The Finite Element Method for the Analysis of Non-Linear and Dynamic Systems

Transcription

1 Swiss Federl Insiue of Pge 1 The Finie Elemen Mehod for he Anlysis of Non-Liner nd Dynmic Sysems Prof. Dr. Michel Hvbro Fber Dr. Nebojs Mojsilovic Swiss Federl Insiue of ETH Zurich, Swizerlnd Mehod of Finie Elemens II

2 Swiss Federl Insiue of Conens of Tody's Lecure Pge 2 Moivion, i overview nd orgnizion i of he course Inroducion o non-liner nlysis Formulion of he coninuum mechnics incremenl equions of moion Mehod of Finie Elemens II

3 Swiss Federl Insiue of Moivion, overview nd orgnizion of he course Pge 3 Moivion i In FEM 1 we lerned bou he sedy se nlysis of liner sysems however; he sysems we re deling wih in srucurl engineering re generlly no sedy se nd lso no liner We mus be ble o ssess he need for priculr ype of nlysis nd we mus be ble o perform i Mehod of Finie Elemens II

4 Swiss Federl Insiue of Moivion, overview nd orgnizion of he course Pge 4 Moivion Wh kind of problems re no sedy se nd liner? E.g. when he: meril behves non-linerly deformions become big (p-δ effecs) lods vry fs compred o he eigenfrequencies of he srucure Generl feure: Response becomes lod ph dependen Mehod of Finie Elemens II

5 Swiss Federl Insiue of Moivion, overview nd orgnizion of he course Pge 5 Moivion Wh is he dded vlue of being ble o ssess he non-liner non-sedy se response of srucures? E.g. ssessing he; - srucurl response of srucures o exreme evens (rock-fll, erhquke, hurricnes) - performnce (filures nd deformions) of soils - verifying simple models Mehod of Finie Elemens II

6 Swiss Federl Insiue of Moivion, overview nd orgnizion of he course Pge 6 Sedy se problems (Liner/Non-liner): The response of he sysem does no chnge over ime KU = R Propgion problems (Liner/Non-liner): The response of he sysem chnges over ime MU&& () + CU& () + KU () = R () Eigenvlue problems: No unique soluion o he response of he sysem Av = λ Bv Mehod of Finie Elemens II

7 Swiss Federl Insiue of Moivion, overview nd orgnizion of he course Pge 7 Orgnision i The lecures will be given by: M. H. Fber nd N. Mojsilovic Exercises will be orgnized/ended by: K. Nishijim Office hours: 14:00 16:00 on Thursdys, HIL E22.3. Mehod of Finie Elemens II

8 Swiss Federl Insiue of Moivion, overview nd orgnizion of he course Pge 8 Orgnision i PowerPoin files wih he presenions will be up-loded on our home-pge one dy in dvnce of he lecures hp:// _ II The lecure s such will follow he book: "Finie Elemen Procedures" by K.J. Bhe, Prenice Hll, 1996 Mehod of Finie Elemens II

9 Swiss Federl Insiue of Moivion, overview nd orgnizion of he course Pge 9 Overview De Subjec(s) Course book Pges: Non-liner Finie Elemen Clculions in solids nd srucurl mechnics Inroducion o non-liner clculions The incremenl pproch o coninuum mechnics Non-liner Finie Elemen Clculions in solids nd srucurl mechnics Deformion grdiens, srin nd sress ensors The Lngrngin formulion only meril nonlineriy Non-liner Finie Elemen Clculions in solids nd srucurl mechnics Displcemen bsed iso-prmeric finie elemens in coninuum mechnics Non-liner Finie Elemen Clculions in solids nd srucurl mechnics Displcemen bsed iso-prmeric finie elemens in coninuum mechnics Mehod of Finie Elemens II

10 Swiss Federl Insiue of Moivion, overview nd orgnizion of he course Pge 10 Overview De Subjec(s) Course book Pges: Non-liner Finie Elemen Clculions in solids nd srucurl mechnics Tol Lngrngin formulion Exended Lngrngin formulion Srucurl elemens Non-liner Finie Elemen Clculions in solids nd srucurl mechnics Inroducion of consiuive relions Non-liner consiuive relions Non-liner Finie Elemen Clculions in solids nd srucurl mechnics Conc problems Prcicl considerions Dynmicl Finie Elemen Clculions Inroducion Direc inegrion mehods Mehod of Finie Elemens II

