144
T«AFSM INTRODUCTION TO THE FINITE ELEMENT METHOD TAYFUN E. TEZDUYAR James F. Barbour Professor of Mechanical Engineering Rice University [email protected] http://www.tafsm.org/~tezduyar Part of the class notes for MECH 454/554 Computational Fluid Mechanics Also listed as BIOE 454/554, CEVE 454/554 Please view these class notes with Acrobat Reader.

INTRODUCTION TO THE FINITE ELEMENT METHODl. FrNtrE ELETqENT MtrrroDs ( eem) FOR L|NEAR STATIG, ANALYSIS PR,ELI'Y I NAR,IE S I{AIN cONSTITUENTS oF FEM FoE T}I€ 3OLT'TION OF A BOUNDARY-

  • Upload
    others

  • View
    2

  • Download
    0

Embed Size (px)

Citation preview

Page 1: INTRODUCTION TO THE FINITE ELEMENT METHODl. FrNtrE ELETqENT MtrrroDs ( eem) FOR L|NEAR STATIG, ANALYSIS PR,ELI'Y I NAR,IE S I{AIN cONSTITUENTS oF FEM FoE T}I€ 3OLT'TION OF A BOUNDARY-

T«AFSM

INTRODUCTION TO THE FINITE ELEMENT METHOD

TAYFUN E. TEZDUYAR

James F. Barbour Professor of Mechanical Engineering Rice University

[email protected]

http://www.tafsm.org/~tezduyar Part of the class notes for MECH 454/554 Computational Fluid Mechanics Also listed as BIOE 454/554, CEVE 454/554 Please view these class notes with Acrobat Reader.

Page 2: INTRODUCTION TO THE FINITE ELEMENT METHODl. FrNtrE ELETqENT MtrrroDs ( eem) FOR L|NEAR STATIG, ANALYSIS PR,ELI'Y I NAR,IE S I{AIN cONSTITUENTS oF FEM FoE T}I€ 3OLT'TION OF A BOUNDARY-

l .FrNtrE ELETqENT MtrrroDs ( eem)

FOR

L|NEAR STATIG, ANALYSIS

PR,ELI'Y I NAR,IE S

I{AIN cONSTITUENTS oF FEM FoE T}I€

3OLT'TION OF A BOUNDARY- UILUE PEOBLE}4 3

( BvP)

O. A sTeorrf(r ( cuasstclu) roRFI oF Tne 6lutEN

BourrueC'- \fALut FroBL€tl

- DIFFESEIfrIAL €truA'lloNs

,-. BouNoAcy CoNDrTtorrt

, ( nrrrrAt co}rDtTror r ) ( rp tr ts AN

|r|rrfAL/ BourrDAQY-rArue Dtott€n )(wEGUTED BE:ToITAL)

l. A hrEAR ( vlerATtoNlt ) forril oF Tu€

PsoBL€JY ' l . (? )

Z. A 6Atclk'|rf FoEH oF T]fC PfsEt€n'r

( = FINFrE DlHcr.r$o$AL \rEtstoN OF TllC

hfEAr, Fo:H ) BY ustN6 Flrlrr€

crcrlENr Fulrs'rrsNs (?)

3 . A F|ATstI noRH oF Tlt€ PEoBrEr-l ( ? )

tezduyar
Rectangle
Page 3: INTRODUCTION TO THE FINITE ELEMENT METHODl. FrNtrE ELETqENT MtrrroDs ( eem) FOR L|NEAR STATIG, ANALYSIS PR,ELI'Y I NAR,IE S I{AIN cONSTITUENTS oF FEM FoE T}I€ 3OLT'TION OF A BOUNDARY-

z.

EXArutpLE : SOLVE, TllE toLtoWlNg Olf E-

D|F|ENSIONAL BOUTTDAR Y -VALU€

PBoBL€t\t

+ + + = o F a E o < x < ld x 1

'

DIFFEEEUTIAL I h, F \ DOmetU OF aUCFquATloN \u'E') ptoB\Er.t nctuDtN6

Tllc ErfD DollfTsg . , r uD l .

u(t) a ? I B*)NDABYv l : : : : : : : ._ (e.c.)

-jlto)= t J couDrrrolrs

4r 0rl : tsrrhrN Fut{cTlgrl AND corrsTA}lT:

IT : UNtsTFh|N VA3IABTC

ReuARr,s

tr'(l)=t ess€rtTlALtsoulrDASt corrDlTlox

NATUSAL

BourlorRY coNDlTloN

.- **t"= tr

tezduyar
Rectangle
Page 4: INTRODUCTION TO THE FINITE ELEMENT METHODl. FrNtrE ELETqENT MtrrroDs ( eem) FOR L|NEAR STATIG, ANALYSIS PR,ELI'Y I NAR,IE S I{AIN cONSTITUENTS oF FEM FoE T}I€ 3OLT'TION OF A BOUNDARY-

3 .E xAr{F\€ P$YS|CAL P?oB\SrS:

.) tl€AT CoND$CnoN PEoB\Etr

DlFFEtenlA( EQUATI0N : ll€tfl ColrurGflott

6:quATlox

+ : H€A? SouBGE ( or srNt )

U '. TEMP€:arueC,

gssEriTlAL B_C: a _ -- TE1v1DG?ATUTE E.l'B.c.: ?,

TEt"f DCrATurE B.!.

NAlttlA\ B.C. , L: H€AT ftUX B. C .

-t Elssrlc 3oD P:,olttFr'l

-;l gTP'lNg PBOBIEH

ttOTAllotl S :

*t = urx

d3. = tlrxxd x a -

tezduyar
Rectangle
Page 5: INTRODUCTION TO THE FINITE ELEMENT METHODl. FrNtrE ELETqENT MtrrroDs ( eem) FOR L|NEAR STATIG, ANALYSIS PR,ELI'Y I NAR,IE S I{AIN cONSTITUENTS oF FEM FoE T}I€ 3OLT'TION OF A BOUNDARY-

r F o < x < l r T l s s A t D r l r A T X € - S L

5-= F, ll

lF o< x < I rr ts sAtD rllAT xe-sl.

JL= Jo, r[-n, , G\ostllgn$\ F--'1 -.r E,, lJ?- : opErrl thmBVAL

( ootn Aril )H r { f , t

STEONS F€E,FT OF TIIC BVP :

R ( eesrur^r )-

Ur r r + f ro oN .SL

u ( r ) r t- u)r(o) = 4

( Vx€sr)af

\For ALL xr N t L i

a'

NOTE AtL DIFFIEB,ENNAL R\IAIIAN S HAUE

Tllg SAtrg GEN€EAL Fol'r-l

R = ot--- gtBrDuAL

rx rrrE AIov€ {ASE R= Urr.*f

tezduyar
Rectangle
Page 6: INTRODUCTION TO THE FINITE ELEMENT METHODl. FrNtrE ELETqENT MtrrroDs ( eem) FOR L|NEAR STATIG, ANALYSIS PR,ELI'Y I NAR,IE S I{AIN cONSTITUENTS oF FEM FoE T}I€ 3OLT'TION OF A BOUNDARY-

TltE ttlAgr soLuTlolr tS TRlul^L

.TO OBTAIN :

x 9u t r ) =

t * A t - r ) 1 t

{ [ t 1 d c d yt r o l

NorE Tf{AT t

^o,O = AE€ DuHorY

VARIABLES.

tezduyar
Rectangle
Page 7: INTRODUCTION TO THE FINITE ELEMENT METHODl. FrNtrE ELETqENT MtrrroDs ( eem) FOR L|NEAR STATIG, ANALYSIS PR,ELI'Y I NAR,IE S I{AIN cONSTITUENTS oF FEM FoE T}I€ 3OLT'TION OF A BOUNDARY-

6 .rrrear ( rrrerc,ril€p BBIDVAL)( Uee|Arrsunl)

re.Fr oF T}Ig DROBL€}I :

B r r . t { r o

wcElntr03RrNc1t0fr

Ita

I *gH5. dx

wc t6tflgD

: o

RCI3IDVA L

WN

BC}aTE,ES !

+ tNsrElD oE SA'tlSF tlrc Tll€ D.E. ATSlq:,v trgtNT lN -fi|, , TltE D.E. lS

sA.ttsFrEO lN A CtoBAL SRISE.

-) Tlill g\olAL CltlrJA'lloN llAt To BE

sATlr,FrS frF. AtJ- */'s ( r^rs,etrflu6

rr)lrcrroN i )'+ A g€lst:,KTtON oN w I w tllrTo

sA'IlsFt 'rut

oF tlrt sslfrtflAL B.<. '!

Fo:ra

tezduyar
Rectangle
Page 8: INTRODUCTION TO THE FINITE ELEMENT METHODl. FrNtrE ELETqENT MtrrroDs ( eem) FOR L|NEAR STATIG, ANALYSIS PR,ELI'Y I NAR,IE S I{AIN cONSTITUENTS oF FEM FoE T}I€ 3OLT'TION OF A BOUNDARY-

T.ErauDLS3 :

u( I ) - I €

u ( I ) = o = E

u ( f ) * bee { )

u(o ) - 5o :D

EXAHDL€-r-

eSrun$ THAT

w(r) r ' o

w t l ) = o

ul ( l ) = o

w(o) a o

ru€ < LAss ot FuucttoNs wlllclf sAllsFT

THF tblro6€\lO\rs fionn oF TB€

BS€NflAL B.c.'5 13 c ALLEI)

Ttrs CLASS OF VAR, lA_fl0N S :

u ( j ) = &THe\J TUF CLABI oF VAElfrlos5

coNslsTs otr FuNcrloul wlrlctfSArlsFY W(f )=o

'(u)

/ V \ - / 1 ' o - l '

t ?ArL BetoNG To tf Dolrr Ber'oll6 To 1I

tezduyar
Rectangle
Page 9: INTRODUCTION TO THE FINITE ELEMENT METHODl. FrNtrE ELETqENT MtrrroDs ( eem) FOR L|NEAR STATIG, ANALYSIS PR,ELI'Y I NAR,IE S I{AIN cONSTITUENTS oF FEM FoE T}I€ 3OLT'TION OF A BOUNDARY-

8.

tF t*€

anrr)

6LotA\ gQuAtlo$ ( ttneEAL

ls sAllsFlED fi)g A\L W I N 'tf

t ?

\ w ( \ , , r { ) d r = og J

T PIi\I TFIS 610IA\ RiJAT ION IS

Vwel|

€ulvAL€\sr To TltE sl6tNAL D.F

Se.ausg Tug G\oBA! R^JA.rfoNt r l l

ls TEST€D Fora F\\ TFs\ ( wner?il6)

FiructtqN: )

FOR FOl,tlrr AL FIOO F

FrRrr DTF|NE Tl^|o FuxcTtoil ctcstgs ( snecgsl

l . TR,IAL SouurroN SPAC,€

= [" t.":tj Y t "' or*G't csNDrrto.'. ]

Z. T€IT

f =

tezduyar
Rectangle
Page 10: INTRODUCTION TO THE FINITE ELEMENT METHODl. FrNtrE ELETqENT MtrrroDs ( eem) FOR L|NEAR STATIG, ANALYSIS PR,ELI'Y I NAR,IE S I{AIN cONSTITUENTS oF FEM FoE T}I€ 3OLT'TION OF A BOUNDARY-

R€-trr|BITE TTC STRoNe FoRr'l :

FrrlD ue h srrtrTHAT

ulrr + f

u ( r )-urr(o)

o

?h

--

a

3

!q.

