9 A C Bam Elmnt Basd on Ovrhausr Intrpolaton Abstract A nw C lmnt s proposd to modl Eulr-Brnoull bams n on and two-dmnsonal problms. Th proposd formulaton assurs C contnuty rqurmnt wthout th us of rotatonal dgrs of frdom, usd n tradtonal lmnts, through th us of an Ovrhausr ntrpolaton schm for bndng dsplacmnts. Th prncpl of vrtual dsplacmnts s usd to dtrmn th qulbrum quatons and boundary condtons for on and twodmnsonal Eulr-Brnoull bams. Th Ovrhausr ntrpolaton s ntroducd and th nw bndng ntrpolaton functons ar dfnd. Fnally, bam and fram problms ar solvd wth th nw formulaton and th rsults ar compard to th tradtonal Eulr-Brnoull lmnt and act solutons. Kywords Fnt Elmnt, ntrpolaton, Ovrhausr, bam. André Schwanz d Lma a Alfrdo Rocha d Fara b a Insttuto Tcnológco d Aronáutca, Dpartmnt of Mchancal Engnrng, São José dos Campos SP, Brasl dlmandr@gmal.com b Insttuto Tcnológco d Aronáutca, Dpartmnt of Mchancal Engnrng, São José dos Campos SP, Brasl arfara@ta.br http://d.do.org/0.590/679-7854 Rcvd 0.06.06 In rvsd form 09..06 Accptd..06 Avalabl onln 9..06 INTRODUCTION In th classcal Eulr-Brnoull bam thory th man assumpton s that th cross sctons rman plan and orthogonal to th bam as aftr dformaton. Ths maks th bam shar undformabl and th shar strsss rman ndtrmnat. Ths thory prsnts good rsults for sotropc homognous thn bams. In ordr to account for shar, th assumpton of orthogonalty to th bam as was rmovd by th Tmoshnko bam thory, but th cross sctons rman plan. Thn, th bam s mad shar dformabl, but th prdctd shar stran and strss provd to b constant through th bam thcknss. Whch s n contradcton wth th prncpl that shar strss s zro at th bam surfac. Thrfor, a shar corrcton factor was ntroducd to account th ffct of ths dfcncy of th Tmoshnko bam thory.
A.S. Lma and A.R. Fara / A C Bam Elmnt Basd on Ovrhausr Intrpolaton 9 Th applcatons of thos two thors n Fnt Elmnt Mthod ar alrady standardzd, as n Rddy (99). Othr bam thors wr formulatd to nclud shar ffcts. Among thos thors s th Rddy thrd-ordr thory whr th applcaton for bams was dvlopd as a spcfc cas of plats. Ths thory was usd by Hylgr and Rddy (988) to dvlop a bam fnt lmnt and study bndng and vbratons of sotropc bams. Rddy (997) latr dfnd lockng-fr lmnts basd on th smplfd thrd-ordr thory, havng th Eulr-Brnoull and Tmoshnko soluton as spcal cass. Such lmnts can b usd to obtan act nodal dsplacmnts and forcs (from th qulbrum quatons of th lmnt) n a fram structural analyss usng on fnt lmnt pr structural mmbr. Whch s not achvd wth th tradtonal qual ntrpolaton, rducd ntgraton lmnt bcaus th lmnts hbt shar lockng for thck mmbrs unlss two or mor lmnts pr structural mmbr ar usd. Polzotto (05) prsntd shar dformabl bams of ncrasng ordr whr a Rddy bam wthout shar concds wth th Eulr-Brnoull bam, and a Rddy bam shar dformabl of nfnt ordr can b dntfd wth th Tmoshnko bam. Nvrthlss, n all prsntd thors and thr drvd lmnts, rotatonal dgrs of frdom ar prsnt n ordr to guarant C contnuty at th pns of ncrasng th numbr of dgrs of frdom (DOF) ncssary to modl bams and frams. Th objctv of ths work s to lmnat th rotatonal dgrs of frdom rlatd to bndng and yt stll satsfy th C contnuty rqurmnt ncssary for th convrgnc of th fnt lmnt mthod appld to bams, rducng th computatonal burdn nvolvd n th soluton. In ordr to achv ths goal an Ovrhausr ntrpolaton schm wll b usd to gnrat a cubc ntrpolaton functons. Th nw formulaton proposd n th prsnt work s dvlopd basd on th Eulr-Brnoull assumptons, whr th rotatonal dgrs of frdom ar rlatd to th transvrs dsplacmnts drvatvs only. Ovrhausr (968) proposd an ntrpolaton schm for curvs and surfacs whr, for curvs, a lnar blndng of mplct parabolas dfns a cubc ntrpolaton btwn ach par of adjacnt ponts, as an clusv functon of th pont s postons. Applyng ths concpt to fnt lmnts mthod, th bam and fram lmnts formulatd n th prsnt work possss cubc ntrpolaton functons to rprsnt bndng basd only n nodal dsplacmnts. But as t wll b pland furthr, th applcaton of th Ovrhausr ntrpolaton s not straght forward and som solutons wr proposd n ordr to ncorporat such ntrpolaton n fnt lmnts formulaton. Th Ovrhausr ntrpolaton has alrady bn plord to solv numrcal mthods problms to guarant C contnuty. In Boundary Elmnts Mthod (BEM), Ovrhausr