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' 9FCFNTAOVANCE5 IN HYBRIDIMIXEI)F|I([TELEMENTS

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:;''7:::_.: I Ma_s_chusetlI;snstituteof Technology ' , O_OINA_ PAGE IS I'

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.........': _mbridge, Ma_. 0X139 U._.A. I l

.;.Y._,;::.:,",_._ ________..___._.__...._. . . , QUALITY .

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,::,',!:h!H :no:.,,'.' ;' S..._a__J.U. , displacement_,

:K'".':',:'

i!:i:,;::::: i, . In two new formulationsof Hybridllllxefdinit_ Many alternativefiniteelementmethods that have {

";:....elementmethodsrespectivelyby the Helllnger.Reis_,xer been developedare bared on the introductionof addl- )

,;,!i,.:i: principle and the Hu-Washizu principle, the stre_$ equt_ , tional field variables. Within the element more than ,

':)..':llbrlum .' equationsare broughtin as conditionsof con- _ one field variablecan be introducedsuch as methods

,,',....... ' iderlvedby the Hellinger-Relssnerprin:lolefor which ,,

)i.ii:,!:.;straintthroughthe introductionof additionalinternalI# {stressesand displacementsare used and methods derivedio '

/i')i')i'i' displacement ,")."_:parameters. The new approachesare more ( Iby the Hu-Washlzuprinciplefor which stresses strains[ ::":"i:if.lexlble 1'..".an:i1Ti' d have better computingefflciencles,A I ' ,

':iY:')roce i'':'dur : efor the choice of assumed stresstermsfor + }and displacementsare used. Such models are named )' "'t::t.'."t): # ' r

'ii_iii 3-0 solids is suggested. Example solutionsare'g".iven'( imtxed mooels. When additionalvariablesare introduce:d."

........... ; 'aS Lagrangemult_pl_ersfor the maintainingcontinuity)

i_ii_[f."or plates and shellsusing the presen:fomulati_s T

........... )conditionsalong _he _nterele_entbounoaryit is la- "

:M'._i:.:Lt_.,i!!la he idea ofndsemiloofelements. (

!j!i!/I._._j{i . | )belledas a hybridmodel. These two models are not i._m

......... ..: elementmethod can be derivedusing di,_ferenvaria- t }:

>:-i"_.... " Fin( o elementmethodscan be consideredas con- _

l:,[',:.:::'ii .... "" " _. itionalformulations. It is, thus,decide_ thek a dis- (

,.:'..-1:1v_nientreth , ods for solvingpartialdifferential)qua- i

itlons. In the case of solidmechan'icsthe most co.-_.oni (cussionof non-primalfiniteelementmethod be _urnoed 1 (togetherunder the name Hybrid/Mixedfiniteelement

......:formula%ionis to use displacementsas fieldvariables' !r._thOd. ' :

,.::::::i. ) .

in the differentialequations, The so-called conven- ( t

.......... ) The assumed stress hybridmethod which makes use

"_.::/St(':/a:nal'_"fi.niteelementformulationis an approxir.,ate ) k

.:.'_:::::_ }of boundarydisplacementsas additionalvariables,and

._.ji.._.'Ti.i)_.) !solutibnbasedon assumeddlsp_acementsand the corres-! ihenceleads to matrix displacementmethods in the fi- _(]

i'_!i'[iii; )pendingvariationalprincipleis the principleof the ! initeelementformulation,can overcomethe shortcominqs_'

:':\"'.'):.'{!'_t:C sta'tionary _ potentialenergy. Such method can thusj ,r •

':'_'")b:e":c'alleda Primal FiniteElementMethod. For solid ( iof the assumeddisplacementmethocsthat were mentienedi'

_::_'1 i'"'"".... '!'_"!'/1_":: _ _above, However it also needs furtherImprovements.

""_"_" ,.,:F.,:.:IUm:::'echanics problems a dual 'formulation can be made using' ',The main ones are (1) the lack of a guideline for the '__'

)_/'j?,stress '_J'.')functions :_i as field variablesand the correspon-, choice of the assumedstressterms, (2) the excessive ":

_6.!!)_))il)) clingcomplementaryon'orgyprinciple, Applicationsof ,'.

computingtime needed in the inversionof a flexibility

iiii):!!_i:_i) Isuchprln:Iplefor the finiteelementmethod have•been

_-:iTi:j Imade,but.are net popular. ) m_trlx during the processof constructingthe element

stiffnessmatrix and (3) the difficultyin the choice

"i')))Y_)))'ii Ate rathere,:'lyage of finite element develop- I

"i'i'.i'iiii'i?i merit,short-cos,lngs " of the assumeddisplacementmethod _ of equilibratingstressesin the case of complex prob-

".".'!M_I" were dlscovered. :'.':'_'I/_ The main one is the difficultyIn _. lems such as shells,

!}'i'))'_),})).i const,uctingshape functionsfor thin platesand shellst The objectivesof the presentlectureare:

....: VC'":_'m'::: ..... C' _ ,(1) TO presenttwo recentlydevelopedHybrid/Mixed

I!" i'.':_::':::{6.'V,. '_'liwhlrh t:fonsraerequire that thecontlnulty-conditlons_ so-calledcompatibleeOth lemeerntobserve- sare i; finite elementformulationsbased,respectivelyon the I'

_.. LL

.....

.::':,.,"::

::

_=

.

;'_ too/rigid :" and, indeed,for some limitingcases, such as!

Hellinger-Relssnerprincipleand the Hu-Washi_uprlncl.)

i._,J_/:_;;: ,ple for which stress equilibriumconditionsneed not be!

,earlyincompressiblmeaterialsor thin plates and" _ [

{_:_!r_hells whichare formuhted by taking transverseshear-: iimposedinitiallybut'arebroughtin as conditionsof '. constraint,instead,by Includlngadditionalinternal '

ing stra':nsinto account,the correspondingfiniteale- .displacementparameters, Thesenew approachesmake the.

_ :meritmodelmay be completelylocked. Anotherarea that {

icallsfor improvementis to avoid the loss in accuracy formulation more flexibleand also improve the comput.

