac approximations, rspctivly. Exprssion (55) can b maniplatd into 1 a = ( K ) ( K a F ), (56) T 1 1 δw = ( δa ) [{ K K ( K ) K } a { F K ( K ) F }] (57) so that th final virtal work xprssion is obtaind by smming th lmnt contribtions of xprssions (57) containing only th paramtrs of intrfac approximations. Aftr solving ths, th paramtrs of lmnt approximations follow from (56). Thn, th comptational work is dominatd by th nmbr of paramtrs of intrfac approximations which roghly coincids with that of th standard mthod. In principl, on may combin a low ordr intrfac approximation with a high ordr lmnt approximation and s a discontinos Galrkin mthod withot mch comptational ovrhad, althogh accracy may b rstrictd by th intrfac approximation. As far as th writrs know this typ of limination procdr has not bn pt into practic althogh th possibility has bn mntiond alrady in []. Th implmntation sd in th application xampls to b discssd latr, maks s of th limination mthod, in which th xtndd principl of virtal work is sd as it is. As an attractiv fatr, th paramtrs of lmnt approximations ar tratd as intrnal variabls not ncssarily visibl to th sr of softwar at all. From th sr viwpoint, th implmntation looks xactly lik th on basd on a continos approximation with additional paramtrs dfining th polynomial dgr of lmnt approximations. 3

12 Figr Transvrs displacmnt / U of a string as fnction of x/ L. Exact soltion in bl lin, lmnt soltion in black lin, and intrfac soltion in black markrs. 33

13 String application xampl As a prliminary xampl, th mthod is applid to th on-dimnsional string problm on Ω = ]0, L[ with a constant S, sinsoidal b and a spcifid val of on th bondary = {0, L}. Exact soltion to th problm is givn by x = sin( π ), (58) U L whr U is th maximm transvrs dflction. Th discrt soltion mthod is basd on th xtndd principl of virtal work and picwis polynomial approximation of dgr p insid th lmnts withot any continity rstrictions on I. Th stabilization paramtr xprssion was chosn as S τ = α, (59) h in which h is th lmnt siz and α > 0. Discrt soltions for som rprsntativ slctions of stabilization paramtr α, symmtry paramtr β, nmbr of lmnts n, and polynomial dgr of approximation p ar compard to th xact soltion in Figr. Th intrfac soltion with β = 1 is xact and thrfor also indpndnt of α, p and n, which was vrifid by solving a st of discrt problms with th xact arithmtic of Mathmatica nvironmnt. Thrfor, larg α dos not man nmrical trobls whn β = 1. For larg vals of α, th lmnt soltion is practically continos no mattr th polynomial dgr p 1 and indpndnt of β. Th nmrical soltion by nonsymmtric formlation β = 1 bhavs mch in th sam mannr, althogh th intrfac soltion is not xact as with th symmtric formlation. Th prcis rols of paramtrs α and β wr stdid from analytical soltions to th discrt problm. As th otcom, th minimm val of α for stability incrass in p according to (sav a positiv mltiplir) αmin = p( p+ 1) β. (60) Thrfor, if β is chosn to b ngativ, any positiv val of α sffics for stability (bt not ncssarily for accracy). Th sam conclsion follows also from a mor carfl stability analysis [5]. Th slction 4 α( p) = 10 max{ p( p+ 1) β,1} (61) basd on (60) will b sd latr in connction with xampls whr th main intrst is not in th ffct of α. Principl of virtal work Som notations W now try to gnraliz th approach dscribd in th prvios chaptr to linar lasticity. Th goal is to s th xtndd principl of virtal work and discontinos 34

