The Finite Element Method

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1 Th Finit Elmnt Mthod Eulr-Brnoulli and Timoshnko Bams Rad: Chaptr 5 CONTENTS Eulr-Brnoulli bam thory Govrning Equations Finit lmnt modl Numrical ampls Timoshnko bam thory Govrning Equations Finit lmnt modl Shar locking Numrical ampl

2 z, w KINEMTICS OF THE LINERIZED EULER-BERNOULLI BEM THEORY q ( ) Strains, displacmnts, and rotations ar small f( ) u z 9 Undformd Bam Dformd Bam dw d dw d Eulr-Brnoulli Bam Thory (EBT) is basd on th assumptions of ()straightnss, ()intnsibility, and ()normality

3 z z y z Kinmatics of Dformation in th Eulr-Brnoulli Bam Thory (EBT) w u σ y σyy σ zz σ z σ y z σ zy dw d σ z σ y σ z Displacmnt fild (constructd using th hypothsis) u(, z) = u z dw, u =, u = w( ) d Linar strains dw d Notation for strss componnts ε γ σ σ = ε = = z d d z z u du d w, u u dw dw = + = + = d d Constitutiv rlations du = Eε = E Ez d = Gγ = z d w, d.

4 Eulr-Brnoulli Bam Thory z, w q() F M L Bam cross sction q() z y M + M c f V V cw f q ( ) σ z z σ σ + Δσ σ z Δ f( ) cw f + Δσ q ( ) V V + ΔV N N + ΔN M M + ΔM f( ) cw Δ f Dfinition of strss rsultants N = σ d, M = σ z d, V = σ d. z 4

5 Eulr-Brnoulli Bam Thory (Continud) Equilibrium quations dn dm dv + f =, V =, + q cfw = d d d Strss rsultants in trms of dflction du d w du N = σ d = E Ez d = E d d d d =, z d =, z d = I du dw dw M = σ z d = E Ez z d = EI d d d dm d d w V = = EI d d d 5

6 Eulr-Brnoulli Bam Thory (Continud) Govrning quations in trms of th displacmnts d du E f, L d d = < < Bars d d w u EI c, + fw q = < < L d d z, w q() F Bams w M ial dformation of a bar L c f Bnding of a bam ial displacmnt is uncoupld from transvrs displacmnt 6

7 Wak Form of th EB Bam Thory d d w EI c, + fw q = < < L d d Wak form vi é b d æ dw ö ù h = v EI c w q ò i f h d a d ç + - d ê ë çè ø úû é b dv dw dw i d æ ö ù é h d ù æ h ö = EI c v w v q d v EI ò f i h i i a d d ç d d ç d êë è ø úû êë è øúû Govrning quation { } st of wight functions Implis that th primary variabl is w (displacmnt) Scondary variabl (shar forc) é b dv d æ d wö ù = EI c vw vq ò d v( ) Q v( ) Q a d d ç d çè ø êë úû f a b 7 b a

8 é b dv dw ù i h = EI c v w v q ò ( ) ( ) + - d v Q v Q a d d - - êë úû b éæ dv ö d w ù i h + EI - ç d çè ø d êë úû Primary Variabl, θ Slop/rotation Wak Form (Continud) f i h i i a i b a Scondary variabl (Bnding Momnt) é b dv dw ù i h = EI c v w v q ò d v ( ) Q v ( ) Q + - a d d - - êë úû æ dv ö æ dv ö i - - Q - ç- Q ç d è ø è ø a f i h i i a i b i 4 ç d é d æ dw ù é h d dw ù ö æ h ö Q = EI V ( ), Q EI V ( ) h a h b d d =- = - d d = ê ç ç ë è øúû êë è øúû a b æ dw ö æ dw ö Q EI M ( ), Q EI M ( ) h h = =- = - h a 4 = h b ç d è ø çè d ø a b b 8