quadrlatral lmnts wr frst appld by Hall and Hbbs (987, 988). Latr, Waltrs and Gpson (994) dmonstratd that Ovrhausr lmnts gv mor accurat rsults thn th standard quadratc and cubc lmnts basd on Lagrang ntrpolaton n BEM and Hadavna t al. (000) dvlopd gnralzd parabolc blndng C lmnts usng Ovrhausr concpt nhancng th accuracy of th soluton ovr non-unform mshng n BEM. Archr (006) proposd contnuous solutons from th Grn lmnt mthod usng Ovrhausr lmnts, as an ffctv altrnatv to th Hrmtan approach prsntd by Tagbnu (998). And n Fnt Elmnt mthod Ulaga t al. (999) appld th Ovrhausr ntrpolaton n contact problms of gars, n ths cas, th contact ara was ap- Latn Amrcan Journal of Solds and Structurs 4 (07) 9-
94 A.S. Lma and A.R. Fara / A C Bam Elmnt Basd on Ovrhausr Intrpolaton promatd wth Ovrhausr spln ntrpolaton functons ovr boundary nods but th gomtry was compltly dcoupld from th fnt lmnt polynomal shap functons. In th squnc of ths work, th fnt lmnt formulaton basd on th classcal bam thory for th bams and frams wll b constructd usng th prncpl of vrtual dsplacmnts (PVD). Th Eulr-Brnoull bam and fram ntrpolaton functons wll b brfly ntroducd. Thn, Ovrhausr ntrpolaton wll b prsntd and th nw bndng ntrpolaton functons usng Ovrhausr schm wll b dvlopd and appld to on and two-dmnsonal formulatons. Typcal bam and fram problms wll b solvd usng th nw lmnts and rsults wll b compard to th analytcal soluton, for th on dmnsonal problms, and to th classcal Eulr-Brnoull bam soluton obtand usng an acadmc lcns of th commrcal Fnt Elmnt Analyss (FEA) softwar ABAQUS from Dassault Systèms. THE EULER-BERNOULLI BEAM FORMULATION Th govrnng quatons of a thr-dmnsonal bam can b obtand applyng th Prncpl of Vrtual Dsplacmnts (PVD). Thn, on-dmnsonal and two-dmnsonal quatons can b gnratd wth th corrct smplfcatons. Th dsplacmnt fld for a thr-dmnsonal bam s rprsntd by: u u( ) y ( ) z ( ) v v( ) z ( ) w w( ) y ( ) z y () Th stran-dsplacmnt rlatons ar: du u d dv yy 0 dy, dw zz 0 dz y z, z y, du dv y z v, z, dy d dv dw yz 0 dz dy du dw z y w, y dz d, () In th Eulr-Brnoull (classcal) bam thory, t s assumd that th plan cross sctons prpndcular to th as of th bam rman plan and prpndcular to th bam as aftr dformaton. Thrfor θz = v,, θy = -w, and as a rsult th trms of th vrtual work rlatd to shar vansh. Thn, th stran-dsplacmnt rlatons bcom: y z du d du dy du dz u, dv d dw d yv, v, w zw v,, w, z,, y z,, y, () Latn Amrcan Journal of Solds and Structurs 4 (07) 9-
A.S. Lma and A.R. Fara / A C Bam Elmnt Basd on Ovrhausr Intrpolaton 95 Anothr assumpton of ths thory s that no normal strsss ar prsnt n th y and z drctons. Along wth th non-dformaton of th cross-scton, w can vrfy that σyy = σzz = τyz = 0. Th prncpl of vrtual dsplacmnt stats: W W 0 (4) Whr δw s th vrtual work of th ntrnal strans and δw s th trnal fforts vrtual work. Th frst trm s dfnd as: T W { } { } V dv (5) Whr {σ} s a column matr n th Vogt notaton of th Cauchy tnsor componnts and {δ} s a column matr n th Vogt notaton of th tnsor [δ], ths tnsor s a quadratc functon of th dsplacmnt gradnts n Lagrangan coordnats. Assumng th dsplacmnt gradnts ar small n modulus, compard to th unty, t s vald to appromat th dsplacmnt gradnt wth th componnts of th Grn Stran Tnsor. Thn, th vrtual work of th ntrnal strsss can b wrttn as T W { } { } V dv (6) Applyng th assumptons of th classcal bam thory for an sotropc matral δw rducs to: l W ( EAu, u, EIyw, w, EIzv, v, GJ,, ) d (7) 0 Th vrtual work of trnal loads for a thr dmnsonal cas consdrng dstrbutd loads, concntratd forcs and momnts s wrttn as: L W quqyvqzw dqu(0) Qyv(0) Qzw(0) Qu( L) Qyv( L) 0 (8) Qw( L) T (0) T ( L) M w (0) M v (0) M w ( L) M v ( L) z y, z, y, z, As th man objctv s to propos a formulaton wthout rotatonal dgrs of frdom for bndng w can conclud that ths formulaton dos not support, at frst, prscrbd momnts as boundary condtons. But latr, w wll s that ths s not ntrly tru. Also, thr-dmnsonal applcatons, whr torson s nvolvd, ar mpractcal. Bcaus n thr-dmnsonal problms bndng coupls wth torson at th junctons of structural mmbrs wth dffrnt orntatons. Thrfor, w cannot lmnat th dgrs of frdom only rlatd to bndng. Rducng th applcablty of a thr-dmnsonal lmnt to structurs composd by only on mmbr. Thn, no advantag arss from th us of th proposd lmnt compard to th tradtonal Eulr-Brnoull lmnt n D problms. Howvr, n on and two-dmnsonal applcatons, ths lmtaton dos not st. Bcaus only on rotaton dgr of frdom s prsnt and t s clusvly rlatd to bndng. Latn Amrcan Journal of Solds and Structurs 4 (07) 9-
