A multiresolution finite element method. based on a new quadrilateral plate element

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1 A multoluton fnt lmnt mthod asd on a nw quadrlatral plat lmnt YMng Xa Cvl Engnrng Dpartmnt Nanjng Unvty of Aronautcs and Astronautcs Nanjng 6 Chna Emal:xym463@sna.com ABSTRACT: A nw multoluton quadrlatral plat lmnt s proposd and a multoluton fnt lmnt mthod s hnc pntd. Th multoluton analyss (MRA) framwork s formulatd out of a mutually nstng dsplacmnt suspac squnc whos ass functons ar constructd of scalng and shftng on th lmnt doman of asc nod shap functon. Th asc nod shap functon s constructd y xtndng shap functon around a spcfc nod. Th MRA ndows th proposd lmnt wth th oluton lvl (RL) to adjust th lmnt nod numr thus modulatng structural analyss accuracy accordngly. As a ult th tradtonal 4-nod quadrlatral plat lmnt and mthod s a monooluton on and also a spcal cas of th proposd lmnt and mthod. Th mshng for th monooluton plat lmnt modl s asd on th mprcsm whl th RL adjustng for th multoluton s lad on th rgorous mathmatcal ass. Th accuracy of a structural analyss s actually dtrmnd y th RL not y th msh. Th ratonal MRA nals th mplmntaton of th multoluton lmnt mthod to mor ratonal and ffcnt than that of th convntonal monooluton plat lmnt mthod or othr corpondng MRA mthods such as th wavlt fnt lmnt mthod th mshfr mthod and th natural lmnt mthod tc. Kywords: Multoluton quadrlatral plat lmnt Multoluton analyss (MRA) Rsoluton lvl (RL) Basc nod shap functon Scalng and shftng Mutually nstng dsplacmnt suspac squnc. Scton: Mathmatcal Physcal & Engnrng Scncs. Introducton Multoluton analyss (MRA) s a popular tchnqu that has n appld n many domans such as th sgnal and mag procssng th damag dtcton and halth montorng th dffrntal quaton soluton tc. Howvr n th fld of computatonal mchancs th MRA has not n n a ral sns fully utlzd n th numrcal soluton of ngnrng prolms thr y th tradtonal fnt lmnt mthod (FEM) [] or y othr mthods such as th wavlt fnt lmnt mthod (WFEM) [ 3] th mshfr mthod (MFM) [4 5] and th natural lmnt mthod (NEM) [6 7] tc. As s commonly known of th FEM owng to th nvaranc of nod numr a sngl fnt lmnt contans th fnt lmnt can rgardd as a monooluton on from a MRA pont of vw and th FEM structural analyss s usually not assocatd wth th MRA concpt.

2 Th MRA sms to rarly usd whn th FEM s mployd to structural analyss. Howvr t s n fact y mans of fnt lmnt modl mshng and r-mshng to modulat analyss accuracy n whch a clustr of monooluton fnt lmnts ar assmld togthr artfcally that th rough structural MRA s xcutd y th FEM. As w can s n ovr whol analyss procss of a structur y th FEM thr s no sold mathmatcal foundaton for th tradtonal fnt lmnt modl mshng and th fnt lmnts ar assmld togthr artfcally. Th tradtonal fnt lmnt modl has to r-mshd untl suffcnt accuracy s rachd whch lads to th low computaton ffcncy or convrgnt rat. Th dfcncy of th FEM coms much xplct n th accurat computaton of structural prolms wth local stp gradnt such as matral nonlnar [8 9] local damag and crack [ ] mpactng and xplodng prolms [ 3]. Th grat fforts hav n mad ovr th past thrty ya to ovrcom th drawacks of th FEM wth many mprovd mthods to com up such as WFEM MFM and NEM tc whch opn up a transton from th monooluton fnt lmnt mthod to th multoluton fnt lmnt mthod faturd wth adjustal lmnt nod numr. Although ths MRA mthods hav llustratd thr powrful capalty and computatonal ffcncy n dalng wth som prolms thy always hav such major nhrnt dfcncs as th complxty of shap functon constructon th asnc of th Kronckr dlta proprty of th shap functon and th lack of a sold mathmatcal ass for th MRA whch mak th tratmnt of lmnt oundary condton complcatd and th slcton of lmnt nod layout mprcal that sustantally rduc computatonal ffcncy. Hnc ths MRA mthods hav nvr found a wd applcaton n ngnrng practc just as th FEM. In fact thy can vwd as th ntrmdat products n th transton of th FEM from th monooluton to th multoluton. Th drawacks of all thos MRA mthods can lmnatd y th ntroducton of a nw multoluton fnt lmnt mthod n ths papr. Wth pct to a quadrlatral plat lmnt n th fnt lmnt stock a nw multoluton plat lmnt s formulatd y th MRA asd on a dsplacmnt suspac squnc whch s consttutd y th translatd and scald von as suspac ass functons of th asc nod shap functon. Th asc nod shap functon s constructd y xtndng soparamtrcal nod shap functon for a convntonal plat lmnt to othr thr quadrants around a spcfc nod and th varous nod shap functons wthn th multoluton lmnt can asly mad up of th scald and shftd von of th asc nod shap functon. It can sn that th shap functon constructon s smpl and clar manwhl th nod shap functons hold th Kronckr dlta proprty. In addton th proposd lmnt mthod posssss a sold mathmatcal ass for th MRA whch ndows th proposd lmnt wth th oluton lvl (RL) that can modulatd to chang th lmnt nod numr adjustng structural analyss accuracy accordngly. As a ult th multoluton quadrlatral plat lmnt mthod can rng aout sustantal mprovmnt of th computatonal ffcncy n th structural analyss whn compard wth th corpondng FEM or othr MRA mthods.. Th asc nod shap functon constructon Corpondng to a quadrlatral lmnt of artrary shap n a Catsan coordnat

