Three-Node Euler-Bernoulli Beam Element Based on Positional FEM

Transcription

1 Avalabl onln at Procda Engnrng 9 () Intrnatonal Workshop on Informaton and Elctroncs Engnrng (IWIEE) Thr-Nod Eulr-Brnoull Bam Elmnt Basd on Postonal FEM Lu Jan a *,b, Zhou Shnj a, Dong Mlng c, Yan Yuqn c a School of Mchancal Engnrng, Shandong Unvrsty, Jnan 56, PR Chna b School of Scnc, Shandong Janzhu Unvrsty, Jnan 5, PR Chna c School of Mchancal & Elctronc Engnrng, Shandong Janzhu Unvrsty, Jnan 5,PR Chna Abstract Basd on postonal FEM, thr-nod Eulr-Brnoull bam lmnt for larg dflcton D fram analyss s rsarchd. Soluton stratgy of gomtrc nonlnar statc analyss wth thr-nod Eulr-Brnoull bam lmnt s ntroducd to th fram structurs, and th program flow chart s gvn, thn Matlab languag s usd to compl program. Smpl xampl s shown at th nd of th papr, comparng th numrcal rsults achvd wth th analytcal and othr numrcal solutons found n th ltratur. Publshd by Elsvr Ltd. Slcton and/or pr-rvw undr rsponsblty of Harbn Unvrsty of Scnc and Tchnology Opn accss undr CC BY-NC-ND lcns. Ky words: postonal FEM; gomtrc nonlnarty; thr-nod Eulr-Brnoull bam lmnt. Introducton Whthr statc or dynamc non-lnar analyss for fram structurs by FEM (fnt lmnt mthod), th calculaton prcson dpnds, to a grat xtnt, on th lmnt typs (Blytschko t al. ). In th past gomtrc non-lnar analyss, tradtonal two-nod bam lmnt s oftn usd (Crsfld 99; Clough 4). Rf. [4] (Coda and Grco 4) prsnts a smpl formulaton to trat larg dflctons by FEM basd on poston dscrpton. Th smplcty of th rsultng formulaton may b consdrd as ts man attrbut. But gt a mor satsfactory accuracy, t must nd mor lmnt numbr, and th dffrnt calculaton rsult of dffrnt lmnt numbr ndcats that t has an mportant nflunc on th rsult accordant to th practcal stuaton to choos th lmnt numbr proprly. So t s ncssary to sarch a * Corrspondng author. Tl.: E-mal addrss: 37976@63.com Publshd by Elsvr Ltd. do:.6/j.prong Opn accss undr CC BY-NC-ND lcns.

2 374 Lu Jan t al. / Procda Engnrng 9 () nw bam lmnt typ that has th followng charactrstcs: hghr prcson and lss dvdd lmnt numbr.. Thr-nod Eulr-Brnoull bam lmnt and stran calculaton Th cntral ln gnral gomtry of a curv ovr a plan (th accptd gomtrc approxmaton) s mappd n Fgur (a). Th curv of Fgur (a) can b paramtrzd as a functon of a nondmnsonal varabl ξ (varyng from to ). In ths study a squar approxmaton for poston n x drcton and a quntc approxmaton for y drcton ar mposd. Cntr nod s nod 3, nowξ =.5, on wrts x = X + cξ + c ξ () y = Y + ξ + ξ + ξ + ξ + ξ () whr l = X X, l = Y Y. x y In ths study th physcal rfrnc confguraton s a straght fram bar, as follows: l x l y X X =, Y 3 3 Y = wth c = l, c =, x = l tanθ x, =-6l tanθ - l tanθ - 8l tanθ + 5l, x x x 3 y = 3l tanθ + 5l tanθ + 3l tanθ - 5l, =-l tanθ -8l tanθ -4l tan θ +6l x x x y x x x y =4l tan θ +4l tan θ +6l tanθ -4l 5 x x x 3 y Applyng Eq. (), for cntral ln on calculats x = X + l ξ (4) x As Eulr-Brnoull hypothss s adoptd, only th longtudnal stran s consdrd. Imagn a fbr paralll to th cntral ln wth an ntal lngth dfnd by ds. Aftr dformaton, ts lngth bcoms ds and th followng non-lnar ngnrng stran rfrrd to th non-dmnsonal spac rprsntd hr by varablξ s dfnd (Ogdn 984): ds ds d s/dξ d s /dξ ε = = (5) ds d s /dξ In th ntal confguraton for th cntral ln passng trough th mass cntr of th bar, on has ds dξ dx dξ dy dξ l l l x y = + = + = (6) whr l s th ntal lngth of th fnt lmnt. A gnral confguraton, for any nstant, s dscrbd by th approxmaton dfnd n Fgur (a). For ths cas th cntral ln auxlary strtch n computd as 3 4 ds dξ = dx dξ + dy dξ = l + + ξ + 3ξ + 4 ξ + 5ξ (7) cntral x Substtutng Eqs. (6) and (7) n Eq. (5) rsults / 3 4 ε = l + ( + ξ + 3ξ + 4ξ + 5ξ ) (8) cntral x l (3)

