MEC-E8001 Finite Element Analysis, Exam (example) 2017

Transcription

1 MEC-E800 Finite Element Analysis Eam (eample) 07. Find the transverse displacement w() of the strctre consisting of one beam element and po forces and. he rotations of the endpos are assmed to be eqal in magnitdes bt opposite in directions i.e. Y Y. Problem parameters E and I are constants. Zz / F F / / X. he XZ-plane strctre shown consists of three beams of eqal properties. Assming that the beams are inetensible in the aial directions derive the anglar speed of free vibrations. Yong s modls E and density of the beam material and the second moment of area I are constants. Assme that A 4I. z 4 z z X Z p X. Determine the (polynomial) eqation giving as its soltion the bckling force pcr of the beam shown. Use one Bernolli beam element. Displacements are confined to the XZ-plane. he cross-section and material properties A I and E are constants. Zz 4. Determine the eqilibrim eqation of the elastic bar of the figre with the large deformation theory. he active degree of freedom is Y and the cross-sectional area and length of the bar are A and withot the po force F acting on node. Constittive eqation of the material is S CE in which C is constant. Use two elements with linear shape fnctions. Yy F X 5. Determine the stationary temperatre distribtion in a Yy thin slab shown. Edge - is at constant temperatre heat fl throgh the other edges vanishes and heat generation rate per nit area is s. Use a rectangle element with bilinear approimation and and as the active degrees of freedom. Consider 4 as known. hickness t thermal condctivity k and heat prodction rate per nit area s are constants. 4 X

2 MEC-E800 Finite Element Analysis; Formlae collection GENERA Displacement: X I Y J ZK i y j zk i vj wk Rotation (small): X I Y J Z K i y j zk i j k i ix iy iz I I j jx jy j Z J J k kx ky k Z K K Coordinate systems: i j k X i Y h Z Strain-stress: yy yy E zz zz y y yz yz G z z E G () or E yy yy [ E] yy ()( ) zz zz zz y y yz G yz z z 0 E [ E] 0 0 0() / 0 E [ E] 0 ()( ) 0 0( ) / inear strain: yy y y zz z z y y y yz y z z y z z z Green-agrange: E E yy y y y y E zz z z z z E y y y E yz y z z y Ez z z y y z z y y y y y y z y z y z z y z y z z z z z y y y y z z y y z y y y z z y z z z y z y z z z PRINCIPE OF VIRUA WORK e W W 0 a W wd ee

3 Bar: w EA et w f ine w A cpl w EA Ω w ()() v v w w CA v w Ω ka p et pω s orsion: GIrr et w w m ine w I rr Bending (z): Bending (y): yy et w w EI w zz et w v EI v w wf z ine w w Aw w I w yy w vf y ine w v Av v I v zz Beam (bckling): w EA GIrr v v EA v EIzzv w w w EI w yy Beam (offset): A S z S y w v E S I I v GI w S w y I yz I yy z zz zy rr f S y f y Sz fz et w v f y w S y f w f v S f Plate (thin-slab): z z z w v y t[ E] v y y v y v w et f v f y ine w t v v E 0 E tc w E yy 0 E yy E 0 0 () y Ey w et g t v g y w cpl Et v y p tk y y p et s Plate (bending):

4 w w t w w yy [ E] w yy w y w y et w wf z ine w t w w wtw w y w y Solid: y v y v V y [ ] y z y z y w v E v v w G v w w w w w z z z z et V f y w v f w f z E E E y Ey V yy [ ] yy yz 4 yz w E C E E G E E E E E zz zz z z g et wv v g y ([ C] [ E]) w g z cpl E wv v y w z V y k y p z z et pv s APPROXIMAIONS (some) N a h Qadratic (line): N N N 4() N ( ) a (bar) Cbic (line): N0 ()( ) N h() N 0 N ( ) N h ( ) 0 z y a () 0 z y (beam z-plane bending) inear (triangle): N y y y y VIRUA WORK EXPRESSIONS X FX X M X et Rigid body: W Y FY Y MY Z F Z Z M Z