14 approximation in th sam mannr as th standard principl of virtal work and continos approximation. A coordinat systm invariant rprsntation basd on vctors and dyads, dnotd a and a, will b sd to kp th xprssions concis. Thr, dobl dots in xprssion a: c mans dobl innr prodct and conjgat dyad a c, dfind by a c = c ac c, rprsnts transpos. If a = ac, th dyad is takn symmtric and, if a = ac anti-symmtric. Th dobl innr prodct of a symmtric and anti-symmtric dyad a and c satisfis a: c = 0. Th sal rls of vctor algbra apply and.g. th dobl innr prodct can b intrprtd as jst two innr prodcts. In a Cartsian coordinat systm, whr i j = δij (Kronckr dlta), th dtaild componnt rprsntations of conjgat dyad and dobl innr prodct ar ac =, ja i ij, (6) a : c = ( a ):( c ) = ac ij c ij, i j ij kl, l k kl ij, ij ij. (63) Th intgration by parts formla, ndd in connction with a linar lasticity problm, ( a ) c dv = n a c d A a :( c ) dv, (64) V A V assms that a and c ar continos in V (or hav vn continos first drivativs in connction with th fndamntal thorm of calcls). In th prsnt contxt, continity assmption of (64) dos not hold as it is, bt V consists of disjoint sb-domains whrin (64) is valid. Thn th intgration by parts formla can b writtn as V\ A A V\ A I ( a ) c d V = n a c d A a :( c ) d V, (65) whr V \ A I is th nion of th sb-domains and A consist of th intrior bondary A I and of th xtrior bondary V. If A I =, as in connction with (64), A= V. Th sm is ovr th sb-domains having a point p A on thir bondaris. On physical sbdomain intrfacs or intrior bondary th nmbr of thos is two and on th xtrior bondary on. In connction with nginring modls and trss-lik strctrs, th nmbr of trms may xcd two. Standard form Th principl of virtal work in its standard form stats that th sm of th virtal works of intrnal forcs and xtrnal forcs acting on a body is zro for any kinmatically admissibl virtal displacmnt. In mathmatical notation INT EXT δw = δw + δw = 0 δ (66) in which th virtal work of th intrnal forcs is givn by (small displacmnt thory is assmd) INT δw σ : δε dv (67) = V c I c c 35

15 and that of th xtrnal forcs is W EXT b δ = δ d V t + δ d A. (68) V Th maning of th notations shold b rathr obvios bt som commnts ar in plac. First, no variations of qantitis calld intrnal work or xtrnal work ar involvd. Th lft-hand sids ar jst short-hand notations for th xprssions apparing on th righthand sids. Scond, qit oftn in th litratr, th mins sign is not sd in th dfinition of th intrnal virtal work. Thn th corrsponding trm appars on th othr sid of virtal work qation. In what to follow, som xprssion can b takn as variations of som othr xprssion and som shold b considrd indpndnt. To mak diffrnc with th two manings of th dlta-symbol, w s th oprator notation δ [ V ] with brackts in th formr cas and omit th brackts whn w man an indpndnt qantity lik δ W. W will rcall shortly how th abov xprssions ar arrivd at. Th wll-known local forms of th balanc laws of momntm, momnt of momntm and th additional displacmnt rstriction on th bondary σ + b = 0 in V, (69) σ σ c = 0 in V, (70) A t t t t = 0 on A. (71) = 0 on A, (7) in which t = n σ and n is th nit otward normal vctor to A, ar takn as th govrning qations. Th balanc law of momnt of momntm xclds volm momnts in its prsnt form (70). W rmmbr th important point that th principl of virtal work is valid for a body irrspctiv of th constittiv law of th matrial of th body. This is rflctd in qations (69) to (71) as no matrial law appars thr. Natrally, to solv an actal problm, w mst introdc at som lvl information abot th matrial bhavior, as w did in qation (4). Prfrably, this shold b don as lat as possibl to kp th formlation as gnral as possibl. Hr w procd ths assming that th strss σ finally dpnds in som way throgh th constittiv law on th displacmnt. Howvr, in most to follow, w do not show dirctly this dpndnc. Also, th constittiv qation is assmd to satisfy qation (70) a priori and thrfor (70) dos not ntr into th discssion. W wold lik to mphasiz, howvr, that w cold wll trat (70) on th sam footing as th othr conditions implid by th basic laws of mchanics. Th stps ar as follows. Eqation (69) is mltiplid by an arbitrary (wighting) fnction δ of position, and intgratd ovr th domain. A scalar qation is obtaind: ( σ) δdv + b δdv = 0. (73) V V 36

