SOME BRIEF FINITE ELEMENT MODELING NOTES

Transcription

1 SOME BRIEF FINITE EEMENT MODEING NOTES (EXCERPTED FROM FINITE EEMENT MODEING NOTES B PETER AVITABIE) SO WHAT AM I TRING TO DO WHEN MAKING A FEM??? CONTINUOUS SOUTION DISCRETIED SOUTION MODEING ISSUES contnos soltons wor well wth strctres that are well behaved and have no geometry that s dclt to handle most strctres don't t ths smple reqrement (except or rsbees and cymbals) real strctres have sgncant geometry varatons that are dclt to address or the applcable theory a dscretzed model s needed n order to approxmate the actal geometry the degree o dscretzaton s dependent on the waveorm o the deormaton n the strctre nte element modelng meets ths need Some Bre Fnte Element Modelng Notes - Rev 00 Peter Avtable

2 FINITE EEMENT MODEING OVERVIEW Fnte element modelng nvolves the descretzaton o the strctre nto elements or domans that are dened by nodes whch descrbe the elements A eld qantty sch as dsplacement s approxmated sng polynomal nterpolaton over each o the domans The best vales o the eld qantty at nodes reslts rom a mnmzaton o the total energy Snce many nodes dene many elements, a set o smltaneos eqatons reslts Typcally, ths set o eqatons s very large and a compter s sed to generate reslts F A, E F θ θ T J, G T ν ν θ E, I θ F F Some Bre Fnte Element Modelng Notes - Rev 00 Peter Avtable

3 FINITE EEMENT MODEING OVERVIEW A TPICA FINITE EEMENT USER MA ASK what nd o elements shold be sed? how many elements shold I have? where can the mesh be coarse; where mst t be ne? what smplyng asspmtons can I mae? shold all o the physcal strctral detal be nclded? can I se the same statc model or dynamc analyss? how can I determne my answers are accrate? how do I now the sotware s sed properly? A THESE QUESTIONS CAN BE ANSWERED, IF the general strctral behavor s well nderstood the elements avalable are nderstood the sotware operaton s nderstood (npt procedres, algorthms,etc) BASICA - we need to now what we are dong!!! IF A ROUGH BACK OF THE ENVEOP ANASIS CAN NOT BE FORMUATED, THEN MOST IKE THE ANAST DOES NOT KNOW ENOUGH ABOUT THE PROBEM AT HAND TO FORMUATE A FINITE EEMENT MODE Some Bre Fnte Element Modelng Notes - Rev 00 Peter Avtable

4 FINITE EEMENT MODEING OVERVIEW Nodes are sed to represent geometrc locatons n the strctre Element bondary dened by the nodes The type o dsplacement eld that exsts over the doman wll determne the type o element sed to characterze the doman Element characterstcs are determned rom Theory o Elastcty and Strength o Materals Fnte element method s a nmercal method or solvng a system o governng eqatons over the doman o a contnos physcal system The bass o the nte element method s smmarzed below sbdvde the strctre nto small nte elements each element s dened by a nte nmber o node ponts assemble all elements to orm the entre strctre wthn each element, a smple solton to governng eqatons s ormlated (the solton or each element becomes a ncton o nnown nodal vales) general solton or all elements reslts n algebrac set o smltaneos eqatons v t s Some Bre Fnte Element Modelng Notes - Rev 00 4 Peter Avtable

5 FINITE EEMENT MODEING OVERVIEW Usng standard nte element modelng technqes, the ollowng steps are sally ollowed n the generaton o an analytcal model node generaton element generaton coordnate transormatons assembly process applcaton o bondary condtons model condensaton solton o eqatons recovery process expanson o redced model reslts FINITE EEMENT MODEING OVERVIEW All strctres are dmensonal n natre bt many tmes smplyng assmptons can be assmed wth no other loss n accracy Elements are typcally categorzed as Strctral Elements Contnm Elements Strctral element ormlatons se the same general assmptons abot ther respectve behavor as ther respectve strctral theores (sch as trss, beam, plate, or shell) Contnm element ormlatons (sch as D and D sold elements) comes rom theory o elastcty A wde varety o derent element types generally exsts n most commercally avalable nte element sotware pacages Typcal strctral elements are mass, trss, beam, membrane, plane stress/plane stran, thn plate, thn shell, thc plate, dmensonal sold wth a varety o shape nctons rangng rom lnear to hgher order polynomal Some Bre Fnte Element Modelng Notes - Rev 00 5 Peter Avtable

6 FINITE EEMENT MODEING OVERVIEW Element Denton Each element s approxmated by where {} δ = [ N]{} x {δ} - vector o dsplacements wthn element [N] - shape ncton or selected element {x} - nodal varable Element shape nctons can range rom lnear nterpolaton nctons to hgher order polynomal nctons A smple llstraton o shape nctons s shown below DISTRIBUTION PATTERN TO BE APPROXIMATED SHAPE APPROXIMATION TOOS AVAIABE Some Bre Fnte Element Modelng Notes - Rev 00 6 Peter Avtable