( s )

(w)

WC NIEt TO SI|OW TNAT A$Cig TO GO9HS

Ars EeurvArcNT_ ( (S)e+ tw) )

Frrtt itrow T]IAT CS) =)(\^r)

f rnrs la Atso rus DGrtvATbN oF (\lr|) ]

l [ *( u;rrrl) l ' = o yw e"lf|trtqgtrnG 3t PArrS ;

( *u , , ) I I - I [ " r , ,u , . l r . t " [w f { rB o

O W :

- u r . ( o ) = { A N D w ( r ) e o

Vwe 1/

\^,EAL Foltrt :

$w€1f

dr + w(.)lt

r r r rD ueA sucnThAT

lS-,,u,. rr = l5 -t

( wu. f ? )

tezduyar
Rectangle
Page 11: INTRODUCTION TO THE FINITE ELEMENT METHODl. FrNtrE ELETqENT MtrrroDs ( eem) FOR L|NEAR STATIG, ANALYSIS PR,ELI'Y I NAR,IE S I{AIN cONSTITUENTS oF FEM FoE T}I€ 3OLT'TION OF A BOUNDARY-

@)(€esrC

T}rgI€ FctC

l[*,. ur, lr B ll*f tt + r^r(o) t

Ntrr tlrsrr r$Ar (w) =+ (S)w€ N$Ft TO Srsyp Tutta rt

u lt A Sotultop oF (t-l) TlrrN

ALSo A So1 r t? ;g11 OC (_ . ) .

r r , ; ' A € u t r ) = A ' /

1b.

Vw elf

lT ti

thr€?f:

sATtsFt ut (rl - o r

w(.)t:,:,1JJ=o

STqD

Tug ts?Y lt rrf,g crN DcFtNtout Duttotg oF

f t

Ar Lor|B AS lxE,IT A}IT r|AY WE

w€ 6tlrt xcrEt To trb\r TIrAT

U r r t r f I ourr (c)+ [ G o

.TO DO TTIA1 r I}ftg6fATF IT I' DArTTlN tC\FllSE I wE e et :

LI*(u, '1 t+) Jr +a o ?

lHow ?

DIooF. B9CAUTE

urAt{' To ltrvg

tezduyar
Rectangle
Page 12: INTRODUCTION TO THE FINITE ELEMENT METHODl. FrNtrE ELETqENT MtrrroDs ( eem) FOR L|NEAR STATIG, ANALYSIS PR,ELI'Y I NAR,IE S I{AIN cONSTITUENTS oF FEM FoE T}I€ 3OLT'TION OF A BOUNDARY-

sor wg tGFnrG w At , "'

\ l r s + ( u r r r * t ).$lFrr ?nG orllT corrDlal$Ut su + !

t. {trf a o L t.rrurr w(r), o]

z . E > o3 . Q ( o ) = o

-A2 | r r . h t-

: \ O ( u r r r * * ) J ' s o $ s r r , + f " o

if ow Tnrr btF ?to\f eD TrrAT Ulrr + I e O

Tlrg ufAt "oth Dsrc,Fl -ro

wt0 [ u.,r to)+ t 1. t tJ =o Vw c,1/

wg cA$ DFFt$g $l ArrJ rrrrY $€ uAlfT :

.rrto) * |

= t [ r r t ) + l r so , /

D OF PToOF

tezduyar
Rectangle
Page 13: INTRODUCTION TO THE FINITE ELEMENT METHODl. FrNtrE ELETqENT MtrrroDs ( eem) FOR L|NEAR STATIG, ANALYSIS PR,ELI'Y I NAR,IE S I{AIN cONSTITUENTS oF FEM FoE T}I€ 3OLT'TION OF A BOUNDARY-

lF u( I ) =

EXAI.IPTFI tog U A N D W

lob.

Lt\tsa,R'SrrNCTloNf

tl --

W =

tA --

\Af 3

l.A a

<- rrOt€ :

+ MrrE It

o|e-

I

T

cl t

u + (t-vr.)x l( r-x) J

o

I

H1I

I

o

\N =.o

tezduyar
Rectangle
Page 14: INTRODUCTION TO THE FINITE ELEMENT METHODl. FrNtrE ELETqENT MtrrroDs ( eem) FOR L|NEAR STATIG, ANALYSIS PR,ELI'Y I NAR,IE S I{AIN cONSTITUENTS oF FEM FoE T}I€ 3OLT'TION OF A BOUNDARY-

l l.NOTATIQN

fl.t w, u) W r t { r * d r{ r t r

= o l

frexp[€ll I

" 5 v" ur* olxl c

u ) u r r \ r d r

a(vru)e a(urv)

II

s " \ w v d rl 1= . \ vw c r

I (wrV)' (v., W)

(sY rrHFTrv )

A r \C l t V . l I =- - \ - J

J

A r \d ( u r Y J =

oBvfouslY ,

(w,+) = ltEXfrnrP\€I

( * r v )

(v, \Al )

oBVtoustY

( sYHu€rE y )

w+ dr

tezduyar
Rectangle
Page 15: INTRODUCTION TO THE FINITE ELEMENT METHODl. FrNtrE ELETqENT MtrrroDs ( eem) FOR L|NEAR STATIG, ANALYSIS PR,ELI'Y I NAR,IE S I{AIN cONSTITUENTS oF FEM FoE T}I€ 3OLT'TION OF A BOUNDARY-

l , , ^

le

= o\

*,. (o u + bv)"* olv

, r=u \ \ t ( ou ) , ' J x

t l ^

+"J wrr (Lv),r etll / ^

= oo)*, l \ * cv

{ - t ' ( * r v r

J rO J

3 a a ( w , u ) + b a ( w , v )

( Lr NEAETTY )

a(w'Hil]

(wro , lA+bv) B

o t*,;;-r L,;j;' Ir€!5

( LlNlEhR lrY )

Tll€t\t TllE \mf rro?H.' (afsm6cf ftr"r)

a (\^l u) = (w, +) + w(o) {. fweU

tezduyar
Rectangle
Page 16: INTRODUCTION TO THE FINITE ELEMENT METHODl. FrNtrE ELETqENT MtrrroDs ( eem) FOR L|NEAR STATIG, ANALYSIS PR,ELI'Y I NAR,IE S I{AIN cONSTITUENTS oF FEM FoE T}I€ 3OLT'TION OF A BOUNDARY-

r3

EXAMPUE : BERNq,LLI- EuuEI, B€AT.I TH€ORY :

; EFT FOTFTUTATION

" (

A .THIT' GSAFI ( BESNu.,LLI - €UL€: TIIGORV )

F ITED AI ONC €}|D A\lD SUBJECTED TO A S}IEAR

FOE3E AND }|O}IENT AT TTIE OTHER END

sTeox6 Fo:Fl oe Tlr€ Pb$fFr :

u

6t\l€N :---

FINDir-D

tFurrlrorlr TM errl Q,

Arrrb lmt

, tr*t| THAT

(rRlrrrcetc eur'lltrun )oN lt = ]o, lLu, rltr = {

tezduyar
Rectangle
Page 17: INTRODUCTION TO THE FINITE ELEMENT METHODl. FrNtrE ELETqENT MtrrroDs ( eem) FOR L|NEAR STATIG, ANALYSIS PR,ELI'Y I NAR,IE S I{AIN cONSTITUENTS oF FEM FoE T}I€ 3OLT'TION OF A BOUNDARY-

l+

BouNDAey coNDtTlo}rS

es:rur fE c . { u ( r ) = O ( z r i n o T G A T B Y s s s G D T S R A G E H G $ ; )

(, u' (r ) - o (z:ro s\oFr ).rrnruf urrr (o) = M ( p:sliccrrED ltoxesr)

t 'c' l"rrrl

(o) = E ( pE€scerBED s]rsAR- )

U'€AT, FOEi.| OB AH€ FTOBLEH

Ftesr DeFtr|g alrc ctAst53 oF TRIAI souuTloNs

AND UATTIATIONS

TLtt cLAlt O? Teltl FuNcrrorrt 1

J> l THt cr.Aslt oF Frn,cnorrS wrfrcH

SAalSF? TllG q1siq;f lAL B.C.'3

u ( r ) = o

U r r ( r ) = O

Tus CrAgS oF UA3ter|o}rl ( tre te rrru6 ?u$cTloruf )

tt : T$G cr.Ass oG Fuxc'rrqoS hrlrrll

sA'rl3F7 Tltg ]lo}|nG3\l9ut FoPFr

oB TltE Fsstilst|AL 8.G'.'t.

tezduyar
Rectangle
Page 18: INTRODUCTION TO THE FINITE ELEMENT METHODl. FrNtrE ELETqENT MtrrroDs ( eem) FOR L|NEAR STATIG, ANALYSIS PR,ELI'Y I NAR,IE S I{AIN cONSTITUENTS oF FEM FoE T}I€ 3OLT'TION OF A BOUNDARY-

I S

lrcnC-

r x ( r ) = o

wrr ( t ) - o

stlrc5 rnG llss€rirnAt B.G.'s AtE

AttlE, ADY t*

$ot og€r'us FoGr :

ru€ Turo CLAttFj AIC

1f = -/)'TltaT 13,

toclfTrcAL

TO oBartN ttr€ wgAra FoEl{ ao

= O

II,TC6'ATE BY FASTS CI3€

j l w( \*r* -{) Jsr- = o if*r eV

( * ' * r r - f )R.TTS IDUA L

tezduyar
Rectangle
Page 19: INTRODUCTION TO THE FINITE ELEMENT METHODl. FrNtrE ELETqENT MtrrroDs ( eem) FOR L|NEAR STATIG, ANALYSIS PR,ELI'Y I NAR,IE S I{AIN cONSTITUENTS oF FEM FoE T}I€ 3OLT'TION OF A BOUNDARY-

l6

(* , * , ** r ) l l - : l wrr urur dr -

' \ * f ar =o

6 t

ao a

=)

Ir.trCerArt lY FnqTS orrc€ moP€

( u, srrr') l: - ( wrr u,r, ) lj - j! w,*r u,rrdr

- " ! * f d x = o

g rw ( r ) = o w r r ( r ) - o

u r " r ( o ) - A U l r r ( o ) 3 M

e +wrr(o) M - !*+{x

=o) 1

oI *tr ur' dl - w(o)

t (

o\ *r'r urtt d x =

I

" l * fa* +w(o) q -\frr(o) M

tezduyar
Rectangle
Page 20: INTRODUCTION TO THE FINITE ELEMENT METHODl. FrNtrE ELETqENT MtrrroDs ( eem) FOR L|NEAR STATIG, ANALYSIS PR,ELI'Y I NAR,IE S I{AIN cONSTITUENTS oF FEM FoE T}I€ 3OLT'TION OF A BOUNDARY-

t?

DEFINE

a( u,v) = :5 {,xrY'r cx

(u.,v) = lt uv Jr ( As B.bee)

+a(w,u) B (wrf) + w(o)q - wr' (o) u

WEIg FO:FI

Q: wHA't oTllEg coNDlTtolrs Do trrEi llAv€

olr rn€ cLAsslF oF R NcTtor|S

couTAtrsrs6 \A, ArrD t

A: ttnn \{r, a}|b \.r )lAug To Bg'f S,quARJE ntrge:'rtLE

n

l c . - l ' r . a

L.e, \ (wr. . )d* <6 J 9 J

Plculouslv ( i.e . Foe llr.. -+ s g )}IAD

\^rg I1rrf To llA\G $l ALID tlrx To EC

squARl htr€gE,AtL€.

tezduyar
Rectangle
Page 21: INTRODUCTION TO THE FINITE ELEMENT METHODl. FrNtrE ELETqENT MtrrroDs ( eem) FOR L|NEAR STATIG, ANALYSIS PR,ELI'Y I NAR,IE S I{AIN cONSTITUENTS oF FEM FoE T}I€ 3OLT'TION OF A BOUNDARY-

I B

6ALER Krrf's AtrpgoxfMATtoN HgrtloD

\^'E SITPIY CANNCTT AFFOR.D ALL T}I€

FuNerroNs fN A At.rD 'lf.

L€T J BG A Fn*rE DrlfENSrouAt

T lltBGFollG uJg N€ED To APPIOXf T'fATCi

/) AND 1' BY FIuItT DIT.IENSIONAL

s\rrstrs oF .,/) AN'Lf .

w€ uss orfrY A PAET oF A AND V,r^,}IIGil W€ gAN AFFP D .

l f n

N!f9 h SuDer$trDT P€F€rrs To HFtr oR--

DtsclETlzATlor| . r.lN

on€OllrENslON,AL

FE HE:uE:, h P,€F€ns To rtlE

€u, }|€rir Lslb llT .

suBter ot At€T 1f

h 8€ A Fnirs DlNEt StoxAt SuBsg oF V

-zJ c, A (i... uh€)h+,^t€ A)ta

V (i... ",t€lrhs*t€ 1f)

tezduyar
Rectangle
Page 22: INTRODUCTION TO THE FINITE ELEMENT METHODl. FrNtrE ELETqENT MtrrroDs ( eem) FOR L|NEAR STATIG, ANALYSIS PR,ELI'Y I NAR,IE S I{AIN cONSTITUENTS oF FEM FoE T}I€ 3OLT'TION OF A BOUNDARY-

l?