'when

•

the ev_luatlonof strainsand hencestressesare ing

(2)

effici

To pre

e

s

ncy.

ent a systematic procedure for the choice ):;

accomplishedby differentiationof the as_ur,.ed of the as;u-nH s+r*_._+,,-_,+h,,_vil .* l "

......... - 1985013377
, ORIGINAL PAG_ I_

' W _2)OR QUAGITY -2-

!

'prevent the appearnace of kinematic deformation modes, With stresses expressed in terms of stress pare•c- "

::'!_ii!ii:::Three dimensional elements are used as examples, tens _,

i,i'!!!!i!!

s);;

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2 _"= "p"_ (5)

! •*t,*_tt¶'-

::,',.:'../:_':!!',,F " and the elemen't displacements u or boundary displace- ,

:::_Y:/,,L:: 2__HvbFjd/Mtxed__Fprmu!ati.onby_H_el]Jngq_r-Retp__nor monte _ interpolated tn terms of nodal displacements _,

;,','.;_:."::,_.,,y.,;:,, Elementstiffnessmatrix can be constructedusing u . N 9 (6)

',i:'..:ilffe, '6::ent : variationalmethods, The initialstep in com-

:::i!)_i',',/men ts to assign nodal displacements q and ¢orrespon- u• c..o (7)

i::i,i,l,i Itlowith na:iiiii! lpner!in¢g:hbIple Ityoriinngesoleme lidntmecs,haTnhiencsEa ysusolution inganfor y vasrlucah- whereH" • /V PTs ....P dV (9))

/.':_,la,_,_probl :,:.emwith prescribedboundaryconditionscan be and G . IV pTN dV by _R (I0)

',/:,,',? )obtained. Such solutionsmay be in terms of displace-

/":!/i:i::_ )ment/s,::)::) stressesand/orstrains. The elementstrain " " aJ'VpT T"ds .... '% (lli /i

"'.':::'!'T":'.'!'!" Tener : gy U can thus be expressed in terms of the nodal 1

! (displacement_s, and the elementstiffnessmatrixk can ment Thestiffness n oyields matrixiobtained nte sofrfq. om tThe he strain resene ulrele- gty

l,

............ be obtained, i :'

:'.... _/',,,.,** +L,!,I:':I" expression is I

//.:////// ) " '/:In,zus: ingthe Helllnger-Relssneprrincipleto .

/:.'T..:".':i6:[ /:.... ,so ./i'-:::lve..the: above bounOaryvalueproblem,the following k..= G.T...H"I G.. (12)

::.;....:::::.)"functionalshouldbe maintainedstationary,

':/::://././.t'_,:' _. oT( = In a new formulation,the stress equilibriumcon- ( V °T" - 9u)]dV " i= ditionsEq. (3) are not satisfiedbut the elementdis- ..,,,

____o____( /__o_ i placements u are expressed fn terms of nodal displace-

.1..,., . $ ?T(_ _) dS (I) mentsq and additionalinternaldisplacementparameters

I I i.e.

:?':.':!w_ hereS is the elasticcompliancematrix and u = Uq +U;k- Nq + MX (13)

'm::::''::; :_:_):_:_:'::_':l :::' " _ T. = .9 .a" (2)" Here _ are compatiblewith the boundarydlsolace..

..... ..-oo°. _ .:

:..:.:._./:'._r":':ela_:.tes the boundary tractions and the stresses, Here me_ts u but u_ are introduced so that u are no lonaer

!!i:."!!_)!:'t_he'! element displacements are assumedto be, tn general, co=patible. By integrating the term with ux and by

;:.:_L/-2:" _.n::o.-t:"compatible with the prescribed boundary displace- recoonizfng that

_iiiii'iment_. If the elementdisplacementsu are compatible ! iwlth_, and thereis no constraintfor'theassumed ' u'X= u - _ on )V

::::.]:/.s:tr::esses .]::::o ;:: then;i!i the resultingelement is identicalto one obtainsfrom Eq, "(I)

"!.!/!.!-!!!_!)!i. (the compatibleelement,as was pointedout by FraeiJs "R " _V['_cTs o Uq) - (DTo)T u) dV (14) 

)):.':')'):)i;)'i)'ii) )de Veubeke,3 If o Satisfythe homogeneousequilibrium,,

I.:::..:I.._ :equation :..2s.T ' If _X are bubble functions,l,e, ux along the

:iii );t • DTo.• 0 (3) eequati'--onosti*

ll holds,It is seen that theOolast term oin

::::::::;::C_,T:::: ) hen the secondterm in Eq, (1)can be changedto the ,E.q.(2)contalnthe ) stress equilibriumconditionwlth uk

:::'_:-;"boundaryIntegral,resultingIn ; as Lagrangemultipliers. Thus, in finiteelementfor-

" conditioninitiallywhile the introductionof the In-

)V * * ; ternaldisplacementsu),will enforcethls condition.

Th ts the negative of the element complementary ener-. With a, _ and ux represented by Eqs, (5) and (13), " :.'..:::g.y::a_._. nd i:s.th:(.e basisfor the initialdevelopmentof the , ,_

_:.::.':.":!;'es ._.:s_ed 'tress hybrid element, 4 which may be consi- _R • " 1⁄2 _BTHB + BTG q - _TR _ (15)

i.:.'::.i:.:':idi_:_!: eredas theresulotf a hybrid variational principle.5 1

;!-:':_;::":::i This :,: also indicates that if the dssumed stresses o tn where H and G are given in Eqs, (9) and (10) and o,,

:-_.::_:..-'£q.(1) sattsfyEq.(,)andu=_theresulttngelement. R • / (DTp)TMdV (16) ";' c:":_F.:.:.::'s,:t_iffnessmatrixby a mixed variationalprlnclpleIs V

",.".':.: Id_n+ic_lto th_,tbv a hvbri_fv_rlationalprinciple, The stationaryconditionof IR then yields
' PAGk;IS -3-

-•,.,,.._.:.:. _ • H - R 2) (17) are coupled in the stress-strain relation it is only ,;..:......,

_:;:":';;:_::': necessaryto u_e the same stressexpansionfor the .... ::'

;}E;;:_;i:-. t normal components, :::::.i/

;y:_.;-::y..:.:. R @m 0 (18). The present formulation by using additional

•tiT.lit, ...... * t" "*''

_:'/_"!_!;_('{llmlnating X and substituting_ into the strainenergy internaldisplacementparametersis a more (]eneralway

•iiir_i}.!,:"_i;eonxperessi obtains on to Introducethe stressequilibriumconditions, For

;;;:::..:_,=: examp1.*the , stresstermsmay be expandedin termsof r_:{':.