9 Bam Elmnt Dgrs of Frdom Gnralizd displacmnts Δ=w ( a ) Δ=w ( b ) Δ=θ ( a ) h Δ=θ 4 ( b ) Gnralizd forcs Q = V( a ) Q = V( b ) Q = M( a ) h Q4 = M( b ) 9

10 FINITE ELEMENT PPROXIMTION: Som Rmarks Continuity rquirmnt basd on th wak form, which rquirs that th scond drivativ of w ists and squar-intgrabl. Continuity basd on th primary variabls, which rquirs carrying w and its first drivativ as th nodal variabls, rquirs cubic approimation w. Post-computation of scondary variabls (bnding momnt and shar forc) rquirs th third drivativ of w to ist. Bams

11 FINITE ELEMENT PPROXIMTION Primary variabls (srv as th nodal variabls that must b continuous across lmnts) dw w, θ = d w ( )» c + c+ c + c Hrmit cubic polynomials w, q w φ a a = + w( )» c + c + c + c º Δ a a a a» º Δ b b b b»- - - º Δ a a a»- - - º Δ b b b 4 w ( ) c c c c q( ) c c c q( ) c c c φ = ( a ) h a h h φ a a = h h φ a 4 = ( a ) a h h 4» = ådj j= w ( ) c c c c f ( ) j

12 HERMITE CUBIC INTERPOLTION FUNCTIONS f i ( ) f ( ) slop = f ( ) h slop = slop = h f ( slop = f ( 4 slop = h slop = h

13 FINITE ELEMENT MODEL K ij = b a 4 w ( )» å Δ f ( ) j j j= 4 Kij j Fi = or [K ]{ } = {F } j= K K K K4 K K K K4 K K K K4 K4 K4 K4 K44 EI d φ d φ i j d d + c f φ i φ j 4 = q q q q 4 + Q Q Q Q 4 b d Fi = φ i qd+ Q i a w Δ w Δ Q, q Q, q θ Δ θ Δ 4 θ Δ Q, q Q 4, q 4 h h

14 For lmnt-wis constant valus of E I and q :(and c f = ): [K ]= E I h Postprocssing Finit Elmnt Modl (Continud) 6 h 6 h h h h h 6 h 6 h h h h h {F } = q h 6 h 6 + h Q Q Q Q 4 M() = EI d w 4 d = EI j= V () = dm d = d EI d w d d σ (, z) = M()z I j d φ j d = EI = Ez d w 4 d = Ez j= j 4 j= j d φ j d d φ j () d 4

15 EI h SSEMBLY OF TWO BEM ELEMENTS connctd nd-to-nd 6 h 6 h h h h h 6 h 6+6 h h 6 h h h h h h +h h h 6 h 6 h h h h h Q L Q Q + Q = q L Q 4 + Q Q L Q 4 U U U U 4 U 5 U 6 h= L/ Q + Q =, Q 4 + Q = Q, q Q, q Q 4, q 4 Q, q Q, q Q, q Q, q Q 4, q 4 5

16 E, EI SIMPLE EXMPLE - L P Eact solution (according to th Eulr-Brnoulli bam thory) PL wl ( ) = EI Givn problm E, EI L U U U, U4, Boundary conditions: U = U =, Q = P, Q = 4 P On lmnt discrtization [ K ]{ D } = { q } + { Q } {F } = q h [K ]= E I h 6 h 6 h + h Q Q Q Q 4 = L 6 h 6 h h h h h 6 h 6 h h h h h 6

17 U SIMPLE EXMPLE (continud) P 6EI L éei 6EI ù ê ú ì U ü ìpü ï ï ï ï 4EI Solution using Cramr s rul L L ï ï ï ï L ( 4PEI / L) PL í ý= í ý U = = = 4 6EI 4EI EI 6EI é( EI ) / L ù EI ê ú ïu 4ï ê ë ú ï ï û ë L L ûïî ïþ î þ L L 6EI 4EI EI L L P L 6EI L -( 6PEI / L ) PL = = =- EI 6EI é( EI ) / L ù ê EI ë úû L L 6EI 4EI L L 4 4 E, EI U wl ( ) = U PL EI PL EI =- dw d = L 4 L 7

