ISOPARAMETRIC ELEMENTS

Transcription

1 5. ISOPARAMETRIC ELEMETS Bce Ion, n 968, Revoltonzed the Fnte Element Method b Intodcng a atal Coodnate Refeence Stem 5. ITRODUCTIO Befoe development of the Fnte Element Method, eeache n the feld of tctal engneeng and tctal mechanc fond cloed fom olton n tem of known mathematcal fncton of man poblem n contnm mechanc. Howeve, pactcal tcte of abta geomet, nonhomogeneo mateal o tcte made of eveal dffeent mateal ae dffclt to olve b th clacal appoach. Pofeo Ra Clogh coned the temnolog Fnte Element Method n a pape peented n 960 []. Th pape popoed to e the method a an altenatve to the fnte dffeence method fo the nmecal olton of te concentaton poblem n contnm mechanc. The majo ppoe of the eale wok at the Boeng Aplane Compan pblhed n 956 [] wa to nclde the kn tffne n the anal of wng tcte and wa not ntended to accatel calclate tee n contno tcte. The ft, fll atomated, fnte element compte pogam wa developed dng the peod of []. It the atho opnon that the ntodcton of the opaametc element fomlaton n 968 b Bce Ion [] wa the ngle mot gnfcant

2 5- STATIC AD DYAMIC AALYSIS contbton to the feld of fnte element anal dng the pat 0 ea. It allowed ve accate, hghe-ode element of abta hape to be developed and pogammed wth a mnmm of effot. The addton of ncompatble dplacement mode to opaametc element n 97 wa an mpotant, bt mno, etenon to the fomlaton [5]. 5. A SIMPLE OE-DIMESIOAL EXAMPLE To lltate the fndamental of the opaametc appoach, the onedmenonal, thee-node element hown n Fge 5. fomlated n a natal coodnate efeence tem. R A( ) 6 R 0 R A 0 A A. GLOBAL REFERECE SYSTEM X ( )/ ( + )/.0.0 B. ISOPARAMETRIC REFERECE SYSTEM Fge 5. A Smple Eample of an Iopaametc Element

3 ISOPARAMETRIC FORMULATIO 5- The hape fncton ae wtten n tem of the element opaametc efeence tem. The "natal" coodnate ha a ange of ±. 0. The opaametc and global efeence tem ae elated b the followng elementa eqaton: ) ( ) + ( ) + ( ) ( ) (5.) ( The valdt of th eqaton can be vefed at vale of, 0 and. o addtonal mathematcal efeence ae eqed to ndetand Eqaton (5.). The global dplacement can now be epeed n tem of the fndamental opaametc hape fncton. O: ) ( ) + ( ) + ( ) ( ) (5.) ( ote that the m of the hape fncton eqal to.0 fo all vale of ; theefoe, gd-bod dplacement of the element poble. Th a fndamental eqement of all dplacement appomaton fo all tpe of fnte element. The tan-dplacement eqaton fo th one-dmenonal element : ( ) d( ) d( ) d ε (5.) d d d Yo ma ecall fom ophomoe calcl that th a fom of the chan le. Fo an vale of the followng eqaton can be wtten: d ( ) (), d (5.a) d d Theefoe: (), () (5.b)

4 5- STATIC AD DYAMIC AALYSIS d( ) d ε (), B( ) (5.5) d d ( ) Fom Eqaton (5.), the devatve wth epect to the global and opaametc efeence tem ae elated b: d (), d ( ) d (5.6) The b element tffne can now be epeed n tem of the natal tem: + K B() EB() ( ) d (5.7) T In geneal, Eqaton (5.7) cannot be evalated n cloed fom. Howeve, t can be accatel evalated b nmecal ntegaton. 5. OE-DIMESIOAL ITEGRATIO FORMULAS Mot engnee have ed Smpon le o the tapezodal le to ntegate a fncton evalated at eqal nteval. Howeve, thoe tadtonal method ae not a accate, fo the ame comptatonal effot, a the Ga nmecal ntegaton method peented n Append G. The Ga ntegaton fomla ae of the followng fom: I + n f ( ) d f ( ) (5.8) The Ga pont and weght facto fo thee dffeent fomla ae mmazed n Table 5.. Table 5. Ga Pont and eght Facto fo mecal Integaton n 0