FoE €rA}1p1g

( essernrnl

( se"rneF|9tr

PltBlwt3l gruGAl FTNS FoR, tlh '

tNs.rEAtr oF U3fN6 -/) ( THs crASS oF ALL

FTTNcTIONS Fe t^rHtGll

q( t ) - t ( ssosnAr B.c. )

.e (Ti.l:'ffi,ilLtl

usEh

A ( THg c't'Ags oF Ptgc€hrfsE LI$AP

I A'�S

. I,*,)r d *

FuNsTroNt ht wrf tC$f

,rt(,) E ?

jt (*1,)'d' <B.C- )

TTTEBT ATTF

DFPIV}TI\'IE ) )t f

tezduyar
Rectangle
Page 23: INTRODUCTION TO THE FINITE ELEMENT METHODl. FrNtrE ELETqENT MtrrroDs ( eem) FOR L|NEAR STATIG, ANALYSIS PR,ELI'Y I NAR,IE S I{AIN cONSTITUENTS oF FEM FoE T}I€ 3OLT'TION OF A BOUNDARY-

2o

stHltARLY :

tr0sTEAlr OF UEIN6

tUNs.llOul FeP

w ( t ) = Q

l " 1

o ) (* ' t) dr

U ( rrrt ctrss oF &LwHtct*

( ?br.roecuoirs Eis6rflAL B.C)

FfPtT UEeluAll\ft )

crAss oF plEc€ rrJts€ssg

LTNqAR,.-

w t ( r ) = el ? e

, \ (* t ,*) -J* ( , co

Furrc'TfoNs FOe UlUtcT

( rrorrooEilorrl €SsErfrat B C.)

( scuers rrJr€iB:ABr,rFregr DEEtvATTtE

)

FtrtGwtrs LlNerA: FurrGTlurl FoB ,nrh:

Uh ( rFE

tezduyar
Rectangle
Page 24: INTRODUCTION TO THE FINITE ELEMENT METHODl. FrNtrE ELETqENT MtrrroDs ( eem) FOR L|NEAR STATIG, ANALYSIS PR,ELI'Y I NAR,IE S I{AIN cONSTITUENTS oF FEM FoE T}I€ 3OLT'TION OF A BOUNDARY-

z l

qullTloN i cAN wt uic 9\lRcwtt€ UNRt

FrNgllet.rs |!af. ru€ 1'tt1.1e U\Ll -E\ILER

Bgerl trQ oqt€'H ?

AuswgR" .. No .

8€Ert|sE r^E l*ED Fuilfn$tr

trrtl€as lRorrg DEeNATIUFS Atf'

lfi, Fcn. P€C€ wftig u Ncre. R NcTlo ut

LW'rrr ls urrognrr€D AT SolfE

Po rllrs .

.'. W€ I,EFD AT L€t\i7 PlEl5trrtst CUBIC FulrGno$S .

SQUAEE IN.rBR, Et \€ .

t . e . l -

^ [(*,lrr)tJ*O J

tezduyar
Rectangle
Page 25: INTRODUCTION TO THE FINITE ELEMENT METHODl. FrNtrE ELETqENT MtrrroDs ( eem) FOR L|NEAR STATIG, ANALYSIS PR,ELI'Y I NAR,IE S I{AIN cONSTITUENTS oF FEM FoE T}I€ 3OLT'TION OF A BOUNDARY-

ziL

( BuEuor\ eeutertx rrgtno

Assun€ THAT U t

ls 6lu€r$. h h

Lgr us (oNsrnvcr J) By g1rN6 U '

.

Drrrwg A Fuucrlo\f d

(a6lvsr$Furrrtail)-

iucrr ruAr 6htr)= t

GrAHP\G fi)e At

( plelc€wrSE Ur$af )

7

f€? grr€Ry

\^,R,IT€

hv c ,

u h =

huh hv * a

htF \lNow c*rcL ro sgE BRoN6s To J

uh(,) = Jg.*dUo as A / l E f J

tezduyar
Rectangle
Page 26: INTRODUCTION TO THE FINITE ELEMENT METHODl. FrNtrE ELETqENT MtrrroDs ( eem) FOR L|NEAR STATIG, ANALYSIS PR,ELI'Y I NAR,IE S I{AIN cONSTITUENTS oF FEM FoE T}I€ 3OLT'TION OF A BOUNDARY-

2 3 .

+

wt'rrt''

:.:;;**

toR'

uhe y ) \

a(*luh) = (*1 i l + *t1o1 / v*IUh

sugstrrurg bQ \th h h' = v + A

VwheUh

2 3 .

Ti{C!3C EgF GATEEtsIN R|?HS

r --. .o)t^$.il--: r r-ALgRtsfN lbct.| ts FllflTg-Dll,lE}lstortA(-rrp t

uESSfoN o9 \$Ar peH.

tt .c. 6AtertN frrH ts AN ApglOIlrrATE

tbtr.rvuATlou OB TltE IrlEAlC frCtf

^ \ L ha ! r h- t v" lt rug uNENowU PABT oF U

tezduyar
Rectangle
Page 27: INTRODUCTION TO THE FINITE ELEMENT METHODl. FrNtrE ELETqENT MtrrroDs ( eem) FOR L|NEAR STATIG, ANALYSIS PR,ELI'Y I NAR,IE S I{AIN cONSTITUENTS oF FEM FoE T}I€ 3OLT'TION OF A BOUNDARY-

3") THg R.H.s.

lNvoL\lEs ALt

( f f t { L )

2A

otr Tffc ABtirg EquATloxqruq'N QuANTt"rFl

REmarr

I F .rh f 'uh +

hTIfAT t$, tF V colrG: FloH * ctAss

wn lcH tt DIFFCB€f.rf THAN TLlg CTASS

TrfAT wh couGls ;EoH , ruE

roat.ru\A'nelr ts cntt€D PlETQov- qALE BtlN

Folrau t' efTIOt.f . Tttll, FtraH ULATI9N

lS uS€D tre FuutD DYNAf-|ICS ptot\Srs.

FOS{.{U\ATION

tezduyar
Rectangle
Page 28: INTRODUCTION TO THE FINITE ELEMENT METHODl. FrNtrE ELETqENT MtrrroDs ( eem) FOR L|NEAR STATIG, ANALYSIS PR,ELI'Y I NAR,IE S I{AIN cONSTITUENTS oF FEM FoE T}I€ 3OLT'TION OF A BOUNDARY-

ts.FrATBlr gguATtolf! : 6TfFFN6S-@ x

sucH T ltAT

wt(x) - Z cA NA tx)A=f

Ltr tft couslsr oF AtL Lrf,ttiAR

Co!4Bt$aTtsfs OF 6IUSN FUWCflONS

NA (x ) ( a. lr zt}t..., . )

TrrAT ts FoR 5acH *h rN lthrHEeS F|(NrS CoNsrA\trs

C , A ( A r f r ? r g r r . r r n )

= C,N,(r) +CaNr(x) +....+CnN"Ut

NA(x) : SHAD€ Fu$cTlar I

(= BAstt Futcrrous )

(= tNTeRPoLATToN Fultcrrour )

tezduyar
Rectangle
Page 29: INTRODUCTION TO THE FINITE ELEMENT METHODl. FrNtrE ELETqENT MtrrroDs ( eem) FOR L|NEAR STATIG, ANALYSIS PR,ELI'Y I NAR,IE S I{AIN cONSTITUENTS oF FEM FoE T}I€ 3OLT'TION OF A BOUNDARY-

ls gcr$,|gF TIrAT

N^(1.) = o

THAT

wh(,) =

( e = l , 2 r . , . , . )

n

U c^N^(r) = oA--f

26.

\S S�TFFIFD

U' ts sAtD To HAVE DluE\lsloN n

rf l6q€e n =$ BFrTe: AFFRoXII{AT|0U

exelrPLE L€r rr= 3ugF prEtwt$, LtNaR, Fuf,icrtaNs

tezduyar
Rectangle
Page 30: INTRODUCTION TO THE FINITE ELEMENT METHODl. FrNtrE ELETqENT MtrrroDs ( eem) FOR L|NEAR STATIG, ANALYSIS PR,ELI'Y I NAR,IE S I{AIN cONSTITUENTS oF FEM FoE T}I€ 3OLT'TION OF A BOUNDARY-

{t.

t/s

wb(r). cr N,(r) +c, Ne(x) +cr N.(u

BY sg\€ertu6 PEOP€e vAtuFS FoR-

Cr r Ctt CI , A\\ PttEr3Fwftt lruFAl?

hrNcrrous F€a xe Lo, r L wrrHEEINES AT X- Vl A1.1D l r Z/3,

t/3

r b .lt

( . , - Vz

lc z= z/C

t c! . t/el .

( . , 9 o

€ ) c a = tL cr= t /Q

tezduyar
Rectangle
Page 31: INTRODUCTION TO THE FINITE ELEMENT METHODl. FrNtrE ELETqENT MtrrroDs ( eem) FOR L|NEAR STATIG, ANALYSIS PR,ELI'Y I NAR,IE S I{AIN cONSTITUENTS oF FEM FoE T}I€ 3OLT'TION OF A BOUNDARY-

24.

{ -1. -

( . , = + |

€ , 1 . . . oILcr s - I

-/)'\rohr, w5 ltAug

fiDe rr HtS wt

So TT{AT fiO..

TO D€HNE

HAu€ To sDrRrFy At

. l

vhe 1J n

V3

.*h= tn*At e -At

D€F|N€ A g![" s*eDe FUNCTIOU

N * , ( x )

Sucu rlf A"

N r * , ( r ) = 1

TFTN WR,IT€

h r r

ai.i= A Nr.rr(x)

tezduyar
Rectangle
Page 32: INTRODUCTION TO THE FINITE ELEMENT METHODl. FrNtrE ELETqENT MtrrroDs ( eem) FOR L|NEAR STATIG, ANALYSIS PR,ELI'Y I NAR,IE S I{AIN cONSTITUENTS oF FEM FoE T}I€ 3OLT'TION OF A BOUNDARY-

2 1 .