18 EXMPLE : dtrminat fram structur Givn structur a b F B C P a b F B C P Finit lmnt discrtization P Pb B F F P Pb P C B F 8

19 EXMPLE (continud) Bar lmnt, B E é -ùì u ü ì Q ü ú ï ï= ï ï a ê í ý í ý ë- úï ûïîu Bïþ ïîq Bïþ Pa u =, QB =-P ub =- E Bam lmnt, B é 6 -a -6 -aùìïw üï ì Q ü ú EI úï ï - ú ï ï ï ï a í ý= í ý 6 a 6 a úï Q - ú ê úï ï ë- ûïïî ïþ ï ïþ w = = Q =- F Q = Pb a B Pb ï ï a a a a ïq ï Q B wb B a a a a qb îq B B, q,, F P Displacmnts at C rlativ to point B P Bar lmnt, BC Pb Bam lmnt, BC u F B C C P F Fb = E é 6 -b -6 -bùìïw üï ì Q ü ú EI úï ï - ú ï ï ï ï b í ý= í ý 6 b 6 b úï Q - ú ê úï ï ë- ûïïî ïþ ï ïþ w = = Q =- P Q = B B ï Bï B b b b b ïqb ï Q C wc C b b b b qc îq C C, qb,, 9

20 EXMPLE : Handling of a vrtical spring z, w q k kw(l ) kw(l ) L k U Q Q U = U = kw( L) Q Q 4 = U 4 ¹ = ku Q =- kw( L) =-ku ltrnativly, é ì s ùï ü ì s - ü ï u ï ï Q í ý= í ï s s s k ú ý, u =, u = U Q = ku ê - úï ï s s u Q ë ûïî ïþ ïî ïþ

21 SOLUTION TO THE SPRING-SUPPORTED BEM é 6 -L -6 -LùìïU = w üï ïì 6 ïü ìïq üï ú ï EI L L L L úï U = q ql L Q - ú ï ï ï- ï ï ï L í ý= í ý+ í ý 6 L 6 L úïu w 6 Q - ú = ê ï -L L L L úï ïu4 = q ï ï L ï ïq 4ï ë ûïî ïþ î þ ïî ïþ Boundary conditions w =, q =, Q =- ku, Q = 4 Condnsd quations for th unknown gnralizd nodal displacmnts éei 6EI ù + k ì U ü ì 6 ü L L ú ql úí ï ï ý= ï í ï ý 6EI 4EI ú úï U 4 -L êë L L úû ïî ïþ ïî ïþ -ku Bams

22 HNDLING OF POINT SOURCES INSIDE N ELEMENT h q = ò q () s f () s ds i i h i ò i i q = q( s) f( s) ds = F f( s ), i =,,,4 q h f = ò q () s f () s ds i =- M d, i i i ds s= s æsö æsö æ sö f() s = - ç +, f() s =-s - èhø èç h ø çè h ø æsö æsö éæsö sù f() s = ç -, f4() s =-s - èhø èç h ø êçèhø hú ë û =,,, 4 F F for F placd q = F q = at s =.5h Fh Fh q =- q = s F s h qs () = Fd( s-s) s M s h qs () = Md '( s-s)