16 Intgration by parts with (64) givs ( σ) δ dv = n σ δ d A σ :( δ ) dv. (74) V A V Th wll-known maniplations basd on th assmd symmtry of σ, division of displacmnt gradint into th symmtric and anti-symmtric parts according to 1 1 δ = [ δ+ ( δ)] c + [ δ ( δ)] c δε + δω, (75) idntity σ : δω = 0, and dfinition of traction t = n σ in (74) giv σ : δε dv + b δ dv + t δda= 0. (76) V c V A Now (76) is sn to b narly th virtal work qation (66). In th convntional application form, on additionally taks th intrprtation δ = δ[ ], so that th additional rstriction (7) imposd on th bondary implis δ = 0 on A, (77) and ss th condition of (71) t = t on A t (78) to obtain th standard form. Th prpos of th slction (77) in (76) is to prvnt th nknown traction on bondary A to appar in th formlation. W s many analogs hr with th maniplations prformd in th arlir chaptrs. Howvr, thr is on important diffrnc. Th principl on virtal work in its gnral form is not a variational principl in th sns that th first variation of a fnctional is st to zro and w ths cannot mak s, say of a principl δ [ W ] = 0. Extndd form Th principl of virtal work will b modifid nxt to tak into accont a possibl discontinity in strss σ and displacmnt insid th domain. W assm that th domain can b dividd into disjoint sb-domains so that conditions for (69) to (7) ar still valid vrywhr xcpt on th sb-domain intrfacs. Displacmnt on th sb-domain bondaris bcoms an additional nknown of th problm, displacmnt insid th sb-domains is dnotd by and a sprscript will b sd to dnot displacmnt in a typical sb-domain E. Th local forms of th balanc laws of momntm, momnt of momntm and th additional displacmnt rstrictions on th bondary bcom now σ + b = 0 in V \ A I, (79) σ σ c = 0 in V \ A, (80) I c 37

17 + t = 0 on A\ A, (81) t = 0 on A, (8) = 0 on A, (83) whr A= A At AI. As th drivation of th balanc law of momntm (79) assms crtain continity on strss and rqir that a matrial is insid th domain, th qation is not valid.g. at points whr an xtrnal srfac load t is acting. Ths points, insid th domain or at th traction bondary, hav bn xcldd from (79) and takn into accont in (81) (implid by th vry sam law) instad. Continity of th displacmnt is rprsntd by (8) and th displacmnt bondary condition by (83). Instad of going throgh in dtail th straightforward bt somwhat lngthy maniplations of th prvios sctions, w start dirctly analogosly from th xtndd principl of virtal work: find, sch that δ W = 0 δ,. (84) Again, th virtal work xprssion consists of th standard, jmp, and last-sqars contribtions ach giving two trms to δ INT EXT JI JB LSI LSB δw = δw + δw + δw + δw + δw + δw, (85) whr th dtaild xprssions ar now chosn to match (79) to (83). Th standard contribtions of intrnal and xtrnal forcs, taking into accont qation (79) and partly qation (81) δw INT I = V\ A EXT δ = δ d + δ V\ A AA \ σ : δε dv, (86) I c W b V t da (87) shold hav rathr clar physical manings. Notation V \ A I indicats that points whr th intgrand may b discontinos hav bn xcldd. Th symmtric jmp contribtions taking into accont partly qation (81) and qations (8) and (83) ar W JI [ t ( δ δ = )]d A, (88) A JB δw = δ[ t ( )]da. (89) A Th gnralizations of (88) and (89) containing th symmtry paramtr β and corrsponding to (47) and (48) shold b obvios. Finally, th last-sqars contribtions associatd with (88) and (89) ar W [( ) ( )]d A δ δ τ = A, (90) LSI 1 38