NOw WE CAN Untt€ :

vh= i d^ NA(x) (ae,nus vLuh)As l

At = Z Nr,+,(r)

Tl{ €lN

h h h , 'lts

'c t'*an - Z- dA k^(r) +

A NvH,(*)

A e l

TtfF coeFFtclEllt s

d ^ ' s ( l = ' l t z t . . o ) n )

AP,g UNENO\IN CottrhNTS.

t{srF TIAT

uh(, )= i d^!e(r ) + ANy,(r ;A r f V l

= A

tS SATISFIED.

tezduyar
Rectangle
Page 33: INTRODUCTION TO THE FINITE ELEMENT METHODl. FrNtrE ELETqENT MtrrroDs ( eem) FOR L|NEAR STATIG, ANALYSIS PR,ELI'Y I NAR,IE S I{AIN cONSTITUENTS oF FEM FoE T}I€ 3OLT'TION OF A BOUNDARY-

30. 1 1 L . hh (F-

w = L co Nn ( Vwheu" G' ALL GoFrtnrf'A=r

-A A \ Tto$s oF cja )

nu^ =

I, d" Ng +

t *n+t ( dE'. AF,b- Trr€

uNrNowtf s )

h

B = l

v" t^SUBSTITUTE, rXESE INTO T}I€ GAtERKIN FORM:

a(*1 ..^t) = (*", () + *to)r{ Ywl1t

, h n

? a(f, .^N^,I ,aeNs *tNn*,) =

wh , h

h

tgc^Nr , ( ) * (*coN^," , l |-Ft

51 h *\oj - FoB AtL GolrBlNrtn$t

otr CA's .

u6fN6 THE ElrlNEARtrY OF a (*rr ) :

tezduyar
Rectangle
Page 34: INTRODUCTION TO THE FINITE ELEMENT METHODl. FrNtrE ELETqENT MtrrroDs ( eem) FOR L|NEAR STATIG, ANALYSIS PR,ELI'Y I NAR,IE S I{AIN cONSTITUENTS oF FEM FoE T}I€ 3OLT'TION OF A BOUNDARY-

3 l=t

({{,*Na, f )---

y y h

(t,oNo(o)t

w\o )

h-a(fc^N^ ,t\,,r,)A=t

- - r

r wh t :::i11,*.,,r lF Cr t

, ha( E crNe

A= l-.-ir-.d

wh

n, : . d " N " )

B: f\--fra-..-,

vh

3ILINEA:FY OF

A N o ( * r * ) !

r( No, n") dB - ( *r^,{) - N^(ollt

+ a(Ne,N, ') t1 = 0J

-> FOt, ALt (oFttl$Aalo$s

o F C r ' s

cQulrtol lt aBu€ Fot" ALt c.or-lElNAAlONt

, wg CAN ALtO WBITE lT AS Fotlorrtt 3

UStNc rn€

a (*i* )

n f n

: c^ l IAs l I B= r

L

slf,||(€ Tlll:

oF Ce's

tezduyar
Rectangle
Page 35: INTRODUCTION TO THE FINITE ELEMENT METHODl. FrNtrE ELETqENT MtrrroDs ( eem) FOR L|NEAR STATIG, ANALYSIS PR,ELI'Y I NAR,IE S I{AIN cONSTITUENTS oF FEM FoE T}I€ 3OLT'TION OF A BOUNDARY-

h

f a (u^ ,NE) d * - (Bgt

3Z

+a(No,N*)?=of , : fp2;. . . t f l

Nsf )- N^ (o) A-t For'

L€:T US DgFiNE

TH5N, ?lrE gQUAT|oN CAN BC WgrTTgN AS.'

oR r r{eg€ €rptrgtTtY :

h

Z K ,sd . = F fBg l

n

I Kr.o J"C=t

. t i t O a

!Il-

L K h B o B8 a t

= F r. . o a a

= F , .

EQuartous

uNtrnldltS

h

h

Fot

K^s

q

= a (NA,n " ) ) FoR ArB = l r?r . . , r h

= (ru^rf ) + NA(o) i-a (Ne,N,,r, )7-f Rlt A: lr?r ...1 l l

n

IB=t

( 1 . 1 1 2 r . . . ) n )-- FAKo" d"

tezduyar
Rectangle
Page 36: INTRODUCTION TO THE FINITE ELEMENT METHODl. FrNtrE ELETqENT MtrrroDs ( eem) FOR L|NEAR STATIG, ANALYSIS PR,ELI'Y I NAR,IE S I{AIN cONSTITUENTS oF FEM FoE T}I€ 3OLT'TION OF A BOUNDARY-

03, eUeN xoBC EIPUIGITLY :

K r r d r + K r a d a * . . . , + K t r . . d n =

F2

Fn

tt

Tuls ts TugP9o BL€I-I

Lgr us AD0DT Tll€

NorATtONS

= t *^o l=Ess MArBu" )

K-

( rrsrrrrrr

) tr f l

^ l f

:

)tt\ llf

K1 d , + K"adr+ . . . . * E rndr . =

t a a ( ) l t a a a a a O a t l a

Kn,d r * Knad" + r , . , . f K r . r . dn =

1a t i a

h uNt,Nowlt :

F,IAT:f I +bEI.l oF "lf E

FOLLOWIN6 TYIATEII

f tt( ' t1 ' ' ' ' | \ ln

h l Z . . . . f , z n

: :t

o t. l^

n n ? , - - - - - ' \ h n

33

nEcuA'?rcNt

tezduyar
Rectangle
Page 37: INTRODUCTION TO THE FINITE ELEMENT METHODl. FrNtrE ELETqENT MtrrroDs ( eem) FOR L|NEAR STATIG, ANALYSIS PR,ELI'Y I NAR,IE S I{AIN cONSTITUENTS oF FEM FoE T}I€ 3OLT'TION OF A BOUNDARY-

F= t t ^ l

K-

d = F- i ,

Fr l

?:E

I t

Ie l

Iv l

a

a

a

a

Iv n

=l

r I t lct = t c l r l =( *J

( 'r Forc€ v.,ror" )

(' urslrAc€H..rr vecroc" )

34.

Eq\fATtOr!

AB€ N

N UNr'NOWNS.

e Tttts fs A LINEA.'SYSTqFI . TU€B€

FqUATIOrIS FOR

Id IS T}IF UNTNOU,N UK-TOR,.

Tffg FrArclr K ANb rlrE vEGTot F AR€- - ,

rlhlollll I THEIS CoFfForrStfS CAlr BE cOr.rPurE'D- r - ' - - '

By u3tlt6 rg€ gIPPrBstoN t D€Rlv€'D DgNl0UsLt.

tezduyar
Rectangle
Page 38: INTRODUCTION TO THE FINITE ELEMENT METHODl. FrNtrE ELETqENT MtrrroDs ( eem) FOR L|NEAR STATIG, ANALYSIS PR,ELI'Y I NAR,IE S I{AIN cONSTITUENTS oF FEM FoE T}I€ 3OLT'TION OF A BOUNDARY-

j 5 .

E IAT"TPLE S :

K r r = a ( N r r r u , ) = ' I

N , , , N , r , d x6 I

= ' (

d \ d ,u , drs J d x d r

l "Kgy = a(Ng,t l*) = ") N.,, Nr,, dx

K r S - K s r ( w H Y ? )

Ft

F7

= ( ru , ,1 ) + N , (o )A-a (Nr r l . r r , . , )p

= o! tl, f .tr + trr,(o) ,. -(j ! Nr,r Nn*r,r Oi t

= (r r r r , ( ) + Nr(o) / r -a (ut ,Nr* , )?

=.! *rf a, +N*p)h fiNr,,

Nn,,,{f

tezduyar
Rectangle
Page 39: INTRODUCTION TO THE FINITE ELEMENT METHODl. FrNtrE ELETqENT MtrrroDs ( eem) FOR L|NEAR STATIG, ANALYSIS PR,ELI'Y I NAR,IE S I{AIN cONSTITUENTS oF FEM FoE T}I€ 3OLT'TION OF A BOUNDARY-

36

Tltg rtNgAg EqoATloN sysTg}4 cAN Bn soLvtD

BV US|N6 AN tQUATlOtl SOLVER-

oss E Tu€ srsr€rr ts soLvgD Fon d , l"

ls TRtutAL tQ 6€T Uh A\rD lrs D€PluATlug .

n

,t(*) = Z dA NA(r) * tNn*,(r) Vx€-n=to,,]

A = l

h . 's(x\r ,.>'doN^(tf, *? Nr*,(t\,A e l

R,EHABK :

K ts sYHr.rqrgrc, BEGAusE a ( Na, N r)

TS A AIHHFNITC. BILINEAN, FOEH :

KAB= fl(l-t^rNa) = a(rua-Ne ) = Ksa' v r t '

TTnAr tS K ' = K

tezduyar
Rectangle
Page 40: INTRODUCTION TO THE FINITE ELEMENT METHODl. FrNtrE ELETqENT MtrrroDs ( eem) FOR L|NEAR STATIG, ANALYSIS PR,ELI'Y I NAR,IE S I{AIN cONSTITUENTS oF FEM FoE T}I€ 3OLT'TION OF A BOUNDARY-

3?.

E X A t r f L E . l : 1 - D E G E J E S . Q F - F B E E D O H

( i... 1 uNtsuorlu )

h = |

w h = c , N r

u h = r t * t t = d , N r * t N z

SHAtr€ FUN(IIONS :

N , ( r ) = l - X

N . ( x ) = X

? Do TI{EY sATtsFY tue

N , ( r ) 3 o

N e ( r ) 3 l

K r r

Ff

d f

xI , o,

Q.Qvttr t.tcNrs ?

Yes

\FS

g = L*,,1F = l t , ]

d = [d,J

tezduyar
Rectangle
Page 41: INTRODUCTION TO THE FINITE ELEMENT METHODl. FrNtrE ELETqENT MtrrroDs ( eem) FOR L|NEAR STATIG, ANALYSIS PR,ELI'Y I NAR,IE S I{AIN cONSTITUENTS oF FEM FoE T}I€ 3OLT'TION OF A BOUNDARY-

38.

K * = a ( N r r x , ) =

I

-' l

r-L

a (NrrNr)y

t r

\ v',' tu,,o ,r-_-

- l + ' t

= ( t , ' r f ) + N , (o ) h -

j l t ,-x)ft*tdx + /t-

It N,,' N1,' Jr

" tn r ) ( - r )dr

tr = .!(r-x )fb)dx!il

NOtJ

uh(*)= Llt (r-t)f(e)c, +ft ril(r-r) + f xNr(r)

= l l , f-x l f tr) dx r lL * t

TltS LIN€AR, q\fATtON S?ST€H

K 1 1 d , = F l

d, = Fr /Kr r =

cofitT:uc? uh :

= 1

+A rt

d x t

,1 outy ox€\

Aro oG

, Nr(r)

EQUATIOU\

UN Ti,OUr|J

dl

tezduyar
Rectangle
Page 42: INTRODUCTION TO THE FINITE ELEMENT METHODl. FrNtrE ELETqENT MtrrroDs ( eem) FOR L|NEAR STATIG, ANALYSIS PR,ELI'Y I NAR,IE S I{AIN cONSTITUENTS oF FEM FoE T}I€ 3OLT'TION OF A BOUNDARY-

; - - . - - e - - - - -7 - - - - - - - - - - - - - - - - -?1 '

i,.'t*r = fr * { (,-x) + [:f,,-t)lt*g](r-') |l - - - - - - - - - - - - - - - - -

dl2

uNtNolr|N

tsNOtrrN

rara5-

R€FrgF{BE? T}f€ FIACT SouvnoU ; ANo coFrtrAP€ ,

u(r) = t+ r { t , -x) * ' t j [ &q de o?

$orrgcCltou SiOLiJTIOr|

P Agrrc,uuaR3 0 L u T l 0 N

oBvtoustY, FEM SorurtoN 6GrS

PACT oF rrrS SOLUTION EXACTTY

Tltg lbrroeeloU)

- ( Vxe [q , ] J

THQQETQ€ r sE ONtY N€ED To CoFttrAEF Tl{E

PAsTlcutAE, SoUW|ONS, T]f€ ACCUIACY OF Tl l€

Fgt*l pAgTlcuteQ SOLuTION , OFCOUPST/ D€FC\DS

oN rHF ForH oF fCr: .

tezduyar
Rectangle
Page 43: INTRODUCTION TO THE FINITE ELEMENT METHODl. FrNtrE ELETqENT MtrrroDs ( eem) FOR L|NEAR STATIG, ANALYSIS PR,ELI'Y I NAR,IE S I{AIN cONSTITUENTS oF FEM FoE T}I€ 3OLT'TION OF A BOUNDARY-

CAEI3 L

PASTICUIAB

ITA<T

Flt Ft :,

, f (x)= P =corrSraNt '

sDrurlolrs:

um.r(x) = ,

(r-x.)

, f * r (x )= ! t r - x )

4 0 .

( w r s l )

( rrueaq ! )

I

)

+ huPl l t = sp^r t X = O A N D X e

AT Tufo [9DgS

AT

( i .e .

l{ow ABour

( * o^"r)r,

( *n ^. r)r,

S0LUrlON

x= iIS EIASf

T}I€ DEPI\IATIUCT ?