96 A.S. Lma and A.R. Fara / A C Bam Elmnt Basd on Ovrhausr Intrpolaton THE OVERHAUSER INTERPOLATION Ovrhausr (968) proposd that for a st of ponts, mplct parabolas ar dfnd usng thr conscutv ponts n a way that ach adjacnt par of ponts blongs to two dffrnt parabolas. Thn a lnar blnd of th two parabolas rsults n a C contnuous cubc functon. In ordr to apply ths da n fnt lmnt formulaton, th ponts coordnats ar rplacd by th nodal transvrs dsplacmnts and th mplct parabolas ar convrtd nto parabolc Lagrang ntrpolatons of thos dsplacmnts, as llustratd n Fgur. v- v va(s) v(s) v+ vb(s) v+ s l 0 l l + l+ + + Fgur : Ovrhausr ntrpolaton schm. Th s ndpndnt varabl orgnally rlatd to th arc-lngth s naturally lnkd to th lngth of th lmnts. And th two parabolas va and vb ar dfnd as: v A( s ) a0 as as, vb( s ) b0 b s bs (9) Whr coffcnts a0, a, a, b0, b, b ar functons of th nodal dsplacmnts v-, v, v+ and v+. Th lnar blndng functons A and B ar dfnd by: s A s dfnd n th ntrval 0 s l. Fnally, th Ovrhausr curv, dfnd n th sam ntrval s:, B A (0) l l v s ) ( s ) v ( s ) ( s ) v ( s ) () ( A A B B Evaluatng th proprts of such curv w notc that, as A and B ar lnar functons of s and va and vb ar quadratc functons of s thus, th ntrpolaton v(s) n Eq. s a cubc functon of s. And f w valuat th valus of v at pont, th ndpndnt varabl assums zro valu and va(s = 0) = vb(s = 0) = v(s = 0) = v. Also, at pont +, s = l and va(s = l) = vb(s = l) = v(s = l) = v+. Latn Amrcan Journal of Solds and Structurs 4 (07) 9-
A.S. Lma and A.R. Fara / A C Bam Elmnt Basd on Ovrhausr Intrpolaton 97 Th frst drvatv of v(s) s gvn by. dv ds dv dv v d v d A B A B A B A B () ds ds ds ds Thn th frst drvatvs of v wth rspct to s at nods and + ar dv ds dva dv dvb 0) ( s 0), ( s l ) ( s l ) () ds ds ds ( s Thrfor, t s sn that dv/ds vars contnuously n btwn ach par of adjacnt ponts. So, th curv v(s), can adquatly rprsnt an nflcton that should occur wthn an ntrval. It should b apparnt that th cubc functon dfnd n th ntrval btwn nods and + dos not pass through ponts and +, ths s an mportant dffrnc n comparson to smpl ft by cubc polynomals. Th mannr of constructon (as a blnd of two parabolas) guarants that spurous wggls wll not b ntroducd, as frquntly happns whn smpl cubcs ar forcd to pass through four ponts of a curv. Howvr for th trmts, Ovrhausr (968) uss th parabolc ntrpolaton and th cubc ntrpolaton drvd appls only n th lmnts that shar two parabolas, what w wll call cntral lmnts (CE). For thos lmnts transvrs dsplacmnt can b wrttn as: vs ( ) v v v v (4) Whr ϕ -, ϕ, ϕ +, and ϕ + ar th ntrpolaton functons for th cntral lmnt (Eq. 8 n Appnd A). Thos functons ar dfnd n th ntrval 0 s l. Assumng all lmnts possss th sam lngth, l- = l = l+ = l, w can plot th ntrpolaton functons for a bttr undrstandng of ts bhavor. Fgur : Intrpolaton functons for th cntral lmnts assumng all lmnts possss th sam lngth. Latn Amrcan Journal of Solds and Structurs 4 (07) 9-
98 A.S. Lma and A.R. Fara / A C Bam Elmnt Basd on Ovrhausr Intrpolaton In Fgur, w s clarly a local support charactr n th ntrpolaton functons, and that th nflunc of th dsplacmnts of th nods outsd th lmnt (- and +) s much smallr than th ons of th lmnt nods ( and +). Now, n ordr to construct cubc ntrpolaton functons for th lmnts on th trmts, som consdratons must b mad. Frst, thr ntrpolatons must satsfy C contnuty wth th Ovrhausr ntrpolaton alrady formulatd for th cntral lmnts. Scond, as t was sad bfor, no rotatonal dgrs of frdom ar prsnt, as a consqunc, no ssntal nor natural boundary condtons of ths charactr can b appld. Nvrthlss, for many bam and fram problms ths nformaton s prmordal. Consdrng thos ponts, a soluton proposd s to buld an lmnt wth a rotatonal dgr of frdom only at th nod not connctd to cntral lmnts (frst or last nod of th structural mmbr;.. nod A n Fgur ). And thn lmnat ths dgr of frdom va statc condnsaton alrady consdrng th ssntal boundary condton. Ths stratgy allows applyng th dffrnt boundary condtons and accountng th ffct of thos boundary condtons n th lmnt stffnss matr. A vc θ- va B s C vb 0 l l l + l+ + + Fgur : Nw ntrpolaton for lft cornr lmnt. For th lft cornr lmnt, th nw curv vc s a cubc dfnd n th ntrval l- s 0 by th followng quaton. v C 0 cs cs cs c (5) whr coffcnts c0, c, c and c ar obtand mposng th condtons, v ( s C v ( s C dv ds C dv ds C l ( s ( s 0) v ) v 0) c c l ) c 0 c c ( l 0 ( s c ( l dv ds A 0) a ) c ( l ) c ( l ) ) c ( l ) (6) Latn Amrcan Journal of Solds and Structurs 4 (07) 9-