18 δw = δ[( ) τ ( )]da, (91) LSB 1 A in which on may assm τ = τ withot loss of gnrality as only th symmtric part c of τ has ffct on th stting. W apply nxt th gnric qations to driv th xtndd virtal work xprssions of bar-trss and bam-trss modls in th sam mannr as is don with th standard principl and mor svr continity assmptions. W ar not awar of a similar consistnt dvlopmnt in connction with discontinos Galrkin mthods. Bar- trss application Som notations Th first application is a trss consisting of bars connctd by frictionlss sphrical or cylindrical joints. Displacmnts at th joints ar continos bt rotations of bars may show jmps thr. Joints ar also abl to transmit forcs bt not momnts. A bar trss can b loadd by distribtd forcs in th dirctions of th bars and point forcs acting on th joints. Althogh th bar-joint systm is an lastic body having jst a complx ntwork gomtry, th joints ar idalizd as small rigid bodis so that th gomtrical axs of bars xtnd to th sam point at a joint. With this idalization, dtaild gomtry of a joint bcoms immatrial. Gomtrically, bars ar slndr cylindrical bodis so that V = A Ω in which Ω is th (mathmatical) soltion domain. Matrial xyz systms ar sd to idntify th particls of bars. Althogh not ncssary, th x axs ar chosn to coincid with th gomtrical axs. A strctral XYZ systm is sd as common rfrnc fram. Th nit vctors of th matrial and strctral coordinats ar dnotd by i, jk, and I, J, K, rspctivly. Th wll-known kinmatic and kintic assmptions of th bar modl ar = xi ( ), (9) σ = σii (93) and intrfac (joint) displacmnt = I + J + K (94) X Y Z may hav on, two or thr componnts dpnding on th trss typ. It is notworthy that th intrfac displacmnt (94) is rprsntd in th basis of th strctral coordinat systm whras th rlativ displacmnt (9) is rprsntd in th basis of matrial coordinat systm. It is notworthy that th displacmnts of bar particls ar dfind by (9) and (94) togthr. By assmption, cross-sctions mov as rigid bodis so that on may attach a matrial coordinat systm to any of thm. Th positions and orintations of matrial coordinat systms ar dtrmind by th intrfac points. Only th rigid body rotation arond th x axis rmains non-spcifid, bt that dos not mattr in th bar modl. 39

19 Bar-trss qations Th diffrntial qation of th bar-trss problm is givn by ( F + B ) i = 0 in Ω \ I, (95) in which F is th strss rsltant, B is th rsltant of th xtrnal volm forcs, and th sprimposd comma mans drivativ with rspct to x. Joints locatd on I ar xcldd as th drivation of (95) assms crtain continity of F. On points of discontinitis, th balanc law of momntm givs instad 0 T + T = on \, (96) in which T nf and n = ± 1 is th nit otward normal to Ω. In th bar-trss modl, th nmbr of bars connctd to a joint is not rstrictd to two and thrfor th sm may hav mor than two trms, too. Th givn xtrnal point forc T is assmd to act on joints whr displacmnt is not spcifid as indicatd by \. Continity of displacmnt at joints is xprssd as ( ) i = 0 on, (97) ( ) i = 0 on, (98) which stand for a st of qations writtn for all th bars connctd to a joint. Th domain notation corrspond to that sd arlir and = t I. Assming that th givn traction t is acting only on th nd plans of bars, th rsltants of xtrnal srfac and volm forcs bcom T = t da, (99) B = b da, (100) in which b is th givn volm forc, and th intgrals ar ovr cross-sctions. Finally, th rsltant of intrnal forcs acting on th cross-sctions is F = i σ da. (101) Eqations abov ar valid no mattr th matrial modl. Virtal work xprssion Th xtndd virtal work xprssions of th bar-trss problm follow from gnric xprssions (86) to (90) whn th kinmatic and kintic assmptions of th modl ar applid thr. Hnc, w procd xactly in th sam mannr as with th (standard) principl of virtal work. 40

20 Figr 3 Axial displacmnt / U of bar as fnction of x/ L. Exact soltion in bl lin, lmnt soltion in black lin, and intrfac soltion in black markrs. 41

21 Th standard contribtions of intrnal and xtrnal forcs, taking into accont qation (95) to and partly qation (96) INT δw = F δ dω, (10) Ω \ d ( ) (103) δw EXT = B Ω δ Ω + \ \ T δ shold b obvios. Th jmp contribtions taking into accont partly qation (96) and qations (97) and (98) ar JI δw = δ[ T ( )], (104) JB, (105) δw = δ[ T ( )] and th last-sqars contribtions associatd with th jmp trms bcom LSI 1, (106) δw = δ[( ) τ ( )] δw = δ[( ) τ ( )]. (107) LSB 1 Mor dtails of th rathr straightforward drivation of ths xprssions ar givn in Appndix A. Linarly lastic matrial modl In linarly lastic, homognos and isotropic cas whn th strss-strain rlationship is givn by th gnralizd Hook s law and th linar strain-displacmnt rlationship sffics, constittiv qation (101) bcoms F = i σ da = iea, (108) in which E is th Yong s modls of matrial. On paramtr family of wightings EA τ = α ii, (109) h of th last-sqars trms (107) and (106) sffics hr, as th main mphasis is on th xplanation of th two-fild formlation. Bar application xampl Th first application xampl is a bar ndr a sinsoidal distribtd load. Lft nd of th bar is fixd and th right nd is fr maning that no xtrnal forc is acting thr. Th xact soltion to th problm is 4

