From Structural Analysis to Finite Element Method

Transcription

1 From Structural Analyss to Fnt Elmnt Mthod Dhman Basu II Gandhnagar Acknowldgmnt Followng txt books wr consultd whl prparng ths lctur nots: Znkwcz, O.C. and aylor, R.. (). h Fnt Elmnt Mthod, Vol. : h Bass, Ffth dton, Buttrworth-Hnmann. Yang,.Y. (986). Fnt Elmnt Structural Analyss, Prntc-Hall Inc. Jan, A.K. (9). Advancd Structural Analyss, Nm Chand & Bros Introducton Analyss of a cvl ngnrng structur, for xampl, a rgd ontd fram s oftn prformd usng cntr-ln lmnt modl, whrn cross-sctonal proprts ar lumpd onto th cntr-ln of th lmnt. Analyss procdur s usually dsplacmnt basd, and drct stffnss mthod s gnrally adoptd. Not that cntr-ln modl s rstrctd to a vry smpl confguraton and cannot b appld n most cass ncludng varyng cross-sctons, wth opnng/dscontnuty and wth two or thr dmnsonal ffcts. In such cass, fnt lmnt mthod (FEM) s wdly usd to captur th rasonabl bhavor. If FEM s appld to thos smpl cass whrn cntr-ln modl provds rasonabl soluton, rsults wll b dntcal. hrfor, som smlarty xsts n prncpl btwn th cntr-ln modl (or convntonal structural analyss) and FEM. Obctv of ths lctur s to xplor that smlarty and llustrat th transton path from convntonal structural analyss to FEM. In what follows nxt, frst qulbrum of a bam lmnt s consdrd and ffct of orthogonal transformaton s llustratd. Scond, convntonal structural analyss usng drct stffnss mthod s dscussd followd by a numrcal llustraton. hrd, convntonal structural analyss s rvstd so as to dscrb th passag to FEM. Fourth, concpt of FEM s brfly dscussd to addrss th smltud wth convntonal structural analyss followd by a numrcal xampl. Fnally, a problm statmnt s mad that dscrb th gnralty of FEM approach.. Elmnt Stffnss Matrx Consdr a two-nodd bam lmnt. Nglctng th axal dformaton for now, ont dsplacmnts (n gnralzd sns, ncludng rotatons as wll) and ont forcs (ncludng momnts also n a gnralzd sns) ar shown n thr postv sns n Fgur. Usng th prncpl of lmntary structural mchancs, qulbrum of th bam lmnt can b xprssd as

2 { q } K { a } = 6 6 Y 6 6 v 4 M EI θ = Y 6 6 v M θ () Axal dformaton, f consdrd (Fgur ), undr th assumpton of small dformaton rmans uncoupld wth rspct to othrs and th qulbrum of th lmnt taks th form as follows: { q } K { a } = EA EA EI 6EI EI 6EI X u v θ u v EI EI EI EI θ 6EI EI 6EI 4EI 3 3 Y 6EI 4EI 6EI EI M = X EA EA Y M () Matrx K s known as th stffnss matrx. Any lmnt, say th, of th stffnss matrx ndcats th forc dvlopd along th drcton of th dgr of frdom du to a unt dsplacmnt along th th drcton of dgr of frdom whl all othrs dgrs of frdom ar hld rstrand. Fgur : Straght bam lmnt wthout axal dgrs of frdom

3 3. Orthogonal ransformaton Fgur : Straght bam lmnt wth axal dgrs of frdom In prvous scton, dgrs of frdom ar chosn along and normal to th axs of th lmnt. Howvr, consttutng lmnts n a structur may hav thr axs orntd along any arbtrary drcton (Fgur 3). In such cass, two dffrnt coordnat systms ar usually consdrd namly, local and global. In local coordnat systm, on of th axs s chosn along th axs of th lmnt and hnc, vry lmnt has ts own local systm. On th othr hand, global coordnat systm s unqu to th ovrall structur and analyss of dformaton s carrd out n ths coordnat systm. Not that, local coordnat systm of any lmnt s rlatd to th global coordnat systm of th structur through a rotaton only. Structural analyss oftn rqurs transformaton of vctor from on coordnat systm to anothr. Fgur 3: Fram wth arbtrary orntaton of lmnt axs Consdr a local coordnat systm whch s rotatd through an angl θ wth rspct to th global coordnat systm as shown n Fgur 4. h coordnat of th pont P s (, ) global systms, rspctvly. By smpl gomtry, t can b shown that ' ' x y and (, ) x y n local and ' x cosθ snθ x y = sn cos y = θ θ ' { } [ ]{ } ' θ θ δ λ δ ' (3) Snc [ λ] [ λ] =, ths s also notd as orthogonal transformaton. 3