A.S. Lma and A.R. Fara / A C Bam Elmnt Basd on Ovrhausr Intrpolaton 99 wth a as th sam coffcnt from th parabola va usd n th blndng of th cntral lmnt to obtan a cubc for th ntrval from to +, and θ- s th rotaton n th pont -. Thn th curv vc can b wrttn as a functon of th nodal dsplacmnts and th rotaton θ- as: v v v v (7) C Whr ηθ, η-, η, and η+ ar th ntrpolaton functons for th lft cornr lmnt (Eq. 9 n Appnd A). Agan assumng all lmnts possss th sam lngth, l- = l = l+ = l, w can plot th ntrpolaton functons to undrstand ts bhavor. Fgur 4: Intrpolaton functons for th lft cornr lmnts assumng all lmnts possss th sam lngth. Agan a local support charactr s obsrvd n Fgur 4, and w s that rotaton nflunc s small, as also th dsplacmnt of th nod outsd th lmnt (nod C n Fgur ). But, as statd arlr, th man propos of ths work s to lmnat th dpndncy of th rotatonal dgrs of frdom. Ths can b accomplshd by prformng a statc condnsaton of th dgr of frdom θ-. Th bndng problm for th lmnt can b wrttn as: [ K ] { v} { f } LCE LCE 4 v 4 v 4 4 4 4 44 LCE v 4 LCE K K K K f K K K K f K K K K f K K K K f (8) Whr [K ]LCE s th stffnss matr of th lft cornr lmnt (LCE) and {f }LCE s th rspctv load vctor, both rsultng from th ntgraton of th PVD wth th ntrpolaton functons ηθ, η-, η, and η+. Latn Amrcan Journal of Solds and Structurs 4 (07) 9-
00 A.S. Lma and A.R. Fara / A C Bam Elmnt Basd on Ovrhausr Intrpolaton Consdrng th ssntal boundary condton of th nod that posssss θ-, thr ar two possblts that mply two dffrnt stffnss matrcs and load vctors. If th nod s clampd θ- = 0, thn w just nd to lmnat th frst row and th frst column of th matr [K ]LCE and th frst componnt of load vctor {f }LCE, rsultng: K K K 4 K K K 4 K K K 4 4 44 LCE v v v f f f 4 LCE (9) But f ths nod s smply supportd or fr θ- s unknown, thn w hav to solat θ- n th frst quaton of th systm prsntd n Eq. 8. f K v K v K v ) (0) ( 4 K Thrfor, t s also possbl to apply a concntratd momnt at th dgr of frdom θ- whn w hav ths typ of ssntal boundary condton. Thn, th stffnss and load componnts rlatd to θ- ar rdstrbutd n an quvalnt problm. Th lmnaton procdur n lmnt lvl s only possbl bcaus th rotatonal dgr of frdom s not shard wth othr lmnts. And as a rsult, th ssntal boundary condton s alrady consdrd n th lmnt lvl, bng smply assmbld n th global problm. θ+ E va* F vb* vd G s l l l l+ + + Fgur 5: Nw ntrpolaton for th rght cornr lmnt. 0 For th rght cornr lmnt an analogous procdur s adoptd to dvlop cubc curv vd and th ntrpolaton functons ar dfnd n th ntrval 0 l+ as prsntd n Fgur 5, rsultng: v D v v v () Th ntrpolaton functons ψθ, ψ, ψ+, and ψ+ prsntd n Eq. 0 of Appnd A and plottd n Fgur 6 (consdrng qual lngth for all lmnts) also guarant C contnuty wth th cntral Latn Amrcan Journal of Solds and Structurs 4 (07) 9-
A.S. Lma and A.R. Fara / A C Bam Elmnt Basd on Ovrhausr Intrpolaton 0 lmnts and orgnat a stffnss matr and load vctor that consdr th ssntal boundary condton at th unshard nod. Fgur 6: Intrpolaton functons for th rght cornr lmnts assumng all lmnts possss th sam lngth. 4 PLANE FRAME FORMULATION For on-dmnsonal problms, th dvlopd functons can b drctly appld. But for twodmnsonal analyss, som consdratons must b mad. Frst, th aal dsplacmnts ntrpolaton functons ar th sam lnar ntrpolaton functons usd n tradtonal Eulr-Brnoull lmnts. s s ( s ) u u l l u () Consquntly, as n tradtonal lmnts, only th dgrs of frdom that blong to th lmnt ar takn n consdraton. Contrary to th nw bndng formulaton that uss nformaton of nods outsd th lmnt. Scond, th dffrnt orntatons of th lmnts must b accountd and cohrnt transformaton matrcs must b laboratd n ordr to corrctly transform th stffnss and load componnts to th global coordnat systm. As thy wr formulatd th Ovrhausr ntrpolaton functons for bndng assum th sam orntaton for all nods prsnt n th cubc formulaton. Thrfor, th transformaton matrcs must act also on th componnts rlatd to th nods trnal to th lmnts for th bndng ffcts. In ordr to llustrat ths concpt w can obsrv Fgur 7 and s how th orntaton of th nods changs from on lmnt to anothr. If w magn that lmnts and ar dfnd as cntral lmnts, w not that for lmnt (Fgur 7.a) th nods, and hav a dffrnt orntaton compard to th sam nods whn thy assocat wth th lmnt (Fgur 7.b). Latn Amrcan Journal of Solds and Structurs 4 (07) 9-
