FINITE ELEMENT METHOD: AN INTRODUCTION Uday S. Dixit Department of Mechanical Engineering, Indian Institute of Technology Guwahati , India

Transcription

1 FIITE ELEMET METHOD: A ITRODUCTIO Uda S. Dt Dpartmt of Mchacal Egrg, Ida Isttt of Tcholog Gwahat-78 39, Ida. Itrodcto Ft lmt mthod (FEM s a mrcal mthod for solg a dffrtal or tgral qato. It has b appld to a mbr of phscal problms, whr th gorg dffrtal qatos ar aalabl. Th mthod sstall cossts of assmg th pcws cotos fcto for th solto ad obtag th paramtrs of th fctos a mar that rdcs th rror th solto. I ths artcl, a brf trodcto to ft lmt mthod s prodd. Th mthod s llstratd wth th hlp of th pla strss ad pla stra formlato.. FEM formlato for a lar dffrtal qato A lar dffrtal qato ca b of th followg form: L + q, ( whr s th ctor of prmar arabls of th problm, whch ar fctos of th coordats, L s th dffrtal oprator ad q s th ctor of kow fctos. Ths dffrtal qato wll b sbjctd to bodar codtos, whch ar sall of two tps- ( th sstal bodar codtos ( th atral bodar codtos. Th sstal bodar codtos ar th st of bodar codtos that ar sffct for solg th dffrtal qatos compltl. Th atral bodar codtos ar th bodar codtos olg hghr ordr drat trms ad ar ot sffct for solg th dffrtal qato compltl, rqrg atlast o sstal bodar codto. For ampl, cosdr th dffrtal qato: d d EA + q. ( d d Ths problm ca b sold compltl dr o of th followg two codtos: ( s prscrbd at both ds. ( s prscrbd at o d ad d/d s prscrbd at th sam or othr d. Howr, th problm caot b sold f ol d/d s prscrbd at both ds. Ths, w srl rqr o bodar codto prscrbg. Thrfor, for ths problm * s a sstal bodar codto ad d/d (d/d * s a atral bodar codto, whr * dcats th prscrbd al. ow cosdr th dffrtal qato d d w EI q, (3 d d Ths dffrtal qato ca b sold compltl b spcfg w ad dw/d at both ds. O ca also spcf d w/d ad/or d 3 w/d 3 as bodar codtos, howr ot of total for bodar codtos, two mst b of o of th followg forms: ( w prscrbd at both ds. ( w prscrbd at o d ad dw/d prscrbd at th othr d. Ths, th prscrbd als of w ad dw/d form th part of sstal bodar codtos ad prscrbd als of d w/d ad d 3 w/d 3 form th part of atral bodar codtos. Two poplar FEM formlatos ar Galrk formlato ad Rtz formlato. I Galrk formlato, th prmar arabl s appromatd b a cotos fcto sd th lmt. Wh th appromat prmar arabl s sbstttd Eq. (, w shall gt rsd dpdg o th appromatg fcto,.., L + q R. (

2 Idall, th rsd shold b zro rwhr. I that cas, appromato bcoms qal to tr al. As t s r dffclt to mak th rsd at all pots, w mak th wghtd rsdal qal to zro,.., wr da, (5 D whr w s th wght fcto. I ordr to wak th rqrmt o th dffrtablt of th appromatg fcto, w tgrat Eq. (5 b parts to rdstrbt th ordr of drat w ad R. I Galrk mthod, th wght fcto s chos of th sam form as th appromatg fcto. Th appromatg fcto s som algbrac fcto. It s commo to rplac th kow coffcts of th fcto b kow odal dgrs of frdom. Ths, tpcall, [ ]{. (6 whr [] s th matr of shap fctos ad { s th odal dgrs of frdom. I Rtz formlato, th dffrtal qato Eq. ( s cortd to a tgral form sg calcls of arato. (Somtms th tgral form tslf ma b asl drabl from th phscs of th problm. Th appromato (Eq. (6 s sbstttd th tgral form ad th form s trmzd b partall dffrtatg wth rspct to {. Aftr obtag th lmtal qatos, th assmbl s prformd. A smpl wa of assmbl s to wrt qatos for ach lmt global form ad th add ach smlar qatos of all th lmts,.., w add th qato mbr from ach lmt to obta th frst global qato, all qato mbr ar addd togthr to g scod qato, ad so o. Th bodar codtos ar appld to assmbld qato ad th ar sold b a stabl solr. Th, post-procssg s carrd ot to obta th drats. 3. Formlato for pla strss ad pla stra Cosdr a lar lastc sold of doma ad hag form thckss bodd b two paralll plas o a closd bodar Γ as show Fg.. Th mag of bodar codtos s plad Fgr. If th thckss z drcto s small compard wth th sz of th doma, th problm ma b appromatd as a pla strss problm. Th followg assmptos ar mad. Th bod forcs, f a st, caot ar th thckss drcto ad caot ha compots th z drcto; th appld bodar forcs mst b forml dstrbtd across th thckss (.. costat th z drcto; ad o loads ca b appld o th paralll plas bodg to th bottom srfacs. Th assmpto that th forcs ar zro o th paralll plas mpls that for pla strss problms th strsss th z drcto ar glgbl small.., σ z σ z σ z (7 Fgr : A sold of doma