4 Fgur 4: Orthogonal transformaton 4. Elmnt Equlbrum n Global Coordnat Systm Consdr an lmnt wth axs orntd at an angl φ wth rspct to th global coordnat systm along th countr clockws drcton as shown n Fgur 5; ovr bar dnots th local coordnat systm. Dsplacmnts and forcs actng at th onts n both coordnat systms ar rlatd as follows: u X cosφ snφ u X v sn cos Y φ φ v Y θ M θ M = u X = cosφ snφ u X v sn cos v Y φ φ Y θ M θ M { a } G { a } G { q } { q } = (4) Employng Eq (4) nto Eq (), qulbrum rlaton n th global coordnat systm can b drvd as follows: { q } = K { a } { q G } = K { a G } { q G } = K { a G } { } { } and G G G G q = K a K = K (5) Not that th sz of th matrcs rmans sam n ths transformaton. 5. Drct Stffnss Matrx Mthod of Analyss Stp-: Elmnt Equlbrum n ocal Coordnat For any typcal lmnt (say th ), dvlop th qulbrum quaton n local coordnat: { q } K { a } = (6) q Consdr { } 6 as th ngatv of th fxd nd forcs du to span loadng. 4

5 Stp-: Elmnt Equlbrum n Global Coordnat Fgur 5: ocal to global coordnat transformaton ransform th qulbrum quaton from local to global coordnat. G G G { q } K { a } = { } { } { } { } G G G K K, q q, a = = = a (7) Stp-3: Elmnt Equlbrum n Expandd Global Coordnat Not that th dgrs of frdom consdrd n Eq (7) ar orntd along th global coordnat systm and hnc, a connctvty matrx must b formd that rlats ach of ths to th global dgrs of frdom. Wth th hlp of connctvty matrx, Eq (7) can b xpandd to th global sz of th problm, whch s 3N and N s th numbr of nods ncludng that ar rstrant. Exp { q G } Exp K G Exp G { a } = 3N 3N 3N 3N (8) Stp-4: Assmbl Elmnt Equlbrum n Expandd Global Coordnat Assmbl Eq (8) for lmnts. t th numbr of lmnts b M. hn 5

6 M M ( 3N 3N N ) Exp G Exp G Exp G { q } = K { a } (9) 3N 3 = = Aftr assmblng Eq (9) and takng nto account th xtrnally appld ont forcs { q } coordnat, t may b wrttn as * 3N n global { q G} { q * G G } K { a } + = 3N 3N 3N 3N 3N () Hr, G K G G, { q } 3N 3N 3 N { a } rprsnt th stffnss matrx, forc vctor assocatd wth span 3 N loadng and dformaton vctor, rspctvly, n global coordnat but wthout accountng for th ffct of rstrant. Stp-5: Effct of Rstrants Idntfy th rstrant dgrs of frdom and rmov th assocatd rows and columns from th stffnss matrx. Also rmov th assocatd lmnts from th forc and dformaton vctors. t th problm b rducd to a sz S and Eq () s mor formally wrttn as { q} [ K] { a} = () S S S S Stp-6: Soluton for Dsplacmnt Eq () can b solvd for th dsplacmnt vctor usng any standard procdur. { a} [ K] { q} = () S S S S Nxt, nsrtng zros nto th computd dsplacmnt vctor for th rstrant dgrs of frdom, th G dsplacmnt vctor { } 3 N a s formd. Stp-7: Soluton for Elmnt Rspons Usng th sam connctvty matrx as usd n Stp-3, nodal dsplacmnt vctor of any lmnt n global G coordnat,.., { } 6 a s xtractd. Assocatd vctor n local coordnat can b calculatd as G { a } { a } = (3) Mmbr nd forc for th lmnt s thn calculatd as { F } = K { a } { q } (4) Not that mmbr nd forc vctor ncluds th ffct of span loadng through th fxd nd forcs. In ordr to calculat th mmbr forcs at any pont wthn th span, suprmpos th span loadng on th nd forcs calculatd through Eq (4). 6