0 A.S. Lma and A.R. Fara / A C Bam Elmnt Basd on Ovrhausr Intrpolaton y y y y + + Y 0 (a) y 4 4 X y y y 0 y y 4 + (b) 4 + Y X Fgur 7: Nods orntaton for dffrnt lmnts. Thn, n lmnt (Fgur 7.a), for nod a stffnss rlatd to th transvrs dsplacmnt y on th lmnt (local) coordnat systm would b transformd nto a aal stffnss n th global coordnat systm (GCS). Ths s not physcally cohrnt nor numrcally, nducng bad rsults. Bcaus as statd bfor, aal bhavor s not affctd by nods outsd th lmnt. In th lght of th abov w propos that, whn w hav lmnts wth dffrnt orntatons sharng a nod (..: lmnts an sharng nod n Fgur 7) w wll trat ach lmnt that shars ths nod as lft cornr lmnts (LCE). Wth all matrcs bng gnratd ncludng th rotatonal dgr of frdom. Thn, thy wll b transformd to th GCS and assmbld n a suprlmnt. Fnally, th rotatonal dgr of frdom can b lmnatd by a statc condnsaton procdur n lmnt lvl analogous to th on prsntd bfor. Ths soluton numrcally assurs C contnuty, avods ncohrnt couplng of stffnss and allows rprsntng structurs such as th on prsntd n Fgur 8, whr lmnts, and 5 shar nod 0. And and 5 ar rotatd of α and α rspctvly, n rlaton to th GCS (lmnt LCS concds wth th GCS). Y 4 4 5 α α 5 6 0 6 X Fgur 8: Eampl of lmnts wth dffrnt orntatons sharng th sam nod. Latn Amrcan Journal of Solds and Structurs 4 (07) 9-
A.S. Lma and A.R. Fara / A C Bam Elmnt Basd on Ovrhausr Intrpolaton 0 Onc th stffnss matrcs and load vctors of ths thr lmnts hav bn transformd to th GCS thy can b assmbld as prsntd n Fgur 9. Fgur 9: Assmbly of th stffnss matrcs and load vctors of, and 5. Thn, th lmnaton of th rotatonal dgr of frdom θ 0 can b prformd wth a statc condnsaton, rsultng th fnal stffnss matrcs and load vctor of th suprlmnt, whch can fnally b assmbld n th global stffnss matr and load vctor of th problm. It s mportant to rmark that th transformatons to th global coordnats systm should hav bn appld bfor th lmnaton and th assmbly, f not, th stffnss rlatd to th θ0 wll b rdstrbutd n a wrong way. 5 NUMERICAL RESULTS Hr w study th bhavor of th Ovrhausr lmnts dscussd on ths papr. Frst, fv ondmnsonal analyss of dffrnt load condtons and ssntal boundary condtons ar tstd and compard to th act soluton accordng Eulr-Brnoull bam thory. Thn, two fram problms ar solvd. For th convrgnc analyss, w adoptd th crtron of % rror n th dsplacmnts of spcfc ponts (not th whol structur) compard to th act solutons for th bam problms, and compard to th Eulr-Brnoull solutons usng ABAQUS for th fram problms. Th bam and fram confguratons tstd ar prsntd n Fgur 0. Fgur 0: Structural confguratons tstd. Latn Amrcan Journal of Solds and Structurs 4 (07) 9-
04 A.S. Lma and A.R. Fara / A C Bam Elmnt Basd on Ovrhausr Intrpolaton For all problms a lnar lastc bhavor was assumd and th followng proprts wr adoptd: E = 0.0 GPa, = 0., L = L = L = m. Th cross scton s crcular (sold) of radus r = 0.054 m and all lmnts possss qual lngth. 5. Bam Problms For th bam problms, th act solutons usng th classcal bam thory accordng to Hbblr (0) ar: P PL v L, 0 L, vma L () 6EI EI L 4, 0 L/, v L/ P PL v ma (4) 48EI 48EI P PL v L 4, 0 L/, vma L/ (5) 48EI 9EI L L, 0 L, v L/ 4 w 5wL v ma (6) 4EI 84EI P 5 PL v L, 0 L / 4, v L / 4 (7) 4EI 6 56EI Whr v s th transvrs dsplacmnt, L s th bam lngth and s th ndpndnt varabl rlatd to th lngth. And P rprsnts th concntratd forc, whr n all cass a forc P = 000 N was appld, cpt for th dstrbutd load cas whr a valu of w = 600 N/m was chosn n ordr to rsult th sam mamum dsplacmnt of th forc P. Th mamum dsplacmnts wr analyzd for th convrgnc tsts, cpt for problm 5 whr th dsplacmnt at = L/4 was montord. Numrcal Rsults Problm 