Finite element method for structural dynamic and stability analyses

Transcription

1 Finie elemen mehod for srucural dynamic and sabiliy analyses Module- Nonlinear FE Models Lecure-39 Toal and updaed Lagrangian formulaions Prof C Manohar Deparmen of Civil Engineering IIc, Bangalore 56 India

2 Linear analysis Displacemens and srains are small ress-srain relaions are linear Equillibrium equaions are derived based on undeformed geomery Principle of superposiion holds Nonlinear analysis Displacemen and srains need no be small; special effors needed o characerize roaions The geomery of he obec, sress-srain relaions, and BCs could change during he process of deformaion. The definiions of sress and srain and he formulaion of he governing equaions need o ake ino accoun hese changes. Necessiaes inroducion of newer measures of sress and srain Principle of superposiion does no hold

3 rain measures Infiniesimal srains: for body under rigid body roaions, he srains would no be zero. New measures needed: Rigid body moions imply zero srains For small srains, he infiniesimal srain definiions are o be resored. 3

4 Green - Lagrange srain measure dx FdX ds d dx dx dx dx dx F FdX dx dx dx F F I dx dx EdX E F F I Green-Lagrange srain measure ds MF E N N & AB cos E M N d Almansi - Hamel (Eulerian) srain dx F dx ds d dx dx dx dx dx dx dx F F dx dx I F F dx dx edx e I F F 4

5 rain measure Engineering Green-Lagrange L L L Lagarihmic L L ln L Exensional raio L L Engineering Green-Lagrange logarhmic Exension raio L L MF L L/L) 5

6 ress measures Cauchy sress ensor: defined wih respec o deformed geomery. This would no be known in advance. Two alernaives: ress as a measure which conugaes wih a measure of srain o produce inernal energy As a quaniy which produces a racion vecor in conuncion wih a normal vecor defined wih respec o a surface elemen 6

7 Cauchy - Euler sress ress a P : n T n n lim a da da JF da N f a Firs Piola - Kirchoff sress (P) Based on deformed configuraion Based on undeformed configuraion sress vecor acing on elemen da wih ouward normal N which produces force = df. T n N df da T da N PN PdA da J F da P J F P is no symmeric; i has 9 independen componens econd Piola - Kirchoff sress () Based on undeformed configuraion Inroduce a pseudo-force vecor dpˆ F df ˆ n dp F df F da F nda JF nf da nda JF F econd Piola-Kirchoff sress ensor is symmeric & JF F PF 7

8 Noaions for configuraions and deformaions x, x, x x, x, x C Q ds P u u u Q x, x, x ds P C Q ds P C Configuraion Coordinaes of a poin olumes Areas Densiy Toal displacemen i Q iq quaniy Q is measured in C wih respec o is value in C i???? Check Noaionson u C C C x x x A A A u u u J N Reddy, 4, An inroducion o nonlinear FEA, Oxford Univ Press, NY, 4 K J Bahe, 996, Finie elemen procedures, Prenice Hall India, New Delhi. 8

9 Deformaion: A paricle in C a x x x x 3 moves o Moion from C o C : u x x x x x x3 ui xi xi i C a ;,,3 and o Moion from C o C : u x x C a x x x x3 ui xi xi ; i,,3 Conservaion of mass : xi xi d Jd d d d x x x x x x3 x x x d Jd wih J F x x x3 J & J x x x x x x3 9

10 rain ensors for C and C configuraions Green Lagrange srain ensor ui uk uk Ei x xi xi x u u u i uk u k Ei x xi xi x Green Lagrange incremenal srain ensor id xid x ds ds ds ds ds ds Ei Ei d xid x ei i d xid x Ei Ei ei i Ei Ei ei i where i

11 d x d x E E d x d x i i i i i i i i i k k k k ei x xi xi x xi x k i xi u uk (nonlinear in incremen u i ) x e d x d x u u u u u u (linear in incremen u ) i

12 Updaed Green Lagrange srain ensor u u i uk u k Ei is helpful in oal Lagrangian formulaions. x xi xi x To faciliae updaed Lagrangian formulaion we inroduce i i d xd x ds ds updaed Green-Lagrange srain ensor. Using he following resuls i k k i i i x xi x k k i i i & ki xi xi we ge i i x x x d x F d x d x ds d x d x u x x wih e x ui xi u u uk u k e xi xi x i u u u u & i k k i i xi xi xi x i,