7 Stp-8: Calculaton of Racton Forcs Instad of rmovng th rows and column assocatd wth th rstrant dgrs of frdom, ntr dgrs of frdom and qulbrum quatons n global coordnat may b rarrangd as qr Krr K rs ar = = q s Ksr K ss a s (5) Hr th subscrpt r and s dnot th rstrant and unrstrant dgrs of frdom. Not, sam as [ K s Eq (). Ractons can now b calculatd as ] S S K ss n Eq (5) s q = K a (6) r rs s whr a s s computd n Eq (). If dsplacmnts along th rstrant dgrs of frdom ar spcfd non-zro, [ ] { } a = K q K a s ss s sr r q = K a + K a r rr r rs s (7) 5. Brf Numrcal Exampl for Drct Stffnss Mthod Consdr th fram as shown n Fgur 6, EA=8 kn/m and EI= knm. Fgur 6: Numrcal xampl 7

8 Elmnt stffnss matrx n global coordnat: K G K = 6 6 = Global ocal Sym K G K = 6 6 = Global ocal Sym Unrstrand global stffnss matrx K G 3N 3N Global Sym ZERO Rmovng th rows and column assocatd wth th rstrant dgrs of frdom, 8

9 Global Sym ZERO K s = Smlarly, fxd nd forcs can b assmbld and accountng for th ont load, global forc vctor n unrstrant and rstrant condtons ar: and Unrstrand dgrs of frdom may b solvd as a s = {-.356,.75, -.58,.78}. G G G Mmbr nd forcs n global coordnats ar obtand as { F } = K { a } { q } For mmbr : {-6.49,.7,, -5.5, 3.93, 7.74} and for mmbr : {5.5, -3.93, -7.7, -5.5, 8.93, -58.}. hs forcs ar shown n Fgur 6d. : 9

10 6. Rvstng Stffnss Matrx for Structural Analyss and FEM In structural analyss, as dfnd arlr, on column of a stffnss matrx s drctly computd by applyng unt dsplacmnt along th drcton of a partcular dgr of frdom whl rstranng th othrs and thn calculatng th forcs dvlopd along th drcton of all dgrs of frdom. In ths scton, w wll rvst th sam concpt from a dffrnt prspctv. 6. Shap Functons Consdrng only flxural dformaton, qulbrum quaton of a bam lmnt of constant flxural rgdty n th unloadd rgon s gvn by 4 v 4 = x (8) h soluton of Eq (8) may b xprssd through a cubc polynomal as ( ) v x = α + α x+ α x + α x (9) whr α s ar arbtrary constants and can b valuatd through th boundary condtons. hs boundary condtons ar: v v= v and = θ at x= x v v= v and = θ at x= x () Utlzng Eq () nto Eq (9) and arrangng th rsultng quatons n matrx form, t may b shown that v α θ α = 3 v α 3 θ 3 α 4 () Invrtng Eq (), 3 α v 3 α θ = 3 = α3 3 3 v α 4 θ { α} [ H]{ a} () Substtutngα s from Eq () nto Eq (9) and thraftr rarrangng th rsultng quaton, t may b shown that

11 ( ) = ( ) + θ ( ) + ( ) + θ ( ) v x v f x f x v f x f x f 3 ( x) ( x) x x = 3 + x x f ( x) = x + f 3 x x = 3 x x f4 ( x) = x + (3) h functons f s ar ssntally th shap functons and drvd from th qulbrum of th unloadd bam lmnt of constant flxural rgdty. Varaton of ths shap functons ovr th lngth s shown n Fgur 7. Fgur 7: Shap functons for a bam wth constant flxural rgdty

