Finite element method for structural dynamic and stability analyses

Transcription

1 Finie elemen mehod for srucural dynamic and sabiliy analyses Module- Approximae mehods and FEM Lecure- Equaions of moion using Hamilon s principle Prof C S Manohar Deparmen of Civil Engineering IISc, Bangalore India

2 Wha is his course abou? FEM in he conex of srucural mechanics problems. Wha are he issues o be deal wih, when he FEM, as developed for analysis of saic problems, is exended o deal wih problems of srucural dynamic and elasic sabiliy analyses New quesions New phenomena New numerical ools Applicaion areas

3 This is no a firs course in FEM Srucural dynamics Srucural Sabiliy analysis Pre-requisies Marix mehods of saic srucural analysis A firs course in heory of vibraions Elemens of elasic sabiliy analysis Marix algebra and ode-s The course will conain brief overviews of he main ideas of hese subjecs 3

4 Differen Faces of FEM A numerical mehod o obain approximae soluions o boundary value problems (ODE-s, PDE-s). A modeling and simulaion ool A ool in compuer aided engineering ha explois he power of compuers Modeling (pre-processing) Numerical soluions Scruiny of resuls (pos processing) Combined wih sensing and acuaion, a ool for Acive conrol of srucures Sysem idenificaion and healh monioring An imporan componen in hybrid esing mehods 4

5 Topics proposed o be covered in his course Approximae mehods and FEM Dynamics of russ and planar frame srucures Damping models and analysis of equilibrium equaions Dynamics of Grids and 3D frames A few compuaional aspecs (soluion of equilibrium equaions, eigenvalue problems, model reducion, and subsrucuring) Dynamic siffness marix and ransfer marix mehods Dynamics of plane sress/srain, plae bending, shell and 3d elemens Applicaions (earhquake engineering and vehicle srucure ineracions) FEA of elasic sabiliy problems Treamen of nonlineariy FE model updaing FEM in hybrid simulaions 5

6 Three inerwining hemes Modeling Numerical soluions Applicaions Targe audience Graduae level sudens Research sudens Engineers who use FEM as a ool in heir work 6

7 Books. M Pey, 990, Inroducion o finie elemen vibraion analysis, CUP, Cambridge.. W Weaver and P R Johnson, 987, Srucural dynamics by finie elemens, Prenice-Hall, Englewood Cliffs. 3. M Geradin and D Rixen, 997, Mechanical vibraions, nd Ediion, Wiley, Chicheser. 4. R W Clough and J Penzien, 993, Dynamics of Srucures, nd Ediion, McGraw-Hill, New York. 5. K J Bahe, 996, Finie elemen procedures, Prenice Hall of India, New Delhi. 6. I H Shames and C L Dym, 99, Energy and finie elemen mehods in srucural mechanics, Wiley Easern Limied, New Delhi. 7. R R Craig, 98, Srucural dynamics: an inroducion o compuer mehods, Wiley, NY 8. C H Yoo and S C Lee, 0, Sabiliy of srucures, Buerworh-Heinmann, Burlingon. 9. G J Simises and D H Hodges, 006, Fundamenals of srucural sabiliy, Elsevier, Amserdam. 7

8 Addiional references. L Meirovich, 997, Principles and echniques of vibraions, Prenice-Hall, New Jersey. O C Zienkiewicz and R L Taylor, 989, The finie elemen mehod, Vols-I and II, 4 h Ediion, McGraw-Hill, London. 3. R D Cook, D S Malkus, and M E Plesha, 989, Conceps and applicaions of finie elemen analysis, 3 rd Ediion, John Wiley, New York 4. J N Reddy, 006, An inroducion o he finie elemen mehod, 3 rd Ediion, Taa-McGraw-Hill, New Delhi. 5. S S Rao, 999, The finie elemen mehod in engineering, 3 rd Ediion, Buerworh-Heinemann, Boson. 6. T J R Hughes, 000, The finie elemen mehod, Dover, Mineola. 7. M Paz, 985, Srucural dynamics, nd Ediion, CBS Publishers, New Delhi 8. W McGuire, R H Gallagher, and R D Ziemian, 000, Marix srucural analysis, nd Ediion, John Wiley, New York. 9. P Seshu, Texbook on finie elemen analysis, 003, Prenice Hall India, New Delhi. 0.G Srang and G J Fix, 008, nd Ediion, An analysis of he finie elemen mehod, Wellesley-Cambridge Press, Wellesley. 8

9 Dynamic loads The magniude, direcion, and (or) poin of applicaion of he load change wih ime. Under he acion of dynamic loads he srucure vibraes, ha is, (a) he srucure develops significan level of ineria forces (b) significan level of mechanical energy is sored as kineic energy Noe: No all ime varying loads need o be dynamic in naure; for example, (a) Load on a dam due o filling of a reservoir (b) Load on a specaor gallery as a sadium ges filled up. 9