13 Euler srain ensor Consider ha he body has reached configuraion C from C in several incremens. The deformaion from C o C could be large. We now wish o move o configuraion C. The incremen from C o C is aken o be small and we could refer o srains wih respec o C. id xid x ds ds k k id xid x ds ds i d x id x xi x x k i i xi We have x x k x u x x x u k k k k k k k k x x x u u u u i k k i x xi xi x x x Euler srain ensor u 3

14 u u u u i k k i x xi xi x The linear par of he Euler srain ensor is given by u u i ei ei x x i This quaniy is called he infiniesimal srain ensor. These srain componens conugae wih Cauchy sress componens o produce he expressions for inernal energy sored. i 4

15 ress ensors Cauchy sress: inernal force reckoned in he deformed configuraion and area reckoned in he deformed configuraion. Configuraion C : Configuraion C : i i i i econd Piola - Kirchoff sress ensor: force in C ransformed o C and area reckoned in C x ˆ n d A d f F d f d f x nˆ uni normal o d A in C. Updaed Kirchoff sress ensor useful in updaed Lagrangian formulaions Consider he poin P x, x, x in C. 3 Cauchy sress ensor a P in C is denoed by i 5

16 i i i Kirchoff sress incremen ensor i i i C C C i i i i i Updaed Kirchoff ress 6

17 i i i Updaed Kirchoff ress i i i ince Kirchoff sress incremen ensor x x i i i J mn xm xn x i i J mn x m x n i x J, we also ge x x i i xm xn x x i i x m x n mn mn Relaion beween Cauchy sress in C and updaed Kirchoff sress xi i x p x q x x i i pq x p x q PK- sress in differen configuraions i x x x i x p x q x pq Relaions beween incremenal sresses xi i x p x q x x i i x p x q x i i x p x pq pq pq pq x q

18 Consiuive relaions Aenion is limied o linear relaions beween conugae sress-srain pairs. Maerial behaviour is elasic: consiuive behaviour is funcion of curren sae of deformaion. Relaion beween second Piola - Kirchoff sress and Green - Lagrange srain C ikl E kl i where C maerial elasiciy ensor ress - srain relaions in incremenal form Kirchoff sress incremen C i ikl kl Updaed Kirchoff sress incremen C i ikl kl C in differen configuraions x i & Green-Lagrange srain incremen i & Green-Lagrange srain incremen i xi xk xl xi xk xl Cikl C ikl & Cikl Cikl xp xq xr xs xp xq xr xs x i 8

19 Principle of virual displacemens um of virual exernal work done on a body and he virual work sored in he body should be zero. Consider configuraion C W : e d f u d u d W i ei d fi ui d i ui d 9

20 Remarks We canno use his equaion direcly since he descripion of configuraion C would no be known. This descripion keeps changing as he deformaion evolves. The assumpion made in he linear analysis ha he configuraion of he body does no change so ha he equaions can be formulaed based on undeformed geomery is no valid in nonlinear analysis. This calls for inroducion of measures of sress and srain which ake ino accoun he changes in configuraion during deformaion. This enables he evaluaion of inegrals in he expression for he inernal work done over known configuraions. We will use ress: he second Piola Kirchoff sress ensor rain: he Green-Lagrange srain ensor

21 Toal Lagrangian approach Curren configuraion Base & reference configuraions coincide (Taken o remain fixed)

22 Updaed Lagrangian approach Base configuraion Curren configuraion Reference configuraion updaed a each incremen while solving he equilibrium equaion

23 Toal Lagrangian approach All quaniies reckoned wih respec o undeformed configuraion (C ). i i i i i i i i i i i i i e d E d f u d f u d u d u d E i d f u d u d i i i i 3

24 E d f u d u d i i i i i i Recall: Consider Recall: i i i E i Ei Ei ei i Ei ei Kirchoff sress incremen ensor E E e i i i i E i i E does no depend on unknown displacemens i 4

25 Consider Recall: i k k k k ei x xi xi x xi x i k k k k ei x xi xi x xi x Consider Recall: e i i u uk x k i xi u u u u u u u u u u u u k k k k i xi x xi x u u u u 5

26 i i i i i i i i i i i E d R Consider he erm R f u d u d E d d i i i ei i d e d i i This represens he virual inernal energy sored in he body in configuraion C. By applying virual work principle o body in configuraion C, one ges R e d f u d u d i i i i i i 6

27 i i d i i d R R i i d d i i Change in virual srain energy due o virual incremenal displacemens u beween configuraions C and C. i irual work done by forces due o iniial sress. i This arises due o change in geomery beween he wo configuraions. Consiuive relaion C i ikl kl ikl kl i i i C d d R R

28 i i i i Assuming he displacemens i ikl kl e i Cikl ekl ei d i i d R R o be small reasonable assumpion for small load incremens d d R R C e C E i ikl kl i u i 8

