CONTINUUM & FINITE ELEMENT METHOD Dirct Approach or Discrt Systms On-Dimnsional Elmnts Pro. Song Jin Par Mchanical Enginring, POSTECH
Dirct Approach or Discrt Systms Dirct approach has th ollowing aturs: It applis physical concpt dirctly to discrtizd lmnts. (.g. orc quilibrium, nrgy consrvation, mass consrvation, tc.) It is asy in its physical intrprtation. It dos not nd laborat sophisticatd mathmatical manipulation or concpt. It is limitd to crtain problms or which quilibrium or consrvation law can b asily statd in trms o physical quantitis on wants to obtain.
Dirct Approach: Exampl Forc Balanc (Linar Spring Systm) Solv th displacmnts at th nods, and th raction orc at th nod numbr. 5 6 5 x 5 6 On lmnt Nod Nod ( ( ) ),
Dirct Approach: Exampl [ ] { } { } K φ Global matrix quation by Assmbly [ ]{ } { } F K whr [K] is calld Stinss Matrix and {F} is calld Rsultant Nodal Forc Matrix. Elmnt Matrix Equation or on lmnt:
Dirct Approach: Exampl Th physical maning o assmbly procdur can b ound in th orc balanc as dscribd blow: () () () () () () ( ) ( ) F () ( ) Extrnal nodal orc applid at nod
Th typical assmbly can b don as shown blow: 6 5 6 5 5 5 5 5 bandd and symmtric matrix Dirct Approach: Exampl
Dirct Approach: Exampl Now, lt us pay attntion to th boundary condition in this particular cas Displacmnts: : spciid,,, 5, 6 : unnown Gomtric condition, Essntial boundary condition Forcs:,,, 5, 6 :spciid : unown raction orc Forc condition, Natural boundary condition
Dirct Approach: Exampl Boundary Condition Rquirmnt Only on o th displacmnt and orc can b spciid as a boundary condition. Natur dos not allow to spciy both th displacmnt and orc simultanously at any nod. I non o th two is nown, thn th problm is not wll posd!
Dirct Approach: Exampl Enrgy Consrvation (-D Hat Conduction) 5 Q Q Q Q Q5 Q6 5 T T T T T 5 T 6 whr Q i rprsnts hat lux input into th systm through nod i.
Dirct Approach: Exampl On lmnt: q q q q T T Enrgy consrvation: Fourir Law: T Q x q q ( T ( T T T ) ) Q ( T ( T T T ) ) with : hat lux ntring th lmnt through nod : hat lux ntring th lmnt through nod
Elmnt Matrix Equation or on lmnt: [ ] { } { } K q q T T φ Global matrix quation by Assmbly: [ ]{ } { } Q T K whr [K] is calld Stinss Matrix and {Q} is calld Rsultant Nodal Enrgy Inlux Matrix. Dirct Approach: Exampl
Dirct Approach: Exampl Th physical maning o assmbly procdur can b ound in th orc balanc as dscribd blow: q () q () K K q () q () T T T Extrnal nrgy inlux ntring th systm through nod
6 5 6 5 5 5 5 5 Q Q Q Q Q Q T T T T T T Th assmbly can b don in th sam mannr as or th prvious xampl: Again, th global stinss matrix is bandd and symmtric. At last on ssntial boundary condition should b assignd in ordr to liminat th singularity. Dirct Approach: Exampl
Dirct Approach: Exampl { Tmpratur : Essntial boundary condition Enrgy lux : Natural boundary condition Thr ar pairs o ssntial and natural boundary conditions or ach nod. Only on o thm should b providd or ach nod.
Dirct Approach: Exampl Mass Consrvation (Flow ntwor, Elctric ntwor) Q sourc Qauct Th problm is to ind watr low rat in th pips and aucts givn th watr low rat rom th rsrvoir.