23 EXMPLE 4: simply-supportd bam (a) Find th cntr dflction using on Eulr-Brnoulli lmnt in full bam é 6 -L -6 -Lù ì U = w ü ì q ü ì Q ü ú ï 8 EI -L L L L úïu = q q Q ú ï ï = ï ï + ï ï L í ý í ý í ý -6 L 6 L úïu = w q Q ú ê ï -L L L L úï ïu4 = q ï ïq 4ï ïq 4ï EI él L ùìu ü FLì ü ú ï ï = ï ï L ê í ý í ý ë L L úï ûïîu 4ïþ 8 ïî- ïþ U FL FL 6EI 6EI =, U4 =- FL ë ûî þ î þ î þ FL Condnsd quations - 8 F L F F q = q = q FL = 8 q 4 FL = w ( ) = Uf( ) + Uf ( ) + Uf ( ) + Uf ( ) 4 4 = U f ( ) + U f ( ) 4 4 ìé æ ö ù éæ ö æ öù ï FL ü = ï í ï ý 6EI ç L ç L ç L è ø è ø è ø ïîê ë úû êë úû ïþ FL æ L Lö FL w( 5. L) = ç - - =- 6EI çè 8 8 ø 64EI 8

24 EXMPLE 4: simply-supportd bam (b) Find th cntr dflction using on Eulr-Brnoulli lmnt in half bam 6 é L Lù ìïu = w üï ìïq üï ú EI ï -5. L 5. L 5. L 5. L úïu = q Q ú ï ï = ï ï L í ý í ý 6. 5L 6. 5L úïu w Q - ú = ê L. 5L. 5L. 5L úï ï U4 = q - ï ï ïq 4ï ë ûî þ î þ Condnsd quations 6EI L U U é 5. L 5. LùìU ü ì ü ú ï ï = 5. F ï ï ê í ý í ý ë5. L 6 û úï ïîu ïþ ïî- ïþ FL 4. 5L FL, EI L 6EI FL 4. 5L FL EI L 48EI = = = = -.5F U Q = =, F F U Q 4 L =, =-.5F

25 EXERCISE PROBLEM Problm: Dvlop wak form and th finit lmnt modl of th following quation, whr w and P ar unknowns: d d w d w EI P, L + = < < d d d Bams 5

26 F EXERCISE PROBLEM d Problm: Us th minimum numbr of EBT lmnts to find th comprssion in th spring, ractions at th fid support, and spring forc. q Rigid loading fram EI h EI h Linar lastic spring, k Bams 6

27 TIMOSHENKO BEM THEORY and its Finit Elmnt Modl Govrning Equations Finit lmnt modl Shar locking Numrical ampl Bams 7

28 Kinmatics of Timoshnko Bam Thory z, w q ( ) f( ) z, u Undformd Bam Dformd Bams dw d 9 dw d Eulr-Brnoulli Bam Thory (EBT) Straightnss, intnsibility, and normality u φ dw d Timoshnko Bam Thory (TBT) Straightnss and intnsibility 8

29 Timoshnko Bam Thory f g Kinmatic Rlations u (, z) = u( ) + zf ( ), u =, u (, z) = w( ) z u du df z d d = = + u u dw d = + = f +, z z w u dw d zf Constitutiv Equations s ædu df ö = E = E ç + z çèd d ø æ dwö sz = Ggz = Gç f + çè d ø 9

30 Timoshnko Bam Thory (Continud) dn dv dm Equilibrium Equations + f =, q+ cfw =, + V =. d d d Bam Constitutiv Equations ædu df ö du N = s d E ò = ò + z d= E ç èd d ø d ædu dfö df M = sz d E ò = ò ç + z zd= EI çèd d ø d æ dwö æ dwö V = Ks sz d GK s f d GK s f ò = + = + ç è d øò èç d ø Govrning Equations in trms of th displacmnts d é æ dwö ù - GK s f + + cfw = q d ê ç d ú ë çè øû () d æ df ö æ dwö - EI + GK s f + = d ç è d ø èç d ø ()