5 ISOPARAMETRIC FORMULATIO ote that the m of the weght facto alwa eqal to. Hghe ode nmecal ntegaton fomla ae poble. Howeve, fo mot dplacementbaed fnte element anal hghe ode ntegaton not eqed. In fact, fo man element, lowe ode ntegaton podce moe accate elt than hghe ode ntegaton. Fo the anal of the tapeed beam, hown n Fge 5., the ame mateal popete, loadng and bonda condton ae ed a wee ed fo the eample peented n Secton.. The elt ae mmazed n Table 5.. Table 5. Smma of Relt of Tapeed Rod Anale ELEMET TYPE Integaton Ode (%eo) σ (%eo) σ (%eo) σ (%eo) EXACT Contant Stan -node opaametc -node opaametc Eact pont pont 0. (-7. %) 0.65 (+0.5 %) (+0. %).67 (+67 %) 0.58 (- %) 0.8 (-7 %).67 (-66 %).0 (-9 %).67 (-6.7 %).67 (-6.5 %). (+5.5 %).76 (+ %) Fom th mple eample, the followng conclon and emak can be made:. Small eo n dplacement do not ndcate mall eo n tee.. Lowe ode ntegaton podce a moe fleble tcte than the e of hghe ode nmecal ntegaton.. If th opaametc element ntegated eactl, the tp dplacement wold be le than the eact dplacement.

6 5-6 STATIC AD DYAMIC AALYSIS. The tee wee calclated at the ntegaton pont and etapolated to the node. Eve compte pogam e a dffeent method to evalate the tee wthn an element. Thoe method wll be dced late. 5. RESTRICTIO O LOCATIOS OF MID-SIDE ODES The pevo eample lltate that the locaton of the md-de node need not be at the geometc cente of the element. Howeve, t locaton not completel abta. Eqaton (5.b) can be ewtten, wth L, L and L, a L ( ) ( ) (5.9) whee the elatve locaton of node, wth epect to the cente of the element. Eqaton (5.5) ndcate that the tan can be nfnte f () zeo. Alo, f () negatve, t mple that the coodnate tanfomaton between and ve dtoted. Fo nfnte tan at locaton can be fond fom: ±, the zeo nglat ± 0, o ± (5.0) Hence, the md-de node locaton mt be wthn the mddle one-half of the element. In the cae of two- and thee-dmenonal element, md-de node hold be located wthn the mddle one -half of each edge o de. At a cack tp, whee the phcal tan can be ve lage, t ha been popoed that the element adjacent to the cack have the md-de node located at onefoth the length of the element de. The tee at the ntegaton pont wll then be ealtc; and element tan eneg can be etmated, whch ma be ed to pedct cack popagaton o tablt [5].

7 ISOPARAMETRIC FORMULATIO TO-DIMESIOAL SHAPE FUCTIOS Two-dmenonal hape fncton can be wtten fo dffeent element wth an abta nmbe of node. The fomlaton peented hee wll be fo a geneal fo-ded element wth fo to nne node. Theefoe, one fomlaton wll cove all element tpe hown n Fge Fge 5. Fo- to ne-ode Two-Dmenonal Iopaametc Element The hape fncton, n the natal - tem, ae a podct of the onedmenonal fncton hown n Fge 5.. The ange of both and ±. All fncton mt eqal.0 at the node and eqal zeo at all othe node aocated wth the element. The hape fncton hown n Table 5. ae fo the bac fonode element. The table ndcate how the fncton ae modfed f node 5, 6, 7, 8 o 9 et.

8 5-8 STATIC AD DYAMIC AALYSIS Table 5. Shape Fncton fo a Fo- to ne-ode D Element ODE SHAPE FUCTIO OPTIOAL ODES (, ) ( )( )/ ( + )( )/ ( + )( + )/ ( )( + )/ ( )( )/ ( + )( )/ ( )( + )/ ( )( )/ ( )( ) If an node fom 5 to 9 doe not et, the fncton aocated wth that node ae zeo and need not be calclated. ote the m of all hape fncton alwa eqal to.0 fo all pont wthn the element. Table wth the ame fomat can be ceated fo the devatve of the hape fncton, and,. The hape fncton and the devatve ae nmecall evalated at the ntegaton pont.