29 Ineria forces i i i i i i u u d u u d u u Equaion of moion i i irs rs i i i R R u u d C e e d d 9

30 Toal Lagrangian formulaion for a D elemen All quaniies measured wr C x, x, x C Q ds P u u u Q ds P Q ds P x, x, x C C x x u u u u x y u v u v ikl kl i i i C e e d d R R

31 Weak form ikl kl i i i e C e e d d R R u u u v v x x x x x e xx v u u v v eyy y y y y y e xy u v u u v v u u v v y x x y x y y x y x u v x x x x x u v u D y y y y y v u u v v y x y x x y y x x y Du u 3

32 e D D u ikl kl i u u D Du C D Du d C C C orhoropically elasic u C e e d e C e d C C C 66 3

33 i i d d u u v v x x x x xx u u v v yy y y y y xy u u v v x y x y u v x x x x u v u y y y y v u u v v y x x y y x x y D u u 33

34 D u D u u u v v x x x x xx u u v v y y y y xy u u u u v v v v x y x y x y x y yy u u C C E xx E yy E xy xx yy C C xy C66 u u v x x x E xx v u v Eyy y y y E xy u v u v u v y x x y y x E 34

35 The resuling forma of i i d d is no sill in a form suiable for developmen of FE soluion. Noe ha he erms are nonlinear in he incremenal displacemen u. We re-arrange he erms as follows: [ xx d u u v v x x x x xx u u v v yy d y y y y xy u u u u v v v v x y x y x y x y u u v v x x x x u u v v yy y y y y u u u u v v v v x y x y x y x y xy ] d 35

36 u xx xy u y xy yy y d v xx xy v x x xy yy u D D u d u u x x v v y y x xx xy xy yy y u & D xx xy v x xy yy y We are now ready o launch he FE formulaion. d 36

37 Incremenal displacemen: Toal displacemen: u u n n u v u v u v u v u v u n u x u n v v x n u x u n v v x n n n v n 37

38 u u u u BL C BL d BL D Du u u NL NL D Du C D Du d D D C D D d wih. e C e d d D D d D D d B B d wih BNL D 38

39 i i R e d e d B L i i i i f d d R f u d u d u d i 39

40 ikl kl i i i C e e d d R R KL KNL F F KL BL C BL d K NL BNL BNL d F f d i x x fy y F BL d ; f ; u d f Remarks L NL K K K K & C are symmeric This is an incremenal formulaion. The siffness marix here is he angen siffness marix. F K NL For a linear analysis,,,& 4

41 u v BL B L B L B L n x x x B L n x x x u u u n u x x x x x x B L u u u n x x x x x x v v v n v x x x x x B x L v v v n x x x x x x 4

42 B NL n x x x n y y y n x x x n y y y KL KNL F F L N L K K K u F F L L N K K K v F F 4

43 Updaed Lagrangian approach All quaniies reckoned wih respec o he laes known configuraion (C ). i i i i i i i i i i i i i e d d f u d f u d u d u d updaed Green-Lagrange srain ensor body force referred o in C. i surface racion referred o in C. i f i d R i i i i i R f u d u d 43

44 We have e i k k i i i xi xi xi x i i xi u u u u e k k k k i xi x xi x i Consider d R Recall u i u x i i i i Updaed Kirchoff ress u u u u i i i i i d R d R d e d R i i i i i 44

45 i i i i d R d e d R i i i i i i i i i Considering he equillibrium of body in C d d R R R e d R f u d u d i i i We have he consiuive relaion ikl kl i i i i i i ikl kl C d d R e d As we did in he oal Lagrangian approach we ake C e & e i ikl kl i i i C 45

46 This leads o he required weak form ikl kl i i i i i The Cauchy sress kl C e e d d R e d Almansi srain componens d x d x i i are evaluaed using C i i ikl kl ds ds ui uk uk i x xi xi x u 46

47 FE model for D elemen based on updaed Lagrangian formulaion K K C e e d d R e d ikl kl i i i i i L NL F F KL B L C B L d K NL BNL BNL d F f d x F B L d ; f ; d f x fy y 47

48 n x x x B L n x x x B NL n x x x n y y y n x x x n y y y 48

49 C C xx yy xy xx yy C C xy C66 u u v x x x xx v u v yy y y y xy u v u v u v y x x y y x 49

50 Follow up Maerial nonlineariy abiliy analysis: inclusion of nonlineariy a differen levels Hybrid esing Bayesian filering Uncerainy modelling and fem Thermal loads: fire Anisoropy 5