Finite element method for structural dynamic and stability analyses

Transcription

1 Fnte element method for structural dynamc and stablty analyses Module- Nonlnear FE Models Lecture-37 Introducton and revew of contnuum mechancs Prof C S Manohar Department of Cvl Engneerng IISc, Bangalore 56 Inda

2 Lnear systems & prncple of superposton x t y t x t y t Addtvty property y t y t x t x t Scalng property ax t ay t for all compelx a

3 Nonlnear system : a system that s not lnear A nonspecfc descrpton y mx c y max c ay amx ac y System s not lnear snce t does not obey scalng property. Zero nput produces zero output n a lnear system. Ths s not satsfed n ths case. 3

4 ;, ;, ;, y y y y y y x t x t ay ay ay ax t ay ay y y y x t y y y y y x t y y y y y x t y y ;, system s lnear y y y y x t y y 3 ;, y y y y x t ; y, y 3 y y y y x t ; y, y y y y y y y y y x t x t y y y y x t x t ; y, y Clearly, 3 y y y system s nonlnear 4

5 Exercse Examne the prevous two examples by ncludng non-zero ntal condtons yt dx x t dt a dx xt axt y t ay t ax t dt Scalng property s satsfed. Addtvty property s not satsfed (verfy) System s not lnear. 5

6 Qualtatve features of nonlnear dynamc response Response to all the loads needs to be analyzed smultaneously Undamped free vbraton: frequency of oscllatons depends upon ntal condtons. Harmonc nputs at can produce harmonc response at frequences prmary, subharmonc and non-harmonc responses super harmonc resonances aperodc responses Recprocty relatons are not vald 6

7 Large responses can occur at frequences other than the drvng frequences Steady state responses depend upon ntal condtons Multplcty of steady state solutons are possble System can possess multple equllbrum states and dsplay a wde range of bfurcatons Concept of normal modes, natural frequences, and natural coordnates no longer applcable Band lmted exctatons can produce responses wth frequency content outsde the bandwdth of the exctaton. 7

8 Example 3 n n x x x ; x A & x x t Acos t The frequency of oscllaton and the nature of oscllatons change. x t x t x t x t x t x t Unknowns:,, &,, x t x t x t n 3 x t x t x t x t x t x t : x t x t : x t x t x t x t 3 8

9 cos x t x t x t Acost x t x t A t A cos t A A Acost cos3t If A,lm xt 4 t 3 3 Ths s physcally not vald A A A x t C snt C cost cos3t 3 3 A x & x x t cos3t cost 3 9

10 3 A x t A t t t 3 cos cos3 cos 3 n A 4 Remark Soluton s perodc (but not harmonc) Perod depends upon ntal condton (hardenng) (softenng) Soluton can be mproved by ncludng hgher order terms

11 Example 3 n x cx x x F cos t ; x x & x x Consder steady state response of the form x t Acost A cost ca snt Acost A cos t 3 3 n F cost cos snt sn A n A A cost c Asnt cos 4 3t 4 A cost B snt 3 3 n A A A & c A B F A c A n 4

12 A n A n

13 Stablty of solutons u t x t t t n x t t c x t t x t t x t t Fcos c t 3 A cos t n Tme varyng system wth perodc coeffcent Use Floquet's theory to nfer stablty of the solutons. 3 Steady state solutons can be multvalued. The steady states are functons of ntal condtons. 3

14 Alternatve approach (A H Nayfeh and D T Mook, 979, Nonlnear oscllatons, John Wley, New York) Detunng x x x x F cos t; x x & x x 3 x x x x k cost 3 Method of multple scales cos x t a t t t k a a snt 3 a a cos t 3 k 8 Let = t 4

15 lma & lm t k a sn 3 k a a 3 k a cos 8 t Mult-valued frequency response functon Stablty Investgate the fxed ponts and stablty of the followng system of equatons: k a a snt 3 a a cos t 3 k 8 5