9 ISOPARAMETRIC FORMULATIO 5-9 The elatonhp between the natal - and local othogonal - tem ae b defnton: ), ( (5.a) ), ( (5.b) Alo, the local and dplacement ae amed to be of the followng fom: ), ( (5.a) ), ( (5.b) To calclate tan t necea to take the devatve of the dplacement wth epect to and. Theefoe, t necea to e the clacal chan le, whch can be wtten a: + + o (5.) The mat known n mathematc a the acoban mat and can be nmecall evalated fom:,,,, (5.) At the ntegaton pont the mat can be nmecall nveted. O: (5.5) The tem the detemnate of the acoban mat and :

10 5-0 STATIC AD DYAMIC AALYSIS (5.6) Fge 5. lltate the phcal gnfcance of th tem at an pont and wthn the element. Smple geomet calclaton wll lltate that elate the aea n the - tem to the natal efeence tem. O: da d d d d (5.7) Hence, all the bac fnte element eqaton can be tanfomed nto the natal efeence tem and tandad nmecal ntegaton fomla can be ed to evalate the ntegal. d d d d d d Aea n - Stem da(,) d d Fge 5. Te Aea n atal Refeence Stem 5.6 UMERICAL ITEGRATIO I TO DIMESIOS mecal ntegaton n two dmenon can be pefomed ng the onedmenonal fomla mmazed n Table 5.. O: f (, ) (, ) d d I f (, ) (, ) j j j j (5.8) ote that the m of the weghtng facto, j, eqal fo, the natal aea of the element. Mot compte pogam e b o b nmecal

11 ISOPARAMETRIC FORMULATIO 5- ntegaton fomla. The fndamental poblem wth th appoach that fo cetan element, the b podce element that ae too tff and the b podce tffne matce that ae ntable, o, ank defcent ng mat anal temnolog. Ung a b fomla fo a nne-node element podce thee zeo eneg dplacement mode n addton to the thee zeo eneg gd bod mode. One of thee zeo eneg mode hown n Fge ode Element b Integaton Zeo Eneg Mode Fge 5. A Zeo Eneg Hogla Dplacement Mode Fo cetan fnte element mehe, thee zeo eneg mode ma not et afte the element tffne matce have been added and bonda condton appled. In man cae, howeve, naccate elt ma be podced f edced ntegaton ed fo old element. Becae of thoe potental poblem, the atho ecommend the e of te two-dmenonal nmecal ntegaton method that ae accate and ae alwa moe nmecall effcent. Theefoe, Eqaton (5.8) can be wtten a f (, ) (, ) d d I f (, ) (, ) (5.9) Eght- and fve-pont fomla et and ae mmazed n Fge 5.5. If 9/9, the eght-pont fomla gve the ame accac a the b Ga podct le, wth le nmecal effot. On the othe hand, f.0 the eght-pont fomla edce to the b Ga podct le. If one want to have the beneft of edced ntegaton, wthot the ntodcton of zeo eneg

12 5- STATIC AD DYAMIC AALYSIS mode, t poble to let ote that the m of the weght facto eqal fo. β β β?.0.0 β 0? / Fge 5.5 Eght- and Fve-Pont Integaton Rle The fve pont fomla ve effectve fo cetan tpe of element. It ha the advantage that the cente pont, whch n m opnon the mot mpotant locaton n the element, can be agned a lage weght facto. Fo eample, f et to /8, the othe fo ntegaton pont ae located at ± 0.6, 0 wth weght of 5/9, whch ae the ame cone pont a the b Ga le. If et to zeo, the fve-pont fomla edce to the b Ga le THREE-DIMESIOAL SHAPE FUCTIOS One can eal etend the two-dmenonal appoach, ed to develop the - to 9- node element, to thee dmenon and ceate an 8- to 7-node old element, a hown n Fge 5.6.