't= - P x l + t ;

f A r- - D l' - . b I

z J

tezduyar
Rectangle
Page 44: INTRODUCTION TO THE FINITE ELEMENT METHODl. FrNtrE ELETqENT MtrrroDs ( eem) FOR L|NEAR STATIG, ANALYSIS PR,ELI'Y I NAR,IE S I{AIN cONSTITUENTS oF FEM FoE T}I€ 3OLT'TION OF A BOUNDARY-

4 l<.As3

PASTKUI A 3

EXACT I

tr€t't :

SOLUTIONS :

u PAEr (r l

t'en:, lr I

( c u 6 r c !\ ---F\

( t , u€ 'D E

= { tr-xt)

= ? t,-r)ETACT AT

( i ' c '

fu) = ?x ( tweae )

)

I 1F€l{ 30LUr IAN tS

D€rlutTtvfi ..

r (eO AuO Ia I

A7 TWO |.0DF:

FE!}l soLunurl I S

ErAer AT x* tfts(* oo.r),,

("L,'),r= - * 1 '"1

Sui"ll-tACY

a .

b .

=-+

hU' l5 EaA'sT AT X=O euo )(= |

FOg €ACII CA:C TIICCS IJ A.r L€D:T

oN€ eotNr rN TllE tll(€\rAL orl A1

wrtrcH *1* ls ExAcT.

tezduyar
Rectangle
Page 45: INTRODUCTION TO THE FINITE ELEMENT METHODl. FrNtrE ELETqENT MtrrroDs ( eem) FOR L|NEAR STATIG, ANALYSIS PR,ELI'Y I NAR,IE S I{AIN cONSTITUENTS oF FEM FoE T}I€ 3OLT'TION OF A BOUNDARY-

4LEXA$A FLE 2- ; 2'Dg6R€g oF F Pgtbor-l

( i . . . L uuENowNs)

c r N r * c " N z

h hu " * t ' = d , N , +drNr * tNt

h = 2

* h =

h| - [ =

StlAP€ FuNcTloNs i

1 .

(.tr.r ,*)=[t;* j ; : : : ' : ]

nr,*r=f 2x 'o{"si} . -

[etr-r1 , *sr{r I

'l;,:;::::1.-l . X

N3(r)

tezduyar
Rectangle
Page 46: INTRODUCTION TO THE FINITE ELEMENT METHODl. FrNtrE ELETqENT MtrrroDs ( eem) FOR L|NEAR STATIG, ANALYSIS PR,ELI'Y I NAR,IE S I{AIN cONSTITUENTS oF FEM FoE T}I€ 3OLT'TION OF A BOUNDARY-

? oo ru€Y sATtsFY

?N{, I ) - ' o.

nN r ( r ) + o .

?Nr ( r ) : I .

s s

"H€ aerplEgs4eNr: ?

't ES

YEI

rs

ST|FFNES YNSLXT FoCC€ VRToR

uif tr|onr$5

K

4 3 .

/ rrnwN

FIE

d-

=h::]=lil

d l

,d'

trl C

'z

E

= [

' I F F I'T

K a l = a ( N r r N . ) = lIn.'N..' dr

- l t t b

/ i - - / td , s

( A r s = l r 1 )

tezduyar
Rectangle
Page 47: INTRODUCTION TO THE FINITE ELEMENT METHODl. FrNtrE ELETqENT MtrrroDs ( eem) FOR L|NEAR STATIG, ANALYSIS PR,ELI'Y I NAR,IE S I{AIN cONSTITUENTS oF FEM FoE T}I€ 3OLT'TION OF A BOUNDARY-

11, rK r r =

_ \ N , r ,q J

. ' . K t f = Z

Sfr.r[AeLY :

Krz = Ket

K a z s 4

!

N,r , dx = t ( ( -z ) ( -z )dx

o )

=, {,= L; ,il[ ] l+

= 2

= - 2

K = 2a

S.TTFFNSS ITS I

FrATt t rpotcE

ro = ( N^,f ) * N^(o) I -l (u^rNg)

= l! *^f J' + N^(o) A -]"[to,' Nr,, ,) f

lrs lNve c Ee

( A ' 1 , 2 )

tezduyar
Rectangle
Page 48: INTRODUCTION TO THE FINITE ELEMENT METHODl. FrNtrE ELETqENT MtrrroDs ( eem) FOR L|NEAR STATIG, ANALYSIS PR,ELI'Y I NAR,IE S I{AIN cONSTITUENTS oF FEM FoE T}I€ 3OLT'TION OF A BOUNDARY-

*2r

case t , f t r )=O ( t toxoearouscesE)

45I

F, = it {.r-2'1f(x) dr + h

! l

Fe = 1 ( t,.f(r)ax *. I z( t-xl ft* ) dr' o J

L

d- r'E= L;i] [;} = ll:11, t(*) - (, * l) N, (r ) + (? * *l^,r

(r, +7 Ndr )

= f (Ar,(rlr Nr(r) +ur(r) ) + ( 1u,t4, ry)

= t + l(rx) <- exlr:t ,roHo6sirout

so,.u:/roN

QT'MARK FE t\{ ALWAYS 6€T3 TIIE STAG'T

uorroGeNpus soLuTloN B€cAus€ EXACT nOnOGStOUS

suwror.r ( urueae ) ls coN"AlN€D lN ltt€ cLAss

oF Tc.lAL soLurtoN5 ( plccgwts€ LlNCrte ) .

tezduyar
Rectangle
Page 49: INTRODUCTION TO THE FINITE ELEMENT METHODl. FrNtrE ELETqENT MtrrroDs ( eem) FOR L|NEAR STATIG, ANALYSIS PR,ELI'Y I NAR,IE S I{AIN cONSTITUENTS oF FEM FoE T}I€ 3OLT'TION OF A BOUNDARY-

p-4

?-ra

:-

?t l

II tt -rb

uprnr , *(r_*")

S a*hr^."= Upoet

46t a

cAsg J(

F , =

F*=

d =

A6Atll, t^lE

souwrous

hu pe.r

ftrl= p = corssmN?-

\ * r + ? + A 1Lget"*? + Atzl

+ L

* 2 3

il \i;il,l =N€€D TO COFIPA:E TX€ PAgTTCqNQ

! x, + tt *, ( rlN€A: .! )

NL'w

Al X=O r

l .

f t ' Q ' A T

(elroeAT lc

* = f , X = I

rlr€ '*,ot

ei' )

: 7

lgQruartues ?DEQ,lvertuES Ar€ trF AT

x = Va AND x* 3/1 ( ExEscls€ :t tbw lT.

tezduyar
Rectangle
Page 50: INTRODUCTION TO THE FINITE ELEMENT METHODl. FrNtrE ELETqENT MtrrroDs ( eem) FOR L|NEAR STATIG, ANALYSIS PR,ELI'Y I NAR,IE S I{AIN cONSTITUENTS oF FEM FoE T}I€ 3OLT'TION OF A BOUNDARY-

4?.

F , = g +' 2 a

F- = -3-*z _ f _

+ d= x l . tp- A -

c^agg cl's /C*l = g x (rrruene- )

,?

9leltcuuat, i, 'LuTte*,i,.

t'*", = !. r,r, * 73- rra (urrrrer | )

eper r = + ( r -x3 ) /6 ( tu8 rc f )

h[ t t- 'peq e UF^.t AT x:o , *=* rx* I

( r'.e . AT nrg rbDt.l )

DgsruATtvES ?

DrEErvrnu€l Ae.g EIAC'r AT Ttrb polNfs ; oNC

rg Tlr€ rrEgEEnAl, Eo,tl , olr€ tr t* , I I .

tL

lz= I 7 + 7 * J '

lztl+a +?+1,

tezduyar
Rectangle
Page 51: INTRODUCTION TO THE FINITE ELEMENT METHODl. FrNtrE ELETqENT MtrrroDs ( eem) FOR L|NEAR STATIG, ANALYSIS PR,ELI'Y I NAR,IE S I{AIN cONSTITUENTS oF FEM FoE T}I€ 3OLT'TION OF A BOUNDARY-

48

Suttt{AE"f/

q,. Thlc Hor{oGti\l9u: soLuTloN 6tu€t\r Bv

TnG FCIYI lS tx{;T Fog At'L CAtts .

'rHls rS BeCnusE Tllrg e?(mT ttox()G€\rous

Sc>urntou ( rru*tcu rS urusre ) <.AN rl€

P.EtrtlSC\tflfD EI!$!!. By THC F"rr\rcTlo*rS

EErougllr6 Tg TrllS TC.|AL F$N('rrouS

CLASS ( wlrlcu tS glc'e€\^rrsF LtNne )

b. Ttttt trAnnrc(r"AB sotuTloN G\v€r{ Bv

TH|E FeI\r l$ €r.AcI AT THE

q}lg$ oF €Acu SuBtr'fi€E\rAL .

C. DtEe\UATru€ OF Tlr|E' F€r.t StlL\rnON

ts e:Acr ar (lr tEAsT) oNE PorNT tN

QrcH susrNTCR\IeL .

tezduyar
Rectangle
Page 52: INTRODUCTION TO THE FINITE ELEMENT METHODl. FrNtrE ELETqENT MtrrroDs ( eem) FOR L|NEAR STATIG, ANALYSIS PR,ELI'Y I NAR,IE S I{AIN cONSTITUENTS oF FEM FoE T}I€ 3OLT'TION OF A BOUNDARY-

4 7 '

flEc|EWISg LIN€AR. FINTTE ELEMENT FUNCTTONS

trABTrTroN Tltts lorurrr ( rurcevru ) ll.lto

n suBDolnl$; ( suB-TNT!!RUALS )

L**" L x^, Xo*,]

( A r l r Z - r . , ) n )

X\ : NODT|L PolNfS

NODTS

JOTNTS

KNOTS

Xt = o. ( uertFrorr r.roDq )

Xn+ r= l . t c t eu rF {os NoDE)

L x^, X o*,J trlNtTE ELCHeT{T Dqpretxs

e LEF{ E$trs

eLEHE\JT r*€-N6Wr r ha = Xo*t - XA * COT.TSTAI\IT

Page 53: INTRODUCTION TO THE FINITE ELEMENT METHODl. FrNtrE ELETqENT MtrrroDs ( eem) FOR L|NEAR STATIG, ANALYSIS PR,ELI'Y I NAR,IE S I{AIN cONSTITUENTS oF FEM FoE T}I€ 3OLT'TION OF A BOUNDARY-

50lF Tll€ eL$ttttST l;GNGTl{ | 5' (SrISTANT

TH€t * hn= l / n * h

3t{Apg FuNcTt0NS

F o R . T N T E R N A * N o D E S ( Z < A S n )

iltsouxrbAeY NGIIHS ( A= i A$r' As h+ i )J

N s ( r ) = ( x r - x ) / h t , X , { x s x a

N n r , ( x ) = ( x - x " ) / h n ) x n < x < X r . * ,

x - X ^ - r ) / h o - , r X A - r ( X <

X e * l - X ) / t * ^ , x A <

t, ELSe\JHgcg j

N^(x) =

Page 54: INTRODUCTION TO THE FINITE ELEMENT METHODl. FrNtrE ELETqENT MtrrroDs ( eem) FOR L|NEAR STATIG, ANALYSIS PR,ELI'Y I NAR,IE S I{AIN cONSTITUENTS oF FEM FoE T}I€ 3OLT'TION OF A BOUNDARY-

5|.

rr xA + l xh

( a o - - o - - ! - f - + = - - - - {' xA-, xl xl*, x.r X"+r

Xn*r Xr! Yn+ r

x l '

Nrt r )

x^-,

' *o{ xe xr*t

€t--Ef}leeff LeNCTbtl lo*r? *Aqs T* *t

rn*,

? mte rHAT TllE S1*&i""

Page 55: INTRODUCTION TO THE FINITE ELEMENT METHODl. FrNtrE ELETqENT MtrrroDs ( eem) FOR L|NEAR STATIG, ANALYSIS PR,ELI'Y I NAR,IE S I{AIN cONSTITUENTS oF FEM FoE T}I€ 3OLT'TION OF A BOUNDARY-

TtlESt liHr\pc Fut-lc'TloNs Ag€ c-euugD u

Fu*,cTroN3 r o* nR**'

RJNCTto$s . Tl{E t

sruFLE $ wTDELY usED.