31 Wak Form of Eq. () WEK FORMS OF TBT ì b d é æ dwöù ü = v ï GK s f cfw qï ò í ýd a d ê ç d ú ï î ë è øû ïþ ì b dv é æ dwöù ü é æ dwöù = ï GK s f cfvw vqïd v GK s f ò í ý - + a d ê ç d ú ê ç d ú ï î ë è øû ïþ ë è øû ì b dv é æ dwö ù ü = GK s ç f + ò í ï ú+ cvw a d çè d f -vq ý ïd ê øú ï î ë û ïþ é æ dwöù é æ dwöù -v( a) - GK s f+ -v( b) GK s f+ ê ç d ú ê ç d ú ë è øû è ø ë û a b ì dv æ dwö ü = ïgk f c v w v qï ò í d -v ( ) Q -v ( ) Q a d çè d ïî ø ïþ s f ý a b b b a

32 Wak Forms of TBT (continud) Wak Form of Eq. () b é d æ df ö æ dwöù = v EI GK s f ò d ê a d ç d ç d ú ë è ø è øû b b édv æ dfö æ dwöù é dfù = EI GKsv f ò + + d - v EI ê a d ç d ç d ú ë è ø è øû êë d úû a b édv æ df ö æ dwöù EI GKsv f æ dfö æ dfö = + + òa d çè d ø çè d ø d -v ( a) -EI -v( b) EI ê ú ç ë d ç d û è ø è ø a b édv æ df ö æ dwöù = EI GKsv f ò + + d -v( a) Q -v( b) Q4 ê a d çè d ø çè d øú ë û Total Potntial Enrgy b EI dφ GKs dw c f Π ( w, φ ) = + φ + + w d a d d b wq d + w( ) Q + w( ) Q + φ( ) Q + φ( ) Q a a b a b 4 b

33 FINITE ELEMENT MODELS OF TIMOSHENKO BEMS Finit Elmnt pproimation w w s s h w h m= n= h w w s s s m= n= h m å w» w y ( ), f» S j ( ) j j j j j= j= { } { } é ù ì ü ék ù ék ù { w} F ê ú ê ú ì ü ë û ë û ï ï í ï ý ï= í ý é { S} K ù ék ù ï ï F êêë úû êë úûúî þ ï ï ë û î þ n å æ dy dy ö ç d K GK c y d, K GK d K a ç çè d d ø a d b b i j i ij = ò ç s + f yy i j ij = s jj = ji ò é b b dj dj ù i j dy j Kij = EI GKsjj ò + i j d, Kij GKs ji d a d d = ò ê ú a d ë û b i = ò yi + yi ( a) + yi ( b) i = ji( a) + ji ( b) Q4 F q d Q Q, F Q a

34 Shar Locking in Timoshnko Bams () Thick bam princs shar dformation, () Shar dformation is ngligibl in thin bams, Linar intrpolation of both w, f : f ¹- w()» w () + w () ()» S () + S () y y, f y y dw d dw f =- d jyj f å jyj j= j= å w» w ( ),» S ( ) w w h S S h Thus, in th thin bam limit it is not possibl for th lmnt to raliz th rquirmnt f =- dw d 4

35 SHER LOCKING - REMEDY In th thin bam limit, φ should bcom constant so that it matchs dw/d. Howvr, if φ is a constant thn th bnding nrgy bcoms zro. If w can mimic th two stats (constant and linar) in th formulation, w can ovrcom th problm. Numrical intgration of th cofficints allows us to valuat both φ and dφ/d as constants. Th trms highlightd should b valuatd using rducd intgration. () () æ b dy dy ö i j () () Kij = GKs cf yi y ò + j d a ç d d çè ø () b dyi () ij = ò s y j = ji K GK d K a d () () é b dy dy ù i j () () Kij = EI GKsyi y ò + j d a d d êë úû 5