4 5 Elmnts to Convrgnc 9 8 40 6 0 vma Eact (0 - m) -4.8555-0.05-0.0759-0.05-0.707* vma Ovrhausr (0 - m) -4.807-0.004-0.075-0.005-0.690* Error (%) 0.9970 0.9970 0.94 0.9794 0.9876* * Th valus for problm 5 rlats to th v( = L/4). Tabl : Numrcal rsults of bam problms. W s n all problms that th Ovrhausr soluton s mor rgd, as pctd of a numrcal appromaton. And dspt bng cubc ntrpolaton functons thy do not rprsnt as wll th soluton as th Hrmt cubc functons. As a consqunc, mor lmnts ar rqurd and th computatonal burdn s bggr (mor dgrs of frdom) than usng Eulr-Brnoull tradtonal lmnts. Latn Amrcan Journal of Solds and Structurs 4 (07) 9-
A.S. Lma and A.R. Fara / A C Bam Elmnt Basd on Ovrhausr Intrpolaton 05 It s also notd that th ssntal boundary condtons nflunc th convrgnc rat. Comparng problms and, whr th natural boundary condtons ar th sam, but n th trmts ar smply supportd and n thy ar clampd, w s that th convrgnc rat rducs by half for problm, vn wth both solutons bng cubc. Ths bhavor s not obsrvd wth tradtonal Eulr-Brnoull lmnts for ths stuaton. But f w compar problms, 4 and 5 wth th sam ssntal boundary condtons w s that th convrgnc rats ar smlar. For problm 4, whr not only th stffnss s ntgratd usng th ntrpolaton functons, but also th load, w s that rsults ar cohrnt, ndcatng good bhavor of th ntrpolaton functons n ths aspct. Fnally, post-procssng was prformd n ordr to tract nformaton about nodal rotaton v () and mamum bndng strss σma() (at th nods and lmnt cntrod) for all ondmnsonal problms. C contnuty was attstd, confrmng th mathmatcal formulaton and also good bhavor was attstd concrnng th crtcal ponts that wr corrctly capturd. But as th dsplacmnt prsntd rrors to th act soluton, thos rrors wr passd to v () and σma() as consqunc. Th rrors to th act soluton as functon of th bam lngth for v(), v () and σma() ar prsntd n Fgur to Fgur rspctvly, on th domans whr th act solutons ar dfnd (Eqs. to 7). Fgur Fgur Latn Amrcan Journal of Solds and Structurs 4 (07) 9-
06 A.S. Lma and A.R. Fara / A C Bam Elmnt Basd on Ovrhausr Intrpolaton Fgur In Fgur and Fgur th arrows ndcat that th rrors trapolat. And for ths rason, w prfrrd to ndcat ts tndncy. It s ntrstng to not that th ssntal boundary condtons ntrfr not only n th convrgnc rat but at th rrors magntud. As t was brfly pland bfor, th clampd and smplysupportd/fr lmnts hav dffrnt stffnss matrcs. Also, w s that for clampd stuatons, th rrors ar bggr at th cornr lmnts dcrasng n th othr lmnts. Whl for th cass whr smply-supportd condton sts, th rrors ar bggr nt to th ponts wr v = 0 (crtcal ponts). Whr, for all problms n whch a crtcal pont occurs n cntral lmnts, th rror n dsplacmnt tnds to %, as prsntd n Fgur. Th sam tndncy s obsrvd n th nodal rotaton rrors prsntd n Fgur. And n problm 5 th rror n v () nt to th crtcal pont rss untl % closr to whr th frst drvatv changs sgn. But as t was sad bfor, th crtcal ponts ar corrctly capturd. Fnally, concrnng strsss, th cornr lmnts ar agan th ons wth th bggr rrors, bng th smply-supportd / fr cornr lmnts th ons that hbt th bggst rrors, n th ordr of 7% for th problms, and 5 whr w hav a concntratd forc and 6% for problm 4 whr w hav a dstrbutd load. Thrfor, w can conclud that both cntral and cornr lmnts prsnt rrors, but t s clar that th proposd soluton for th cornr lmnts s mor crtcal, spcally whn t coms to th scond drvatv of th ntrpolaton functons. 5. Plan Fram Problms For th fram problms th global coordnat systm XY was dfnd n Fgur 0 and th structural mmbrs wth lngth L ar vrtcal (paralll to Y) and th ons wth lngth L ar horzontal (paralll to X). Th L-fram of problm 6 and th U-fram of problm numbr 7 both hav clampd bass and lmnts wth qual lngth wr usd. Both structurs wr frst submttd to concntratd forcs at th junctons of th structural mmbrs that producd smpl tracton to th vrtcal sgmnts n ordr to valdat th suprlmnt soluton accordng to th prncpls of th classcal bam thory. Accordng to ths thory, for Latn Amrcan Journal of Solds and Structurs 4 (07) 9-