22 x = sin(7 π ), (110) U L in which U is th maximal displacmnt. Th discrt soltion mthod is basd on th xtndd principl of virtal work and picwis polynomial approximations of dgr p withot any continity rstrictions at joints or at th lmnt intrfacs. Th displacmnt bondary condition is satisfid in th strong sns. Th discrt soltion with th symmtry paramtr β = 1 and som rprsntativ vals of stabilization paramtr α, nmbr of lmnts n, and polynomial dgr p ar compard to th xact soltion in Figr 3. From th nmrical viwpoint, an lastic bar problm dos not diffr from th string problm and th discssion and conclsion thrin applis also hr. Z p 3 z x p 1 z x 1 X p Bar-trss application xampl Figr 4 Trss consisting of two bars. Th scond application xampl is a bar-trss consisting of two lastic bars connctd by frictionlss sphrical joints shown in Figr 4. Th joints of th initial gomtry ar at points p 1 = (0,0,0), p = ( L,0,0), p 3 = (0,0, L) on th X and Z axs of th strctral coordinat systm. Bars 1 and of th trss hav nd points (p 1,p ) and (p 3,p ), rspctivly, in which th ordr of points indicats th positiv dirction of th x axis. Joint at p is fr to mov and th rmaining two ar fixd. Th crosssctional ara A and Yong s modls E ar constants. Bar 1 is sbjctd to a distribtd sinsoidal loading acting in th x dirction so that th xact soltion is 1 = sin(5 π )( I + K), (111) U 1 U x = sin(5 π ). (11) L 43

23 Th rfrnc val U controls th maximm displacmnt. Althogh bar dos not ndrgo a lngth chang and = 0, its orintation changs, as both bars ar connctd to th joint at p. Again, th discrt soltion mthod is basd on th xtndd principl of virtal work and picwis polynomial approximations of dgr p withot any continity rstrictions at joints or lmnt intrfacs. Th displacmnt bondary condition is satisfid in th strong sns. Stabilization paramtr xprssion (61) is sd to obtain a nar-continos discrt soltion. Th soltion obtaind with th rprsntativ vals of symmtry paramtr β and polynomial dgr p ar compard to th xact soltion in Figr 5. Th discrt soltion bhavs in th sam mannr as in th bar application. Figr 5 Axial displacmnt / U of bar 1 as fnction of x/ L. Exact soltion in bl lin, lmnt soltion in black lin, and intrfac soltion in black markrs. Bam- trss application Som notations Th scond nginring modl application is a bam-trss. In th sam mannr as with th bar modl, th soltion domain is a D or 3D ntwork of bams connctd by sphrical, cylindrical, or tc. joints transmitting forc and momnt componnts in som combination. Th trss can b loadd by distribtd xtrnal forcs and momnts acting on th bams and point forcs and momnts acting on th joints. Th coordinat systms coincid with thos of th bar-trss problm. Th kinmatic and kintic bam assmptions ar = 0 ( x) + θ( x) ρ, (113) σ = σ ii + σ ij + σ ji + σ ik + σ ki, (114) xx xy yx xz zx 44

24 in which 0 ( x) and θ ( x) dnot th translation and (small) rotation of cross-sctions and ρ ( y, z) = jy + kz is th position vctor rlativ to th intrsction of cross-sction and th x axis. Intrfac displacmnt and rotation = XI + YJ + ZK, (115) θ = θxi + θyj + θzk (116) may hav on, two or thr componnts dpnding on th trss typ. In th prsnt cas, th nknowns of th problm ar th displacmnt and rotation componnts which shold b accontd for in th intrprtation δ = δ[ ]. Also now, th displacmnts of bam particls ar dfind by intrfac and lmnt displacmnts togthr so that th positions and orintations of matrial coordinat systms ar dtrmind by (115), and (116) and rlativ displacmnts by (113). Timoshnko modl Th strong form of a Timoshnko bam trss problm consists of diffrntial qations F + B = 0 in Ω \ I, (117) M + i F + C = 0 in Ω \ I, (118) in which F and M ar th strss rsltants, B and C ar distribtd forcs and momnts. Whn writtn for joints and othr possibl points of forc and momnt rsltant discontinitis, th balanc laws tak th forms + T = 0 on \, (119) T 0 S S = on \ θ, (10) in which T nf, S nm, and n = ± 1 is th nit otward normal to domain Ω. Th givn xtrnal forcs and momnt T and S ar assmd to act on joints whr displacmnt or rotation ar not givn. Continity of displacmnt and rotation at joints can b xprssd as = 0, (11) θ θ = 0 on θ, (1) θ θ = 0 on, (13) = 0 on. (14) Assming again that th givn traction t is acting only on th nd plans of bams, th rsltants of xtrnal volm and srfac forcs bcom 45