36 STIFFNESS MTRICES OF TIMOSHENKO BEM ELEMENT (for constant EI and G) Rducd intgration linar lmnt (RIE) Linar approimation of both w and é 6 h 6 h ùì w ü ì q ü ì Q ü f ú EI h h h hz úï ï f q Q - ú ï ï ï ï ï ï m 6 h 6 h í ý= í ý+ í ý - h úï w q Q ú ï -h hz h h úï ïf q êë úï ûî ïþ ïî 4 ïþ ïîq ï 4ïþ EI = L, z = 5. -6L, L =, m = L GKh s Consistnt intrlmnt lmnt (CIE) Hrmit cubic approimation of w and dpndnt quadratic approimation of f é 6 -h -6 -h ùì w ü ì q ü ì Q ü ú EI h h h h úï ï f q Q - S Q ú ï í ï ý= ï í ï ý+ ï í ï m 6 h 6 h h ý - úï w q Q ú ï -h hq h hs úï ïf q êë úï ûî ïþ ïî 4ïþ ïî Q ï 4ïþ EI S =. + L, Q =. -6L, L =, m = + L GKh s Bams 6

37 E, EI L U U U, U4, F Eact solution (according to th E-B bam thory) wl ( ) = FL EI On lmnt discrtization using th RIE lmnt é 6 -h -6 -h ùì w ü ì q ü ì Q ü EI -h h h hz f q Q ú ï ï ï ï ï ï m 6 h 6 h í ý= í ý+ í ý h úï w q Q - ú ï ê-h hz h h úï ïf ï q 4 Q ë úî û þ ïî ïþ ïî 4þï EI = L, z = 5. -6L, L =, m = L GKh Boundary conditions: N EXMPLE of TBT s U = U =, Q = F, Q = 4 7

38 N EXMPLE (TBT) (continud) EI é 6 LùìïU üï ìïfïü m L FL L FL (. + L) 5 6 ú ï ï= ï ï U = = ml ê í ý í ý L L úïu ï ï 4 ï EI 6L 9L 6EI( L) ë ûïî ïþ ïî ïþ ( - ) Whn EI 5. FL FL L= = U = = GKsL 6EI 4EI ( too stiff ) FL (. 5+ 6L) FL Whn L¹, thn U = = (. 75+ L) 6EI EI L= EI ( + n ) H ( + ) H. H H = = n æ ö æ ö 6. æ ö = = GKL LK 6K ç èl ø 5 çèl ø çèl ø s s s 8

39 N EXMPLE of TBT On lmnt discrtization using th CIE lmnt é 6 -L -6 -Lùì w ü ì Q ü ú EI L L L L úï ï f Q - S Q ú ï ï ï ï ml í ý= í ý 6 L 6 L úïw Q - ú ï ê L L L L úïf Q ë- Q Súï ûî ï ïþ ïî 4 ïþ EI S =. + L, Q =. -6L, L =, m = + L GKL Condnsd quations for th unknown displacmnts s EI é 6 L ù ìu ü ì Fü ml FL S mfl S ú ï ï= ï ï U = = ml ê í ý í ý ël SL úï ûïîu 4ïþ îï þï EI ( LS-9 L ) EI( S-9) 9

40 N EXMPLE (TBT) (continud) mfl S FL Whn L= S= and m = ; thn U = = EI( S-9) EI mfl S FL ( + L )( + L) FL Whn L¹, U = = = ( + L) EI( S- 9) EI ( + L) EI ( n) ( n). EI + H + æh ö æh ö æh ö L= = = 6. = = GK L L K 6K ç èl ø 5 çèl ø èçl ø s s s 4

41 SUMMRY In this lctur w hav covrd th following topics: Drivd th govrning quations of th Eulr-Brnoulli bam thory Drivd th govrning quations of th Timoshnko bam thory Dvlopd Wak forms of EBT and TBT Dvlopd Finit lmnt modls of EBT and TBT Discussd shar locking in Timoshnko bam finit lmnt Discussd assmbly of bam lmnts Discussd ampls 4

Nonlinear Bending of Strait Beams

Nonlinear Bending of Strait Beams Nonlinar Bnding of Strait Bams CONTENTS Th Eulr-Brnoulli bam thory Th Timoshnko bam thory Govrning Equations Wak Forms Finit lmnt modls Computr Implmntation: calculation of lmnt matrics Numrical ampls