A.S. Lma and A.R. Fara / A C Bam Elmnt Basd on Ovrhausr Intrpolaton 07 such loadng, only aal dformaton of th vrtcal sgmnts and smpl translaton of th horzontal sgmnts should occur. What ndd happnd, valdatng not only th soluton, but th mplmntaton of th aal bhavor as wll, obtanng th sam rsults of Eulr-Brnoull tradtonal lmnts wthn ABAQUS. Th sam dos not happn whn w us cntral lmnts to modl th dg. And a transvrs dflcton appars n both sgmnts. As an ampl, th dsplacmnts of L-fram structur wth th forc F = 0 4 N at th juncton of th structural mmbrs (s Fgur 4) ar prsntd n Tabl. Th structur was modld wth s lmnts (thr pr structural mmbr) usng th suprlmnt (SE) soluton and also usng cntral lmnts (CE) at th juncton. F 4 5 6 L L Y X 0 Fgur 4: L-fram wth forc F at th juncton. As statd bfor, modlng th junctons usng cntral lmnts nducs couplng btwn aal and transvrs bhavor that s not n accordanc wth th classcal bam thory. Ths can b sn n th dffrnt nodal dsplacmnts on Y drcton of th nods of th horzontal mmbr. But th suprlmnt soluton provd to b consstnt wth th problm physcs. Nod Wth SE Wth CE X (0-6 m) Y (0-6 m) X (0-6 m) Y (0-6 m) 0 0.0000 0.0000 0.0000 0.0000 0.0000 7.84 0.0000 7.84 0.0000 5.669 0.0000 5.669 0.0000.4944 0.0000.4944 4 0.0000.4944 0.0000.58 5 0.0000.4944 0.0000 9.57 6 0.0000.4944 0.0000 46.9887 Tabl : Nodal Dsplacmnts for L-fram wth forc F at th juncton. Latn Amrcan Journal of Solds and Structurs 4 (07) 9-
08 A.S. Lma and A.R. Fara / A C Bam Elmnt Basd on Ovrhausr Intrpolaton For th L-fram othr two loadng stuatons wr tstd, as shown n Fgur 5. F L F L Y X Fgur 5: Othr load cass for L-fram. Th convrgnc crtron was % rror n both X and Y dsplacmnts of th fr nd. Frst a forc F = 000 N was appld. Ths problm rsmbls th cantlvr bam problm analyzd bfor. Convrgnc was ndd achvd usng th sam nn lmnts n th vrtcal sgmnt (8 lmnts for th whol structur, s Fgur 7) as n th cantlvr bam and no aal dsplacmnt was hbtd by th nods n th vrtcal mmbr, as th classcal thory dctats. Also, th dsplacmnts n th nods of th horzontal mmbr (nods 9 to 8 for th convrgd soluton) n th X drcton ncras lnarly, as shown n Fgur 6. Also n accordanc wth th classcal bam thory. Fgur 6: Nodal X-Dsplacmnts n th horzontal mmbr. Fgur 7: Msh convrgnc tst for L-fram wth load F at th fr nd. Latn Amrcan Journal of Solds and Structurs 4 (07) 9-
A.S. Lma and A.R. Fara / A C Bam Elmnt Basd on Ovrhausr Intrpolaton 09 Scond, th horzontal forc F was rplacd by a vrtcal forc F = 00 N at th sam pont. Inducng bndng n both mmbrs and aal dformaton n th vrtcal mmbr. Th msh convrgnc tst n Fgur 8 shows that th soluton achvd th convrgnc crtron wth alrady ght lmnts (four pr structural mmbr). Also t rvals that th X-dsplacmnt of th ndnod s th sam that th on obtand usng Eulr-Brnoull tradtonal lmnts vn wth s lmnts. Fgur 8: Msh convrgnc tst for L-fram wth load F at th fr nd. Comparng th two load cass and thr rspctv convrgnc analyss (Fgur 7 and Fgur 8), w s that, curously, th dsplacmnts n th drcton of th concntratd forc ar th ons that prsnt rror. And stran nrgy prcnt rror of th structur concds wth th prcnt rror n th dsplacmnt n that drcton. Ths ndcats that bndng s th domnant phnomna n trms of nrgy. Also, th dffrncs n th convrgnc rats ndcat that, th soluton s somhow snsbl to th drcton of th forc and th prsnc of bndng n on or mor mmbrs. Fnally, th U-fram undr multpl forcs accordng to Fgur 9 was analyzd, whr F = -500 N and F = 000 N ar n th Y drcton and F = 000 N s n th X drcton. In ths cas, all mmbrs ar undr bndng and aal solctatons. F F F L L L Y X Fgur 9: U-fram multpl load tst. Latn Amrcan Journal of Solds and Structurs 4 (07) 9-
0 A.S. Lma and A.R. Fara / A C Bam Elmnt Basd on Ovrhausr Intrpolaton Th convrgnc crtron adoptd was % rror to th Eulr-Brnoull tradtonal lmnt n both X and Y dsplacmnts of th juncton whr F and F wr appld. Soluton convrgd usng 60 Ovrhausr lmnts (0 pr structural mmbr), as llustratd n Fgur 0, much mor than both stuatons of th L-fram. Fgur 0: Msh convrgnc tst for U-fram wth multpl loads. Also, w can not that th rror assocatd to th X-dsplacmnt s bggr, and ths s th drcton whr bndng dsplacmnts ar mor prssv, as t happnd n th L-fram tsts. Concrnng th stran nrgy, agan w attst that bndng s th domnant phnomna, thrfor th prcnt rror n nrgy concds th prcnt rror n X-dsplacmnt (prncpal bndng drcton). Fnally, for all fram tsts that nvolvd bndng, th sam bhavor of bggr rrors n th nodal transvrs dsplacmnts closr to th clampd dgs s hbtd, as t happnd n th clampd bam problms. Rvalng that dspt th cubc ntrpolaton functons, th Ovrhausr soluton s lss accurat, spcally for th clampd cornr lmnts. 