25 B = b da, (15) C = ρ b da, (16) T = t da, (17) S = ρ t da, (18) in which ρ = yj + zk is th rlativ position vctor, b is th givn body forc, and th intgrals ar ovr cross-sctions. Thrfor, th momnt rsltants C and S ar sn to dpnd on th slction of matrial coordinat systms. Finally, th rsltants of intrnal forcs acting on th cross-sctions ar F = i σ da, (19) M = i σ ρda, (130) T = nf, (131) S = nm. (13) Eqations abov ar valid no mattr th matrial modl which will again b introdcd as lat as possibl to kp th formlation as gnral as possibl. Brnolli modl In th Brnolli bam-trss modl, qations (117) and (118) ar jst rwrittn as i ( F + B ) = 0 in Ω \ I, (133) M i B + ( C ) = 0 in Ω \ I. (134) M + i F + C = 0 in Ω \ I, (135) and th shar forc componnts F y and F z ar takn as constraint forcs to b solvd from (135). Also, th corrsponding kinmatical so-calld Brnolli constraints θ =, (136) y z y z θ = (137) ar sd to liminat th rotation componnts θ y and θ z from constittiv qations to b discssd latr. In short, th main diffrnc btwn th two modls ar th Brnolli modl xprssion δθ = δθ i δ j + δ k, (138) x z y 46

26 F= Fi M j+ M k x z y valid whn C y = C z = 0. Th intrfac displacmnt as with th Timoshnko modl. and rotation θ (139) ar th sam Virtal work xprssion Th (xtndd) principl of virtal work follow from th gnric xprssions and th Timoshnko bam assmptions (113) and (114). Hnc, w procd xactly in th sam mannr as with th (standard) principl of virtal work. Th wak form of th Brnolli bam modl follows from that of th Timoshnko modl, whn xprssions (138) and (139) ar sbstittd thr. Th standard contribtions of intrnal and xtrnal forcs, taking into accont qations (117), (118) and partly (119), (10), bcom INT δw = [ F δ + M δθ + ( F i) δθ ]dω, (140) Ω \ I EXT δw = ( B δ+ C δθ )dω Ω \ I + ( T δ ) + ( S δθ ) \ (141) \ θ Th jmp contribtions taking into accont qations (11) to (14), ar δw JI δ [ T ( )] δ [ S ( θ θ = + )], (14) θ δw JB [ T ( )] [ S ( )] δ δ θ = + θ, (143) and th last-sqars trms ar LSI 1 [( ) ( δw = δ τ )] 1 [( ) θ ( δ θ θ τ θ θ )], (144) LSB 1 [( ) ( δw = δ τ )] 1 [( ) θ ( δ θ θ τ θ θ )] θ. (145) Som dtails of th drivation ar givn in Appndix B. Linarly lastic matrial modl In th linarly lastic, homognos and isotropic cas, whn th strss-strain rlationship is givn by th gnralizd Hook s law and th linar strain-displacmnt 47