16 Importance of nonlnear analyss Earthquake engneerng: Structures are to be desgned to dsplay controlled nelastc responses; there are certan preferred modes of falures and certan falure modes are not preferred. Use of snubbers, restrant devces, solators, and nonlnear energy dsspaton devces (e.g., slotted bolt connectons) Wnd engneerng: wnd structure nteractons; across wnd oscllatons; gallopng. Materals lke concrete and sol dsplay nonlnear behavour even at low values of strans. Dfferng behavour n tenson and compresson. Response depends upon entre tme hstory, duraton over whch the load s appled and ambent effects such as temperature. Vbraton of cracked structures. Study of falures. Loss of stablty Prototype testng usng nonlnear FE models (e.g., crash analyss n automotve desgn, smulaton of drop test n electroncs ndustry) 6

17 Sources of nonlnearty Nonlnear stran-dsplacement relatons (geometrc nonlnearty) Nonlnear consttutve laws (nonlnear stress-stran relatons) Nonlnearty assocated wth boundary condtons Nonlnear energy dsspaton mechansms 7

18 Stress Materal nonlnear; small dsplacements and small strans Stran Materal lnear/nonlnear; large rotaton and small strans 8

19 Materal lnear/nonlnear; large rotaton and large strans Nonlnear boundary condtons Stress Stran

20 Restorng Force, kn Nonlnearly elastc systems and systems wth heredtary nonlneartes,,, ; ; x,, Nonlnear functon of nstantaneous values of & xt mx cx kx g x t x t t h x x t f t x() & specfed g x t x t t x t h x, x ; t Nonlnear functon of response tme hstores up to tme t...5 Restorng Force Vs. Dsplacement Dsplacement,mm

21 Nonlnear effects n wnd nduced oscllatons e.g., Across wnd oscllatons of chmneys mx cx kx -k x x F t Lmt cycles Entranment a

22 Objectves A bref revew of background concepts Present a flavour of treatment of nonlnear structural mechancs problems usng fnte element method. Focus on geometrcally nonlnear problems

23 Planar beam element Large transverse dsplacements Small strans Moderate rotatons Changes n geometry due to deformaton s not accounted for n defnng stress. Euler-Bernoull beam: Upon deformaton remans straght and normal to the neutral axs and ts length does not change. dw u u x z dx u, u, u3 dsplacement along x, y, z respectvely u u x axal dsplacement of a pont on neutral axs u3 w x w x axal dsplacement of a pont on neutral axs j u u j u u x x x x m m j j mn m n 3

24 dw u u x z ; u ; u3 w x dx u u u u 3 du d w dw z x x x x dx dx dx 33 u u u u 3 x x x x 3 x3 x3 x3 3 3 x x x x x x x x x x3 x x3 x x3 x x3 3 u3 u u u x3 u u u u u u u u u x u u 3 x 3 3 u u u u u x u x u u u u u u u x x x x u x u x 3 gnored (small strans) : rotaton ncluded 4

25 , ;, u x u x t w x w x t w u x t u x t z u u3 x t w x t x,, ; ;,, u w w z x x x E U E dv & T u w dv A l V u E{ x V l u w w U E z dadx x x x A ww z x x 4 w w u w u w z z x 4 x x x x x } dadx 5

26 U l 4 u w z u w w u w E{ z x x 4 x A x x x x ww z } dadx x x Consder beam to possess symmetrc cross secton. l l u w x x U AE dx EI dx l 4 l w u w + AE dx AE dx 8 x x x New terms due to presence of nonlnearty T u w dv m u w dx V l 6

27 l l l u x x w L m u w dx AE dx EI dx l 4 l w u w - AE dx AE dx 8 x x x u t, P t u t, P t 5 5 u t, P t EI, AE, m, l, c u t, P t 4 4 u t, P t u t, P t 7

28 6 u x, t u t x 5 4 w x, t u t x 3 3 x x x x x 3 ; x x ; l l l l 3 3 x x x x x 3 ; x l l l l x x x ; x l l d L L,,,,6 dt u u 8