11 Swiss Federl Insiue of Moivion, overview nd orgnizion of he course Pge 11 Overview De Subjec(s) Course book Pges: Dynmicl Finie Elemen Clculions Mode superposiion Dynmicl Finie Elemen Clculions Anlysis of direc inegrion mehods Dynmicl Finie Elemen Clculions Soluion of dynmicl non-liner problems Soluion of Eigen vlue problems The vecor ierion mehod Soluion of Eigen vlue problems The rnsformion mehod Mehod of Finie Elemens II

12 Swiss Federl Insiue of Inroducion i o non-liner nlysis Pge 12 Previsously we considered d he soluion of he following liner nd sic problem: KU = R for hese problems we hve he convenien propery of lineriy, i.e: KU = λr U = λu If his is no he cse we re deling wih non-liner problem! Mehod of Finie Elemens II

13 Swiss Federl Insiue of Inroducion i o non-liner nlysis Pge 13 Previsously we considered d he soluion of he following liner nd sic problem: KU = R we ssumed: smll displcemens when developing he siffness mrix K nd he lod vecor R, becuse we performed ll inegrions over he originl elemen volume h he B mrix is consn independen of elemen displcemens he sress-srin mrix C is consn boundry consrins re consn Mehod of Finie Elemens II

14 Swiss Federl Insiue of Inroducion i o non-liner nlysis Pge 14 Clssificion of non-liner nlysis Type of nlysis Descripion Typicl formulion used Sress nd srin mesures used Merilly-nonliner only Infiniesiml displcemens nd Merillynonliner-only Engineering srin nd sress srins; sress rin relion is nonliner (MNO) Lrge displcemens, lrge roions bu smll Displcemens nd roions of fibers re lrge; bu fiber Tol Lgrnge (TL) Second Piol- Kirchoff sress, Green-Lgrnge srins exensions nd ngle chnges srin beween fibers re smll; sress srin Upded Lgrnge (UL) Cuchy sress, Almnsi srin relionship my be Lrge displcemens, lrge roions nd lrge srins Mehod of Finie Elemens II liner or non-liner Displcemens nd roions of fibers re lrge; fiber exensions nd ngle chnges beween fibers my lso be lrge; sress srin relionship my be liner or non-liner Tol Lgrnge (TL) Second Piol- Kirchoff sress, Green-Lgrnge Upded Lgrnge (UL) srin Cuchy sress, Logrihmic srin

15 Swiss Federl Insiue of Pge 15 Inroducion i o non-liner nlysis Clssificion of non-liner nlysis σ Δ P 2 σ = P / A E L ε = σ / E 1 P 2 Δ=ε L ε < 0.04 ε L Liner elsic (infiniesiml displcemens) Mehod of Finie Elemens II

16 Swiss Federl Insiue of Pge 16 Inroducion i o non-liner nlysis L Clssificion of non-liner nlysis Δ P 2 P 2 σ = P / A ε ε < σ σ σy E E T Y = σ σ Y 1 E P / A 1 E T ε L Merilly nonliner only (infiniesiml displcemens, bu nonliner sress-srin relion) Mehod of Finie Elemens II

17 Swiss Federl Insiue of Inroducion i o non-liner nlysis Pge 17 Clssificion of non-liner nlysis y x L ε < 0.04 Δ= ε L Lrge displcemens nd lrge roions bu smll srins (liner or nonliner meril behvior) Mehod of Finie Elemens II

18 Swiss Federl Insiue of Inroducion i o non-liner nlysis Pge 18 Clssificion of non-liner nlysis Lrge displcemens nd lrge roions nd lrge srins (liner or nonliner meril behvior) Mehod of Finie Elemens II

19 Swiss Federl Insiue of Inroducion i o non-liner nlysis Pge 19 Clssificion of non-liner nlysis P 2 P 2 Δ Chnging boundry condiions Mehod of Finie Elemens II

20 Swiss Federl Insiue of Inroducion i o non-liner nlysis Pge Exmple: Simple br srucure E = 10 N / cm R Are = 1cm Secion L = 10cm 2 u Secion b L = R 5cm σ σ Y 1 E 1 ε = E T ET = 10 N / cm σ Y :yield sress ε :yield srin ε Y ε Mehod of Finie Elemens II