.:;.;:':':':'.".;::Yu::k.::::'"_GT'_'IHG- GTH'IR (RT'IHR)"I RTH'IG (lg) natur,1 c'oordlnate_ such as Isoparamotrlc coordinates |,ii_

!!!!_!!!!:'!i" ' for qualrtlatoral and hexahedral elements and area and

.....,,,,,.,,..

'._,_iiiiiiiii

the stress

(1;

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,

_

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used,F:'"

;;!i!;i:!;i!_'_ ......,..,...'i'iin'ii'i': term/'iicas::2n°,ttfb°nSeCide artenstified ian c°°dir rdectnates' tly. sFouchr thcoinnstraiplatnt es egand ua" , Isnatithe sfiedse ,'asappesroxtheimatstreelyss butequthtletbrtumformulaequati tion onosf theare i"'!I i

"°;

W;:,:.::.: ,shell.::: sfor which compatibleshapefunctionsare dlffi- problemis simplified, It shouldbe remarkedthat in = '

Eiii!!!iul.!!!!! t to c:onstruct,_c of Eq, (4) is used by the exist- the formulationby the Hellinger-Reissnerprlnciple .,v.,

";i_:{{Y_ng:h_y:b_ri:"dstressmethod, Tong has suggestedthe use the equilibriumconditionsneed not be satisfiedfor :iiii'.

_!;.';:::i!'_',i.Oufncoupledstresses whileadding a constraintterm :....

'.:'_ii;i'::::-ii'_i!_:::'BTR_ to Eq...:::., (4).=;:6:. The corresponding varlational rune- the assumedstresses, .. 'it':/,::::

::_::"':',:_.:::::_::':-t _aylonbeal writtenes 3 H_brld/Mixedrormulationb.v _':,:'

_'F._.:'_,:,:.:.:.: ; Hu-Washizu Prin;_ple :'"'

_:';i:'Y.".':'):':::'l_'R= /V[" cTs o - (DTo)T uX] dV + / TT_ dS i::/'.

" " av" " znusinthe g Hu-W.hipri =uncipletosol ethe :"

.............. :j::

",:*.:.::',"," "". t

.y.-:..:andi.:.th:eir::esultirgele_.entstiffnessmatrix is the same above boundaryvalue problemthe followlngfunctional ".':

:.'6_:'a:s.'.[q::, (..Ig:),'::T:'.': he motivationby Tong is that when the shouldbe maintainedstatlor,_ry, i:'_' :.'::.:::::... _.:_.

matrixin arenotcoupledthe,inversiofon 1⁄2 i;. :.::_;M.. the:!MflexibilitymatrixH can be slmnlified7,8. For _HW = ; [ C s - + oT(o u)] dV ,i.:

'example,in the case of 3-D solid,with _ = {¢x Oy az

_o......... ':"_'M'M:Ox '_'':y:':_y'Z':::M:': z OXZ'':';':':if':" the expansion -Pl_ for all stress . rayTT(u"" " "_) dS (25) :...:'_

_%.:!;._:.,'com _;:.._!p:o' nentars e of ioenticalform then the H-matrix

........... ' " $ 'In1973,Wolf7 used the Hu-Washlzuprln:iplefor it"

:.:._:"_:'.:'Eq ":'::.....(.._. g)..will take tl'eform ..:;

":-:".':::':.. finiteelementformulations,but the resultingmethod E_,

,"_Cii!" ;:::;.:::;;:,';;iiiiiI,, ' Sll$, " * . , 516¢_ has not been used widely. L.R. Tang and his ELL ;..

T•'.,'T.i._,:':,H :='.:: J (21) colleagues 10 at the DallanInstituteof Technology it":

i , . . . : ';::

_:i:_:!!:_:(!:_::!:!:_ S;6¢ ' $66¢" have initiatedthe so-called quasi-conformlng elementsit:.:

)_':q::_:l;_iM_ _ 0r string-netelementsfor the constructionof element _._:

t ".::

"_'m%l_:'m_'_'_'_':_: where S is i;heelasticcompliancematrixand stiffnessmatricesand have discussedthe connection __"

•:'.i.ii:'ii.!_.i_iii " of their method with the Hu-Washizu principle ll An :_!"

//:':M:':':'':::::'::'/': ; £ • $ PITPIdv (22) interpretationof theirmethod Is that in applying i..:

T.';::'::::*: #

•:.:........ .. , the Hu-Washizuprincipleeach stresscomoon_ntand :

_6"!'C:-.)'_ i: The inversionof H then becomesslmply ' ¢orrespondlngstraincomponentare expressedin terms •

............... of the same Oolynomialexpansion,and the r.=sultinq •

............. [ I

:V_::_M_)) :'_::":':'"""_' ell* " ' " ' C16._ ', matrixto be invertedthen becomesone with only it}

,;:.=.;;;;-;;..-*';,' HI•:_"::")))) " .'..::i"//, ::/ (23). diagonalsubmatrices,This approachCan aaain,be 'HI:_ M

=':':::::"= C16_ 66_ I c°mblnedwlth the use °f addlti°nalInternaldisplace"If)

"_'::_-!i'g'(:C"-." , ment parametersto enforcethe stressequilibrium

6::E':;',_• :,:/:: conditions, Aqaln,using the 3-D problemas an

E':,' _ _'C"-":w_here C • S-I and _ • _-I . (24).& [:: ;;_'_il.'_ " " ' . '-example,tn the finiteelementimplementation,let

_::_._:.;m::_:_d_'}_' Although in using uncoupled stresses the size of ¢ - P _ (26) i.:...