AEFIABE,S :

'1. Na (Xe) = Eo'= l:

i.s. TlfE SFAPE EUrleTfON

C EBTAII| }\lOD€ IIAS A

tF A--B ?t

r F A + B J T oeoPR€SEo$DlUOf A

VALU€ OF I. AT

5Z

ltAT tt

ag€

LEFI qlQFErIt

Rt6tf" E\€i{ar

z .

THAT NODE 't HAS A VALUE OF O. AT

ALL OTH€E NObgS.

Tl*E s[AFg FUNeTTON COtRtSSPOtrDlN6 TO

A CligTAlN NOD! hAS A NoN-tClGl VALUE

oNry INTHF (eeN) suBDoHAlNs (etsrrElfG)

oNg To rt{€ \€FT A}tD o}l€ To TllE E,|6TT

6- aTA? rglC. tN o.rueB $oBDls, lT lrAs A

Nofrt€F rfALtt srLy tN TH€ (onetu) sugDotrAml

t€\SrG$S) T*A-t coNlrru THlr $oDe.

i .€ .

Na(*) f o ONIY FCT

X e J X^-r ,Xet (-

+re I Xr , xe*rf

Page 56: INTRODUCTION TO THE FINITE ELEMENT METHODl. FrNtrE ELETqENT MtrrroDs ( eem) FOR L|NEAR STATIG, ANALYSIS PR,ELI'Y I NAR,IE S I{AIN cONSTITUENTS oF FEM FoE T}I€ 3OLT'TION OF A BOUNDARY-

53PRODGETTES OF tlG S'nFFNGSS F4ArElI s

h.tArn, ot THe, ENTnttns oB K AEs rcco.

l rKet =o\ No,t Nao dx

FOT. A 6I\IEN }FDC

\IIIAT VALUES OF B

Fr B= A- l

Ket *. o

Foa lrr A

l(er * O

A I LET US INsDFT ftI,

wE 6ET Kes * o

BS A-2 : KeB - O.

f -l A-l A-l

A-1 A-l

A-f A A+t

,,^.1

La/

Page 57: INTRODUCTION TO THE FINITE ELEMENT METHODl. FrNtrE ELETqENT MtrrroDs ( eem) FOR L|NEAR STATIG, ANALYSIS PR,ELI'Y I NAR,IE S I{AIN cONSTITUENTS oF FEM FoE T}I€ 3OLT'TION OF A BOUNDARY-

stroR. B= Atl

(eB +o

B> A+2.

Krt =r O

r,r= Krz

K =-

A I L

\ \

+ 5 l s BAND€b.

Page 58: INTRODUCTION TO THE FINITE ELEMENT METHODl. FrNtrE ELETqENT MtrrroDs ( eem) FOR L|NEAR STATIG, ANALYSIS PR,ELI'Y I NAR,IE S I{AIN cONSTITUENTS oF FEM FoE T}I€ 3OLT'TION OF A BOUNDARY-

55.

ssg:l. Tltc tt\frx€s oF E uJ$rcr wF Elnbt to B€ tteGto

lrr AutANtg ( rrrrflforrlf GALGI IATING K ), AEiE_,./r a

W i lN TH€ CgoATlort SowEI TllCt

AgG }FT OD€CATFb gN.

2. tT CA\t EE Sllobfir TnAT TnE STIFFhIESS FrA'rtlI

K ls FO3trlvE -DtEFllrrtc_'

' J l -

MA?II€F4ATKAL ABFECTS OF THg tr€7l SOLUTIO\I

l. rT €AN BE s$owN TIIAT THC F€Fr sotunotlrI

ts aAgT trt Tlf€ NobEs . l.c .

uh( xr) = U (Xe) ( e= ) ,2 r . . . , n+ l )

Z. |t CA\I 3€ ShOTTI}r T$T TTCEE II AT LEAT

oit Folrtr tN c\fEtY 3g6gr$ Ar br|tCH

THs pEEruATy€ oB rlfG Fst{ Sorunou ( "1. )

- f3 ErAgT.

tezduyar
Rectangle
Page 59: INTRODUCTION TO THE FINITE ELEMENT METHODl. FrNtrE ELETqENT MtrrroDs ( eem) FOR L|NEAR STATIG, ANALYSIS PR,ELI'Y I NAR,IE S I{AIN cONSTITUENTS oF FEM FoE T}I€ 3OLT'TION OF A BOUNDARY-

% .

DCFTNTTIONS

t Ge*gG. rN @ AT Follt ot

e ( d ) = u h ( a ) - i l ( q )

t eetot ltl rlr€ lrEc fvATlvS Fr pofilt {

e,r r td) = r { r to) -urr

-) oRlee of Acutacy ( olDrc oF cotfvGee€$(G )

rr|SDTECjT ]low €rr (q ) t1g€tt To tC:O

A3 Tlrt Grgllcrlr tGtSrH hA GD|E.ll To

ircQo) . THlrt t3 r As

hA-r O

brrs err(a) = d(he) -t oo? gr (d) = o( hi) - r o

oBrfrousrr g(h: ) GoGts \tD z€Bo ruITIIAN g(hA

) . ttl €rNGEAt \E"

9rr(c) = g( ll I 3r k : oruEQ- oF AGG.tct

oF €rr(q).

tezduyar
Rectangle
Page 60: INTRODUCTION TO THE FINITE ELEMENT METHODl. FrNtrE ELETqENT MtrrroDs ( eem) FOR L|NEAR STATIG, ANALYSIS PR,ELI'Y I NAR,IE S I{AIN cONSTITUENTS oF FEM FoE T}I€ 3OLT'TION OF A BOUNDARY-

57.P.CF|A:ES :

l . rT C^A}T GE STOUTs 'HAT

€rr(s) = g(ttil Foe

eri (d)

SECOND O:D'E?

s th^ ) Fo?

o( = L--Xnrr2

( gu€r.rFllr FrlDDolrr)

AeculeTE

s * I a lX r+ l

lfz

T

2 .

( t}| CCNtSAL )

!n F tes" oarGe ACCuIATE .

lr cAN B? strorrx T]IAT tF rrrG €rAc"

SoLuTtgN tt $rADtftlG, ?HCN lT€

ls:ruFtug oe Tlr€ FgM solurtorr

ls €t3Aet xr Tlf€ €u€mertf nrtDF!|ls

- - t l -

tezduyar
Rectangle
Page 61: INTRODUCTION TO THE FINITE ELEMENT METHODl. FrNtrE ELETqENT MtrrroDs ( eem) FOR L|NEAR STATIG, ANALYSIS PR,ELI'Y I NAR,IE S I{AIN cONSTITUENTS oF FEM FoE T}I€ 3OLT'TION OF A BOUNDARY-

5E

TllE ttL€lr€NT CohtcgPr

MFSII :-- l l l

a. -a a

a

I 2 3 , ' g "'.-

\-1 hcl

xr Ir Xr 14 : XA xe*i: I.-, xo Y.*,\ t

o , o - - t '

€: €LCFTENa -iruFrQC;

nef ; \lurtlC: oF G\Gf\iertTS ( t <g S nel )A : NED? NUrtCqR

Foe Arr ELCmINT g

xe+t

F€3 TTl3 Slr"r?ta ql€-DtltCNStoNft CASE

Tt+ccc ts A slFtDLE gcvlrrcN tc"rt.r€cN TlrF

?gff.rCt'Sr hluirtER AwO rrE NoDE Nut.rBtES

oF A 6l\lQN gL€S't€l$r.

ICFT roOE, uus^lGt A e €

P.r3$" rrooc $uFrlct A+l = e'+l

xA

Page 62: INTRODUCTION TO THE FINITE ELEMENT METHODl. FrNtrE ELETqENT MtrrroDs ( eem) FOR L|NEAR STATIG, ANALYSIS PR,ELI'Y I NAR,IE S I{AIN cONSTITUENTS oF FEM FoE T}I€ 3OLT'TION OF A BOUNDARY-

5q

Foa Gxarret5 Goe gt€l'r€flr NuFrQgC, 3

T]tg \Gtrr rrlolG Hur'ltlGR tS 3 ArlD lnt,

e,t3rrr silC lS I .

gE,FnA3E3

l. Fot A ClU€il t \€t{C}lr e, T$€rrroO€

A AlrD A+l AtE cA\t'EDNUFTBC3S

Ct€lf€r|" rroDE lluFrt€tS.

2 - Tlr€ Cn ffrcil Bcrwggu Tllg €\E|.crlr

NuF4tgt Arro guE+r€rSf uoDtF xorat€tS

t3 rulr so $Fr"!E Fo? T\uo (oe THrgE)

Drr.r€ltSrouAL FIC:$f lE:l .

Fot' A Orrx!,N ?LeFrgtt8? e' I wf tsrotrf rnC

t.CFr AND gtSrlT NODE r||XyrC€93 . Trt€a€For€ l|oc

trprrr TFrr cooSDrlJrfns Otr TTC LcFr Atn Bl5lf7

rus. r^rg tArt(l glg|:lEr, TO Tfl€ \TFT NODS A:

NolG t AuO TO TtrG RfShT lror, RS rNDe 2.

Page 63: INTRODUCTION TO THE FINITE ELEMENT METHODl. FrNtrE ELETqENT MtrrroDs ( eem) FOR L|NEAR STATIG, ANALYSIS PR,ELI'Y I NAR,IE S I{AIN cONSTITUENTS oF FEM FoE T}I€ 3OLT'TION OF A BOUNDARY-

Cp

e

xi ;,e x;

xr€ s XA Le;T G.FaurDl't

xf t X r*t ctsrr (sotol$AtG

he = x;- xi q\G'ftErr LtarG.rrll

+ r{ob, DeFlr*, A LaAL Gseu}lAT€ q( A TnANsForHAtloN )

q (x ) = ( 2r- *i- rl )/he

+ Tlrrs rs A L!sA? R€affnoN

- ) t ,= E(x ,e) = - l

h "= h (x f ) = + l

Page 64: INTRODUCTION TO THE FINITE ELEMENT METHODl. FrNtrE ELETqENT MtrrroDs ( eem) FOR L|NEAR STATIG, ANALYSIS PR,ELI'Y I NAR,IE S I{AIN cONSTITUENTS oF FEM FoE T}I€ 3OLT'TION OF A BOUNDARY-

61 .-a T$|F rN\fGEt€ e.|ElATrOu ( 'rEAllsFOrnantor

)

13 LINEE?- T€.

x ( \ ) = ( h t q + x i * x ; ) l z

-) nor€ TIIAT

x ( q , ) s x C , ) = x r e

x ( \ r ) = x ( + r ) = X " '

+ E\EH€rr (OoBDINATE

FPACTE

t t*r (Locnt cooeDtNAT€FtanG )

-) €tEHENr (OO?DllrATrE F!'AH€ 13 TFE SAF{€FeC AtL CFI€t{ENTS.

/

/

lr=-l O l. E

Page 65: INTRODUCTION TO THE FINITE ELEMENT METHODl. FrNtrE ELETqENT MtrrroDs ( eem) FOR L|NEAR STATIG, ANALYSIS PR,ELI'Y I NAR,IE S I{AIN cONSTITUENTS oF FEM FoE T}I€ 3OLT'TION OF A BOUNDARY-

poDEgroBAr ^

gtEI.{ENr

A, B rc , . . o

c r b r c r . . .

N"(l) = i ( r+ ge t ) ( o = l , L )

Nur{BCPlN6

NoD€ NulrB€Fl$6

62.-t Tll|r'T3ArrsFolFrSltla,

g\!E|.{(?rr <ooeDts xB

Coo?Dt\lATtet UARfE:

gL€HENT.