7 FINITE EEMENT MODEING OVERVIEW Stran Dsplacement Relatonshp The stran dsplacement relatonshp s gven by where {} ε = [ B]{} x {ε} - vector o stran wthn element [B] - stran dsplacement matrx (proportonal to dervatves o [N]) {x} - nodal varable Mass and Stness Formlaton The mass and stness relatonshp s gven by where T [ M] = [ N] ρ[ N] V V T [ K] = [ B] [ C][ B] V V [M] - element mass matrx [K] - element stness matrx [N] - shape ncton or element {ρ} - densty [B] - stran dsplacement matrx [C] - stress-stran (elastcty) matrx Some Bre Fnte Element Modelng Notes - Rev 00 7 Peter Avtable

8 Some Bre Fnte Element Modelng Notes - Rev 00 8 Peter Avtable FINITE EEMENT MODEING OVERVIEW Coordnate Transormaton Generally, elements are ormed n a local coordnate system whch s convenent or generaton o the element Elemental matrces are transormed rom the local elemental coordnate system to the global coordnate system sng { } [ ]{ } x T x = OCA SSTEM GOBA SSTEM φ Assembly Process Elemental matrces are then assembled nto the global master matrces sng { } [ ]{ } g x c x = where {x } - element degrees o reedom [c ] - connectvty matrx {x g } - global degrees o reedom x x X X X X

9 FINITE EEMENT MODEING OVERVIEW Bondary Condtons Elemental matrces are then assembled nto the global master matrces sng K x = F [ ]{ } { } n n n [ Kaa ] [ Kab] xa Fa = [ K ] [ K ] x F ba bb b b where 'a' dentes solton varable and 'b' dentes a bonded do The eqaton or solton s K x + K x = F [ aa ]{ a} [ ab]{ b} { a} [ K ]{ x } = { F } [ K ]{ x } aa a a where the eqaton or the reacton loads s K x + K x = F ab [ ]{ } [ ]{ } { } ba a bb b Bondary Condtons - Method - Decople Eqatons Set o-dagonal terms to zero [ Kaa ] xa F = [ ] Kbb xb b b { a} [ Kab]{ xb} [ K ]{ x } Bondary Condtons - Method - St Sprng Apply st sprng to bonded dos (approx zero o-dagonal) [ Kaa ] [ Kab] xa { Fa } = [ ] [ ] Kba Kbb + Kst xb Kst Fb bb b [ ]{ } Bondary Condtons - Method - Partton Eqatons Partton ot bonded dos K x = F [ ]{ } { } aa a a Some Bre Fnte Element Modelng Notes - Rev 00 9 Peter Avtable

10 FINITE EEMENT MODEING OVERVIEW Types o Bondary Condtons X FREE - FREE EFT END - FREE RIGHT END - FREE X SIMPE SUPPORT EFT END - X=0, =0 RIGHT END - =0 X CANTIEVER EFT END - X=0, =0, R=0 RIGHT END - FREE X BUIT IN BOTH ENDS EFT END - X=0, =0, R=0 RIGHT END - X=0, =0, R=0 X BUIT IN BOTH ENDS - HAF MODE EFT END - X=0, =0, R=0 RIGHT END - X=0, R=0 Some Bre Fnte Element Modelng Notes - Rev 00 0 Peter Avtable

11 FINITE EEMENT MODEING OVERVIEW Solton Technqes Statc Soltons typcally nvolve decomposton o a large matrx matrx s sally sparsely poplated maorty o terms concentrated abot the dagonal Egenvale Soltons se ether drect or teratve methods drect technqes sed or small matrces teratve technqes sed to extract a ew modes rom a large set o matrces Propagaton Soltons most common solton ses dervatve methods stablty o the nmercal process s o concern at a gven tme step, the eqatons are redced to an eqvalent statc orm or solton typcally many tmes steps are reqred FINITE EEMENT MODEING OVERVIEW - THE EEMENTS TRUSS TORSIONA ROD STRUCTURA EEMENTS D BEAM PATE CONTINUUM EEMENTS DEGREES OF FREEDOM maxmm 6 do can be descrbed at a pont n space nte element se a maxmm o 6 do most elements se less than 6 do to descrbe the element characterstcs Some Bre Fnte Element Modelng Notes - Rev 00 Peter Avtable