Dirct Approach: Exampl On lmnt: q P P q L, D µ :watrviscosity Mass consrvation: Fluid mchanics> ( P q 8qLµ P ) πd ( P P ) with πd 8Lµ q q ( P ( P P ) P ) : masslow ratntringthlmntthroughnod : masslow ratntringthlmntthroughnod (Not that q q or mass consrvation)
Dirct Approach: Exampl Elmnt Matrix Equation or on lmnt: [ ] { } { } K q q P P φ Global matrix quation by Assmbly: [ ]{ } { } Q P K Th assmbly procdur is idntical to th prvious two xampls.
Dirct Approach: Exampl F F 5 7 i : lmnts i : nods y x F 6 5
Dirct Approach: Exampl On lmnt: Forc-Dormation Law: v, y u, x Elastic Elongation: F AE L L y v, y x u, x E, A, L Elmnt Matrix Equation or on lmnt: [ K ] u v u v x y x y
Dirct Approach: Exampl In ordr to ind th coicints o th lmnt stinss matrix, K ij, considr th ct o displacmnt v o unity with u u v or K i. F L v sin sin F AE sin L v F L x y x y F cos K, K, K, K F sin x y
Dirct Approach: Exampl Similarly, on can obtain th othr coicints, rsulting in th ollowing lmnt stinss matrix: : symmtric (Not that Fx Fy M ar satisid automatically.)
Dirct Approach: Exampl Global matrix quation by Assmbly:. Idntiy th global nodal numbr i and j corrsponding to th two nd-nods and o th lmnt to b addd.. Add x lmnt matrix componnts to th corrsponding rows and columns o i-, i, j-, j in th global stinss matrix.
[ ]{ } { } F K Dirct Approach: Exampl yj xj yi xi j j i i v u v u i- j- i- j- i j j
Dirct Approach: Exampl Stinss Matrix, Flxibility Matrix For th dormation problms, thr ar two inds o approachs dpnding on whichvariabl is considrd unnown to b solvd or. i) [K]{x}{F} with [K] bing stinss matrix K ij : inlunc coicint which rprsnts orc F i du to unit displacmnt o x j Displacmnt-basd FEM ii) [I]{F}{x} with [I] bing lxibility matrix I ij : inlunc coicint which rprsnts displacmnt x i du to unit orc o F j Forc-basd FEM
Coordinat Transormation Coordinat Transormation or Dirct Approach Global Coordinat Systm > Local Coordinat Systm Local Coordinat Systm : Much asir to dtrmin stinss matrix
Coordinat Transormation y y u v u v u u x u v u v Global systm u u ' ' Local systm x
Coordinat Transormation i) Vctor Transormation in -D y y (x,y) or(x,y ) V xi yj x i y j x cos x sin y y sin x cos y j j i i x x x cos y sin x cos y sin sin x cos y sin x cos y or
Coordinat Transormation ii) Transormation o stinss matrix Suppos th lmnt stinss matrix is rprsntd by th ollowing quation: [ K ] { x } { b } { x } { } Th vctor transormation o and b btwn th local and global coordinat systm might b { x } [ Φ]{ x} { b } [ Φ]{ b} thn [ K ] [ Φ]{ x} [ Φ]{ b} [ K ] [ Φ] [ K ] [ Φ]
Coordinat Transormation Exampl v, u, y v, y u, x E, A, L y u, u, x u, u, x L u u σ Eε, ε L L EA σa ( u u ) L σa EA [ K ] L
Coordinat Transormation Exampl sin cos sin cos u u v u v u [ ] Φ sin cos sin cos v u v u u u [ ] Φ ([ ] [ ] ) T Φ Φ
Coordinat Transormation Exampl cos sin cos sin [ ] [ Φ] [ ] [ Φ] EA K K cos sin L cos sin Finally, on can obtain th stinss matrix as blow [ ] K cos AE sin cos L cos sin cos sin cos sin sin cos sin cos sin cos cos sin cos sin cos sin sin cos sin
Coordinat Transormation Exampl You hav to rcogniz th ollowing acts: [ ][ ] [ ] x I Φ Φ whil [ ] [ ] [ ] x sin sin cos sin cos cos sin sin cos sin cos cos I Φ Φ But rcognizing that [ ] [ ] Φ Φ y x y x y x y x tc., sin cos y x tc.) (i.., sin cos sin cos cos x x x y x