3 systm shown n Fg. a soparamtrc quadrlatral lmnt n a natural coordnat systm shown n Fg. s adoptd wth th transformng rlatons from th Catsan coordnat systm to th natural on( ) dfnd as follows: 4 4 m m () x N x y N y whr x y ar th Catsan coordnat valus at th -th nod. N m ar th convntonal shap functons for th dffrnt nod whch ar dfnd on th doman of [] wth th -lnar functons as follows N N N N m m m 3 m 4 () zw y y x (4)(x 4 y 4 ) x (3)(x 3 y 3 ) () (x y ) ()(x y ) Fg. A quadrlatral plat ndng lmnt Th drvatv rlatonshp twn th two coordnat systms xsts as follows: x J (3) y x/ y/ n whch J x/ y/ s th ft ordr Jaco matrx. Th dsplacmnt of a classcal ndng lmnt shown n Fg. can asly acqurd and concsly xpsd n trms of natural coordnats as follows [4]: 3

4 4 4 4 x x y y (4) w N w N N whr w s th transv dsplacmnt n th z axs drcton at an artrary pont of th lmnt. w ar th transv rotatonal dsplacmnts at nod of th lmnt x y pctvly n th Catsan coordnat systm. N N N ar th convntonal shap x y functons at th nod whch ar dfnd on th doman of [] as follows N X Y X Y X Y X X YY N X Y X Y X Y X X YY N X Y X Y X Y X X YY 3 N X Y X Y X Y X X YY 4 N X Y y X X y YY x 4 N X Y y X X y YY x 3 N X Y y X X y YY x N X Y y X X y YY x N X Y x X X x YY y 4 N X Y x X X x YY y 3 N X Y x X X x YY y N X Y x X X x YY y (5) (6) (7) (=~4j=~4). whr X X Y Y x x x y y y j j j j (4) (3) () () Fg. Th nod shap functon doman Fg 3. Th xtndd shap functon doman for of a classcal quadrlatral plat lmnt th nod () 4

5 As to th proposd lmnt th shap functons rgardng th nod () as shown n Fg. ntally dfnd on th doman of [ ] should xtndd to th doman of [ ] y mans of shftng th lmnt around th nod () vrtcally horzontally and olquly pctvly to th othr thr quadrants thus covrng th ght nods adjacnt to th nod () as shown n Fg 3 th asc shap functon for th nod () at th pont coordnat of ( ) can fnally dfnd as follows: 4 N N : N3 N x Nx Nx3 Nx4 N : y N y 3 N y3 N y4 N : (8) (9) () Wth hlp of q.(3) t can sn that th Kronckr dlta proprty holds for th asc nod shap functons 3 systm: wth pct to th Catsan coordnat 5