21 Swiss Federl Insiue of Inroducion i o non-liner nlysis Exmple: Simple br srucure ε Are = 1cm Secion L = 10cm 2 u =, εb = L + σb = σ R A σ A u Secion b L = u L b R 5cm σ σ Y R E 1 ε = ε = (elsic region) E σ σ Y Δσ ε = εy + (plsic region) i ) Δ ε = (unloding) E E T Mehod of Finie Elemens II E T 7 2 E = 10 N / cm 5 2 ET = 10 N / cm σ : yield sress Y ε Y : yield srin ε Pge 21

22 Swiss Federl Insiue of Inroducion i o non-liner nlysis u Exmple: Simple br srucure ε, Are = 1cm Secion L = 10cm 2 u Secion b L = R 5cm σ σ Y R E 1 1 ε = E T E = 10 N / cm ET = 10 N / cm σ Y : yield sress ε : yield srin Y ε = εb = L R+ σ A= σ A b σ ε = E ε = ε + Δ ε = Y u L b (elsic region) σ σy E T Δσ (unloding) E Pge 22 (plsic region) Boh secions elsic 1 1 R = EA u ( + ) u = L L 310 R R 2 R σ =, σb = 3 A 3 A b 6 Mehod of Finie Elemens II

23 Swiss Federl Insiue of Inroducion i o non-liner nlysis u Exmple: Simple br srucure 7 2 E = 10 N / cm ε, Are = 1cm Secion L = 10cm 2 u Secion b L = R 5cm secion b will be plsic when R = σ u u = E, b ET( Y) Y L σ = L ε σ b EA u ET A u R = + ETεYA+ σya L L σ σ Y R E 1 ε = E T 5 2 ET = 10 N / cm σ Y : yield sress ε : yield srin Y ε Secion is elsic while secion b is plsic T Y Y Mehod of Finie Elemens II b 2 σ A 3 Y R / A + E ε σ R u = = E/ L + E/ L b 2 = εb = L R+ σ A= σ A b σ u L b Pge 23 ε = (elsic region) E σ σy ε = εy + (plsic region) ET Δσ Δ ε = (unloding) E

24 Swiss Federl Insiue of Inroducion i o non-liner nlysis Pge 24 Wh did we lern from he exmple? The bsic problem in generl nonliner nlysis is o find se of equilibrium beween exernlly pplied lods nd elemen nodl forces R F = 0 R = R + R + R B S C We mus chieve equilibrium for ll ime seps when incremening he loding Very generl pproch F = F R I = B τ dv m V ( m) ( m) T ( m) ( m) includes implicily lso dynmic nlysis! Mehod of Finie Elemens II

25 Swiss Federl Insiue of Inroducion i o non-liner nlysis Pge 25 The bsic pproch in incremenl nylsis is R F = 0 +Δ +Δ +Δ ssuming h R is independen of he deformions we hve +Δ F = F+ F We know he soluion F ime nd F is he incremen in he nodl poin forces corresponding o n incremen in he displcemens nd sresses from ime o ime +Δ his we cn pproxime by F = KU Tngen siffness mrix Mehod of Finie Elemens II F K = U

26 Swiss Federl Insiue of Inroducion i o non-liner nlysis Pge 26 The bsic pproch in incremenl nylsis is We my now subsiue he ngen siffness mrix ino he equlibrium relion KU = R F +Δ U = U+ U +Δ which h gives us scheme for he clculion l of he displcemens he exc displcemens ime +Δ correspond o he pplied lods +Δ however we only deermined hese pproximely s we used ngen siffness mrix hus we my hve o iere o find he soluion Mehod of Finie Elemens II

27 Swiss Federl Insiue of Inroducion i o non-liner nlysis Pge 27 The bsic pproch in incremenl nylsis is We my use he Newon-Rphson ierion scheme o find he equlibrium wihin ech lod incremen K Δ U = R F (ou of blnce lod vecor) +Δ ( i 1) ( i) +Δ +Δ ( i 1) U = U +ΔU +Δ () i +Δ ( i 1) () i wih iniil condiions U = U ; K = K ; F = F +Δ (0) +Δ (0) +Δ (0) Mehod of Finie Elemens II