:i:._:_::!:.!_iH: is increased and the inversion of another matrix must i:.

"_j_,._:':.'.;b'.'e:;':performedwhen Eq. (lg) is used to derive the .. - ..

_:._:;_::_'_'ele ::: mentstiffness matrix,therecouldbe considerable' with identicalP for stressesand strainsbut the ii

-::j:,-,:-::,:/ r

..::..=._.::;i.,.:::: _:s.;av::in:gs:"In computationwhen the ordersof ¢ are much e)p_nsionsfor dTfferentcomponentsmay be entirely . =-;.:

_;._':_Y,smal :((l:er. than thatof the H matrixof a coupledstress Iniependent.With elementdisplacementsagain ."

":":""':'" • ,'.. '. -== ,, ,.,_...., ._,,,.._-' (e_+-nole ev,_re_edbv Eo. (13),e_uation(25) then becomes .

I IIII I II - - .....................
, • a ameer@ISAaTI,Y -4-

1⁄2_TJ _TR _Tr, _TR How considera 20-nodebrickele=entwith the x, ,_j rHW , _ - . _ + , q_- , X (2B) y and Z axes coincidewith the axes of symmetV _s

i:_i!:;_!} :i whore J . / pTc P dV (29) shownin Figure 1, The element displacements can be .:

t*..._..* * i

.'.":.'." ', (30) ';

:;/._._ and G and R are given in [qs. (10)and (IG).

The stationaryconditionof _Hklwith respectto ;:

= andS. then leads to three sets of equations, which x _o,_ 2a _"y

'_i::J,!ca.n.::... be use.d...... to obtain= in terms of q, The resulting " _I"- .,

;:.:,:.:,V, elementstiffnessmatrix is ' ;

v,v,v,::,_ _ Figure 1 20-NodeBrick Element

:.'.K:. k - _-'-G' - _" R _'" " RTw G expressedin termsof the nodaldisplacementsuI vI

,::::,:'::=T =TO (T,,:)-I (31) andwi (i- ItoZO)as

I _: l

:,...;l;

.,....:'" Ut ,'

':::.v,wh:.ere W = R'Ij R"l (32) u 20

:.':::i.'::" u .= v = Ni(x,y,z,)vi (34)

'.-"i/.A.s:.s;h"-. own.i.n..Eq......(30..).... ,Rmatrixconsistsof only diagonal w i I wl

'.IC"CCs.ubm .'C". atrl: ces.Thus, its inversionalways involvesonly

...:;..,the:'..:'::' inversionof Hi even if the materialpropertyis One can concludethat the shape,functionsNl(x,y,z) ;

:...::=a... nisotropic, can be representedby 20 independentand simple '

!!'_! For generalquadrilateraland hexahedralelements oolynomIaltermsas follows: _ ,

'_!:i_ it iS more convenient to express strains, stresses and Constant, x, y, z, x2, y2, Z.2, xy, yZ, xz, x2y, '

:ii:!:i';!:!:!d: isplacement.- in the iso:)_.rametric coordinates. Xn x,v_ .Y_.z.,yZ2. X2_, _2, #,YZ, ,x2b'=,-V_ and ,xy=Z ! /

:".'_:"that ;" case, the Jacobian !Jlappearsin the area or

':::::':" For a regularbrickalienS, then,all posslbleelement' :.!::. the volumeintegrals, here,however,the strainterms

can be expressedas disnlecenentscan be re_resentedby GO independent [.

..... _.. displacement_odes. Here,the six rigid body displace-i:

::: :v IJI ._ • P .= (33) ment modescorrespond to zero strains, hence, can not E

.... , .o _;

.;.:.:::- and for propUrly chosenP, the res,.'lttno subr,,a:rices be suppressed by any stresses and one only needs to ...:

:_!ii!!{ Hi may all _ecomediagonalmatricesand R will also be considerthe remainin_54 displacements. ,.... :.:.::../:-

'-:.i_iiii a smplediagonalmatrix. It is seen that each displacementmode may T':"

......... ;,-;

"_........ .!_"i: correspondup to threenon*zerostrains. Let _ be }i'!

":'::;:Y:':," one of these strains then, to suppress a particular ,,, ..i

_:...:.:.:.:.: . I!.!I :

::'.(..:(.._;:._ mode it _s necessarythat a stressterm o existsfor

'_i'-.._i\.'-.-'_4 ChoSceof /_ssumedStress Terms the stress componentcorresponding to c such that the I_{{

,°.,,,..,

however, can be used to suppress only one individual

.':"::::' The guidelinesfor the assumedst'essesin the'

.......... displacementmode. An idealsituation,thus, would be i_ii *i

:_://:.." hybrid/mixed finite element formulations are to include to choose 54 assumedstress terms each of which is _':"',, :_

:{_'::"::;"aoequatetermsto suppressall posslblekinematic t::::

.....• |;:°t

•::'-," ' associated with one individualdisplacement mode. _/:::_ .

:<".:::_'idefor : mation modesand to avoid excessive stress terms

......... , One of the posslblechoicesiS shownin Table 1, _ii!_ . i

.;_.:.::";i.Ii.i;n orderto preventthe formingof an overly rigid Here, in the variousblockspacesthe polynomial F:::

.::_.'..:'_e.:lement. For hexahedral elements of general distorted I:;"/:::

.':":':";/ expansions for the slx stresscomponentsand the _IC_::CI

:':-'..g'eo:_i'_ metrykinematicdeformatio_modes can, _n general,• correspondingdisplacementsto be suppressedAre i::;_

"::ii!iii_ii ::::It" . be'suppressedwhen the numberof stress termsis equal indicated, Thereare G4 independentB-terms, artmany!:..!i.}

__

•::::::::. or larger than the totalnumberof degreesof freedom _:'::"

I..:_"=_!::.F. other blocks stress terms are introduced to matntai_ i_:_i_

::.:._.minus ._. six. However,for regularbrlck-shapedelements the equilibriumconditions.They are not used tO r.....