12 FINITE EEMENT MODEING OVERVIEW - THE EEMENTS TRUSS BEAM TORSION D SOID PATES SHES D SOID slender element (length>>area) whch spports only tenson or compresson along ts length; essentally a D sprng slender element whose length s mch greater that ts transverse dmenson whch spports lateral loads whch case lexral bendng same as trss bt spports torson element whose geometry denton les n a plane and appled loads also le n the same plane plane stress occrs or strctres wth small thcness compared wth ts n plane dmenson - stress components assocated wth the ot o plane coordnate are zero plane stran occrs or strctres where the thcness becomes large compared to ts n plane dmenson - stran component assocated wth the ot o plane coordnate are zero element whose geometry les n the plane wth loads actng ot o the plane whch case lexral bendng and wth both n plane dmensons large n comparson to ts thcness - two dmensonal state o stress exsts smlar to plane stress except that there s a varaton o tenson to compresson throgh the thcness element smlar n character to a plate bt typcally sed on crved srace and spports both n plane and ot o plane loads - nmeros ormlatons exst element classcaton that covers all elements - element obeys the stran dsplacement and stress stran relatonshps Some Bre Fnte Element Modelng Notes - Rev 00 Peter Avtable

13 EEMENT TPES TRUSS slender element (length>>area) whch spports only tenson or compresson along ts length; essentally a D sprng F A, E F The trss stran s dened as ε= d dx The trss stness and lmped/consstent mass matrces are / / / / 6 / 6 AE [ ] = ;[ m ] = ρa ;[ m ] = ρa l c / TORSION smlar to trss bt spports torson θ θ T J, G T The torsonal stness matrx s JG [ ] = t Some Bre Fnte Element Modelng Notes - Rev 00 Peter Avtable

14 EEMENT TPES BEAM slender element whose length s mch greater that ts transverse dmenson whch spports lateral loads whch case lexral bendng Beam assmptons are constant cross secton cross secton small compared to length stress and stran vary lnearly across secton depth 4 4 The beam elastc crvatre de to lateral loadng s satsed by EI d υ / dx = q The longtdnal stran s proportonal to the dstance rom the netral axs and second dervatve o the elastc crvatre gven as ε = yd υ/ dx ν ν θ E, I θ F F The stness and consstent mass matrces are EI [ ] = ; [ m] ρa = Some Bre Fnte Element Modelng Notes - Rev 00 4 Peter Avtable

15 EEMENT TPES BEAM The ll beam stness matrx can be assembled sng the trss, torson and two planar beam elements (one on plane and one ot o plane) AE AE EI 6EI EI 6EI EI 6EI EI 6EI JG JG 6EI 4EI 6EI EI 6EI 4EI 6EI EI AE AE EI 6EI EI 6EI EI 6EI EI 6EI JG JG 6EI EI 6EI 4EI 6EI EI 6EI 4EI Some Bre Fnte Element Modelng Notes - Rev 00 5 Peter Avtable

16 FINITE EEMENT MODEING OVERVIEW A smple sprng example s sel to llstrate the nte element process Consder the sprng system shown below each sprng element s denoted by a box wth a nmber each element s dened by nodes denoted by the crcle wth a nmber assgned to t the sprngs have a node at each end and have a common node pont the dsplacement o each node s denoted by wth a sbscrpt to denty whch node t corresponds to there s an appled orce at node FINITE EEMENT MODEING OVERVIEW The rst step s to ormlate the sprng element n a general sense p p p the element label s p the element s bonded by node and assme postve dsplacement condtons at both nodes dene the orce at node and node or the p element Applcaton o smple eqlbrm gves p p = = p p ( ( ) = + ) = p p + p p Some Bre Fnte Element Modelng Notes - Rev 00 6 Peter Avtable

17 Some Bre Fnte Element Modelng Notes - Rev 00 7 Peter Avtable FINITE EEMENT MODEING OVERVIEW Ths can be wrtten n matrx orm to gve = p p p p p p Now or element = And or element = The eqlbrm reqres that the sm o the nternal orces eqals the appled orce actng on each node FINITE EEMENT MODEING OVERVIEW Three eqatons can now be wrtten as = + = + + = or n matrx orm = + Now applyng a bondary condton o zero dsplacement at node has the eect o zerong the rst colmn o the K matrx whch gves three eqatons wth nnowns Solvng or the second and thrd eqaton gves = + 0

18 FINITE EEMENT MODEING CONSIDERATIONS TRUSS EEMENTS ASSEMBED TOGETHER AE / -AE / -AE / AE / AE / -AE / -AE / AE / AE / -AE / -AE / AE / AE / -AE / -AE / AE / + AE / -AE / -AE / AE / + AE / -AE / -AE / AE / EEMENT ASSEMB the elements can be assembled nto one matrx Some Bre Fnte Element Modelng Notes - Rev 00 8 Peter Avtable