12 Now consdr, for xampl, v =., θ =, v =, θ =, whch corrsponds to v( x) f( x) that, ths s also th cas of gnratng th frst column of stffnss matrx. hrfor, f ( ) =. Rcall x rprsnts th dsplacmnt profl whn unt dsplacmnt s appld along th frst dgr from frdom whl othrs ar hld rstrant. Smlar ntrprtaton holds for othr shap functons also. Hnc, dsplacmnt at any pont ovr th span can b radly calculatd f th nodal dsplacmnts and all th assocatd shap functons ar known. 6. Castglano s horm Consdr an lastc systm subctd to a st of consrvatv forcs P, =, n. t Δ dnots dsplacmnt along th drcton of P and at ts pont of applcaton. t us consdr two dffrnt cass. In cas-, only th forc P s appld and { } Δ dnots th rsultng dsplacmnt at all th n ponts but ar appld and { Δˆ } along th drcton of forc as dscrbd n Δ. In cas-, all th forcs xcpt th dnots th rsultng dsplacmnt at all th n ponts but along th drcton of forc as dscrbd n Now, f cas- loadng s appld frst followd by cas-, th work don on th body or stran nrgy s gvn by n ˆ U ˆ = PΔ + PΔ + PΔ (4) = Δ. If an nfntsmal vrtual dformaton profl δ a (compatbl wth th rstrant) s appld n btwn cas- and cas-, thn th stran nrgy wll b gvn by n ˆ U ˆ = PΔ + Pδ a + PΔ + PΔ (5) = hrfor, chang n stran nrgy s ΔU ΔU = Pδa P = (6) δ a Consdrng t δ a, U P = a (7) Eq (7) s known as th Castglano s frst thorm. 6.3 Applcaton of Castglano s horm Assumng only flxural dformaton, stran nrgy of th bam lmnt shown n Fgur can b xprssd as

13 U EI v = x Notng dx (8) ( ) ( ) θ ( ) ( ) θ ( ) v x = v f x + f x + v f x + f x (9) '' '' '' '' '' 3 4 and mployng Castglano s thorm w may wrt, for xampl, Y EI dx EI v f x f x v f x f x f x dx U v v '' '' '' '' '' = = = ( ) + θ ( ) + 3( ) + θ 4( ) ( ) v x v x '' '' '' '' '' '' '' '' 3 4 = v EI f ( x) f ( x) dx+ θ EI f ( x) f ( x) dx+ v EI f ( x) f ( x) dx+ θ EI f ( x) f ( x) dx (3) = K v + K θ + K v + K θ 3 4 Smlarly, othr thr nodal forc can also b drvd ladng to th th lmnt of stffnss matrx as '' '' ( ) ( ) (3) K = EI f x f x dx Usng th shap functons dvlopd n Eq (3) nto Eq (3) and carryng out th ncssary ntgraton ovr th lngth, on may drv th stffnss matrx whch s sam as that shown n Eq (). For xampl, '' '' 3 3 x x x x 6 x EI K = EI dx= EI + dx= 3 3 (3) 6.4 Applcaton of Raylgh Rtz Mthod Exprsson of stran nrgy nvolvs drvatv/ntgraton of dsplacmnt profl and hnc that of shap functons, whch bcoms complcatd for mor complx fnt lmnts. o ovrcom such problm, Raylgh Rtz mthod s llustratd blow wth rspct to th sam bam lmnt. Assumng th sam cubc polynomal dsplacmnt profl as consdrd bfor and substtutng ts scond drvatv nto th xprsson of stran nrgy, w may wrt 3 '' ( ) = α + α + α + α ( ) = α + 6α v x x x x v x x EI U = + x dx= EI ( α3 6α4 ) ( α3 6α3α4 6α4 ) (33) Eq (33) s quadratc of th coffcnt α s and can b xprssd as 3

14 α U = = 4EI 6EI α { α α α3 α4} { α} k { α} α 3 3 6EI EI α 4 (34) Hr, lmnts of k ar obtand through k U = α α (35) Substtutng { α } from Eq () nto Eq (34), stran nrgy can b xprssd as { } ([ ] [ ]){ } U = a H k H a (36) Anothr way to formulat th stran nrgy s to us th nodal forc and dsplacmnt vctors as follows: Y M U = v v = a K a Y M { θ θ } { } [ ]{ } (37) Hr [ K ] s th lmnt stffnss matrx, whch can b xprssd by quatng th stran nrgy from Eq (36) and Eq (37) as = [ K] [ H] k [ H] Substtutng [ ] obtand as H from Eq () and k (38) from Eq (34) nto Eq (38), lmnt stffnss matrx can b [ K ] = = EI 6EI EI EI EI (39) 4