5 J ine W y m y y J yy y J z z z zz z Bar: W EA h W et fh W ine Ah 6 W cpl EA P ka h P et sh orsion: W GIrr h W et mh W ine Irrh 6 Bending(z): W z 6h 6h z y EI yy 6h 4h 6h h y z h 6 6 z y y h h 6h h 6h 4h W et z 6 y z h f h z 6 y h W ine z 6 h 6 h 56 h 54 h z y I yy h 4h h h Ah h 4h h h y () 0h 6 h 6 h h 56 h h h h 4h h h h 4h z z y y W sta z 6 h 6 h z y h 4h h h y N z 0h 6 h 6 h z y h h h 4h y h N EA() Beam (y-bending): W y 6h 6h y z EIzz 6h 4h 6h h z y h 6h 6h y z 6h h 6h 4h z W et y 6 z f yh h y 6 h z

6 W ine y 6 h 6 h 56 h 54 h y z Izz h 4h h h Ah h 4h h h z () 0h 6 h 6 h h 56 h y y z h h h 4h h h h 4h z W sta y 6 h 6 h y z N h 4h h h z y 0h 6 h 6 h y z h h h 4h z h N EA() CONSRAINS Frictionless contact: n A 0 Jo: B A Rigid body (link): B A A AB. B A MAHEMAICS i Polar representation: e cos isin sin i i sinh cos i cosh i Eigenvale decomposition: A XλX Matri fnction: If A XλX then f ()() A Xf λ X Newton s method: If R() a a a ()() R a G(a) then R() a 0 a aylor series: n d i ( n) n [ a] i0 i d n f ( a)(a)()()a f f!( )! INEGRAION IN IME (free vibrations) Crank-Nicholson: ( i)() 0 I I / I I / a0 a a t / / α I α I a t i α M Kt Disc. Galerkin: ( i)() 0 α I α / 0 I a0 a a t / / I α α I I a t i α M Kt

7 Find the transverse displacement w() of the strctre consisting of one beam element and po forces and. he rotations of the endpos are assmed to be eqal in magnitdes bt opposite in directions i.e. Y Y. Problem parameters E and I are constants. Zz / F F / / X Soltion Virtal work epression of the ernal forces W z 6h 6h z y EI yy 6h 4h 6h h y z h 6 6 z y y h h 6h h 6h 4h is available in the formlae collection. However eternal part for forces is given only for a constant distribted force and po forces acting on the nodes. Here the po forces are acting inside the element. heir contribtion follows from the definition of work bt the virtal displacement need to be epressed in terms of the displacement and rotation of nodes by sing the cbic approimation for bending. he active degrees of freedom are the rotations which satisfy Y Y ()( ) 0 () Y w ( ) ( ) 0 ( ) Y Y. w ( ) Y At the pos of action of the forces and w ( ) Y Y and 9 w ( ) Y Y. 9 herefore the virtal work epression of the po forces et 4 W Y F YF Y F Virtal work epression of the ernal forces simplifies to 0 6h 6h 0 Y 6h 4h 6h h EI Y EI W 4 0 6h 6h 0 Y 6h h 6h 4h Y Y Y

8 Principle of virtal work and the fndamental lemma of variation calcls imply et EI 4 W W W Y (4) 0 Y F 9 F Y. 9 EI F w ( ) 9 EI