3 Fgr : Spport codtos Pla stra s dfd as a dformato stat whch thr s o dformato z-drcto ad dformatos othr drctos ar fctos of ad bt ot of z. Ths, sta compots ε z ε z ε z. I pla stra problms o-zro strss compots ar σ, σ, σ adσ z. Howr, σ z s ot a dpdt compot ad ca b obtad f σ ad σ ar kow. Ths maks th FEM formlato for pla strss ad pla stra problms smlar. Ol dffrc s th costtt matrcs for both problms. I ths scto FEM formlato for pla strss ad pla stra problms wll b dscssd. Th gorg qatos for th pla lastct problms ar g b σ σ + + f ρ (8 t σ σ + + f ρ (9 t whr ƒ ad ƒ dot th bod forcs pr t olm alog th ad drctos, rspctl ad ρ s th dst of th matral. σ, σ ar th ormal strsss ad, ar th dsplacmts ad drctos rspctl, σ s th shar strss o th z ad z plas. Stra-dsplacmt rlatos ar g b ε, ε, ε + ( For pla strss problms, strss ad stra ar rlatd b th costtt matr D, th followg mar: σ d d ε σ d d ε ( σ d33 ε whr d j (d j d j ar th lastct (matral costats for a orthotropc matral wth th matral prcpal drctos cocdg wth th co-ordat as (, sd to dscrb th problm. For a sotropc matral pla strss d j ar g b E E E d d, d d, d 33, ( (+ whr E s Yog's modls of th matral ad s Posso's rato. For pla stra problms: E( ν Eν E d d, d d, d33. (3 (+ ν ( ν (+ ν ( ν (+ For th g problm, sstal or gomtrc bodar codtos ar ad atral bodar codtos ar, o Γ ( 3

4 whr, t σ + t o Γ t (5 σ t σ + t o Γ t (6 σ ar th compots of th t ormal ctor o th bodar Γ. Γ ad Γ t ar portos of t, t ar spcfd bodar strsss or tractos, ad, ar th bodar Γ (Γ Γ U Γ t. spcfd dsplacmts. Ol o lmt of ach par, (, t ad (, t ma b spcfd at a bodar pot. I th Rtz FEM mthod, th arabls whos als ar to b dtrmd ar appromatd b pcws cotos polomals. Th coffcts of ths polomals ar obtad b mmzg th total pottal rg of th sstm. I FEM, sall, ths coffcts ar prssd trms of kow als of prmar arabls. Ths, f a lmt has got ods, th dsplacmt fld ca b appromatd as (7 whr ar th odal dsplacmts -drcto ad ar th shap fctos, whch ar fctos of coordats. For pla lastc bod, th total pottal rg of a lmt s g b (sg d otatos, Π dv f dv t ds (8 * σjεj V V Γ whr, V dots th olm of lmt, Γ s th bodar of doma, σ j ad ε j ar th compots of strss ad stra tsors, rspctl ad ƒ ad t ar th compots of bod forc ad bodar strss ctors, rspctl. ot that σ σ, σ σ σ σ (9, f f f, t t, t f, t ( Th frst trm qato (8 corrspods to stra rg stord th lmt, th scod rprsts th work pottal of th bod forc, ad th thrd rprst th work pottal of srfac forcs. For pla strss problms wth thckss h, t s assmd that all qatts ar dpdt of th thckss co-ordats z. Hc, Π h ( σ ε + σ ε + σ ε d d ( Γ h ( f + f dd h ( t + t ds whr ƒ ad ƒ ar th bod forcs pr t ara, ad t ad t ar bodar forcs pr t lgth. Eqato ( ca b rwrtt as T ε σ T T f t Π h ε σ dd h dd h ds ( f t ε σ Γ