15 h stffnss matrx s sam as that obtand usng lmntary structural mchancs or by usng Castglano s thorm. Whl drvng th stffnss matrx usng thr Castglano s thorm or Raylgh Rtz mthod, axal dformaton s not consdrd. Undr th assumpton of small dformaton problm, axal dgrs of frdom ar uncoupld wth flxur and shar, and can b ncludd n th stffnss matrx wth an addtonal par of shap functons. 7. Fnt Elmnt Mthod: A Prlmnary Rvst 7. Dsplacmnt Functons Consdr for smplcty a plan strss (or two dmnsonal n a loos sns) problm as shown n Fgur 8. h structur s dscrtzd nto svral ara lmnts and a typcal lmnt s dnotd as. Nodal ponts of th lmnt ar ndcatd by,, m tc. A typcal nod has two dsplacmnts, on along ach orthogonal drcton, and for th th nod, t s dnotd as a { u u } =. Hr ovr bar dnots a x y vctor quantty to mak dstncton from a scalar. Smlarly, th dsplacmnt at any pont wthn th u = u x, y u x, y. { x y } lmnt may b dnotd as ( ) ( ) Dsplacmnt u may b approxmatd as a ˆ u u = Nkak N N... a = = Na (4) k.. h functons othr words N, N should b chosn as to gv approprat dsplacmnts at th rspctv nods. In (, ) N x y = = δ = (4) Undrstandng that nodal dsplacmnt at a partcular nod s a vctor quantty nvolvng dsplacmnt along any st orthogonal drctons, thr spatal drvatv tc, w may now drop th ovr bar from th rprsntaton of vctor quantts and Eq (4) s rstatd as u uˆ = Na (4) Furthr, unlss othrws spcfcally statd, w do not rstrct th dscusson wthn th doman of plan strss problm. 5

16 Fgur 8: A plan strss problm 7. Stran-Dsplacmnt Rlaton In most cass, usng th prncpl of mchancs, t s possbl to xprss th stran tnsor at any pont wthn a fnt lmnt n trms of th spatal drvatv of dsplacmnts at that pont. In a plan strss problm, for xampl, stran vctor (consttutd from th ndpndnt lmnts of th stran tnsor) can b xprssd as { ε} u x x x ε xx u u y x = ε yy = = y y uy ε xy u u x y + y x y x (43) Hr, u x and uy dnot th dsplacmnt at th pont along two orthogonal drctons. In gnral, Eq (43) may b wrttn as { ε} { ˆ ε} [ S]{ u} = (44) and upon substtutng Eq (4) 6

17 { ε} { ˆ ε} = [ S]{ u} = [ S][ N]{ a } = [ B]{ a } [ B] = [ S][ N] (45) 7.3 Consttutv Rlaton Assumng lnar lastc bhavor, gnral strss stran rlaton can b xprssd as { σ } [ D]{ ε ε } { σ } = + (46) Hr, { σ } s th stran vctor (consttutd from th ndpndnt lmnts of th strss tnsor) and, { σ } and { ε } stand for th rsdual strss and ntal stran, rspctvly. For a plan strss problm, ν and xy ( ν) σxx E { σ} = σ yy [ D] = ν ν τ (47) whr, E and ν ar th Young s Modulus and Posson s Rato, rspctvly. 7.4 Extrnal oadng hr dffrnt typs of loadngs ar consdrd. Frst, dstrbutd body forcs of ntnsty b ; ths forc s consdrd actng on pr unt volum of th matral. Scond, dstrbutd tracton of ntnsty t ; ths forc s consdrd actng on pr unt surfac ara. hrd, xtrnal concntratd forcs actng on th nodal ponts. 7.5 Drvaton of Elmnt Equlbrum h smplst procdur of stablshng lmnt qulbrum s to apply an admssbl vrtual dsplacmnt δ a b th vrtual dsplacmnt vctor at th and quat th xtrnal and ntrnal work don. t { } nods. At any pont wthn th lmnt, th assocatd dsplacmnt and stran vctors may b xprssd as { δu} [ N]{ δa } and { δε} [ B]{ δa } = = (48) Work don pr unt volum by th ntrnal strss and body forc may b xprssd as { } δε { σ } { δu} { b}. Furthr, work don by th surfac loadng pr unt surfac ara s { δu} { t}. Notng that total work don ovr th ntr lmnt s zro, w wrt { δε } { σ } { δu} { b} dv { δ u} { t } da = (49) V Substtutng Eq (48), Eq (49) can b smplfd to A 7