29 l l l u w x x L m u w dx AE dx EI dx l Lead to the structural element mass and stffness matrces deduced earler 4 w u w - AE dx AE 8 x x x Newer terms due to nonlnearty l l 4 l 4 w Consder L AE dx AE u t x dx 8 8 l x L AE u tx xdx u t u t u t I u I k jmk l l l dx k j m jmk j m AE x x x x dx; k,,3,4 j m k 4 9

30 uw Smlarly, consder L AE dx x x L AE u t x u t x dx l 6 4 L k u k l AE u t x u t x x dx; k,,3,4 l u t u t AE x x x dx; k,,3,4 j j k 5 j j l u t u t J k j jk jk j k l,,3,4 J AE x x x dx; 5,6; j, k,,3,4 3

31 6 4 5 L AE u t x u t x dx l 6 4 L k uk 5 l l r s r s AE u t x u t x x dx; k 5,6 AE u t u t u t x x x x dx; k 5,6 l r s r s k u t u t u t K r s rsk K AE x x x x dx; r, s,,3,4;, k 5,6 rsk r s k 3

32 Form of the element equaton of moton M u K u u t Vector of quadratc and cubc terms n e e e e e Remarks Assembly of element level matrces and vectors can be done as before to obtan the global equatons of moton. Dervaton of external forces and mposton of BCs agan follows the earler developed procedure. The resultng equatons of moton would be of the form Mu Cu Ku g u F t ; u & u specfed. These equatons can be ntegrated usng numercal procedures dscussed earler (see Lecture 6). 3

33 Tmoshenko beam element u u x z x u u w x u u u u du d dw 3 z x x x x dx dx dx u u dw x x x dx 3 3 U V 3 3 dv 33

34 How about a more general theory? Allow measures of stran and stress to be defned consstent wth deformatons. Allow for materal nonlnearty 34

35 References T Belytschko, W K Lu, and B Moran,, Nonlnear fnte elements for contnua and structures, Wley, Chchester. O C Zenkewcz and R L Taylor, 99, The fnte element method, 4 th Edton, Vol II, McGraw-Hll, London. J N Reddy, 4, An ntroducton to nonlnear fnte element analyss, Oxford Unversty Press, New York. J N Reddy, 8, An ntroducton to contnuum mechancs, Cambrdge Unversty Press, New York. C S Jog, 7, Foundatons and applcatons of mechancs, Vol I, Contnuum mechancs, nd Edton, Narosa, New Delh. G A Holzapfel,, Nonlnear sold mechancs, Wley, Chchester. [Professor C A Felppa, Unversty of Colorado at Boulder) W F Chen and D J Han, 8, Plastcty for structural engneers, J.Ross publshng/cengage Learnng, New Delh. 35

36 Notatons Indcal notatons Algebrac notatons Matrx notatons Tensor notatons 36

37 Indcal notatons A set of varables x, x,, x s denoted as x. The range of values taken by the ndex, needs to be specfed. Typcally,,,3. Repeated ndces mples summaton = n ax j 3 n s wrtten as = a x,,, n Note: = a x a x n n U kjuu j s wrtten as U kjuu j, j,,, n Kronecker delta ds dx dx dx s wrtten as ds dx dx Permutaton symbol j f j f j jk s s j j 3 37

38 a a a3 A a a a ; A a a a a 3 a3 a33 dentty: 3 jk j k3 Dfferentaton f f x, x,, x f f f f df dx dx dxns wrtten as df dx,,, n x x x x The, symbol Consder jk rst js kt jt ks n,,,,,,,, f f x x x f f x x x f f x x x f , j f x Smlarly j j, k j x k n 38

39 Algebrac notatons Vectors and tensors are represented by sngle letters (bold face) x x, x, x 3 y y, y, y 3 jk j k x y x y x y x y e e e 3 z x y x x x e z e z e z z x y y y y 3 grad e e e e 3 x x x3 f f x, x, x scalar functon f f f grad f = e e e e f e f e f f f x x x x3 x 39

40 e e e e x x x x 3 3 Consder a vector valued functon F dv F. F e e e e F e F e F F F F F x x x3 x x x x3 e e e 3 z=curl F F e z ez e3z x x x z jk F x k j 3 F F F 3 3 4