6 CONCLUSIONS In ths papr, w formulatd C bam fnt lmnt basd on Ovrhausr ntrpolaton and th Eulr-Brnoull bam thory that gos wthout th us of rotatonal dgrs of frdom to rprsnt bndng. Th man objctv of such ntrpolaton schm was to rduc th computatonal burdn assocatd wth th prsnc of th rotatonal dgrs of frdom, but as obsrvd n all th numrcal tsts prformd, th Ovrhausr soluton dmands mor computatonal ffort bcaus dspt havng only dsplacmnt dgrs of frdom, mor lmnts ar rqurd to achv an accptabl rsult. Latn Amrcan Journal of Solds and Structurs 4 (07) 9-
A.S. Lma and A.R. Fara / A C Bam Elmnt Basd on Ovrhausr Intrpolaton Also w hav sn that soluton s much mor nfluncd by th ssntal and natural boundary condtons compard to th tradtonal Eulr-Brnoull lmnts, spcally th cornr lmnts. It s not a trval quston to b answrd why ths nflunc of th boundary condton occurs and why th cubc ntrpolaton dos not rprsnt th soluton as wll as th Hrmt cubcs. But th artcl brngs ths mportant brakthrough n undrstandng such ntrpolaton as ths s stll a dffculty to b ovrcom by th proposd ntrpolaton class. A futur work suggston s to nvstgat th us of dffrnt lmnt szs for th trmts, to valuat th nflunc of such lmnts n th bhavor of th whol structur. Fnally, th applcaton of such formulaton s rstrctd to on and two-dmnsonal problms bcaus torson cannot b rprsntd n th classcal bam thory, wthout th us of a rotatonal dgr of frdom, that n compl fram structurs, coupls wth bndng rotatons. Rfrncs Archr, R. (006). Contnuous solutons from th Grn lmnt mthod usng Ovrhausr lmnts. Appld Numrcal Mathmatcs 56:-9. Hadavna, H., Travs, R.P., Fnnr, R.T. (000). C-contnuous gnralsd parabolc blndng lmnts n th Boundary Elmnt Mthod. Mathmatcal and Computr Modllng :7-4. Hall, W.S. and Hbbs, T.T. (987). Th tratmnt of sngularts and th applcaton of Ovrhausr quadrlatral boundary lmnt to thr-dmnsonal lastostatcs, In IVTAM Sympo. on Advanc BEM, San Antono, TX. Hall, W.S. and Hbbs, T.T. (988). C contnuous quadrlatral boundary lmnts appld to thr dmnsonal problms n potntal thory, In Boundary Elmnt X (Edtd by C.A. Brbba) :8-9. Hylgr P.R. and Rddy, J.N. (988). A hghr-ordr bam fnt lmnt for bndng and vbraton problms. Journal of Sound and Vbraton 6():09-6. Hbblr, R. C. (0). Mchancs of Matrals, 8th dton, Parson Prntc Hall, NJ. Ovrhausr, A.W. (968). Analytcal dfnton of curvs and surfacs by parabolc blndng (Tchncal Rport No: SL68-40), Ford Motor Company (Darborn, Mchgan). Polzzotto, C. (05). From th Eulr Brnoull bam to th Tmoshnko on through a squnc of Rddy-typ shar dformabl bam modls of ncrasng ordr. Europan Journal of Mchancs - A/Solds 5:6-74. Rddy, J.N. (99). An Introducton to th Fnt Elmnt Mthod, nd dton, McGraw-Hll, Nw York. Rddy, J.N. (997). On lockng-fr shar dformabl bam fnt lmnts. Computr Mthods n Appld Mchancs and Engnrng 49: -. Tagbnu, A.E. (998). Numrcal stablty charactrstcs of a Hrmtan Grn lmnt modl for th transport quaton. Engrg. Anal. Boundary Elmnts :6 65. Ulaga, S., Ulbn, M., Flaškr, J. (999). Contact problms of gars usng Ovrhausr splns. Intrnatonal Journal of Mchancal Scncs 4:85-95. Waltrs, H.G. and Gpson, G.S. (994). Evaluaton of ovrhausr splns as boundary lmnts n lnar lastostatcs. Engnrng Analyss wth Boundary Elmnts 4:7-77. Latn Amrcan Journal of Solds and Structurs 4 (07) 9-
A.S. Lma and A.R. Fara / A C Bam Elmnt Basd on Ovrhausr Intrpolaton APPENDIX A Intrpolaton functons for th cntral lmnts: vs ( ) v v v v l l s s s ll ll ll ll ll ll l l ( l l) l ( ll ) l l s ll l l s ll ll l ll ll ll l l l ( l l) l ll l l ll l l s l l ( l l) l s ll ll ll ll l ll ll ll l l ( l l) l l ll l l ll l l s l s s ll ll ll ll s (8) Intrpolaton functons for th lft cornr lmnts: v ( s) v v v C s s l l l l l l l l l s s s ll( l l) ll( l l) ll( l l) l l l l l ll l s s s ll( l l) l l( l l) l l( l l) s s l l l l l l l l l l l l l l ( ) ( ) ( ) s (9) Intrpolaton functons for th rght cornr lmnt: v ( s ) v v v D s s l l l l s s s l l( l l) l l( l l) l l( l l) l l l l l l ll l l l s s s ( l) l l l l l lll l s s s l l( l l) l l l l l l l l l l l l l l (0) Latn Amrcan Journal of Solds and Structurs 4 (07) 9-