18 { } [ ] { } { } [ ] {} { } [ ] = δa B σ δa N b dv δa N { t } da V = a B N b dv N t da { δ } [ ] { σ} [ ] {} [ ] { } V A A (5) hrfor, lmnt qulbrum can b xprssd as [ B] { σ } [ N] { b} dv [ N] { t } da = (5) V A Furthr substtutng th consttutv rlaton Eq (46) and subsquntly Eq (45), qulbrum quaton can b rwrttn as = [ B] [ D]{ ε} dv [ B] [ D]{ ε } dv [ B] { σ } dv [ N] {} b dv [ N] { t } + da V V V V A (5) = [ B] [ D][ B] dv { a } [ B] [ D]{ ε} dv [ B] { σ} dv [ N ] {} b dv [ N ] { t } + da V V V V A h lmnt qulbrum quaton can b xprssd n mor formal way as follows: { q } K { a } = = V [ ] [ ][ ] K B D B dv { } = [ ] [ ]{ ε} [ ] { σ} + [ ] {} + [ ] { } q B D dv B dv N b dv N t da V V V A (53) 7.6 Ovrall Analyss Eq (53) s of th sam form as dscussd n convntonal structural analyss (Eq () or Eq ()). hrfor, all th stps dscrbd n drct stffnss mthod of convntonal structural analyss ncludng local to global transformaton, xpanson n global coordnat, assmbly (takng nto account th xtrnally appld concntratd nodal forc vctor { q * }), and applyng physcal rstrants to th global dgrs of frdom wll b followd aftr Eq (53) to gt th complt soluton for unrstrant nodal dsplacmnts. Onc th nodal dsplacmnt vctor { a } assocatd wth an lmnt s xtractd, strss vctor at any pont wthn th lmnt may b calculatd as { σ } [ D][ B]{ a } [ D]{ ε } { σ } = + (54) hrfor, FEM follows ssntally th sam prncpl as that of th convntonal structural analyss but th procdur of formulatng govrnng qulbrum quatons, for xampl stffnss matrx tc., s dffrnt. Us of approxmat shap functons whl formulatng th qulbrum quaton allows FEM to 8

19 b applcabl to all possbl problms, whras convntonal structural analyss uss xact shap functon and ts applcablty s rstrctd to such smpl cass for whch xact shap functons xst. 7.7 Fnt Elmnt Mthod wthout Assmblng Elmnt Equlbrum Unlk drct stffnss mthod, qulbrum of th ntr structur wthout assmblng th lmnt qulbrum quatons s oftn consdrd n convntonal mthod. Such s a cas, usually appls to a rlatvly smpl confguraton, whrn stffnss matrx of th ntr structur basd on unrstrant dgrs of frdom s drvd n global coordnat. FEM can also b drvd from that prspctv wthout assmblng th lmnt qulbrum quatons. In ordr to llustrat that, a st of admssbl dscrtzaton as n th prvous cas s consdrd. All th concntratd loads ar assumd to b appld through th nodal ponts only. t { a} dnots a vctor lstng th dsplacmnt of all th nodal ponts and N dnots th assocatd shap functons such that dsplacmnt at any pont s gvn by = { u} N { a} (55) hs shap functons ar dffrnt than what wr assumd prvously. Consdr, for xampl, as a nodal pont common to a st of lmnts { }. Shap functon assocatd wth th nodal dsplacmnt at s { } N. If th pont at whch dsplacmnt to b approxmatd ls outsd all th lmnts ncludd n, thn N =. Othrws, N functon of th = N whr th pont blongs to th lmnt and th nod of lmnt, as dfnd n prvous approach. As bfor, N s th shap { } = [ S]{ u} = [ S] N { a} = B { a} ε (56) W now drop th ovr bar wth an undrstandng that th quantts lk shap functons tc. ar dfnd ovr th whol rgon. Applyng any admssbl vrtual dsplacmnt { δ a} and quatng th xtrnal and ntrnal work don, w may wrt * { δa} { q } + { δu} { b} dv + { δu} { t } da = { δε} { σ} dv (57) V A V Now takng th varaton of Eq (55) and Eq (56) for th vrtual quantts, and substtutng nto Eq (57), w wrt * { δa} { q } [ N ] {} b dv [ N ] { t } da + + = { δa} [ B] { σ} dv (58) V A V Cancllng out th vrtual dformaton and, substtutng th consttutv rlaton and Eq (56), 9