S tlAF€ F\tNgTtONS :

v̂A

6LOBAL

xe*,

X(q) tT. rne,TO TTC GLOBAT

Feota Eteugrrr To

tf,rl

- ff.r-fLOCaI

N'(t ) = ? = !z ( t - r ) = *C r +q ,E )

N a ( l ) = ? = * t r + q ) = * ( r * t " t )

NG'TATIO$ I

!,(t)

Page 66: INTRODUCTION TO THE FINITE ELEMENT METHODl. FrNtrE ELETqENT MtrrroDs ( eem) FOR L|NEAR STATIG, ANALYSIS PR,ELI'Y I NAR,IE S I{AIN cONSTITUENTS oF FEM FoE T}I€ 3OLT'TION OF A BOUNDARY-

63 '-t Nolf€ THAI Falr A 5|\IEN t'.GcAL

cooebtuATe q rtnEQE tg ACoe?€tiDollDllsS

6Lo3AL COO?DI}|A're x

x - x ( f )A N D

\(xf = ilh (t )

-) l{roul6lER T$lS lS hfr T?UE ftt T$C

DtglvA'nvtg OB TllE Stf ADg Fu$cTrot'ft

I* l . c .

Ss(x) *+(E)d x ' c l t

+ usg €.FA111 trut€

dN.- =

d x

€ Nor* = N..rt Er,

{3d x

d uq

Page 67: INTRODUCTION TO THE FINITE ELEMENT METHODl. FrNtrE ELETqENT MtrrroDs ( eem) FOR L|NEAR STATIG, ANALYSIS PR,ELI'Y I NAR,IE S I{AIN cONSTITUENTS oF FEM FoE T}I€ 3OLT'TION OF A BOUNDARY-

4 .- l

Er" r

Err =

(x)

+x(r)IxrtlXri =

*x; )/rJddtt [t h't +x,o

_ he'T

I

r hQ '? - 'l - l =t - Z J

oa a

2-

h€

* [t.r- x,e-x.t)/ht]

t J tHF sArrg

oR.

r F d h / . , \a r x b S 5 t ^ ) =

( * * )

=

aO t Nor, = \lorl

ft

Page 68: INTRODUCTION TO THE FINITE ELEMENT METHODl. FrNtrE ELETqENT MtrrroDs ( eem) FOR L|NEAR STATIG, ANALYSIS PR,ELI'Y I NAR,IE S I{AIN cONSTITUENTS oF FEM FoE T}I€ 3OLT'TION OF A BOUNDARY-

65.

€oeel\tA:iluEs oF THE STAD? FsNcrlON fN tOcAL

coC,E DINATC s?lrrE ..

No, r r 5 [Ne( r )a5 |L t tl+ i . t l

No,l G i t. (rourterr)

Nrr i =

i r t , t =

LOCAL -TO- STOBA\ COO3DIUAT€ TIIATTSFOIT-IATIOT

+- I

z+Jz

3Y SIIADC FuNC.tlONS z

Tll€ 'ttAt tFgLFrA'noN Ftot{ T}lC \OCAL

c.ooruxATG t

To Tllt etote\ co:DtilAT€ x

wHrc,lt wrt DgrBarilxED 'tA E€

x e x (t ) = ( h. q r-x,e+ x: ) /z

cA u AL30 Be €X PrEsseD llr T€eFrs oF

Thq Sh ADE FoucTl0ltl ArlD 'Tlg IDDAt

coocbtuATEs.

Page 69: INTRODUCTION TO THE FINITE ELEMENT METHODl. FrNtrE ELETqENT MtrrroDs ( eem) FOR L|NEAR STATIG, ANALYSIS PR,ELI'Y I NAR,IE S I{AIN cONSTITUENTS oF FEM FoE T}I€ 3OLT'TION OF A BOUNDARY-

e 3 .

x(f ) = N,(t) x, '* N.(rt) Xf, L ae : No(l) x:q B l

v€elFrCATlOlr :

x ( t ) s * ( r - l ) x F

N,(q )

r - i ( r * l ) X.€

ue( l )

,o* ": ] /z= [t *.t- x,e) q + x- --_a,

hc

( h ' l + r , Q * x ; ) / z

(rus srrE AsBgFoA!' )

- - ' E -

Page 70: INTRODUCTION TO THE FINITE ELEMENT METHODl. FrNtrE ELETqENT MtrrroDs ( eem) FOR L|NEAR STATIG, ANALYSIS PR,ELI'Y I NAR,IE S I{AIN cONSTITUENTS oF FEM FoE T}I€ 3OLT'TION OF A BOUNDARY-

6'ln i l

DISDLACC}T€TF F $$CTIONtr

cLoEAL PANTPErl

LO(AL FrCTtrCg

GLOBAL :

LO(/AL

\IERtFlCATlOt.r : Fl lr.rsDes'ftoN r'

d*t

d,e

xe+ttl

x;

uh1r1 = N^(r)d^ * N$) C^*,

uh(q; = Nr (\ tdi * Nr(t) dr'L

= :- No(t)d:Cls I

d;

xAI t

X,'

- l

t ,o t q

r.

|NT€BFoTATION otr TH€ !6erAcgt{Elr Furrrnorr:

aa

Page 71: INTRODUCTION TO THE FINITE ELEMENT METHODl. FrNtrE ELETqENT MtrrroDs ( eem) FOR L|NEAR STATIG, ANALYSIS PR,ELI'Y I NAR,IE S I{AIN cONSTITUENTS oF FEM FoE T}I€ 3OLT'TION OF A BOUNDARY-

6g

R,EmAe'E,

T}Ig LOCAL INTEEOOIATIO}I OF THCIt tf

DtsDtAC€ r-rErtr FUI CTlOrl

rrAs Tltg setr€ Fol,H A3 Ttlg LgcAL-To- GUDlAt

ccor?DtNATC'T3AN !$ tsgpt ATTOX

,rh(h ) =?.z

q G l

No(E) d:

x(h) =2_z

cr=lN.(t) x:

Tlll3 ls THt DeFlNL"lql, oF AN

rSoPae,anefEfc e L€r.rENTt l

Page 72: INTRODUCTION TO THE FINITE ELEMENT METHODl. FrNtrE ELETqENT MtrrroDs ( eem) FOR L|NEAR STATIG, ANALYSIS PR,ELI'Y I NAR,IE S I{AIN cONSTITUENTS oF FEM FoE T}I€ 3OLT'TION OF A BOUNDARY-

6l

ELSFIENT STIFFNESS T"TATR.I X AND

FORc'E VECTOR

R.EFrlFrqC. T}IC GLOBAL SrIFFNESS T'TAT:,IT

ANO Foteg \rCCrOR

? euoBAL lNTEstlTlgalS cAN

A3 STJWTS OF ITTTE3SAqONS

K r s = a ( N A r N s )

FA = (N^,t) *$g{ - a(ru^,N"*')/6o.

rh Csr

hIRI.ITEN

R. E,tElfChIrS

,C J-st-5st

L

rlcl,- s- z _

€=l

BC

o\f €

-

J

dL€LOBAL

DOI|IAIN

EtElilsrDorqet rt

tezduyar
Rectangle
Page 73: INTRODUCTION TO THE FINITE ELEMENT METHODl. FrNtrE ELETqENT MtrrroDs ( eem) FOR L|NEAR STATIG, ANALYSIS PR,ELI'Y I NAR,IE S I{AIN cONSTITUENTS oF FEM FoE T}I€ 3OLT'TION OF A BOUNDARY-

7I).-rHeN, rNc CAN wR,rTc :

nela(t,x) = t ' a (*, *)u

€ = l

(x, x) = f (x,* )u€ a l

wlrtE€

a ( il, *)t = It*),, ( x I,. cxge

( * ,x ) = ! , * ) ( r ( ) dxje

DcFtlrE GtE F{Ea$ L€ltE L srlFFNE:s ltliJF4CC ugcToR

r i u = INo,r N"r, Jr G O(N^rNJ

Ff = (*^,ff + 5",6o, /t-l(N^,'jf

tezduyar
Rectangle
Page 74: INTRODUCTION TO THE FINITE ELEMENT METHODl. FrNtrE ELETqENT MtrrroDs ( eem) FOR L|NEAR STATIG, ANALYSIS PR,ELI'Y I NAR,IE S I{AIN cONSTITUENTS oF FEM FoE T}I€ 3OLT'TION OF A BOUNDARY-

ur*EcE ? | '

(rue,{) '= I ru^{J:r-ge

-9'e = [x,trx;] (ewrrorrDo]rAN)

couslD€t T]lE FoLLowlNG €tt Hqflf I

f EtEHST NSHBCR'e-

e,t

,..* uoDENur{ B€f

FoL F^'

Q,+l

t-n t6HT NolC

NurqBgl

OFA- wtLL BE Fq\rAL Tg tcn,o uNttss

A fS Entt€P, Tlr€ L€91 o3't l lF E,l6lf l

ruoD€ ot rllAT eLelfealf .

i . e . rF ( A*e AND A t e+ r ) EN= ! 7 - - -

trl= ot A

tezduyar
Rectangle
Page 75: INTRODUCTION TO THE FINITE ELEMENT METHODl. FrNtrE ELETqENT MtrrroDs ( eem) FOR L|NEAR STATIG, ANALYSIS PR,ELI'Y I NAR,IE S I{AIN cONSTITUENTS oF FEM FoE T}I€ 3OLT'TION OF A BOUNDARY-

' 13^ ._

?FrrE. K^-B

-

oKai wttt- B€ lGso oNlsss

t

(A tS "mHE? tll€ LCPT oe. ttlC

uoDE oF THAa g\€r'l Gtsr J

(g rS r . . r !D t t . - , . ,,,

t

L.C .

:tc|lT

AND-

IF-\,

Tlrr\frr-p

lr-

oR,

)

( t * * e( B t e

AND h* e+r

AND B * e+l

K;s r' o

)

)

n t

eK =|A.

e€+l

Fe'! r FI T

-

n I n FtA.|Brt o l l

tezduyar
Rectangle
Page 76: INTRODUCTION TO THE FINITE ELEMENT METHODl. FrNtrE ELETqENT MtrrroDs ( eem) FOR L|NEAR STATIG, ANALYSIS PR,ELI'Y I NAR,IE S I{AIN cONSTITUENTS oF FEM FoE T}I€ 3OLT'TION OF A BOUNDARY-

73.

cof,r SILC?

euD rnc

Eue:YT]ltN6 Ci'ts€

ilc80.

TflC lruCrtYtATR lI

il* lerz)L " l

suB.\|ccrol,? . t

l * l' l r i ( z r t 2L !

lsKE-

I N)

ANo F-_

SUB. FI ATII I PCCAtL

TllA'1

THtS

IS

2r t

eq b =

iI-

2x1

N., Nr)a(

54c

a r b t lrL

Nor, Nbrr Jr

srlQ- ucgtot +'€ALL TIIE

TIIAT 13

_

[ 5 . . { i . = r 1I o i e. 22.-7tlel-l f

t- t l. i €ro3f Jf: = Ir.rJ& +

a,t lra g,C

tezduyar
Rectangle
Page 77: INTRODUCTION TO THE FINITE ELEMENT METHODl. FrNtrE ELETqENT MtrrroDs ( eem) FOR L|NEAR STATIG, ANALYSIS PR,ELI'Y I NAR,IE S I{AIN cONSTITUENTS oF FEM FoE T}I€ 3OLT'TION OF A BOUNDARY-

74.At33ur-rtN6 TIIAT NE'U€ COr-rnrtgD

'r]tG gr'gHFNT LG.G\ t',a?Ptx [rr-l' - r ' r r r ' r L * t J

ArrU trt etc FrtrrT tc\rEt uec.ra

= ke'-

bl = f t,

}IOl^| E l^JC FTACE T}IEH IN T}I€

6LoBAt' HAT: U, K ANo Tlr€ etotAL-t

vecroR. F ?-

gXAFIPLE

€ I t\GHill? tntFrlc? r ?

ESIES rrobcNUit!€3 r Q,el

CTOBAL NODC--

NUFrB€l : A= 7

CTENETT' NOD€Irr-

NUFrt€R as 2.