6 Anods x Anods Anods y x y Anods x Anods Anods y x y 3 3 3Anods x 3 3Anods 3Anods y x y whr th notaton Anods s rfrrd to th ght nods adjacnt to th asc nod () Th asc nod shap functons 3 systm ar llustratd n Fgs.4 ac pctvly () n th natural coordnat a.. c. 3 Fg 4. Th asc nod shap functons 3 on th doman [-] [-] Ovously th asc shap functons 3 pctvly. ar contnuous Bass functon constructon for a dsplacmnt suspac squnc formng a nw MRA In ordr to carry out a MRA of a thn plat structur th mutual nstng dsplacmnt suspac squnc for a plat lmnt should stalshd. In ths papr a totally nw tchnqu s proposd to construct th MRA whch s asd on th concpt that a suspac squnc (mult-oluton suspacs) can formulatd y suspac ass functon vcto at dffrnt oluton lvls whos lmnts-scalng functon vctor can constructd y scalng and shftng on th doman[] of th asc nod shap functons. As a ult th 6

7 dsplacmnt suspac ass functon vctor at an artrary oluton lvl (RL) of (m+) (n+) for a quadrlatral plat lmnt s formulatd as follows: Ψ Φ Φ Φ () / n / m Φ s th scalng ass functon whr 3 = ( m r n s) = ( m r n s) 3 = 3 ( m r n s) m vctor n dnotd as th postv ntg th scalng paramt n drctons pctvly. r s as th postv ntg th nod poston paramt that s r 3 m s 3 n Hr m r n s. It s sn from Eq. () that th nods for th scalng procss ar qually spacd on th doman [] wth a stp sz of /m n and / n n drctons pctvly. Scalng of th asc nod shap functons on th doman of [- ] (prcsly on th doman of r s m m n n ) and thn shftng to othr nods m n on th lmnt doman of [ ] wll produc th varous nod shap functons that ar shown n th Fg 5. at th RL of a. (RL= ). (RL=3 3) Fg 5. Th scald and shftd von of th asc nod shap functon on th doman [] [] Snc th lmnts n th ass functons ar lnarly ndpndnt wth th varous scalng and th dffrnt shftng paramt th suspacs n th suspac squnc can stalshd and ar mutually nstd thus formulatng a MRA framwork that s W V... Vj... V V j : span j : j Z V V V V V V j ( j) j ( ) j j ( )( j) whr Z dnotd as th postv ntg Ψ (3) V j as dsplacmnt suspac at th oluton 7

8 lvl of (+) (j+). Thus t can found that th mutually nstng dsplacmnt suspac squnc W can takn for a sold mathmatcal foundaton for th MRA framwork and V s quvalnt to th dsplacmnt fld for a tradtonal 4-nod plat lmnt that s th rason why th tradtonal quadrlatral plat lmnt s rgardd as a mono-oluton on and also a spcal cas of th multoluton quadrlatral plat lmnt. Basd th MRA stalshd th dflcton of a quadrlatral plat lmnt n th dsplacmnt suspac at RL of (m+) (n+) can dfnd as follows w Ψ a (4) a w... w... w w x y ar whr x y x y x y r s th transv and rotatonal dsplacmnts pctvly at th lmnt nod m n. It s ovous that th proposd mult-oluton lmnt s a mshfr on whos nods ar unformly scattrd nod numr and poston fully dtrmnd y th RL. Whn th scalng paramtr m=n=(rl= ) that s a tradtonal 4-nod quadrlatral plat lmnt q. (4) wll rducd to q. (4). T 3 Multoluton quadrlatral plat lmnt formulaton Accordng to th classcal assumpton of th thn plat thory th gnralzd functon of potntal nrgy n a dsplacmnt suspac at th oluton lvl (m+) (n+) for a quadrlatral plat lmnt s a T a V D dxdy qw dxdy Qw (5) whr w x w y w xy D C / C 3 Eh E s th matral Young modulus h th thcknss of th lmnt th Posson s rato q dstrutd transv loadngs Q th lump transv loadngs. B B B a (6) 8

9 whr B Φ Φ Φ x y xy Φ / x Φ / Φ / / y J Φ ' ' / / / / xy Φ Φ Φ Φ J T x/ y/ x/ y/ x/ y/ x/ y/ x/ x/ y/ y/ x/ y/ y/ x/ ' s th scond ordr Jaco matrx. y x / / / J x x x x y y y y. ' y/ x/ / J Susttut Eq.(4) Eq.(6) nto Eq.(5) th concs xpson can otand aftr rassmlng as follows: T T T pv aka af af (7) whr K s th lmnt stffnss matrx f th lmnt dstrutd loadng colu vctor F th lmnt lump loadng colu vctor. Accordng to th potntal nrgy mnmzaton prncpl lt V lmnt qulrum quatons can otand as follows K a f F (8) th plat p Th lmnt xpson of th stffnss matrx vctof F can gvn as follows: K and th loadng colu K k k k... k k k... k k k j (9) 9