5 5 Th ft lmt modl of th pla lastct qatos s dlopd sg th matr form (. Th dsplacmts ad ar appromatd b th Lagrag faml of trpolato fctos (shap fctos. Lt ad ar appromatd or b th ft lmt trpolatos, (,, ( (3 whr s th mbr of ods rprstg th lmt, ar th dsplacmt shap fctos, ad ar th odal dsplacmts - ad - drctos rspctl. Th dsplacmts ad stras or lmt ar g b [ ]{ ( ad ]{ ][ [,{ ]{ [ { B D B σ ε (5 whr ] ][ [ ] [ T B s calld Gradt matr ad [T ] s th matr of dffrtal oprators. Sbstttg ths prssos for th dsplacmts ad stras to ( T T T T T T T { ([ ] [ ][ ]{ d d { [ ] d d { [ ] d { ([ ] { { { T f h B D B h f t h s k f Q t Γ Π (6 Mmzg ths,.., dffrtatg th abo prsso wth rspct to {, w gt { { { ] [ Q f k + (7

6 whr [ k ] h [ B T ] [ D ][ B ] dd, f T t T { f h [ ] dd, { Q h [ ] ds (8 f Γ t Th lmt stffss matr [k ] s of ordr ad th lmtal load ctor [ F ] { f + { Q (9 s of ordr whr s th mbr of ods of th lmt. Shap fctos or trpolato fctos ar sd th ft lmt aalss to trpolat th odal dsplacmts of a lmt to a pot wth ach lmt. Th trpolato fctos for th for oddd qadrlatral lmts show Fg. 3 ar ( ξ ( η (+ξ ( η (+ξ (+η,, 3, ( ξ (+η (3 whr ξ ad η ar th atral co-ordats for th phscal co-ordats ad, rspctl. I atral coordat sstm, th coordats of for ods ar (-,-, (,-, (, ad (-,. O of th arlst ft lmts s a thr oddd traglar lmt show Fg.. A arbtrarl locatd pot P dds a tragl --3 to thr sb-aras A, A, ad A 3. Th, th atral coordats of th pot P ar dfd as ratos of aras: A A A3 ξ ξ ξ3 (3 A A A whr A s th ara of tragl --3. Sc AA + A + A 3, th ξ ar ot dpdt. Th satsf th costrat qato ξ + ξ + ξ3 (3 For ths tragl, shap fctos trms of atral coordats ar g as ξ ξ 3 ξ3 (33 It ca b show that dsplacmt fld obtad sg ths shap fctos prods costat stra sd th traglar lmt. Ths lmt s, thrfor, calld costat stra tragl (CST. Fgr 3 A qadrlatral lmt 6

7 Fgr : A traglar lmt Th alato of th lmt matrcs qato (8 s do b sg mrcal tgrato tchqs. For all ara ad l tgrals Gass-Qadratr rl s sd. All phscal doma tgrato s trasformd to th (ξ,η pla as show Fg. 5. As a rslt, f (, dd f ( ξ, η J dξdη (3 whr J s th dtrmat of th Jacoba matr of th trasformato ad s g b J J, ξ, ξ J (35 J J 3, η, η Fgr 5: atral co-ordat sstm. whr s th mbr of ods of th lmt, ( ξ, η s th shap fcto corrspodg to od, (, s th phscal co-ordats of ods. I wrtg qato (6, th soparamtrc formlato has b sd,.., th trpolato fctos sd for th gomtr arabls (, ad fld arabls (, ar sam. Also, th spatal drats ar trasformd to th (ξ,η pla sg 7