'_'i':':::!!._he . possibledisplacement_des are of simple shapes, C::'::: ,,_

"'::::".' suppressany of the 54 displacementmodes, _:..._.) :

:...::"not all assumedstress termsare usefulfor suppress-

.E:!:L It should be notedthat some displacementterms i.ili_

._;:,_.;.;tri ._.g:._the:;. kinematic modes., Thus, their selections become may each correspond to three strains and somestrain ":.... t

"C.::':::':e: very Importantstep in the finiteelement formulA-

"";:',:;:: termsmay be arrivedfrom more than one dl-_plAcementi:i_"

•:::::..,: _,'.',::_:
' jm._aa s

., Table I • _isplacementrodes , n,aybe suppressedby alternative

i _&ressTermsand Corresponding'Displacement Modes sl;resstermsand thereexistm_ny other combinationsof

:"::: to be Suppressedfor 20-NodeBrick Element

;:"://'_i I : I_ stress termsthat can suppressall possible

::':::: I kinematic modes,

":":T'_!'L_:::_ .....;a.":"/ "_2_V.... ,"_ ....... 13Xy - avZ_ ,_'0" , Anotherobservatl_n Is that the stressexpansion

,,':.;;M,,;,:;';"" , ' _I 3 ].£ ............... 5 16 iS complete' only up to linear terms, hence, the element

:%,::>J,";Cons:U=w v.y w_2 lUy,v*Y v.7.,_yu"=._x propertieswill be dependenton the referenceaxe_ f_r

,;T.:'.:';,_i.x '.-BI4 _9 611 PI3. _16 '_Ig2 the as;_umedstresses, SpllkerQt al,12 has shown that :;;',W%

....... B21 v.xy w-xz v'x_ _,xy _x a hybrid stre_sale,antmay behaveexcellentlyunder

:'>%...... :": one co'ordinatesystembut will containa kinematic

:i_ii",'!',:.iu.xy "_'_8 w_yz u.y- w=y" U=YZ deformationmode underanotherreferencesystem, For

..:,,:._., i a regularbrickelement,the referenceaxes should

):!!i_)_i z B8 _i0 "_17 _"15 6182 _2i2 obviously be l_he axes of syr_etry. For elements of

;_3ii_:_i u,xz v.yz "_I9 IV.=XZ V=Z- U_Z

I:"FC"! " xy B22 _27 B._2 _39 _41 generalgeometricshapeIt is stlllrequiredto choose ..... ::: a localreferencecoordinatesystemIn order to achieve

; ,....,j" u,x2yJv,xy2 l WC'Xy z W'X_ 2 _l=X2y an o_timalelementbehavior, ,

:::.::::, yz E2B _33 , P42 It is also no_ed that thereare alter,,:tlve

.... .'.:.:::: :!I''!i!iii:iu=B23xyz v=yZZ.. w:Yz2 JU=y2z u=yz2 choicesof the stress termsfor maintainingthe stress

:;:.'I!C:"_ XZ £24 '529 _34 =38 540 e:utltbrium conditions. In Table 1 co,'nplete symmetric

:'Y-'/:"/_ u=x2z v-xyz w=xz2 _v=x2z v=xz2 condltlons , are not maintainedIn this particularstress

/I::.'Y.':':" ....

".'::.{{".,>>_ .::.. $30 E35 "_27/Z • .634/2 selection, _f such conditionsare to be maintained

:i_i:,.1i ,.-%: x2 ar.d all the stressequillbrium'equa_ionarse to be

"'.::':::':1;'_ v'xZY w=X2Z "£a0/2 "5Y9/2 satisfied the number of independent stress terms will

i/}."."'?_i , y2 _2B 2 _3_ *=_22/ZI"633/_ necessarilyexceed5_. Alternatively,some stress '::::..::_./::.u=_y :........ w=)'_z"_2/2. "E4_/_ equilibriumconditionscan be relaxedin order to

;'.... ji:!._.:Z.2 E262 E312 "E28/2 '-"_24/_ ma_ntalnco,P.pl_¢esyr_,etricconditions, It can be

.;::,.:., u:xz- v=yz "P'3_/2"_3"/? seer,that in "[ableI if we excludeall the unoerlined

.... x.. termwhi s chare ntroduc. tooaintathe in e uilibrium

...... t u=x YZ v"xY2: _=xYz2 conditions of all cubic stress tern=s, the sy,,_,etry

.i::".".:.::x>':v" ., 3B52 w_4721 conditionis satisfied, An examplesolutionof a t: ..::'::..:. _ xy z spherewith a hole at its centersubjectedto varying ) ,... :::' • ,,,

.TT:LIT'.I " temc,eraturesalong the radialdirectionshows, in fact,i

'"":;:;:'"xZ'v' BAg "3_52 "_51/2 that by using fewerstresstermsin such a way the i

:*'::'='" w=x2yz resultsare betterthan thoseobtainedby using all _

:: :::::'/) .......... !:

}'.i:1 '_iiiiYZ2 _43 ;3_53 . terms in TableI, and even betterthan that by the ._

;.::..._..._u,xy;2 i_ conventionaldisplacementmethod, It has been found, ;.:

i_-_ j_z2 however,that the equilibriumconditionsfor the lower.}}!

%:: ........ .':'.% B442 "_4_ g/2 _"3B53 order stress terms should be maintained in order to ;.:::..

-T''"::::::'':: ::: .....u=xy-z achievean optimalelementbehavior, Also excellent :"" i_::

:,v/ XZ2

:i::'i:i:i:: B4_ 2 "B5__2 "3_54 results can be obtained by using an element with 57 ii:

•:..::. v=xyz -- _*terms obtained by replacing the _:_, _¢_ and _ca fi_

termsin Table 1 by o_ " _=oxy2 + _xz 2, o_ = B_ax2yi';".

:.L:.?.:::: :::':: ---- v=x2yz + Essyz', oz • B_6Y.z + _5?x'z without any shear ._{!.*C:

..... :.:v. ..... stress terms for maintainingequilibriumconditions,!!_:.'i

:.:'./C.IC x3 ,..... "':::':: _52 Detailsof these solutionsare givenin Ref, 13. !:::_':::'.' '.7_:?'T(. ':':'.:

S...... ..":':_-_. u=x2 , .. For brick elements with 8, 12 and 16 nodes the ,:_' .':...._X

_":"'=',_:y:3..:::,':::"':: independent displacementmodescan all be identified.......

