Summary of the Class before Exam1

Transcription

1 uar o the lass beore Ea Builing a FEA Moel Ingreients o a FEA sotware pacage teps in builing a FEA oel Moeling consierations D pring/truss Eleents ingle D spring/truss eleent Global stiness atri; properties Bounar conitions hape unctions an their properties D an D truss eleents; rotation atri Bea eleents D bea eleents D bea eleents Distribute loa Fraes Bea eleents with bening an aial loas D bea eleents with bening, aial loas, an torsion

2 Builing A FEA Moel i basic ingreients o a FEA sotware pacage. Tpe o analsis. Geoetr (eine through noes). Eleents. Material properties 5. Bounar conitions. Tie unctions

3 Builing A FEA Moel teps in builing a FEA oel. Discretiation o the geoetr into an equivalent sste o inite eleents with associate noes an choosing an appropriate eleent tpe to closel oel the actual phsical behavior.. election o an appropriate isplaceent (interpolation) unction within each eleent.. Deining the strain/isplaceent an stress/ strain relationships or eriving the equations or each inite eleent. Derivation o the eleent stiness atri an equations. The noal orces are relate to the noal isplaceents using the irect equilibriu etho, or energ ethos or the weighte resiual etho. 5. Assebling the global stiness atri ro the eleent stiness atrices. The introuction o the bounar conitions. 7. olving or the unnown egrees o reeo (or the generalie isplaceents) 8. olving or the eleent stresses an strains 9. Interpretation o the results or use in the esign/analsis process

4 Builing An FEA Moel The ierences between a phsical oel an an FEA oel Developing a phsical oel is a process that sipliies a real-worl proble into a proble that is suitable or FEA. onsierations in builing a phsical oel. The nature o real-worl proble. The cost o conucting FEA onsierations in builing an FEA oel. The accurac o the FEA oel. The cost o conucting FEA Factors that aect the cost:. Degree o Freeos (DOF): p. Nuber o Noes : N. How the noes are nubere. Nuber o Integration Points in Each Eleents 5. Nonlinear Analsis How to estiate the sie (the eor nee) o an FEA oel?

5 Eleent tiness Matri an Global tiness Matri Eleent stiness atri hape unctions, or interpolation unctions: approiation o isplaceent iel Noal orce-noal isplaceent relationship Matri or o above relationship Global tiness Matri Direct stiness atri etho oorinate Transoration e [ ] e K [ ] [ K] [ T ] T T Write own global isplaceent vector Use this vector as an ine or DOFs Airline ticets Bounar onitions oorinate rotation or sewe support ross out soe ters olve global stiness atri olve noal reaction orce (eternal orce) 5

6 pring/truss Eleents For the spring to be in equilibriu, + { } []{ } e [ ] e stiness atri

7 pring/truss Eleents T T u ε u? u + u N N + u [ N N ] hape unctions 7

8 Bea Eleents V In bea theor V In FEA Mechanics o Beas v v V v V ( ) V v w( ) w( ) v w I ( ) 8

9 Bea Eleents v Bea Eleents V ( ) In general, V ( ) v ( ) a + a + a + a Interpolation unction or election. Bea eleent Bounar conitions ( ) a v v ( ) a ( ) a + a + a a v + v Wh we use a thir orer polnoial? ( ) a + a + a 9

10 Bea Eleents: isplaceent unction a v ( ) a + a a ( ) + ( + ) In atri or [ N ]{} + a + a a ( ) ( + ) a v [ ] [ N N N N ] N {} [ ] T ( ) N ( + ) N + N + ( ) N ( ) Note: N ( ) N ( ) ( ) Bea Eleents N N ( )

11 Bea Eleents: tress an train ( ) v ( ) v V Bea Eleents [ ] e K [ ] K e

12 Torsion in FEA onsier torsion onl J G M ( ) GJ GJ Fraes

13 onsier shear, bening, an torsion GJ Fraes

14 Fraes Bea In D pace tart ro ocal oorinate onsier bening in ŷ an onsier shear in ŷ an ẑ ẑ, an torsion in, an aial in,,,,,,

15 5 Fraes GJ AE GJ GJ AE AE Bea In D pace, ocal oorinate

16 hape Functions hape unctions escribe the shape o isplaceent iel an are one o the eterinant actors in governing the eicienc an accurac o FEA. A shape unction taes the value o ONE at the noe associate with it, an taes the value o ZERO at other noes. The choice o shape unctions epens on: The nuber o noes in the eleent. The unction shoul be able to escribe rigi bo otion, an constant strain. Two noes truss: N N Two noes bea bening: ( ) N ( + ) N + N + ( ) N ( )

17 oorinate Transoration Rotation Matri ŷ cosθ sinθ sinθ cosθ θ or cosθ sinθ sinθ cosθ cos θ sin θ sin θ cos θ { } [ T ]{ } { } [ ] T { } [ T ] [ T ] T θ Measure ro to counterclocwise θ Measure ro to counterclocwise 7

18 8 Rotation Matri oorinate Transoration

19 oorinate Transoration oorinate transoration atri ŷ ẑ cos (, ) cos (, ) cos(, ) cos (, ) cos (, ) cos(, ) cos (, ) cos (, ) cos(, ) ŷ ẑ 9

20 oorinate Transoration D an D eleents: Eleent stiness atri in global coorinate { } [ ] T e [ T ] K [ T ]{} T [ ] e K [ ] [ K] [ T ] T

21 pring Eleents Global tiness atri: irect stiness atri etho Eleent I i j N b N atri i j N ( I ) ii ( I ) ij ( I ) ij ( I ) jj [ K] (I ) ii (I ) ij (I ) ij (I ) jj i j N

22 Global tiness atri: Properties. It is setric K ij K ji Global tiness Matri. The eleent on the iagonal o the atri is alwas positive. K ii >. The prouct o the i-th row o the global stiness atri an the global isplaceent atri gives the eternal orce on the i-th DOF o the sste.. Kij is equal to the reaction orce on the i-th DOF ue to a unit isplaceent on the j-th DOF whereas all the other DOFs are ie. 5. The global stiness atri is singular.

23 Bounar onitions Bounar onitions F P P + F F F + P P F How to in? F This etho is applicable to other eleents. Wh we nee B?

24 Incline or ewe upports oeties, a bounar conition can be ore convenientl applie in a local coorinate. α In the local coorinate sste: In the global coorinate sste: cosα sinα sinα cosα

25 5 [ ] [ ] [ ] [ ] [ ] [ ] [ ][ ] [ ][ ] [ ] T K T K T T K K K K K F F F T T Fraes Incline or ewe upports M F F M F F M F F

26 Bea Eleents Distribute loas: Wor Equivalence Metho W istribute w ( ) v ( ) W iscrete ( ) w N ( ) w N w( ) N w( ) N Note, reers to local coorinate, it taes values between an.