10 Examples:. Earhquake loads on buildings, bridges, dams, power plans ec.. Wind loads on long span bridges, all chimneys ec. 3. Loads due o blas and impac. 4. Running machineries in buildings. 5. Vehicle moving on a bridge. 0

11 Degree of freedom (DOF) Number of independen coordinaes required o specify he displacemens a all poins in he sysem and a all imes. k m u f L g y m x y L x

12 Two-dof sysems z Paricle in space DOF=3 x z y x Rigid body in space DOF=6 y

13 This chunk has mass, siffness and damping DOF N w x, a x DOF N n n n Unknown Known funcion funcion of ime of x Two-dof sysem 3

14 Remarks: The DOF is no an inrinsic propery of a given srucure. I is a choice exercised by he modeler. SDOF sysems: sysems wih one DOF. MDOF sysems: sysems wih DOF> MDOF sysems could be discree (finie DOFs) or coninuous (infinie DOFs) The coordinaes need no have direc physical meaning (generalized coordinaes) DOF=a number or he coordinae FEM: replaces a sysem wih infinie dofs by a sysem wih finie dofs. 4

15 Eniies in a simple mahemaical model for a vibraing sysem Whenever a srucure vibraes, i no only displaces, bu also acceleraes. Besides, he mechanical energy is convered o hea and/or sound. Siffness: resiss displacemen Ineria: resiss acceleraion Damping: dissipaes energy Inpus: exernal forcing and/or iniial condiions; supply mechanical power Mass: offers resisance o acceleraion, sores kineic energy Damper: dissipaes mechanical energy Spring: offers resisance o displacemen, sores poenial energy Force and/or iniial displacemen, velociy: supplies mechanical power 5

16 Posiion of he mass when he spring is no sreched k m u f c Mass: offers resisance o acceleraion, sores kineic energy Damper: dissipaes mechanical energy Spring: offers resisance o displacemen, sores poenial energy Force and/or iniial displacemen, velociy: supplies mechanical power Mass (kg) KE= mu Force= Spring (N) PE= ku Force= Damper (Ns/m) DE= cu Force=c mu ku u 6

17 u Mass kg m mu Force= mu KE= T mu Spring N/m u k u Force= k u u PE= V k u u Damper Ns/m c u u Force= c u u DE c u u 7

18 How does he moion ake place? Hamilon's principle q n vecor of sysem dof-s q q T q q V q q Lagrangian : L,,, T q, q oal kineic energy of he sysem V q, q oal srain energy of he sysem Among all he dynamic pahs which saisfy he boundary condions (on prescribed displacemens) a all imes, and wih he acual values a wo arbirary insans of ime and a every poin of he body, he acual dynamic pah minimizes he funcional,, T q q V q q d 8

19 u u u u 9

20 Remarks Funcional: funcion of funcions Domain: se of admissible funcions A funcional can also be viewed as a funcion of infinie se of variables,, T q q V q q d Opimizaion of funcionals is sudied in he subjec of calculus of variaions. Presenly we have no included exernal forces and damping forces. Immediae objecive: Gain familiariy wih he applicaion of he principle by considering equaions of moion of simple oscillaors and srucural elemens 0

21 Example- k u m Hamilon's principle: Minimize he funcional L d T V d mu ku d

22 u () u u() u () u u u u Admissible funcions Unknown opimal soluion Variaion u u & u u arbirary real number arbirary fucnion 0 sufficienly smooh, reasonably small, & 0 bu oherwise arbirary

23 u u Admissible funcions Unknown opimal soluion Variaion x x & x x arbirary real number arbirary funcion 0 mu ku d L u, u d L u L u d 0 0 for 0 d u u 0 L L d u u L d L L d u d u u d L L d u u 0 0 because is arbirary 0 d

24 d L L 0 d u u L d L L L u d u u mu ku 0 mu ku mu; mu; ku d L L The equaion 0 is called Lagrange's equaion. d u u For discree sysems, wih n dofs, Lagrange's equaion can be generalized as d L L 0; i,,, n d ui ui where LL q i, qi ; i,,, n.