'"'"::" B53 from the 60 modesof the 20-node element. Thus, the :.:':_-

_",".'/':": V=y2 ..... ....... corresponding_tresstermscan also be identified i_..:.:._"} :.

:, '.., | J'**'

.:-..-, Z3 i,_,edlatelyfrom Table 1 For example,the indepen°;.':'"

...... . B54 ' ,, :..:

•:...,.;. dent displacementsfor the B-nodebrick elementare r...-.

......;.. w_?2, i_i.:i

. .',T:: | ".":'

: .:.:. u,V,W = constant, x,y,z,xy,yz,xz,xyz _::.;.::

.....:.... and the correspondingassumedstressesare ,.:.:_
6.lb *_ !

ox r _l + 6_y + _3Z + _4yZ .................... ":.

•: , ._OJ'(S) I' _i

:'": °z " _9+ elOx + _'IIy + el2xy l , ,)o. _r.Nm_") i)

.!6i:_ OXy m,13|3+ _14Z ' ' ;I' , _ "'"I-e1_nl }

"-i _yz" el_+ BI6x 13_) O- Na,ly_

st.- "; °XZ m l_l7 gl@Y ¢ Zo O-- NO,iS.Fele_rt

.*# I ,T

"" In this case the thruequadratlcstresstermsare

t;'

.'. necessaryin order to suppressthe displacementmDdes

•,,: u, v, w • xyz. It is also noted that even the linear

::: 10 -,

,:o ' termsare not completehere. In fact,if the linear

.(_' termsare completewith altogether24 B's, the result- _

:! Ing elementwill be almostas rigidas an assumed-

" ,displacementelementbased on trlllnoarshapefunctions. '4 _t: If'

,::L;: In using the presentmethodsby rR for Isetroplc o ':

40 80 _'|_0 'ZOO (:.:'.ma' terials,the stresse_panston of the tnreenormal

: m : : Jstress componentsmust be identicalwhilethe snear-

'.:'_ tng stress componentsmay be kept the same as those

'Vt

":: :;; , in Table 1" For a formulationsimilarto tl'e57-B -10

::,. elementdescribedearlier,the size of the H-matrix ii

;c.,.:'is 75 x 75 and the evaluationof an elementstiffness :..'_

iS'.:'..matrix ... v'lllrequirethe inversionof one 18 x 18 )( •:': ' I :;

..., , • .

.::::.i(RTH"I. .R)matrix,one 17 x 17 ¢-matrlx.and threet_x 8 Fig, 2 Center Deflectionof ClaPpedSquarePlate bnder ::...

•":. , matricesinsteadof the inversionof a 57 x 57 flexi- a ConcentratedLoad ; a t"*

'L!} Ibllitymatrix. When the formulationis by 'HW and the )i:

":::" :as shownby the solutions (Fig. 2) of,the center deflec-, "'" stress termsare the same as that by "R' tlleresult

..._ will be identicalto that by _R' However,in this 'tionof a clampedsquareplate undera concentrated :*

•:i,ii case the stress expansionsfor differentnormal load. The use of four additionalmomentterms improves_::

"::: Components'heed not be the same and one may use fewer the accuracy of the solution. The addition of eight F:'

:._) stress terms than that by _'R' : momentterms,however,makes the elementtoo rigid. _.6

m:/_ , _ 'Two16"dofsemiLoofelementsof rectangulars_lapehave [ii" :_

:i;;:.'_)i:- 5 Ser.ILoofElementsfor Plates,. and Shells : also been formulatedusing 13- and 17-_ terms. These Iilii

M'_:t:::> " ' elementsare necessarilymore flexlblethan the uorres-,I'ii

::'_: For thin plates and shells,Irons has pointed . pondlngtriangularelementsas shownby the results ; _i

,_';_L.....,. out.; the advantageof using semtLoof elements2, for given in Figure2, I[_';

:?.!,-.whi::ch the derivativesof lateraldisplacement w are For shellelementsthe equilibriumequations_ill i""i_

:.:'::n]ot maintainedcompatibleat the cornersof the alwayscouple the membranestressesand the moment '°".'"t ;

.,::::.: .: :., e:l;ementswhile normal derivativesW n at some points stresses, However,for Isotroplcmaterials,they ere ":i':

;t:: along the Interelementboundaryare kept the same, not coupledin the stress-stralnrelations, Thus, in ".C

•":' By using the assumeddisplacementapproachthe derive- ;"

::: this case,by using the.newapproachby _'Rthe H-matrlxi'(i,:i '..':.'ti:oi,n of a semlLoofelementis very complicated, ,,,.:.:;

i can alwaysbe partitionedinto four sub-matricesalong _'.':

_'?. ,i,

:ii'i;;' However,it Is ratherstraightforwardusing the assumed the diagonal, In the new approachby _HW' of course, i". i

derivativesof boundarydisplacementw are not contln- are always decoupled. Anotheraddedadvantageof the [._}i!:(

r uous at the corner nodes but the tnterelement compat- new.formulations is that the resulting H matrices by ;,;t.

Ibllitycon'dltlonasre completelysatisfied. ; ;:-:."

'bothvariationalprinciplesdo not contalnthe geometry i(!'!

Triangularplate elementswith 12 D.O.F.as shown of the shell,making the implementationsimplified. ' '::."/- :::?

_!i') Inindfigure ependent 2,mhave omentbeen terms. derived The using Loof-n9,odes13arande loc17ated element Bothshtriangular ave been coand nstruct quadrilateral edfor semlsLoo hallow fbounda ,hellry ]_: !!!!);'

at I/3 points. The 9 momentterms are the complete nodes by the present methods based on both the r/.:.f...