10 whr th supcrpt dnotd as th row numr of th matrx and th suscrpt as th algnd lmnt nod numrng (r s). In trms of th proprts of th nod shap functons w hav k kcd cr ds () k kcd whn c r d s k n whch cd s th coupld nod stffnss matrx rlatng th nod (c d) to (r s). T cd cd D k B B J dd () T f Ψ q J T F Ψ P d d () whr Ψ s th shap functon matrx P s th lump loadng vctor. 4 Transformaton matrx In ordr to carry out structural analyss th lmnt stffnss K th loadng colu vcto f F should transformd from th lmnt local coordnat systm (xyz) to th structural gloal coordnat systm (XYZ). Th transformng rlatons from th local to th gloal ar dfnd as follows: K T K T (3) T l f T f (4) T l F T F (5) T l whr K s th lmnt stffnss matrx f F th lmnt loadng colu vcto undr th gloal coordnat systm. T s th lmnt transformaton matrx dfnd as follows; T λ... λ j... λ cos zz cos cos cosyx cos yy j xx xy whr s th ntcton angl twn th local and th gloal coordnat axs. λ

11 Th structural gloal stffnss K and th gloal loadng colu vcto f F can otand y splcng K f F of th lmnt pctvly 5 Numrcal xampl Exampl. As shown n Fg.6 a two oppost dg smply supportd and othr two fr 6 skw plat wth th gomtrc confguraton of lngth L and th Posson s rato.3 s sujctd to th unform transv loadng of magntud q. Evaluat th dflcton at th cntr pont of th plat. Th dsplacmnt ponss ar found y th proposd quadrlatral plat lmnt modl th tradtonal 4-nod quadrlatral plat lmnt modl and th wavlt lmnt modl asd on two-dmnsonal tnsor product B-spln wavlt on th ntrval (BSWI) [3] pctvly. Th BSWI s chosn caus t s th st on among all xstng wavlts n approxmaton of numrcal calculaton [5] and drctly constructd y th tnsor product of th wavlts xpansons at ach coordnat. Th cntral dflctons of th plat ar summarzd n tal.. L 6 L Fg. 6. A skw plat L L L L a. A multoluton modl. A monooluton modl Fg. 7 Th fnt lmnt modl for th skw plat

12 Tal.. th cntr pont dflcton (w/ ql 4 /D ) Elmnt typ Th proposd (RL) Th convntonal (msh) dflcton On BSWI [3].795 Analytcal [6].7945 Th multoluton modls composd of on proposd multoluton quadrlatral plat lmnt wth th RLs of and 7 7 as shown n Fg.7.a ar adoptd th monooluton modls composd of mshs of 8 8 and 6 6 as dsplayd n Fg.7. ar also mployd and th wavlt modls mad up of on D BSWI lmnt (B-spln wavlt on th ntrval lmnt) of th jth scal=3 th mth ordr =4 ar usd pctvly arvatd as BSWI43 wth th DOF of. Th RL of ach proposd and th corpondng mshs of th convntonal ar compard. It can sn that th analyss accuracs wth th proposd quadrlatral lmnt ar gradually mprovd pctvly wth th RL rachng hgh. Although th BSWI43 s of hgh accuracy whn compard wth th proposd th dfcncs of th BSWI lmnt ar ovous as follows. In lght of tnsor product formulaton of th multdmnsonal MRA framwork [5] th DOF of a mult-dmnsonal BSWI lmnt wll so drastcally ncrasd from that of a on-dmnsonal lmnt n an rratonal way ultng n complx shap functons and sustantal rducton of th computatonal ffcncy. Scondly du to th asnc of Kronckr dlta proprty of th tnsor-product constructd shap functons th spcal tratmnts should takn to dal wth th lmnt oundary condton whch wll rng aout low computatonal ffcncy. Thrdly thr xsts no such a paramtr as th RL wth a clar mathmatcal sns. In addton th RLs of th proposd and th corpondng mshs of th convntonal ar dsplayd n Tal. It can found that th analyss accuracs wth th proposd and th convntonal ar gradually mprovd pctvly wth th RL rachng hgh and th msh approachng dns. Howvr th RL adjustng s mor ratonally and ffcntly to mplmntd than th mshng and th r-mshng for th followng two rasons. Ftly th RL adjustng s asd on th MRA framwork that s constructd on a sold mathmatcal ass whl th mshng or rmshng whch orts to th mprcsm has no MRA framwork. Scondly th stffnss matrx and th loadng colu vcto of th proposd lmnt can otand automatcally around th nods whl thos of th tradtonal 4-nod quadrlatral plat lmnts otand y th artfcally complx rassmlng around th lmnts. Thus th computatonal ffcncy of th proposd lmnt mthod s hghr than th tradtonal on. In ths way th proposd plat lmnt xhts ts strong capalty of accuracy adjustmnt and ts hgh powr of oluton to dntfy dtals (nods) of dformd structur y mans of modulatng ts oluton lvl just as a multoluton camra wth a pxl n ts takn photo as a nod n th proposd lmnt. Thr appa no msh n th proposd lmnt just as no grd n th photo. Thus an lmnt of supror analyss accuracy surly has mor nods whn compard wth that of th nfror just as a clarr photo contans