28 Swiss Federl Insiue of Inroducion i o non-liner nlysis Pge 28 The bsic pproch in incremenl nylsis is I my be expensive o clcule he ngen siffness mrix nd; in he Modified Newon-Rphson ierion scheme i is hus only clculed in he beginning of ech new lod sep in he qusi-newon ierion schemes he secn siffness mrix is used insed of he ngen mrix Mehod of Finie Elemens II

29 Swiss Federl Insiue of The coninuum mechnics incremenl equions Pge 29 The bsic problem: We wn o esblish he soluion using n incremenl formulion The equilibrium mus be esblished for he considered body in is curren configurion In proceeding we dop Lgrngin formulion where rck he movemen of ll pricles of he body (loced in Cresin coordine sysem) Anoher pproch would be n Eulerin formulion where he moion of meril hrough sionry conrol volume is considered Mehod of Finie Elemens II

30 Swiss Federl Insiue of The coninuum mechnics incremenl equions Pge 30 The bsic problem: x 3 δu 1 δu = δu2 δu 3 Configurion corresponding o vriion in displcemens +Δ δu u Configurion ime +Δ Surfce re Volume +Δ V +Δ S x 2 Configurion ime 0 Surfce re Volume 0 V 0 S Configurion ime Surfce re Volume V S x (or x, x, x ) 0 + Δ Mehod of Finie Elemens II

31 Swiss Federl Insiue of The coninuum mechnics incremenl equions The Lgrngin formulion We express equilibrium of he body ime +Δ using he principle of virul displcemens +Δ +Δ +Δ τδ ed V R +Δ V +Δ +Δ ij = +Δ ij +Δ +Δ j x 3 x (or x, x, x ) 0 + Δ δu1 δu = δu2 δu 3 Configurion corresponding o vriion in displcemens +Δ δu u 0 Surfce re S 0 Volume V x 2 Pge 31 Configurion ime + Δ +Δ Surfce re S +Δ Volume V Configurion ime Surfce re S Configurion ime 0 Volume V τ : Cresin componens of he Cuchy sress ensor 1 δu δ u i j δ e = ( + ) = srin ensor corresponding o virul displcemens 2 x x i δui : Componens of virul displcemen vecor imposed ime +Δ +Δ xi : Cresin coordine ime +Δ +Δ V : Volume ime +Δ +Δ +Δ B +Δ +Δ S S +Δ R = f δ ud V = f δ u d S V Mehod of Finie Elemens II i i i i S +Δ +Δ f

32 Swiss Federl Insiue of The coninuum mechnics incremenl equions The Lgrngin formulion We express equilibrium of he body ime +Δ using he principle of virul displcemens R= f δud V = f δu d S +Δ +Δ B +Δ +Δ S S +Δ i i i i where f +Δ +Δ : exernlly pllied forces per uni volume V S f +Δ B i +Δ S fi +Δ S f : surfce ime +Δ S +Δ δui : δui evlued he surfce Sf x 3 x (or x, x, x ) 0 + Δ : exernlly pllied surfce rcions per uni surfce δu1 δu = δu2 δu 3 Configurion corresponding o vriion in displcemens +Δ δu u 0 Surfce re S 0 Volume V x 2 Pge 32 Configurion ime + Δ +Δ Surfce re S +Δ Volume V Configurion ime Surfce re S Configurion ime 0 Volume V Mehod of Finie Elemens II

33 Swiss Federl Insiue of The coninuum mechnics incremenl equions Pge 33 The Lgrngin formulion We recognize h our derivions from liner finie elemen heory re unchnged bu pplied o he body in he configurion ime +Δ Mehod of Finie Elemens II

34 Swiss Federl Insiue of The coninuum mechnics incremenl equions Pge 34 In he furher we inroduce n pproprie noion: Coordines nd displcemens re reled s: x = x + u 0 i i i x = x + u +Δ 0 +Δ i i i Incremens in displcemens re reled s: +Δ ui = ui ui Reference configurions re indexed s e.g.: f +Δ S 0 i where he lower lef index indices he reference configurion τ = τ +Δ +Δ ij +Δ ij Differeniion is indexed s: u +Δ 0 +Δ i 0 m 0 u i, j=, x = 0 +Δ m, n +Δ xj xn x Mehod of Finie Elemens II