41 e e e e x x x x 3 3 Consder a vector valued functon F F F F grad F=, F e e e 3 x x x3 e e F e F e F e e F e F e F x x e e F e F e F x3 Thus grad F can be descrbed by the followng matrx F F F 3 x x x F F F 3 x x x F F F 3 x3 x3 x3 4

42 Matrx notatons xx xx yy yy x t zz zz t t ; ; ; ; zy zy x 3 zx zx x x r x x U C xy xy No explct menton of connectve symbols (multplcaton) Tensor notatons ndces not shown applcable to Cartesan and other coordnate systems xy x.y dot denotes contracton of nner ndces A B A: B colon denotes contracton of a par of repeated ndces j j c C: j jkl kl 4

43 t.k. = K = K Tensor Matrx Indcal t :C: C Cj j Tensor Matrx Indcal j, j t b D b Indcal j j t C C j Cm mnc jn Matrx Indcal Matrx x x 3 b x3 3 b 3 x x x b3 x x x 3 Full notaton Notaton of last resort 43

44 Contnuum hypothess Matter s nfntely dvsble. Each nfntesmal element retans all the propertes of the materal. Newtonan mechancs s drectly applcable. Calculus works (governng equatons can be derved as PDE-s or ODE-s; varatonal approaches can be adopted to descrbe system behavor) -6 Attenton s lmted to characterstc dmensons > about cm (dameter of a water molecule -8 cm) The theory s vald for both solds and fluds Notons of densty, temperature, pressure, etc., at a pont make sense. Prmary am: to model macroscopc behavor of solds and fluds. Ignores the atomc structure of matter. (Matter conssts of dscrete partcles whch are perpetually n moton) Questons on treatment of molecular, gran or crystal structure are not addressed. 44

45 Three themes Knematcs : Moton and deformaton Knetcs: Concept of stress Balance laws (common to fluds and solds) Understandng of rotatons alternatve defntons of stress and stran treatment of materal nonlnear behavor 45

46 Knematcs : Study of moton and deformaton wthout concernng wth causes of moton and deformaton. Body B at tme= P x X, t x3, X3 X x p Body B at tme=t e 3 O e x, X x, X e 46

47 Reference frame: orgn O; orthonormal bass: e, e, e 3 Body B occupes dfferent regons,, at tme nstants,, t. The regons,, occuped by the body at dfferent tme nstants,, t are known as confguratons of the body at the respectve tme nstants. tme t = ntal state of the body; ntal confguraton. Could also be taken as the referec n e confguraton wth respect to whch moton s descrbed. Undeformed confguraton: s an dealzaton. boundary of the ntal confguraton. tme t current state of the body; current (deformed) confguraton. boundary of the current confguraton. 47

48 Euleran (spatal) and Lagrangan (materal) coordnates In Lagrangan descrpton we take X, X, X, t as ndependent varables. 3 3 In Euleran descrpton we take x, x, x, t as ndependent varables. X X e Poston vector of a materal pont n the ntal confguraton. Ths does not change wth tme. Labels all materal ponts. X e X e X e x x e x e x e x e Provdes the poston of a pont n the current confguraton. Changes as confguratons evolve n tme. In problems of sold mechancs, Lagrangan descrpton s often used. Lagrangan descrpton s also known as materal descrpton and Euleran descrpton s also known as spatal descrpton. 48

49 Moton,, x X, X, X, t 3 x X, t X, t x X, t,,,3 x X, X, X, t 3 x X, X, X, t When reference and ntal confguratons concde we get x X X X X x X, X, X, s an dentty transformaton. Materal coordnates Velocty : v X, t Acceleraton : a X, t,,,, X, t u X, t u Dsplacement : u X t x X X t X X t X t t v X, t u X, t u t t 49

50 Spatal coordnates D dx xt, Dt t x dt a 3 v v dx j v dx v j ;,,3 t x dt t x dt v a v grad v t j j j j j 5

51 Deformaton gradent Body B at tme= Q dx P x X, t u Q x3, X3 X P X Q u P q Body B at tme=t e 3 O e x, X x q x p p dx x, X e 5