25 Remarks T u, u V u, u d is a funcional mu ku d d The condiion d for minimum of 0 : one parameer family of funcions 0 is no a sufficien condion. The discussion on wheher his condiion implies minimum, maximum, or or saionary value of is no considered here. We will refer o (0) as being he exremum and he opimizing funcion as he exermizing funcion. Even when (0) is a minimum, i need no be he absolue minimum. Refer, for a furher discussion on hese issues, R Weinsock, 974, Calculus of variaions, Dover, NY. 5

26 Remarks (coninued) The condiion for saionariy of he funcional L u, u d d L L is 0 d u u The condiion for saionariy of he F u v funcion, is F F 0 & 0 u v 6

27 Example- u u m k k m n ; DOFs: u & u T m u m u V ku k u u Lu, u, u, u mu mu ku k u u d L L 0; i, d ui ui mu ku k u u 0 m 0 u mu k u u 0 0 m u k k k u k k u7 0

28 Example-3 n ; DOFs: u & k M g L m u X Y T() Mu mx my V Ku mgy n ; q u, X u Lsin ; Y L cos ; X u L cos ; Y L sin 8

29 T () Mu mx my n ; q u, V Ku mgy X u Lsin ; Y L cos ; X u L cos ; Y L sin cos T Mu m u L cos m L sin V Ku mgl L L u,, u, Mu m u L cos m L sin cos Ku mgl L

30 L Mu m cos sin cos u L m L Ku mgl d L L 0 M m u ml cos sin ku 0 d u u d L L 0 ml mlu cos mlu sin mglsin 0 d Remark The approach can handle nonlinear sysems 30

31 Example-4 m M J Mr k k r x x x x T mx + Mx + J Mr = mx+ Mx 4 Mr = mx + 4 V k x k x x Mx x x r No slip r = x x x r x 3

32 Mr x x + L mx Mx k x k x x 4 r d L L M 0 mx x x kx k x x 0 d x x d L L d x x m Mx x x k x x M M x M 3M x Noe T x Mx V x Kx M 0 0 k k k x 0 k k x 3

33 Disribued parameer sysems : Axially vibraing rod x L, AE x, m x u x, Kineic energy: T m xu x, dx Poenial energy: V AE xu x, dx 0 L 0,,, L L 0 Lagrangian L T V m xu x, dx AE xu x, dx L L F u x u x dx 0 0 L 33

34 Le a, u x, u x &, u x, u x Admissible funcions: saisfy he above condiions u x, u x, x, Admissible Unknown opimal Variaion funcion soluion x, is such ha x, x, 0. We will shorly come o he quesion of wha condions x, mus saisfy a x 0 and x L for &. L L m x u x, dx AE x u x, dxd

35 d d L 0 0 a 0 [ m x u x, x, AE x u x, x, ] dxd 0 0 Consider m x u x, x, d L Similarly AE x u x, x, dx m xu x, x, m xu x, x, d L L AE xu x, x, AE xux, x, dx 0 x 0 35

36 d 0 a 0 d L 0,,,, L u AE xu x, mx x, dxd 0 x 0 Since x, is chosen arbirarily, we ake ha each of hese erms are separaely equal o 0. Tha is L 0 L 0 m x u x x dx AE x u x x d x u AE x u x, m x x, dxd 0 m x u x x dx,, 0 AE x u x, x, d 0 L 0 L 0 36

37 Consider L 0 x u AE x u x, m x x, dxd 0 u x, is arbirary AE xu x, m x 0 x Consider Valid over he lengh of he bar,,,,,, m x u x x m x u x x m x u x x x x mxu x x,, 0,, 0 37

38 Boundary condions Consider he erm L,,,, 0 0, 0, AE x u x x AE L u L L AE u 0 The wo erms on he RHS need o be zero individually (again, because he variaion is arbirary) Two siuaions arise: The unknown funcion, ha is, u x,, is no specified a he boundaries (as in he case of a free end of he rod). Here i is "naural" o expec ha he erm AE L u L, or AE 0 u 0, is zero. When he unknown funcion, ha is, u x,, is specified o be zero (as in he case of a clamped end), he variaion, in order ha i confirms wih he sipulaed geomeric consrains, needs o be zero. 38

39 u u AE x mx ; IC-s: u x,0 & u x,0 specified x x Boundary condiions AE 0 u 0, 0 & AE L u L, 0 AE 0 u 0, 0 & u L, 0 u 0, 0 & AE L u L, 0 u 0, 0 & u L, 0 Geomeric, forced, or kinemaic boundary condiion: u x, 0 on he boundary x Free or naural boundary condiion: AE x u, 0 on he boundary 39

40 Remarks AE x u x, a x 0 & L represens he axial hrus a x 0 & L respecively. From a physical sand poin, i is clear ha if he displacemen is specified o be zero, ha is he end is clamped, here would be an axial hrus, represening he reacion, a he boundaries. For a free end, displacemen a he end is no specified. The boundary condiions ha we obain a ends where he unknown field variable is no specified are called he naural boundary condiions (also called addiional or dynamical boundary condiions). 40

41 Remarks (coninued) The boundary condiions ha we obain a ends where he unknown field variable is specified are called he geomeric boundary condiions (also called he essenial or imposed boundary condions). The variaional mehod idenifies he required boundary condions consisen wih he physics of he problem along wih he governing field equaion. 4