13 mor pxls. Exampl. A crcular rng sla s sujctd to th unform transv loadng q as shown n Fg.8 wth ts oundary condtons as: th nnr dg s fr and th outr dg s fully clampd and ts gomtry and physcal paramt as: th nnr radus th outr radus a th thcknss t th lastcty modulus E th Posson s rato. Fnd th dsplacmnt around th nnr fr-dg rng of th sla. Y a X Fg. 8. A crcular rng sla Y Y x y a 3 a 4 X X a A.multoluton modl. A monooluton modl Fg. 9. Th fnt lmnt modl for th /4 crcular rng sla To calculat th dsplacmnt ponss symmtry condtons ar xplotd and only th ft quadrant of th plat s dscrtzd. th mutoluton plat modl s hrn composd of four multoluton quadrlatral plat lmnts 34 wth th lmnt RL of 5 3 hnc th /4 sla RL of 5 9 as shown n Fg. 9a. and th monooluton on composd of th msh of 4 8 dsplayd n Fg.9. In th analyss procss ths four multoluton lmnts ar splcd togthr along th common ntcton oundary and th analyss accuracy can modulatd y mans of adjustng th RL. Wth pct to th convntonal monooluton th structur s mshd nto a group of monooluton lmnts and th analyss accuracy s mprovd only y mans of r-mshng. It can sn that th RL adjustng s mor ratonally and asly to 3

14 mplmntd than th r-mshng caus th proposd multoluton lmnt modl of th crcular rng plat structur contans much lss lmnts than th monooluton hnc rqurng much lss tms of th transformaton matrx multplyng whch ults n much hghr computatonal ffcncy for th proposd lmnt mthod than that for th tradtonal lmnt mthod. Th dsplacmnt around th nnr fr-dg rng of th sla s summarzd n Tal.. Th RL of th ntcton oundary should th sam as that of th adjacnt lmnt just as PS (Photoshop) four photos. Tal.. Th maxmum dsplacmnt (w/ qa 4 /Et 3 ) around th nnr fr-dg rng of th sla Elmnt typ Th proposd (RL) Th convntonal (msh) dflcton Analytcal [7] Dscusson From th two numrcal xampls aov t s shown that asd on th multoluton quadrlatral plat lmnt formulaton a nw multoluton fnt lmnt mthod s ntroducd whch ncorporats such man stps as RL adjustng lmnt matrx formaton lmnt matrx transformaton from a local coordnat systm to a gloal on and gloal structural matrx formaton y splcng of th lmnt matrcs. Owng to th xstnc of th nw MRA framwork th RL adjustng for th proposd mthod s mor ratonally and asly to mplmntd than th mshng and r-mshng for th tradtonal 4-nod quadrlatral plat lmnt mthod. Du to th asc nod shap functon th stffnss matrx and th loadng colu vcto of a proposd lmnt can automatcally acqurd through quadraturng around nods n th lmnt matrx formaton stp whl thos of th tradtonal 4-nod quadrlatral plat otand through complx artfcally rassmlng of th lmnt matrx around th lmnts n th r-mshng procss whch contruts a lot to computaton ffcncy mprovmnt of th proposd mthod. Morovr snc th multoluton quadrlatral plat lmnt modl of a structur usually contans much lss lmnts than th tradtonal monooluton lmnt modl thus rqurng much lss tms of transformaton matrx multplyng th computaton ffcncy of th proposd mthod appa much hghr than th tradtonal n th stp of lmnt matrx transformaton. In addton caus of th smplcty and clarty of th shap functon formulaton wth th Kronckr dlta proprty and th sold mathmatcal ass of th nw MRA framwork th proposd mthod s also supror to othr corpondng MRA mthods n trms of th computatonal ffcncy th applcaton flxlty and xtnt. Hnc takng all thos causs nto account th concluson can drawn that th multoluton quadrlatral plat lmnt mthod s mor ratonally asly and ffcntly to xcutd whn compard wth th tradtonal 4-nod quadrlatral plat lmnt mthod or othr corpondng MRA mthods and th proposd plat lmnt s th most accurat on formulatd vr snc. 7 Conclusons 4

