Module 3: Element Properties Lecture 5: Solid Elements

Transcription

1 Modue : Eement Propertes eture 5: Sod Eements There re two s fmes of three-dmenson eements smr to two-dmenson se. Etenson of trngur eements w produe tetrhedrons n three dmensons. Smr retngur preeppeds re generted on the etenson of retngur eements.fg..5. shows few ommon used sod eements for fnte eement nss. Fg..5. Three-dmenson sod eements Dervton of shpe funtons for suh three dmenson eements n Crtesn oordntes re ger qute umersome. Ths s oserved whe deveopng shpe funtons n two dmensons. Therefore the shpe funtons for the two s eements of the tetrhedr nd preeppeds fmes w e derved usng ntur oordntes. The ponom epresson of the fed vre n three dmensons must e ompete or nompete ut smmetr to stsf the geometr sotrop requrements. Competeness nd smmetr n e

2 5 ensured usng the Ps prmd whh s shown n Fg..5.. It s mportnt to note tht eh ndependent vre must e of equ strength n the ponom. Fg..5. Ps prmd n three dmensons The foowng -D qudrt ponom wth ompete terms n e pped to n eement hvng 0 nodes. f h = h h h h However the geometr sotrop s not n souterequrement for fed vre representton to derve the shpe funtons..5. Tetrhedr Eements The smpest eement of the tetrhedr fm s four node tetrhedron s shown n Fg..5.. The node numerng hs een foowed n sequent mnner.e n ths se nt-owse dreton. Smr to the re oordntes the onept of voume oordnteshs een ntrodued here. The oordntes of the nodes re defned oth n Crtesn nd voume oordntes. Pont P nd s shown n Fg..5. s n rtrr pont n the tetrhedron.

3 Fg..5.Fournode tetrhedron eement The ner shpe funton for ths eement n e epressed s N.5. Here re the set of ntur oordntes nsde the tetrhedron nd re defned s foows.5. Where s the voume of the su eement whh s ound pont P nd fe nd s the tot voume of the eement. For empe m e nterpreted s the rto of the voume of the su eement P to the tot voume of the eement. The voume of the eement s gven the determnnt of the nod oordntes s foows:.5. The retonshp etween the Crtesn nd ntur oordntes of pont P m e epressed s.5.5 It m e noted tht the dentt nuded n the frst row ensure the mtr nverte.

4 7.5. The nverse reton s gven.5.7 Here s the voume sutended from fe nd terms nd represent the proeted re of fe on the oordnte pnes respetve nd re gven s foows:.5.8 w e n order.e.. The voume oordntes fuf nod ondtons for nterpoton funtons. Therefore the fed vre n e epressed n terms of nod vues s f f f f f =.5.9 Though the shpe funtons.e. the voume oordntesn terms of go oordntes s ger ompeut the re strghtforwrd.the prt dervtves of the ntur oordntes wth respet to the Crtesn oordntes re gven.5.0 Smr to re ntegr the gener ntegr ten over the voume of the eement s gven s r q p s r q p d s v r q p..5. The four node tetrhedr eement s ner funton of the Crtesn oordntes. Hene the frst prt dervtves of the fed vre w e onstnt. The tetrhedr eement s onstnt strn eement s the eement ehts onstnt grdents of the fed vre n the oordnte dretons. Hgher order eements of the tetrhedr fm re shown n Fg..5.. The shpe funtons for suh hgher orderthree dmenson eements n red e derved n voumeoordntes s for hgher-order two-dmenson trngur eements. The seond eement of ths fm hs 0 nodes nd u form forthe fed vre nd nterpoton funtons.

5 8.5. Br Eements rous orders of eements of the preepped fm re shown n Fg..5.. Fg..5. shows the eght-node r eementwth referene to go Crtesn oordnte sstem nd then wth referene to ntur oordnte sstem. The ntur oordntes for the r eement n e rete Crtesn oordnte sstem nd.5. Here nd re the ength heght nd wdth of the eement. The oordnte of the enter of the r eement n e wrtten s foows: nd 5.5. Thus from eq..5. nd eq..5. the nod vues n ntur oordnte sstems n e derved whh s shown n Fg..5.. Wth the ove retons vrtons of h & w e from - to. Now the nterpoton funton n e derved n sever proedures s done n se of two dmenson retngur eements. For empe the nterpoton funton n e derved nspeton n terms of ntur oordnte sstem s foows: N 8 h = hh.5. Fg..5. Eghtnode r eement B usng fed vre the foowng terms of the ponom m e used for dervng the shpe funton for eght-node r eement. = fh h h h h.5.

6 9 The ove equton s nompete ut smmetr. However suh representtons re qute often used nd souton onvergene s heved n the fnte eement nss. Agn the shpe funtons for three dmenson 8-node or 7-node r eements n e dervedusng grnge nterpoton funton. For ths we need to ntrodue nterpoton funton n the ζ-dreton. Thus for empe the grnge nterpoton funton for three dmenson 8 node r eement n e otned from the produt of pproprte nterpoton funtons n the ξ η nd ζ dretons. Therefore the shpe funton w eome N h = f f hf Where =. n-node.5.5 Thus usng the grnge nterpoton funton the shpe funton t node n e epressed s - h-h - h = h = - h-h - - h- - N f f f = = - -h Usng n of the ove onepts the nterpoton funton for 8-node r eement n e found s foows: N= - -h - N = -h - N= h - N = - h - N5= - -h N = -h N7 = h N8= - h.5.7 The shpe funtons of retngur preepped eements wth hgher nodes n e derved n smr mnner stsfng neessr rter.