Finite element method for structural dynamic and stability analyses

Transcription

1 Finie elemen mehod for srucural dynamic and sabiliy analyses Modules- & 3 Finie elemen analysis of dynamics of planar russes and frames. Analysis of equaions of moion. Lecure-7: FE modeling of planar srucures: sysem wih consrains, shear building models, modeling of sress field. Models for damping Prof C S Manohar Deparmen of Civil Engineering IISc, Bangalore India

2 iv EIv y, v x, AEu mu 0 Modeling sysems wih consrains x, u x, l, 0, sin vl, cos 0,0 0 ;,0 0,0 ;,0 0 0 EI, AE, m, l u 0, 0; v 0, 0; v 0, 0 EIv u l mv AEu l, cos EIv l, sin 0 u x u x u x u x v x v x v x v x A B Reacion parallel o AB=0 Translaion normal o AB=0 How o allow his in FE modelling?

3 Ball and socke join 3

4 Beam on an inclined roller wih an inermediae hinge BCS: u u u 0 Consrains u u u u u u EI, AE, m, l co 9 EI, AE, m, l 8 Hinge 5 6 A 5 7 B 7 u sin u 7 5 cos E 0 GPa; =7800 kg/m ; B=0. m; D=0.3 m; l m; l 3 m; =40 3 0

5 u u co, u u, u u u u u co u u u 6 u7 u 8 u 9 U I GI GII 0 GIU I GIIU II 0 U II U G G U II II I I 5

6 U G G U II II I I I U 9 U I U U 6 GII G I T V U MU U I U M U wih M M I U KU U I I I U K U wih K K I M U K U I I I MU I KU I F 6

7 D beam elemen u, P u, P 5 5 u, P 3 3 M u, P EI, AE, m, l, c 6 6 u, P u, P l l ml 0 l 4l 0 3l 3l l 0 56 l 0 3l 3l 0 l 4l 4 4 7

8 K EI l l l r r 0 6l 0 6l 0 6l 4l 0 6l l ; r l l r r 0 6l 0 6l 0 6l l 0 6l 4l 3 I A 8

9 A A

10 G I = ; M 0 Na freq HZ K

11 Mode Mode Mode 3 Mode 5

12 Neglec axial deformaion Hinge A u6 cos u7 sin u u4 u7 u9 u9 u u u0 u5 u7 u u7 u u8 u8 u4 u u u B A B

13 Shear building models Popularly used in earhquake response analysis of buildings Rigid slab Flexible columns EI K 3 l M Mass of slab + a fracion of mass of columns How o obain a sdof model bu sill be able o include join roaions? 3

14 Sraegy: Relae roaions o ranslaions hrough relaions valid under saic condiions 3 4 6l 6l EI K l l l 3 l 6 8 6l l 8l u4 0 7 EI 3 l 4 6l 6l u f 6l 8l l u 0 6l l 8l u 3 0 4

15 kuu ku ut f k u k u 0 k u k u f k u uu T u k u u T u 0 k k uut k u k k k u f uu T u u T k k k k k eq uu u u SDOF approximaion Mu k u f eq Remark The sraegy of relaing cerain dof-s o ohers hrough relaions valid under saic condiions is known as saic condensaion echnique. Dof-s reained: maser dof-s. Dof-s eliminaed: slave dofs. 5

16 4EI EI k ; uu k u l l 3 l l k u EI 6l EI 8l l ; 3 k 3 l 6l l l 8l 3 EI EI l 8l l EI 4 6l k 6 6 eq l l l l EI l 8l l 6l EI l SDOF approximaion EI EI Mu 6.8 u f insead of Mu 4 u f 3 3 l l 6

17 m m 5 m Approximaions Lump masses a slab level Neglec axial deformaions u u u u u u 0 u u ; u u ; u u Remaining dof-s=8-6-3=9 Relae roaions o ranslaions as if saic equillibrium relaions are valid. 7

18 K k uu ku 4 7 u 3 6 k k 9 k u u k u u 5 8 kuu ku um PM k k us 0 k u k u P uu M u S u M S k k u 0 S u M kuu ku k k u um P m 0 0 u u 0 m 0 u kuu ku k k u u P 0 0 m u u