19 FINITE EEMENT MODEING CONSIDERATIONS SOME COMMON MATERIA TERMS/DEFINITIONS Elastc or ong's Modls (E) gves a drect ndcaton o stness and s the rato o stress to stran Shear Modls (G) or Modls o Rgdty s the raton o shear stress to shear stran Mass densty (r) s the weght densty dvded by the acceleraton de to gravty Posson's Rato (n) s the raton o lateral stran to extensonal stran near Isotropc materal has materal constants o elastc modls, shear modls, Posson's rato and thermal expanson whch are all constant propertes whch are ndependent o the coordnate system o the element near Ansotropc materal has materal constants dened by a 6x6 symmetrcal matrx and 6 terms or thermal expanson whch are dependent on drectonal orentaton n the materal near Orthotropc materal s a specal case o Ansotropc materal whch contans 4 ndependent constants FINITE EEMENT MODEING APPROXIMATIONS Approxmaton o the bondary condton s appled n the nte element model at the node ponts and not along the srace o the element Dstrbted orces are appled n an approxmate sense at the nodes o the model and not actally dstrbted as n the real world sense Some Bre Fnte Element Modelng Notes - Rev 00 9 Peter Avtable

20 FINITE EEMENT MODEING CONSIDERATIONS COMMON MODEING BUNDERS nconsstent set o nts (e, materal n ps - model n eet) weght densty sed nstead o mass densty polar moment o nerta (J) sed nstead o torsonal constant (J) beam orentaton - and - swtched aspect rato ncorrect symmetry bondary condtons ncorrectly speced never se smple model rst to assre closed orm solton can be obtaned or nderstand the sage o the modelng technqe parts o the model not hooed together msnterpretaton o local/global coordnate systems a ner mesh never sed to assre convergence o the model relctance to read ser & theoretcal manals assme sotware shold behave a certan way becase o amlarty o how a derent sotware pacage behaves gnorance o warnng and error messages snce they appear to be wrtten n a oregn langage Some Bre Fnte Element Modelng Notes - Rev 00 0 Peter Avtable

21 FINITE EEMENT MODEING MATAB SCRIPT FIE Peter Avtable - Modal Analyss & Controls aboratory Unversty o Massachsetts owell Ths MATAB le s sed to develop the reqences and mode shapes or a cantlever beam sed or ME40 Fnal Proect The model s dened wth 0 beam elements wth do/node (shear/rotary) The parameters are dned below (node nmbers) ----x----x----x----x----x----x----x----x----x----x O (mass at tp o beam) =============================================================================== ncrementers and conters =============================================================================== nel = 0; nmber o beam elements nodes = nel + ; total nmber o nodes ndpn = ; nmber o DOF per node n = nodes*ndpn; total nmber o DOF n model beore BC added n = n - ndpn; total nmber o DOF ater blt-n BC added nnc = ; ncrement or beam element assembly n mass and stness matrces =============================================================================== physcal parameters =============================================================================== E = 0e6; ong's Modls (ps) b = 0998; beam dmenson (nch) h = 05; beam dmenson (nch) I = /*b*h^; area moment o nerta (nch**4) length = 75; total length o beam rom constrant (nch) len = length/nel; length o ndvdal beam element (nch) rho = 0/864; mass densty (not weght densty) A = b*h; cross sectonal area (nch**) m_acc = 0/864 assme accelerometer weghts 00 lb Some Bre Fnte Element Modelng Notes - Rev 00 Peter Avtable

22 FINITE EEMENT MODEING MATAB SCRIPT FIE (CONTINUED) =============================================================================== Setp and Assemble Mass and Stness Matrces =============================================================================== Kn = zeros(n,n); setp ntal matrx space or stness Mn = zeros(n,n); setp ntal matrx space or stness =============================================================================== ndvdal element characterstcs =============================================================================== element = beam(e,i,len); ==>> ==>> EXTERNA SCRIPT FIE NEEDED!!! melement = mcbeam(rho,a,len); ==>> ==>> EXTERNA SCRIPT FIE NEEDED!!! =============================================================================== assemble ndvdal elements nto matrces =============================================================================== [Kn] = assemble(kn,element,[,,,4],nel,nnc); ==>>SCRIPT FIE NEEDED!!! [Mn] = assemble(mn,melement,[,,,4],nel,nnc); ==>>SCRIPT FIE NEEDED!!! [Mn] = assemble(mn,m_acc,,,); add accel mass at do= at tp do =============================================================================== constran system by removng eqatons assocated wth bondary =============================================================================== Kn = Kn(:n,:n); remove rst two eqatons rom stness matrx Mn = Mn(:n,:n); remove rst two eqatons rom mass matrx =============================================================================== perorm egensolton to obtan reqences and mode shapes =============================================================================== [shapes,req] = egen(kn,mn); ==>> ==>> EXTERNA SCRIPT FIE NEEDED!!! gre() plot(shapes((::(n-)),(::))); plot all three modes ttle('mode Shape - rst three modes - 0 elements') req(::) Some Bre Fnte Element Modelng Notes - Rev 00 Peter Avtable