9 he XZ-plane strctre shown consists of three beams of eqal properties. Assming that the beams are inetensible in the aial directions derive the anglar speed of free vibrations. Yong s modls E and density of the beam material and the second moment of area I are constants. Assme that A 4I. z 4 z z X Soltion As the beams are inetensible in the aial directions the active degree of the strctre is Y. et s start with the element contribtions. Since the beam is assmed to be massless only the virtal work epressions of the ernal forces (available in formlae collection) is needed. Z W z 6h 6h z y EI yy 6h 4h 6h h y z h 6 6 z y y h h 6h h 6h 4h W ine z 6 h 6 h 56 h 54 h z y I yy h 4h h h Ah h 4h h h y () 0h 6 h 6 h h 56 h h h h 4h h h h 4h z z y y Element contribtions simplify to ( A 4I ) 8 W W W Y (4) EI Y 0 Y. Virtal work epression of strctre is the sm of element contribtions. W Y () EI Y 5 Y. Finally principle of virtal work and the fndamental lemma of variation calcls implies the ordinary differential eqations: EI Y 5 Y 0. Anglar speed of free vibrations k / m 5 EI

10 p X Determine the (polynomial) eqation giving as its soltion the bckling force pcr of the beam shown. Use one Bernolli beam element. Displacements are confined to the XZ-plane. he cross-section and material properties A I and E are constants. Soltion he degrees of freedom of the strctre are y Y z Z and X. In this case the normal force in the beam N p can be dedced directly and it is enogh to consider only the bending and copling terms of the virtal work epression. As bckling is confined to the XZ plane Zz W EI p () Z Z Y Y W Z EI 6 p 6 Z () Y Y. Principle of virtal work and the fndamental lemma of variation calcls imply 6 6 () Z Y where p. 0EI A non-trivial soltion is obtained only if 6 6 det()( 6)(4 4)(6 ) p cr EI

11 Determine the eqilibrim eqation of the elastic bar of the figre with the large deformation theory. he active degree of freedom is Y and the cross-sectional area and length of the bar are A and withot the po force F acting on node. Constittive eqation of the material is S CE in which C is constant. Use two elements with linear shape fnctions. Soltion Virtal work density of the non-linear bar model Yy F X w v v w w CA v w ()() is based on the Green-agrange strain definition which is physically correct also when rotations/displacements are large. he epression depends on all displacement components material property is denoted by C (constittive eqation S CE ) and the sperscript in the cross-sectional area A (and in other qantities) refers to the initial geometry (strain and stress vanishes). Otherwise eqilibrim eqations follow in the same manner as in the linear case. For element the active dof y Y. As the initial length of the element h linear approimations to the displacement components w 0 and v Y v Y. When the approimation is sbstitted there virtal work density of the ernal forces and thereby the virtal work epression (density is constant) simplify to Y Y CA w () Y W CA. () Y Y For element the active dof y Y. As the initial length of the element h linear approimations to the displacement components w 0 and v () Y v Y. When the approimation is sbstitted there virtal work density of the ernal forces and thereby the virtal work epression (density is constant) simplify to Y Y CA w () Y W CA. () Y Y

12 Virtal work epression of the force is W F Y. Virtal work epression of the strctre is obtained as sm over the element contribtions [()() CA Y ] CA W Y X F. Principle of virtal work and the fndamental lemma of variation calcls imply that () Y F F 0 Y. CA CA

13 Yy Determine the stationary temperatre distribtion in a thin slab shown. Edge - is at constant temperatre heat fl throgh the other edges vanishes and heat generation rate per nit area is s. Use a rectangle element with bilinear approimation and and 4 as the active degrees of freedom. Consider as known. hickness t thermal condctivity k and heat prodction rate per nit area s are constants. 4 X Soltion he density epressions associated with the pre heat condction problem in a thin slab are p tk y y and p et s in which is the temperatre k the thermal condctivity and s the rate of heat prodction (per nit area). For a thin-slab element element contribtions need to be calclated from scratch starting with the densities and approimation. he shape fnctions can be dedced from the figre. Approimation ( /)( /) y ( /)( /) y y y () ( /)( /) y ( /)( /) y 0 y and y 0 y (variation of vanishes). When the approimation is sbstitted there density simplifies to et y y y p p p tk s tk s. Element contribtion is the egral of the density epression over the domain occpied by the element: 0 0 [() ]. P p ddy tk s Variation principle P 0 a and the fndamental lemma of variation calcls imply that

14 tk() 0 s s. tk