19 General forma for saic condensaion MX CX KX F X M Maser dof-s X X S Slave dof-s P 0 Slave dof-s are no exernally driven F KMM KMS X M P KSM K SS X S 0 X S KSS KSM X M X M I X X X S KSS KSM M M M TX M M MTX CTX KTX F T MTX T CTX T KTX T F M M M R M R M R M R M X C X K X F More on his laer. 9

20 Modeling of sresses s s 3, u x u, x l u l u x u x u u, 0, u u 3 l l x x l u l, u u 0, x 0 Elemen Elemen 0

21 Elemen Elemen x x u x, u u x u l u u3 u xx x, xx x, l l u xx xx x, l, E E u u l l, xx xx x, 0, 3 l l E E 3 3 u u l u u l u u u l, E 0, E xx even when 3 xx l l Elemen Elemen E E x

22 Sress Disconinuiy inroduced by FEM x Displcemen based FEM inroduces disconinuiies in spaial variaion of quaniies which are ruly coninuous.

23 s s Recall, x x x x 4 v x u x i 3 x x 3 l l 3 x x x ; l l 3 x x 3 ; l l 3 x x l l i i 3

24 Elemen, v x u x u x x x x x u 3 u 3 l l l l x 3x x 3x v x, u 3 u 3 l l l l 3x 3x v x, u 3 u 3 l l l l 3 3 v x, u u 3 l l v l, u ; v l, u ; 6 4 EIv l, EIu EIu l l 6 EIv l, EIu EIu 3 l l 4

25 Elemen, v x u x u x 3 3 x x x x u 3 3 u x l l l l x 3x x 3x v x, u 3 3 u l l l l 3x 3x v x, u 3 u 3 l l l l 3 3 v x, u u 3 l l v 0, u ; v 0, u ; 6 4 EIv 0, EIu EIu l l 6 EIv 0, EIu EIu 3 l l 5

26 Elemen vl, u ; v l, u ; 6 4 EIv l, EIu EIu l l 6 A he common node EIv l, EIu EIu 3 l l ranslaion and roaion Elemen are he same bu he v0, u ; v0, u ; BM and SF differ 6 4 EIv 0, EIu EIu l l EIv 6 0, EIu 3 EIu l l 6

27 Dynamic response analysis 0 0 MU CU KU G U, U, F U(0) U ; U 0 U Frequency domain mehods Time domain mehods Response specrum based mehods Linear ime invarian sysems Time varying sysems Nonlinear sysems Quaniaive mehods Direc mehods Mode superposiion mehods Qualiaive mehods Bifurcaions and sabiliy 7

28 Review of soluion of equaion of moion for discree MDOF sysems MX CX KX F X 0 X ; X 0 X 0 0 NPTEL Video course Sochasic Srucural Dynamics [C S Manohar] Lecure 3 Slides 3-50 Lecure 4 Slides -4 hp://npel.iim.ac.in/courses/ / 8

29 Inpu-oupu relaions for linear ime invarian sysems f () LTI x ) h( ( ) f ( ) d 0 f ( ) F( ) x( ) X ( ) h( ) H ( ) F() LTI () LTI h() X ( ) H( ) F( ) exp( i) LTI H( )exp( i) 9

30 Linear Damping models Viscous Srucural Classical Non-Classical Classical Non-Classical Classificaion ino viscous and srucural depends upon behavior of energy dissipaed under harmonic seady sae as a funcion of frequency. Classificaion ino classical and non-classical depends upon orhogonaliy (or lack of orhogonaliy) of damping marix wih respec o undamped normal modal marix. 30

31 Equllibrium equaion: mx cx kx P cos mx cx kx x P x Power balance equaion: cos As /, energy disspaed in a cycle, ED cx d 0 lim x( ) X cos lim x( ) X sin SDOF sysems D / E c X E D 0 sin ( ) (/ )( / ) d c X cx This conradics he experimenal observaion ha E D is consan wih respec o driving frequency 3

32 Remedy : adop an equivalen damping model such ha C eq c Ceq mx x kx P cos( ) Ceq ED X (/ )( / ) C X eq E D is independen of driving frequency (as is observed in experimenal sudies). This damping model is called he srucural damping model. 3