20 q N b dv N t da B D B dv a B D dv B dv * { } + [ ] {} [ ] { } [ ] [ ][ ] + = { } [ ] [ ]{ ε} + [ ] { σ} V A V V V or, B D B dv a = q + N b dv + N t da+ B D dv B dv V V A V V or, K a q q * [ ] [ ][ ] { } { } [ ] {} [ ] { } [ ] [ ]{ ε} [ ] { σ} (59) * [ ]{ } = { } + { } Clarly, Eq (59) rprsnts th smlar form that obtand aftr assmblng Eq (53) for all th lmnts, whrn ) { q} s sam as th assmbld quvalnt nodal forc vctor, ) assmbly of ntgraton ovr lmnts s sam as th ntgraton ovr th whol rgon and ) [ K ] s sam as th assmbld lmnt stffnss matrcs. 7.8 Fnt Elmnt Formulaton from th Mnmzaton of otal Potntal Enrgy Vrtual work prncpl s usd n formulatng th FEM n prvous two cass. In ths scton, admssbl vrtual dsplacmnt s consdrd as th varaton of th ral dsplacmnt. hrfor, { δ a} and hnc, { δu} and { δε } ar th varaton of ral quantts. Dnotng W and U as th potntal nrgy of th xtrnal load and stran nrgy, rspctvly, Eq (57) can b rwrttn as δ + + = δ ε σ V A V or, W = U U + W = = * { a} { q } {}{} u b dv {} u { t } da { } { } dv (6) δ δ δ( ) δ( Π ) Hr Π s th total potntal nrgy, whch s statonary pr Eq (6). hrfor, fnt lmnt formulaton can also b drvd by sttng Π Π Π =.. = a a a (6) hs s th wll known Raylgh Rtz mthod and llustratd arlr n contxt wth convntonal structural analyss of a bam lmnt. 7.9 Exampl of FEM Formulaton on a Bam Elmnt Assumng only flxural dformaton govrns, strss-stran rlaton can b consdrd n as gnralzd dv sns as momnt-curvatur rlaton. Hnc, gnralzd stran s ε κ = and gnralzd strss s dx dv σ M = EI. Clarly, σ = Dε wth D = EI. h bam s dscrtzd and consdr an lmnt dx wth nods and. Dsplacmnt at any pont wthn ths lmnt can b approxmatd as { u} [ N]{ a } =. Snc th stran nvolvs scond drvatv of dsplacmnt, t s ncssary that both v

21 and dv dx b contnuous btwn th lmnt, whch can b nforcd by ncorporatng thm nto th nodal dv dx dsplacmnt vctor. hrfor, { a } = v = { v θ } Snc, th lmnt has two nods ach wth two varabls, assumng a cubc polynomal dsplacmnt profl, as shown n Eq (9), th shap functons can b drvd as shown n Eq (3). hrfor, ( ), ( ), ( ), ( ) N = f x f x N = f x f x 3 4 Consquntly, B '' ( ) '' ( ) '' ( ) '' = f x, f x, B = f3 x, f4( x) '' '' '' '' Notng that [ B] = B B = f ( x) f ( x) f ( x) f ( x) by = = 3 4, th lmnt stffnss matrx s gvn [ ] [ ][ ] [ ] ( )[ ] K B D B dv B EI B dx V Any lmnt of ths s gvn by '' '' '' '' [ ] ( )[ ] ( )( ) ( ) ( ) ( ) K = B EI B dx= f x EI f x dx= EI f x f xdx hs s sam as what has bn drvd wth convntonal structural analyss. 8. Gnral Problm Statmnt Consdr a doman Ω nclosd by a boundary Γ. A st of coupld dffrntal quatons of th form ( ) { ( ), ( ),...} Au = A u A u = {} wthn th doman Ω dfns th problm statmnt togthr wth crtan boundary condtons of th form ( ) { ( ), ( ),...} Bu= B u B u = {} dfnd on th boundary Γ. h obctv s to fnd an unknown soluton u that satsfs th dffrntal quatons and th boundary condtons. FEM attmpts to fnd th soluton of ths problm usng a varty of tchnqus..