15 A nw multoluton fnt lmnt mthod that has oth hgh powr of oluton and strong flxlty of analyss accuracy s ntroducd nto th fld of numrcal analyss. Th mthod posssss such promnnt fatu as follows:. A novl tchnqu s proposd to construct th smpl and clar asc nod shap functon that holds Kronckr dlta proprty.. A mathmatcal ass for th MRA framwork that s th mutually nstng dsplacmnt suspac squnc s consttutd out of th scald and shftd von of th asc nod shap functon. Th MRA framwork ndows th plat lmnt wth th RL to adjust th lmnt nod numr modulatng th analyss accuracy of structur accordngly. Hnc th tradtonal 4-nod quadrlatral plat lmnt and mthod s a monooluton on and also a spcal cas of th proposd. An lmnt of supror analyss accuracy surly contans mor nods whn compard wth that of th nfror. 3. Th RL adjustng for th multoluton plat lmnt modl s lad on th rgorous mathmatcal ass whl th mshng or rmshng for th monooluton s asd on th mprcsm. Hnc th mplmntaton of th proposd lmnt mthod s mor ratonal and ffcnt than that of th tradtonal or othr MRA mthods such as th wavlt fnt lmnt mthod th mshfr mthod and th natural lmnt mthod tc. 4. A qut nw concpt s ntroducd nto th FEM that th structural analyss accuracy s actually dtrmnd y th RL-th dnsty of nod unform dstruton not y th msh. 5. Wth advnt of th multoluton fnt lmnt mthod th ratonal MRA wll fnd a wd applcaton n numrcal soluton of ngnrng prolms n a ral sns. Th upcomng work wll focusd on th tratmnt of ntrfac twn multoluton lmnts of dffrnt RL. Th ntrfac may xtndd to th rdgng doman n whch th transtonal lmnt (xpandd Srndpty lmnt) could usd just as PS photos of dffrnt RL. 8 Acknowldgmnt Th author would lk to thank th rf for thr valual commnts also Prof ShaoLn Chn and assocat Prof Gan Tan for thr assstanc. Appndx Th quvalnt nod loadng f of a unform dstrutd loadng q ovr or th quvalnt nod loadng F of a lump loadng P appld at th cntr of a cll whch s a 4-nod sudoman n an lmnt can rad as follows: F P /4 y y4/6 x x4/6 /4 y y3 /6 x x3 /6 /4 y34 y4/6 x34 x4 /6 /4 y34 y3 /6 x34 x3 /6 T f q /8Z T x T y Z T x T y Z T T Z T T 3 x3 y3 4 x4 y4 5 T

16 x y x 3 B C y3 A B C T y x A B C x3 A B C x C T y x4 34 B C y4 A B C T y4 y34 A B C y3 A B C /4 /4 /4 Z A B C T y A B C y A B C T x A B C x A B C Z A B C T y A Z A B C T y A B C y A B x 3A5B3C x 5A3B3 C Z 36A36B8 C T y 3A A x y x y B x y x y C x y x y Appndx Start Data nput ncludng lm RL Form quvalnt nod loadng vctor Form nod stffnss matrx Elm loop Form lm stffnss matrx Nod loop Form lm transformaton matrx End loop Splc quvalnt lm nod loadng vctor to form quvalnt structural nod loadng vctor Splc lm matrx to form ovrall structural matrc Apply oundary condtons End loop Equlrum quaton soluton Data output Cod flow chart End 6

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