I linear terms,hence, resultingelementby the hybrid Hel}Inger-RelSsneprlncip r loand the Hu-Washlzuprlncl- U_:

':::'!_*.,._}'rmetraheodli_istl.iW,h_k_ denticaTM. lto tIh+e(_eq,_*._ uilibri ,_-llv umelev_vment¢byI_._I. ple15. The assumedstress termswhich for the two i_:Y;_L, i
OIU Itl,PAI,ts -7- .

Oirlolt QItALtrY

pri,,clpland e the Hu-Washi_uprincipleto finite

t .,o,.... .

.:::,_::,',, o,_ elementformulation_have led to a more flexibleand ,:?::;;;

"{"'{:";:-;. (l-,,et,rilng#11rel,men, ;.::::,',

........... ;}Ci_Ci!i_ _ (,_,_I) most efficientfiniteelementm_thodby the assumed ;::YV."

!.......... _... roct_n@uellJe,ment sl;resses,The possibilityof usingnatu-alcoordinatesi:-c.

ii!))iii!_!.._ '(h_'_'_) 0,! and the relaxationof some equilibriumconditionsare L._'_,'._

':.."4;'..'$ V" ;."_ ,,-.-,-(Indllyti¢i(olllutlon ;':':::!.

_})',.'.'_'._ :_'t'! , (FIU@go) the flexibilityof the new hybrid/mixedstressmethod, !./::_:

;;:U"'_;_'!.,.,...,,,. The decoupllngof the flexibilitymatrixH lo_ds to a ,,. •

./,L:.":C:.;:,._ ;()_:II_:: -- _c .;::.: a savingin computingtime, A syst_.matlprocedure c for :.:i!;".

";""." the choiceof stressterms in certainhybrid/mixed _';"

_/_::i::"_:'':v ;:finite elementshas boon os=ablishedb=_d on the f::;i}!!::::,

;;i_i:_i:i!':::::';: ,requlrementfor suppressingthe kinematicdeformation I!!'.,

........... C ; : _- o modes Numericalresults indicate the advantage of i::,"_',

•.,o,,,,,,,, S_ -

% ¶,,,;.,,,

i:,:./(.ii_. 4 ' of maintainingthe equlllbrlumof the lowerorder [.::.;:,:

..... stresstermswhile relaxingthat o/ the higherorder '......

".",i, ,., _'.;'i,i,i.i .I0 terms Hybrid/Mixedformulationsare suitableto • ,o'/,'.'. ,., , ..,

,:,;, .,.; @

':,:,!':i::.!!!! -IS platesand shellsusingser,,iLoelofements, ],?:,

':'.',;'.r','.','.,", _cknowledgment:This paperwas preparedunder the ',:.:,

!i!i_!/!/!! S supportof NASA GrantNAG 3-33 by the NASA-Lewls _;.;'_::::_::"..

..'./:.:.::.'.':: ,,_JG.,O. ..... :.._

'_'.::..:,:...;..:,:.:..).'. ResearchCenter, The assistanceof ProfessorDa-Peng iiL.):,:.i

•:.).:./.;:.;.}}}} v.,,.'

..,..... .... C o Chert,and Messrs,KiyohideSu_,ihar,and DavidKang _,,,...v

.i',_'/'_:i'{'_'_'i'.i_ .$ in the preparationof this paper is acknowledged, .".:':/':

"_ -I0 References I i:',:',':i.'.,:

".":':' ' :'F'" .;:'::::;

"_;Yi:':;i" -IS I, T,H,H,Plan, "On Hybridand MixedFiniteElement :.:':.

i'/:':.6;":"::}!:. ( Methods",paperpresentedat CMES,CSTAM,CMS :..,,....

•:1'..::,:-.:,.;1 -re Symposiumon FiniteElementMethods,May 15-23, _ 'IN ::' ":"

..... I 1981, Hefei,China.

•:...:..;.r.. Fig. 3 :Stres ._. sDistributionof a PinchedCylinderby 2. B.M, Irons,"The Se_iLoofShellElement",Finite _!".?!!:

!:':':":'' SemiLoqfElements' i Elementsfor Thl. Shellsand CurvedNe_ers, Edited :"'"'

"':::"::.i'::. I I by D,G. Ashwelland R.H, Gallagher,Wiley, 197-222 i.:i.:':Y

:._'..e:_.' lement, ::.:cb-_i.nsistof 37 independent_'s initially,and ' ,'}:})}i)

";;:::??;:'" , •.....

.i:.ii12:,:const i:.!iit'alntequati._ for 6 are then introduced : (19761, ;i....-::

::.:...;.-: 3, B.M. Fraeijsde Veubeke,"Displacementand Equlll- F=:"/Y"

..,.....reducing :".:-.".;the:.::number .:.::of ,,,dependentB's to 25, For • ;..:.... .r_':

;-7.';.:;.:::: brlumModels.inFiniteEle¢,ent Method";_ .L':'._::

"' ,:::",'.1:::,:'1'1"th:.1:"e::::32-0of , quadrilateralshe • s...::..:.; ll element 55 independent Analysis,D,C. Zienkiewiczand G.S. Hollster(Eds) :...='..

.......... _ " ' :..::;.E:.:

•':._.:E::;.B':;_S were used initiallyand 22 constraintequations Wiley, 1945-197(19651 "":":::

•":t":'.:_' for 5:a;'_re"then introducedreducingthe numbernf o*.'::.:.:::'.

?_"_']"_']"i'i independent 'i B's to 33. I 4, T.H,H,Plan, "Derivationof ElementStiffness _:_;;';iF

:Y:'.Y':;':':':': Matricesby AssumedStressDistributions",AI_A ..:::::::':./

!'.' .......... .'::.:'_A 'p!_i" roblemof the pinchedcylindricalshellwith , Journal,2, 1333-1336(1964), ' _::'': ":'m

";.;:::: free; ly supportedends was analyzedusing 4 x 4 mesh, - ",:.'.