3 Lu Jan t al. / Procda Engnrng 9 () Followng usual ngnrng procdurs, for Eulr-Brnoull (orgnally straght) fram lmnt, th stran at a fbr whch s z dstant from th cntral ln can b wrttn as ε = ε + z r (9) cntral whr / r s th xact curvatur of th cntral ln dpctd n Fgur (a). Ths curvatur s gvn as a functon of ξ only, as on can s n (Wyl and Barrtt 995): 3/ = + Rplacng th known approxmatons,.. Eqs. ()-(4), Eq. () bcoms dx d y dx dy r dξ dξ dξ dξ = l ( + 6ξ + ξ + ξ ) l + ( + ξ + 3ξ + 4ξ + 5ξ x x ) () r Placng Eqs. (8), () and (9) togthr, and rmmbrng that th ntal confguraton s a straght lmnt, on has nonlnar ngnrng stranε, corrspondng to th ngnrng strssσ. 3. Th smpl postonal formulaton Adoptng lnar consttutv rlaton for hypr-lastc matrals, th stran nrgy can b wrttn for th rfrnc volum as U u σε ε () = d = d = d (spc) In ordr to calculat th stran nrgy t s ncssary to ntgrat th spcfc stran nrgy u ( spc ) ovr th ntal volum of th analysd body, as th proposd stran masur s, by natur, a Lagrangan varabl. Substtutng Eq. (9) nto Eq. () rsults n E E u = z z z spc ε + = ε + ε + cntral cntral cntral (3) r r r It s ncssary to ntgrat th spcfc stran nrgyu, Eq. (3), ovr th bar volum. Intgratng t ovr th cross-scton ara rsults n ( spc) EA EI u = ( ε ) + cntral (4) r Now on ntgrats th stran nrgy pr unt of lngth, Eq. (4), along th orgnal lngth of th bar,.. EA EI U = d d ε + l ξ l u ξ cntral = r (5) Th potntal nrgy of appld consrvatv concntratd forcs s wrttn as W= FX + F + Mθ + F X + FY+ Mθ + FX + FY+ Mθ (6) or W = FX whr x y x y x3 3 y F rprsnts forcs (or momnts) appld n drcton and of th pont whr th load s appld. () (7) X s th th coordnat paramtr

4 376 Lu Jan t al. / Procda Engnrng 9 () Th total potntal nrgy s wrttn as = l udξ FX F Mθ F X FY Mθ FX FY Mθ x y x y x3 3 y (8) Fgur: (a) Curv n a D spac; (b) Flow chart of th nonlnar analyss for bam structurs; (c) Rlaton btwn th appld bndng momnt and th rotaton at th loadd nd. whr ( X, Y,, X, Y,, X, Y, ) (,,,,,,,, ) θ θ θ ar nodal postons and thr conjugat forcs ar F F M F F M F F M. As thr s no sngularty n th stran nrgy ntgral on X Y X Y X 3 Y3 3 can drv Eq. (8) rgardng nodal postons. Th nxt quaton shows ths procdur for poston X. u = l ξ F = x X X or n a compact notaton: d (9) whr = g p j = f p F j = p p s a gnralzd paramtr and ndcs ar rlatd to nodal postons by ( X, Y,, X, Y,, X, Y, ) (,, 3, 4, 5, 6, 7, 8, 9 ) θ θ θ = In a vctor rprsntaton on has g( pf, ) = f ( p) F = () () ()