27 rlationship sffics, th Timoshnko bam constittiv qations (19) and (130) bcom F = i σd A = iea + jga( θ ) + kga( + θ ), (146) x y z z y M = i σ ρd A = ig( I + I ) θ + jei θ + kei θ yy zz x zz y yy z (147) in which E and G ar th Yong s and shar modl of matrial, rspctivly. In th bam modl, th gomtry of th cross-sctions is dscribd by th momnts A= da, (148) Sr Irs = rda, (149) = rsda, (150) whr rs, { yz, }. Th simpl constittiv qations assm that th matrial coordinat systm is chosn so that th first momnts Sy = Sz = 0 and th scond momnts I = I = 0. Th wightings yz zy 1 τ = α ( iiea + jjga + kkga), (151) h 1 τθ = αθ [ iig ( I yy + Izz ) + jjeizz + kkei yy ] h (15) of th last-sqars trms (145) and (144) sffic for dmonstrativ prposs. Exprssions can b obtaind by considring polynomial approximations and th cass of pr bnding, torsion tc. sparatly and combining th rslts. Brnolli modl constittiv qations follow from (146) and (147) whn Brnolli constraints (136) and (137) ar sd thr (actally, th Brnolli constraints follow from th constittiv qations (146) and (147), bt this thm is somwhat ot of scop). Whn combind with xprssion (139), th otcom is F = iea jei kei, (153) x yy y zz z M = ig ( I + I ) θ jei + kei. (154) yy zz x zz z yy y Th wightings of th last-sqars trms ar obtaind also hr by considring first th cass of pr bnding, torsion tc. sparatly and combining th rslts. Th otcom is τ = α ( iiea + jj EI ) yy + kk EI zz, (155) h h h 1 τθ = αθ [ iig ( I yy + Izz ) + jjeizz + kkei yy ]. (156) h 48

28 Figr 6 Dflction w/ W and rotation θ / Θ as fnctions of x/ L. Exact soltion in bl lin, lmnt soltion in black lin, and intrfac soltion in black markrs. 49

29 Bam bnding application xampl Th first application xampl is pr bnding of a simply spportd bam in th XZ plan ndr a sinsoidal distribtd load. Cross-sctional ara A, scond momnt of ara I, Yong s modls E, and shar modls G ar constants. Th crosssctional ara is takn to b larg so that w W x = sin(5 π ), (157) L is th xact soltion to th Brnolli and Timoshnko modls. Rotation follows from θ = w. Th discrt soltion is basd on xprssions (140) to (145), and polynomial approximation of dgr p for displacmnt in th Brnolli modl and p, q= p 1 for displacmnt and rotation, rspctivly, in th Timoshnko modl. Stabilization paramtrs α = α( p) and αθ = α( q) according (61) striv for a nar continos discrt soltion. Figr 6 shows th discrt soltions to w and θ with som rprsntativ vals of th symmtry paramtr β, nmbr of lmnts n, and polynomial dgr p. Th discrt soltions by th two modls practically coincid and β has a ngligibl ffct d to th larg α. Intrfac displacmnt and rotation ar xact whn β = 1, p 3 and q p 1. A convrging soltion is obtaind also whn p =, bt accracy of th intrfac soltion is lost and th vals of α and αθ bcom significant. Whn p = q= 1, th Timoshnko modl soltion shows svr shar locking. Z p 3 z x z x p p 1 X 1 Bam-trss application xampl Figr 7 Trss consisting of two bams. Th scond application xampl is a trss consisting of two lastic bams shown in Figr 7. Th gomtry is th sam as with th bar-trss application xampl. Th cross-sctional ara A, scond momnt of ara I, Yong s modls E, and shar modls G ar constants. Cross-sctional ara is larg so that th xact soltions to th Brnolli and Timoshnko modls practically coincid and axial displacmnts ar 50

30 ngligibl. Bam 1 is sbjctd to a distribtd transvrs sinsoidal load, bam is clampd at p 3, bam 1 rotats frly at p 1 and displacmnts and rotations ar continos at p. Th xact soltion to th Brnolli bam modl is 1 w W 1 x 3 π x x = s in(3 π ) + ( )[( ) 1], (158) L L L w 9π = ( x ) ( x ). (159) W L L Th rfrnc val of dflction W controls th maximm displacmnt. As th crosssctional ara is considrd vry larg, bams ar inxtnsibl in th axial dirctions and joint at p may ndrgo a rotation bt not displacmnt. In th discrt soltion, th displacmnt and rotation bondary conditions ar satisfid in th strong sns. Discrt soltions to th dflction and rotation of bam 1 ar compard to th xact soltion (158) in Figr 8. Stability paramtr xprssion was according to (61) i.. α = α( p) and αθ = α( q). Th intrfac soltions by th symmtric formlation ar xact and thrfor indpndnt of α and p 3. Th discrt soltion by th Brnolli and Timoshnko modls practically coincid and Figr 8 srvs for both cass and also β = ± 1 d to th larg α sd. Th discrt soltions to bam of th trss coincid with th xact soltion in all cass of Figr 8. 51