8 whr J J [ J ] J 3 J Fall, th Gass-Qadratr schm gs,, [ J J J J J ] f ( ξ, η dξdη w w r s r s, ξ, η f ( ξ (37 r, ηs whr mbr of Gass pots ξ drcto, mbr of Gass pots ηdrcto, w r, w s wghts of corrspodg Gass pots. If th dsplacmt fld wth ach lmt s assmd to b blar th, Gass qadratr actl tgrats all trms of th lmtal stffss matr. ow, cosdrg th alato of bodar tgral of th tp whr Γ (36 Q q ( sds (38 q s a kow fcto (hr bodar strss of th dstac s alog th bodar Γ. It s ot cssar to compt sch tgrals wh a porto of Γ dos ot cocd wth th bodar Γ of th total. Ths s bcas for a tror bodar th strsss from adjact lmts cacl ach othr. Th -D l tgral (, pla s trasformd to -D l tgral th atral co-ordat pla b sg th fact that alog a lmt sd o of th atral co-ordats s costat. Ths, s f (, d s f ( ξ J dζ (39 d d whr th bodar Jacoba Jb + dζ dζ. Oc th lmt matrcs ar obtad, th ar assmbld to form th st of lar smltaos qatos, th solto of whch lds th dsplacmt fld. Th assmbl s basd o th prcpl of matag th cott of th prmar arabl, ths cas dsplacmt ad th qlbrm of th scodar arabls, hr forcs ad tractos. Th two tps of bodar codtos ar sd:. Esstal or gomtrc bodar codtos whch ar mposd o th prmar arabl lk dsplacmts, ad. atral or forc bodar codtos whch ar mposd o th scodar arabl lk forcs ad tractos. Th forc bodar codtos ar mposd drg th alato of th lmt matrcs tslf whl th prscrbd dsplacmt bodar codtos ar mposd aftr th assmbl of th lmt matrcs. Th th global sstm of lar qatos ar sold b a mrcal tchq to gt th dsplacmts at global ods. I ft lmt calclatos, o oft has a d for accrat stmats of th drats of th prmar arabl. For ampl, pla strss or pla stra aalss, th prmar kows to b comptd ar th dsplacmt compots of th ods. Howr ma cass th stras ad strsss ar th prm mportac, whch ar comptd from th drats of th dsplacmts. As th ft b 8

9 lmt soltos s ol a trpolat soltos, t was act at th ods ad appromat lswhr. Sch accrac s rar bt, gral, o fds that th comptd als of th prmar arabls ar most accrat th ods pots. Ths, for th sak of smplct t s assmd that th lmt s odal als ar act. It was obsrd that drats stmats ar last accrat at th ods, grall ad most accrat at th Gass pots. Ths pots ar also calld as Barlow pots or optmal pots. Ths, th ctr of th lar lmt s tak as th optmal posto for samplg th frst drat. As a ampl of fdg ot th strss whol doma, cosdr th problm of a plat wth th hol loadd b forml dstrbtd tsl load (Fg. 6. Ths problm was sold o ASYS, commrcal FEM softwar. Bcas of th smmtr of th problm, ol a qartr plat ds to b aalzd. O th l of smmtrs, tractos ad ormal dsplacmt compots wll b zro. Fgr 7 shows th ft lmt msh ad bodar codtos. Hr, qadrlatral lmts ha b sd. Fgr 8 shows th cotors of logtdal strsss. Appld tsl strss was MPa. ot that th doma far awa from th hol strsss ar clos to ths al. Thr s a rgo of hgh strss coctrato ar th hghst rtcal pot of th crcl, th strsss thr bg arod 336 MPa. O ca obta th cotors of othr strss compots ad qalt strsss also. Fgr 6: A plat wth a hol loadd b forml dstrbtd tsl load o both sds Fgr 7: Ft lmt msh for solg qartr plat problm 9

10 Fgr 8: Cotors of logtdal strsss. Coclsos I ths artcl a brf trodcto to ft lmt mthod ad ft lmt formlato of pla strss ad pla stra problms has b dscrbd. Smlar procdr s mplod for th FEM formlato of th othr problms. Followg ar th stps of th ft lmt formlato: ( prprocssg that clds msh grato ( obtag th assmbld sstm of qatos, for whch th lmtal matrcs ad ctors d to b alatd ( applg th bodar codtos ( solg th lar sstm of qatos ad ( post-procssg. Th tm-dpdt problms ar sold b tratg th drat wth tm b th ft dffrc appromato. Th drats wth rspct to spatal arabls ar tackld th sal wa of FEM. For frthr dtals radrs ca rfr th lctr ots of th athor o Ft Elmt Mthods Egrg. Ths ots ar aalabl o