13 ISOPARAMETRIC FORMULATIO Cone ode t 8 9- Edge ode -6 Cente Face ode Cente of Element Fge 5.6 Eght- to 7-ode Sold Element Thee-dmenonal hape fncton ae podct of the thee bac onedmenonal fncton and can be wtten n the followng fom: G,, t ) g(, ) g(, ) g( t, t ) (5.0) ( The tem, and t ae the natal coodnate of node. The onedmenonal fncton n the, and t decton ae defned a: g g g g(, ) ( + ) f ± g(, ) ( + ) f 0 0 f node doe not et (5.) Ung th notaton, t poble to pogam a hape fncton botne dectl wthot an addtonal algebac manplaton. The fndamental eqement of a hape fncton that t ha a vale of.0 at the node and zeo at all othe node. The node hape fncton the bac node hape fncton g coected to be zeo at all node b a facton of the bac hape fncton at adjacent node. The hape fncton and 8 fo the 8-cone node ae:

14 5- STATIC AD DYAMIC AALYSIS g g g / g / 8 (5.a) E / F 7 The hape fncton 9 and 0 fo the -edge node ae: g g g / (5.b) F / 7 The hape fncton and 6 fo the 6 cente node of each face ae: g / (5.c) g 7 The hape fncton fo the node at the cente of the element : 7 g 7 (5.d) The tem the m of the g vale at the thee adjacent edge. The tem g E the m of the g vale at the cente of the thee adjacent face. g F The 7-node old element not ed etenvel n the tctal engneeng pofeon. The majo eaon fo t lack of pactcal vale that almot the ame accac can be obtaned wth the 8-node old element, wth the addton of coected ncompatble dplacement mode, a peented n the net chapte. The nmecal ntegaton can be b b o b b a pevol dced. A nne-pont, thd-ode, nmecal ntegaton fomla can be ed fo the eght-node old element wth ncompatble mode and, gven b: 0?, 0 / 8 and (5.) The eght ntegaton pont ae located at ±, ± and t ± and the cente pont located at the cente of the element. If 0 0 the fomla edce to the b b. If 0 6/ the othe eght ntegaton pont ae located at eght node of the element, ± and /. 5.8 TRIAGULAR AD TETRAHEDRAL ELEMETS The contant tan plane tangla element and the contant tan old tetahedal element hold neve be ed to model tcte. The ae

15 ISOPARAMETRIC FORMULATIO 5-5 nmecall neffcent, compaed to the comptatonal eqement of hghe ode element, and do not podce accate dplacement and tee. Howeve, the -node plane tangla element and the ten-node old tetahedal element, hown n Fge 5.7, ae accate and nmecall effcent. The eaon fo the cce that the hape fncton ae complete econd ode polnomal. A. SIX-ODE TRIAGLE B. TE-ODE TETRAHEDRAL Fge 5.7 S-ode Plane Tangle and Ten-ode Sold Tetahedal Element The ae ed etenvel fo compte pogam wth pecal meh geneaton o atomatc adaptve meh efnement. The ae bet fomlated n aea and volme coodnate tem. Fo the detal and bac fomlaton of thee element ee Cook [5]. 5.9 SUMMARY The e of opaametc, o natal, efeence tem allow the development of cved, hghe-ode element. mecal ntegaton mt be ed to evalate element matce becae cloed fom olton ae not poble fo nonectangla hape. Element mt have the appopate nmbe of gd-bod dplacement mode. Addtonal zeo eneg mode ma cae ntablte and ocllaton n the dplacement and tee. Contant tan tangla and tetahedal element hold not be ed becae of the nablt to capte te gadent. The -node tangle and ten-node tetahedal element podce ecellent elt.

16 5-6 STATIC AD DYAMIC AALYSIS 5.0 REFERECES. Clogh, R The Fnte Element Method n Plane Ste Anal, Poc. ASCE Conf. On Electonc Comptaton. Pttbg, PA. Septembe.. Tne, M.., R.. Clogh, H. C. Matn and L.. Topp Stffne and Deflecton Anal of Comple Stcte,. Aeonat. Sc. V.,. 6. pp Sept... lon, E.L. 96 Fnte Element Anal of Two-Dmenonal Stcte, D. Eng. The. Unvet of Calfona at Bekele.. Ion, B. M. and O. C. Zenkewcz The Iopaametc Fnte Element Stem A ew Concept n Fnte Element Anal, Poc. Conf. Recent Advance n Ste Anal. Roal Aeonatcal Socet. London. 5. lon, E. L., R. L. Talo,. Dohet, and. Ghabo. 97. "Incompatble Dplacement Model," Poceedng, OR Smpom on mecal and Compte Method n Stctal Mechanc. Unvet of Illno, Ubana. Septembe. 6. Cook, R. D., D. S. Malk and M. E. Pleha Concept and Applcaton of Fnte Element Anal. Thd Edton. ohn le & Son, Inc. ISB