31 Figr 8 Dflction w/ W and rotation θ / Θ of bam 1 as fnctions of x/ L. Exact soltion in bl lin, lmnt soltion in black lin, and intrfac soltion in black markrs. Conclding rmarks In this work th standard principl of virtal work was xtndd to th cas, whr th sal continity assmptions ar not valid. Th xtndd principl of virtal work was drivd dirctly from th basic balanc laws of mchanics taking into accont discontinitis. As a novlty, a two-fild rprsntation was applid to trat th conditions for domains of smooth bhavior and thos for discontinitis. Th xtndd principl of virtal work for 3D continm can b sd in th sam mannr and for th sam prposs as th standard on. First, th corrsponding principl for an nginring modl follows whn kinmatical and kinmatic assmptions ar sbstittd thr as indicatd by th bar- and bam-trss application xampls. Althogh th xtndd principl of, say, for th lastic Timoshnko and Brnolli bam modls can b drivd from th bondary val problm dirctly, th s of th gnric form is mor straightforward. Scond, th diffrntial qations and th corrct jmp conditions follow from th xtndd principl for an nginring modl in th wllknown mannr. Third, th xtndd principl srvs as a consistnt and rady-to-s wak form of a discontinos Galrkin mthod. Th principl of virtal work is attractiv from an nginring viwpoint as th trms hav clar physical manings. Th first two trms of th xtndd xprssion practically coincid with thos of th standard xprssion. Th jmp trms combining th virtal work of intrnal forcs in jmps and continity constraint of displacmnt, 5

32 ar niq p to th symmtry paramtr β of th displacmnt constraint. Arbitrary mltiplir α of last-sqars trms brings anothr paramtr that can b sd to tn th mthod to match th approximation sd. Th application xampls indicat that a symmtric formlation is xcptionally accrat. In connction with bar and bam modls, th intrfac soltions or nodal vals wr xact, and thrfor indpndnt of α and polynomial dgr p. Also, if α is chosn larg, th discrt soltion bcoms insnsitiv to β. Implmntation and sability isss ar important in practic. Implmntation on a standard finit lmnt framwork with varios convntions concrning concpts and algorithms shold also b possibl. On of th major advantags of standard Galrkin mthods ovr discontinos Galrkin mthods has bn comptational simplicity. Th ways to liminat ithr th intrfac or lmnt qantitis prsntd hr aim to mak th mthod comptitiv with th standard on. Th first option sd.g. in [5] sms not to hlp mch in this rspct. Anothr novlty of th prsnt stdy is th s of th intrfac approximation as th primary nknown. Thn th (asymptotical) comptational complxity in arithmtic oprations bcoms th sam as with th standard mthod no mattr th lmnt approximation, th mthod fits wll in th xisting implmntations, and additional assmptions ar not ndd. Th last itm is important in trss applications or whn diffrnt nginring modls ar sd to modl a strctr. As far as th writrs ar awar, this typ of limination procdr has not bn pt into practic althogh th possibility has bn mntiond in []. As an illstrativ xampl, limination mthod givs th lmnt contribtion T z1 z1 δ hδθ y1 EI 1 6 4(1 3 ε ) 6 (1 6 ε ) hθ y1 z 1 Bh δw + = δ 3 z h 1ε z hδθ y 6 (1 6 ε) 6 4( 1+ 3ε) hθ y 1 for a Timoshnko bam in xz plan bnding and a distribtd constant load, whn β = 1, p 3 and q p 1. It is notworthy that th wighting of th last sqars trms or th dgr of polynomial sd in th lmnt approximation do not affct th xprssion. Paramtr ε = EI /( h GA) gos to zro in th Brnolli limit and th lmnt contribtion is sn to coincid with th wll-known xprssion of th Brnolli bam. On may spclat that limination along th sam lins cold b sd to gnrat lmnt contribtions for modls lik Rissnr-Mindlin plat or shlls sffring from locking phnomna in th sam mannr as th Timoshnko bam modl. Excptional accracy of th intrfac soltion is obtaind as th Grn fnction of th adjoint problm (slf-adjoint in this particlar cas) is polynomial which blongs to th approximation spac whn p is chosn larg nogh. Whn th Grn fnction is not polynomial as.g. in connction with bam on an lastic fondation, sing a larg p is advantagos as th Grn fnction can b approximatd bttr and thrby also th intrfac soltion bcoms mor accrat. Th ability of th lmnt approximation to b discontinos at th intrfac points is ssntial in both cass. Th priz of th 53