;":':':":":?:" 5. T.H,H, Plan and P, Tong, Basisof FiniteElement L::'::"

"ii'_;'It:_was :'-;............ foundthat the displacementsand stresses Methodsfor Solid Continua",Int. O. Num, Moth. !.'/_;:i:_:;i:

I L..,,,'..:

":::._'obtai .::)C.C nedby:::t:hese ::._two variationalmethodsdifferso i Engng,,_, 3-28 (1969),' F.:::....<:

:::'.:,,'::'.'1,1::t::tl:,::e:;:. the the resultingplotsof the membraneand li:;::/C::

:'..!........ ,:...;'.....:m'o::ment i:i'i_i_' d stri:but":ions(Fig,3) can not be distinguished 6, P, Tong, "A Family of HybridPlateElements",to i.w ...... :.:::. . :;

"_:::::':':from one to the other, An analyses of the computihg be published in the Int. g. Num.Meth. Engng. _-:'"::":

:_'_L::;S , 7, G, Horrignoe,"FiniteElementInstabljityAnalysis II:'C;I'_

.::.._,,_ effo:rt:",sfor thesetwo approachesindicatesthat the r.,m;,;::

,:,:;.r-:i:.=_:fo'.r:.mu_.lati:on:,_ by' the Hu*Washizuprinciple has an edge _ of Free-FormShells",NorweigianInstituteof Tech-[...::::.

':':::::'-':" ' nology, Dtv. of StructuralMeChanics,ReportNo, F:'":2::Z ". Cv:.:.:':,:ove :Y'Yr/.............that by the Hellinger-Relssnerprinciple. ' _,_.,:..._

'........... _ 77-2, (Ig77), •

_)iJ.l_ h methodsare simplerto formulatethan the original i_Z' . _.::.:'T::._! .,,..,:

"::':.'_i':_i:_hyo.:rtd stress method for which the membraneand moment B, J.-L. Batoz, K.-J. Bathe and L.-W. He. "A Study of _.);)'.:_::2_

:):::;;stress _1' termsare coupledby the equilibriumequations. Three-NodeTriangularPlateBendingElements",Int. _'":';/-:

T/Y,"/,::: I _ _. Num, Meth, Engng 15, 1771-1812 (Ig80) i_;_;_;'

":',:.:-",:.": ' I ' _ ' ' _":",'.

............ t F':::::,:

,;:.::,'::::': $ Conclusions 9. J.P. _olf, "Generalized Hybrid Stress Finite r":.::.7.

)":_"'"_' " ' I. I: Ele,_ent_,odels", AIAA Journal,--II,386-388,(1973).|.',_'_;

" ::::R''!':_'.:!; New applicationsof the Hellinger*Reissner ' ti_ii! I
' _U, L,M, Tang, W,_I, Chenand Y,Xo ltu, "String Net . -B-

";:;:' FunctionApproximationand Quasi-ConformlngTech-

._:;::::; nlqwe",Paper presentedat the InternationalSym- ,

•..,;:::.;', podium on Hsb,'idand Mlxod Finite[lementMethods,

¢ f

,._i!!_!_!( Fprll B-IO, 1981,Atlanta,GeorgiaU.5,A, i

:>";:;;"..11.., , L... ,M, Tang, W,O, Chenand Y.X, Liu, "Quasi-Conform.;

;.,[;C£ %..... '__I; in@ E%omentand G_nerali_edVariationalI1lagali-_ •..,.,,,,

',*,"..":", 1;ies",Paper presentedat Ct4{$CSTAHCf4SSymposium

: +:.,;:_; w o

:_.'::;;,:;'i_on('na. ;I_" ;hniFltol,.,;' Elemen ::".tHe_,t_,ho,d_, ,_?May IP-23,1_(_I,Hefel,I

i;iiiiii: 12, R,L, Spilker,S,M. Ma_keriand {, Kanla,"Pla,e i

,iiHH: l_oparameIri¢Hybrid*Stre_s_lements: Inwriance : "._'_.

'{.,;,:.!{! and OptimalSampling",Int, J, Num. Moth,Engng,, !

',?::}2i!i 17, 1469-1469,(IgSl),

...._: 13. T.H.H. Plan, D.P Chen'and D, Kang, "A flow Formu-

::;';: I lationof H#brld/MixedFinite Elc_nent"P,a_er_;o I

;i::'i_ii}i be presentedat the symposiumin Advancesand i

•:.;::,,:: Trends in Structura%_nd Solid HechI_nics,Oct, 4-7

":i!•/';i, IgB2, Washington,D.C.

:,.:.);.i:1a4.,.:B-..M.:igrael ::';:js:'":' ,,de V;eubekeand G. Sander,"An EQulII-i

)'::'i'.:'i';brium 'i, _odel for Plate Bending",Int.J. Solids

•_i_!ii_iJJ Structures,4, i

_ii! !15. _'.H.H.Plan and K. Sumihara,"HybridSemlLoof

.........i Elemen_sfor Platesand Shells",to be published,

.,,....... ,

:ii%i.i ' ,

o.......

l

;".":.'.i':r '

.';ii".:-:',!,.:.

....... ,....

Ji-_::.ii:'i

:::;}::.:'..:}

.:,*?.*:?:':.'.

..,.,,..., .

.,,,,,,..,,

":.:.:.'.._:.:.:.:,:. OI¢I(_II_FAI, tI_IR I_

:.,......-:....::C.::;;:;c.;lcc: Oolt; 'AMI'Y

:.9:::.:::::,

.,....., ....

......... °,,

/:"c:)?i':!:2 ..;-.. :;';.':"

,..,. ,,,,.,

":::.:-:_,'";':'_:' "1 ......... - _ _- -

-----. .......

+ . . _

_-,

...... o.,o°_

:L:: ::: :-:;':.: ......... ,°.,,,..,,. ......... .o,°.._,, .,.°,..,.,,,..o.,,_o°_..°,, :;.*.: ;-:%_;:.:;;;,°.....,;,.,L.°_,_ ....... .-.,.,° ••::.::_._:;.'_:- V"i,",'.. _.' ": ..° ;:::" ....................... _:_:;_`_:_:_`_._._*_`_.``_:-_-.;_:`_::_:._`_:_:Y:_*_L_t_

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....... ;...... ' ,,,,,,i,,,i i ii iii lllllllllllllmll i I I IIII i