5 Lu Jan t al. / Procda Engnrng 9 () Th vctor functon g p s non-lnar rgardng nodal paramtrs ( p and F ). To solv Eq. () on can us th Nwton-Raphson procdur, flow chart of th nonlnar analyss for bam structurs s as fgur (b). A, E, I and L rprsnt cross-sctonal ara, Elastc Modulus, nrtal momnt, cantlvr bam lngth rspctvly. ElmntNum, ps, ItrMax rspctvly rprsnt total structur lmnt numbr, convrgnc standard and maxmum tratng tms (whn traton numbr mor than ths, th nd). n and ar traton tms and lmnt numbr. X and F ar + 3 matrxs, rspctvly rprsnt poston and forc vctor. Norm (.) s th calculaton vctor norm. 3. Numrcal xampl Th numrcal xampl s an Eulr bam ntally horzontal, clampd at on nd and subjctd to an appld momnt at th othr. It has bn prsntd n Rf. [4] (Coda and Grco 4). Th adoptd 6 4 physcal proprts ar: L = 5cm, A = cm, E = N / cm and I = cm. Sx load stps and two fnt lmnts wr usd to run ths problm untl an ntr lap of th gomtry. Som mportant valus ar compard wth Rf. [4] (Coda and Grco 4). In Fgur (c) rsults for ths study and Rf. [4] (Coda and Grco 4) valus ar compard wth th analytcal soluton for bndng momnt/rotaton rlaton. On can s that th global structur usually nds to b dvdd nto mor dvdd lmnt numbr by usng tradtonal two-nod bam lmnt than thr-nod bam lmnt to achv th sam prcson. Mor dvdd lmnt numbr mans ordr numbr of systm balanc quatons ncrasd twofold; t must brng a lot of dffcult to solv. Thrfor, th prcson of thr-nod bam lmnt s much hghr than on of two convntonal two-nod bam lmnts n nonlnar analyss. 4. Conclusons In gomtry nonlnar statc and dynamc analyss of Structurs that xhbt larg dsplacmnt, only smply ncrasng dvdd lmnt numbr must brng a lot of solvng dffcult, and cannot solv all problms, spcally n nonlnar dynamcs. Although thr-nod ncrasd computatonal complxty of lmnt lvl, th computrs tratmnt polynomal functon ffcncy s vry hgh, and n th ovrall lvl, not ncrasd amount of calculaton. Thrfor, thr-nod bam lmnt has a grat advantag ovr two-nod on, and ths artcl provds a nw solvng mthod to th larg dsplacmnt gomtry nonlnar statc and dynamc analyss. Rfrncs [] Blytschko T, Lu W K, Moran M. Nonlnar fnt lmnts for contnua and structurs. Nw York: John Wly & Sons Ltd, [] Crsfld M A. Non-lnar Fnt Elmnt Analyss of Solds and Structurs - olum : Essntals. Chchstr, UK: John Wly & Sons, 99 [3] Clough R W. Spch by profssor Clough R W arly hstory of th fnt lmnt mthod from th vw pont of a ponr. Intrnatonal Journal for Numrcal Mthods n Engnrng. 4, 6(): [4] Coda H B, Grco M. A smpl FEM formulaton for larg dflcton D fram analyss basd on poston dscrpton. Computr Mthods n Appld Mchancs and Engn rng. 4, 93(): [5] Ogdn R W. Non-lnar Elastc Dformaton, Ells Horwood, England, 984. [6] Wyl C R, Barrtt L C. Advancd Engnrng Mathmatcs, sxth d., McGraw-Hll, Nw York, 995.