33 Consider P exp( i) mx cx kx P exp( i) lim x( ) m ic k Ceq P exp( i) mx x kx P exp( i) lim x( ) ( ) Define * k k iceq * mx k x P exp( i) lim x( ) * m k m k iceq P exp( i) We can alk of complex valued siffness in he conex of srucural damping models. 33

34 C eq mx x kx P i mx k x P i * exp( ) OR exp( ) Remarks Energy dissipaed per cycle becomes consan wih respec o driving frequency. This is no an equaion in ime domain: i does no make sense o alk of free vibraion; response o ransien loads canno be described. I can be shown ha he sysem is no causal. Srucural damping model: mahemaically no sound bu explains experimenal observaions on dependence of energy dissipaed per cycle as a funcion of driving frequency. Viscous damping model is mahemaically sound bu no saisfacory in erms of explaining experimenal observaions 34

35 Analysis of MDOF sysems wih classical viscous damping 0 0 TZ () MX CX KX F X 0 X ; X 0 X X () MTZ CTZ KTZ F() T MTZ T CTZ T KTZ T F() MZ CZ KZ F M, C,& K srucural marices in he new coordinae sysem. F ( ) force vecor in he new coordinae sysem Quesion Can we selec T such ha M, C,& K are all DIAGONAL? If yes, equaion for Z ( ) would hen represen a se of uncoupled equaions and hence can be solved easily. 35

36 How o selec T o achieve his? Consider he seemingly unrelaed problem of undamped free vibraion analysis MX KX 0 Seek a special soluion o his se of equaions in which all poins on he srucure oscillae harmonically a he same frequency. Tha is k k exp ;,,, X R i R n X ir exp i & X R expi MR KRexpi 0 x r i k n or, exp where is a vecor. 36

37 MR KR i RM KR K K ; M M exp 0 KR MR This is a algebraic eigenvalue problem. Noe K is posiive semi-definie M is posiive definie Eigensoluions would be real valued and eigenvalues would be non-negaive. 0 37

38 KR MR K M R Le K K M R 0 exis. K M K M R IR 0 R 0 If exiss, =0 is he soluion. Condiion for exisence of nonrivial soluion is ha K M M K M n should no exis. 0 This is called he characerisic equaion. This leads o he characerisic values R, R,, R. n and associaed eigenvecors 0 38

39 Orhogonaliy propery of eigenvecors Consider r- h and s-h eigenpairs. KR KR r r r s s s () R R KR s s r r s r () R R KR MR MR r r s s R MR R MR s r s s r r s s r r () () (3) (4) Transpose boh sides of equaion (4) R K R R M R Since K K & M M, we ge Rs KRr s RsMRr (5) Subsrac (3) and (5) R MR s 0 R MR 0 r s s r R KR 0 r s s r Normalizaion R MR s R KR s s s s 39

40 Inroduce R R R ( nn) Diag n n Orhogonaliy relaions M I K Selec T 40

41 Consider Undamped Forced Vibraion Analysis 0 0 MX KX F X 0 X ; X 0 X X () Z M Z KZ F() r r r r M Z KZ F() IZ Z F() z z f ; r,,, n How abou iniial condiions? X(0) Z 0 MX (0) M Z 0 Z 0 Z 0 MX (0) & Z 0 MX (0) 4

42 z 0 z z f d r r r 0cos r sin r sin r r r 0 r X Z k kr r r n x z n zr 0 kr zr 0cosr sinr sinr fr d r r 0 r 4

43 How abou damped forced response analysis? 0 0; 0 0 Z M Z C Z K Z F() IZ C Z Z F() MX CX KX F X X X X X () M Z CZ KZ F() If C is no a diagonal marix, he equaions of moion would sill remain coupled. 43

44 Classical damping models If he damping marix C is such ha C is a diagonal marix, hen equaions would ge uncoupled. Such C marices are called classical damping marices. Example Rayleigh's proporional damping marix C M K C M K I 44

45 C M K T T C [ M K] T T I K [ I] Diag[ i ] c c n n n n n n n How o find and? We need o know damping raios a leas for wo modes. For example, n Knowing and, solve for and 45

46 ea mass and siffness proporional siffness proporional mass proporional frequency rad/s 46