6rotAL NoDg

i . iunr0CR , A= l0

ASistl|C

k Q =-

IIAUF

+',b

cOc{FufgD

3 lfflIXA? t^lE

f t.t, k.i 'l

L tl, ti' J

Page 78: INTRODUCTION TO THE FINITE ELEMENT METHODl. FrNtrE ELETqENT MtrrroDs ( eem) FOR L|NEAR STATIG, ANALYSIS PR,ELI'Y I NAR,IE S I{AIN cONSTITUENTS oF FEM FoE T}I€ 3OLT'TION OF A BOUNDARY-

trr G- .F.Flo +

l -I

Ia l a

e

.rfi'it t o

f}K r . +

K g , ,o s

K , g g +

Klo,to +-

\rprri\

- t l

a t ,- - . . ^ r l -'-i ttl

'r- -.tl- l \- '

a '

o - 'i t \

€ t ,

\e,e

? 5 .

,( esuruDrrwlfcu K

i,

ts sY lt til

a a a

t?i"t a -

r t t \ a

) l

v - ^ l

l r l

a c

-a

r o

l(,':. .:'l'r {- r -t I 'L-. -! - l l - - a\ - -

\ - ! -

.

i d ' t

1t:,E- t o

aa

+

b -)

e

a , t J

l(.^q'l r f-( +I ! I r \ - - / f ^

\ r . - \ t\ -

\ Ot _

Page 79: INTRODUCTION TO THE FINITE ELEMENT METHODl. FrNtrE ELETqENT MtrrroDs ( eem) FOR L|NEAR STATIG, ANALYSIS PR,ELI'Y I NAR,IE S I{AIN cONSTITUENTS oF FEM FoE T}I€ 3OLT'TION OF A BOUNDARY-

76.

|MPL€}'r€NTATION :

ASSEF|BLY OF

(g= lr-, Dct)

IK.!

F

K A N D F :- -

It

LM Ag3.A7

THE tfTFOBHA'TIOX

€ f,fT|tTES OF

A r o E .

Jf '-

k".t

IF-

Trf€ AREAY

NEgDED

ke ANo-

a t

LM CONTAI}IS

TO PLACE THE

fe rrrA

K.ry

ELc'mgNT suBR'ourlNe

Ltt I ASSE]"fBL.I, INFO--

L0CATION l'lATtlI= =

ASSEl,rgtY suBtoUrNE

Page 80: INTRODUCTION TO THE FINITE ELEMENT METHODl. FrNtrE ELETqENT MtrrroDs ( eem) FOR L|NEAR STATIG, ANALYSIS PR,ELI'Y I NAR,IE S I{AIN cONSTITUENTS oF FEM FoE T}I€ 3OLT'TION OF A BOUNDARY-

77.l r l r

DIHEUSrcNS OF THE LiYI ABEAY .'

h e n x D e [rrutrBGf of J t- NulrEEB oFELEI.ISNT €L€Ff GNTgNODES

EIAFIFLE I- D FTOSL€I.I I-

hgn = 2.

n e [ r n

6tvEN eL€ttlirT Nt5.tCR g

6luFN E tCr.rcNT NOb€ NUI*rtgg A,

F|ND THC SLoBAL EC)uATlOtf NoFrt€R By

A = L M ( " r e )

Fot TUE l-D pgO g LCH

LM (ore) = { e rF (f'= | ( L€Fr NoDE)

\(. "*l rF 4=z ( ererr ""D9

J

LM (2rnel) € ? = O ( s€sAus€ to EQuATloilr-oc THA.I $otg )

Page 81: INTRODUCTION TO THE FINITE ELEMENT METHODl. FrNtrE ELETqENT MtrrroDs ( eem) FOR L|NEAR STATIG, ANALYSIS PR,ELI'Y I NAR,IE S I{AIN cONSTITUENTS oF FEM FoE T}I€ 3OLT'TION OF A BOUNDARY-

\s'*ETTHEN?NODE I}

1

ET.EH€T$NoD€UtJr{ttG

Et EtjlEl[?*

EOBALrloD€ N 1

6IOEALEeutnorrB

I

(oNSEqUE\rCE

2

T}IE LF4

3 ' i .. llel-Z hcl-l Del

net ne{-l Oel a t.t

4 h'L fFl n h+l

h.a l\-l n

-f a.ATTAYF€3 ?HC l-D egggteH

l3 As FoLLOWi ..

z

2

32

3z

4sz o

a a , , t ' a . . a \ a o - l . - a a

d- -. . .>i ; i . . - .$:-- - :)i l . __ : . )

wTAT tS Ta€ OF LM (z,rs1)=o

Page 82: INTRODUCTION TO THE FINITE ELEMENT METHODl. FrNtrE ELETqENT MtrrroDs ( eem) FOR L|NEAR STATIG, ANALYSIS PR,ELI'Y I NAR,IE S I{AIN cONSTITUENTS oF FEM FoE T}I€ 3OLT'TION OF A BOUNDARY-

.7?.

ACC }IOT

taa

oa

r ? n CK - A N D T ,A- ,g

. t - l \

I e t . .

:{lj,iii

E?i l t r ,a gg u

.526 3A

( mor

. . . t t t t t i t t . a . .

tr.,.'.:.tti.:.,111,{i:,t t r '

' ' " ' - - t c t " '

Keii;g'' "' | .1i:,1j.:,- ' l l !

\ r r r \ r r r . . - . r " "

Fs. + frit' l e . i \ r - - - - - - - - J . r o .

Fe

? E ('5 i l t tr'Z di t' rl

k [ ' , k I ' , k ] ' , +n ' lAss erlBLgD.

ASS€nnBtY OF TIIE TNTPIE] OF

K"r"*,

K.*,, e <E

IK*,c+f +- *E{,rfi-i

+- K':'.. /:<.'+fe i ' ie. ;l ^ - . 't . . i . i , - - c o t

i - - a o -

\ . t '

+

PE RbCl{ED IF K-

ts s.tHr{GrPc )a o - - l

o o e \" + k ; "_ j.fjr,ia.]a

s t ) o a

<-

Page 83: INTRODUCTION TO THE FINITE ELEMENT METHODl. FrNtrE ELETqENT MtrrroDs ( eem) FOR L|NEAR STATIG, ANALYSIS PR,ELI'Y I NAR,IE S I{AIN cONSTITUENTS oF FEM FoE T}I€ 3OLT'TION OF A BOUNDARY-

8o.START

DATA INPUT

SET UP LM

INITIALIZE K AND F-

FoR,M ke AND f e. usE "LM'To ADD

r - -

b. AND f t rNTo AppRopRtATE

LocATloNs oF TllE E AND E

e + Q + l

YES

ASSEMBLY AL6OR.ITHM

Page 84: INTRODUCTION TO THE FINITE ELEMENT METHODl. FrNtrE ELETqENT MtrrroDs ( eem) FOR L|NEAR STATIG, ANALYSIS PR,ELI'Y I NAR,IE S I{AIN cONSTITUENTS oF FEM FoE T}I€ 3OLT'TION OF A BOUNDARY-

e l}{CITATION i

EIEF,CISE I

r ^ € l CETE LTCTT COFTPUTATION OF K

- EXb T .

- - t -

pRELlfr,ltNARy FOEHULAS :

dx (tt

THE ASSrlrBtY ALGo:,ITU|{ CAN EE R€PBEStNTED

BY AN AESEHBLY OPIETCTOR At t

nc [K = A k e' - J l q '

€rg I

nelE = A j ?. - J l . : -

€ e l

1dr (;i:.,)

('.1i1 )

rz, Q..

, , I f ( r )dx =*,J+(x(r))

& f(x(t)) = dl(x(l) drdt

Page 85: INTRODUCTION TO THE FINITE ELEMENT METHODl. FrNtrE ELETqENT MtrrroDs ( eem) FOR L|NEAR STATIG, ANALYSIS PR,ELI'Y I NAR,IE S I{AIN cONSTITUENTS oF FEM FoE T}I€ 3OLT'TION OF A BOUNDARY-

82 .COHFU?ATION OF pe

a t -

k l t = l4,9

* ) (= -,1d r

* l r - - . - l

= _, l .Nnt(t),nttt t l ( x,*At)

( - , ) t ( - r )b

2 Z

q,+b + |

l :g (crz h e - r l I

T

No,r(r) Nrrr(r) dx

dq

dl2aIIl

he

s.+b- ( - t )- !:rEr

he

=)

k " =-

!he ll :l

Page 86: INTRODUCTION TO THE FINITE ELEMENT METHODl. FrNtrE ELETqENT MtrrroDs ( eem) FOR L|NEAR STATIG, ANALYSIS PR,ELI'Y I NAR,IE S I{AIN cONSTITUENTS oF FEM FoE T}I€ 3OLT'TION OF A BOUNDARY-

coFreuTATlON oF83.

BOUNDASY T€RlrS

PCsG LE}r TO

FO:, EACH

+ ; = tg,C

N.(rl ft*l dr +

gur f txl cHAN3rErr Fror-r

PTOBT,E}I. DO WE R.C PBO6:AH

f t x l ?

ft*l -:- f1', ( uxeerlrl.l€:,Fot tflorl )

a.

Ia3f

(n ,

t nt.

+ frN.

+.+- IT'TEEEOLATION

{

{,e

T}IE LINEAS

t3 LI$€AE,

s x i

NTEEDoLATICI ls EIAg? lF fttl

t . e . f t r l . a r +b

{f i*l

Page 87: INTRODUCTION TO THE FINITE ELEMENT METHODl. FrNtrE ELETqENT MtrrroDs ( eem) FOR L|NEAR STATIG, ANALYSIS PR,ELI'Y I NAR,IE S I{AIN cONSTITUENTS oF FEM FoE T}I€ 3OLT'TION OF A BOUNDARY-

t4.

TTICN i

I N,,tr) fl'l lx-1e

.ge

= sNo

I N.(r) ft*t tr g

gc

heez

N,(h)N,( t )d1 = f tr-t) Jt

Ltr) Z Nr( '){s dx

b=t

q , - * l-

*, [ '*ott)Nb(t"{fu

?+ l

-,1 *(t-t)+ l

-, 1

N,(t) N.(i) J1 = .5 t(r-i)) t(r+il.r,1

zr-

3

J3

::I- t Nr(t) N.(i)d1 a

-'l

t(,*t[t+t)dt = 3

Page 88: INTRODUCTION TO THE FINITE ELEMENT METHODl. FrNtrE ELETqENT MtrrroDs ( eem) FOR L|NEAR STATIG, ANALYSIS PR,ELI'Y I NAR,IE S I{AIN cONSTITUENTS oF FEM FoE T}I€ 3OLT'TION OF A BOUNDARY-

ES.

BouuuAt,yTgE f.l S

BoUNDAR,Y

TTStnS

slt ll llj+

fg Jt+' + +, I {FT t +, *z+.[ I

= D cr-

-} rrl r

-

ED- p C

+f g

At

- f l -

Page 89: INTRODUCTION TO THE FINITE ELEMENT METHODl. FrNtrE ELETqENT MtrrroDs ( eem) FOR L|NEAR STATIG, ANALYSIS PR,ELI'Y I NAR,IE S I{AIN cONSTITUENTS oF FEM FoE T}I€ 3OLT'TION OF A BOUNDARY-

86 .FOET.TULATTON OF TWO OR, T}IN-EE

DIHCN�TONAL BoUNDAEY-VALU€ PBOBTEM S

xe! =

[::]

PROBL€M S}rAIN

BoUNDABY oT THg DoHAIN

PAgr oF TUC IOUNOAIV Wll€tE

EISQNAIAL GOTTNDAIV @Nolrloll lS

specrFrsD ( 6-

ryFE touNDABy coxD.)

Dlcr oF lllg buNDAr- trlllgRE NAIUIAL

BourDAEy coNornoN rt sfiFrFtgD ({-noe)

NG|fATION :

-sL

r

\

I

r t :

tezduyar
Rectangle
Page 90: INTRODUCTION TO THE FINITE ELEMENT METHODl. FrNtrE ELETqENT MtrrroDs ( eem) FOR L|NEAR STATIG, ANALYSIS PR,ELI'Y I NAR,IE S I{AIN cONSTITUENTS oF FEM FoE T}I€ 3OLT'TION OF A BOUNDARY-

x =l I r? -- -?DINATE v€croR'

A?'

- (r.! : c'oosDlNATE vEcTroR'

R 3|:t l : uNrr NelrAL . lEcrce,'A tnu)

PSELIFITITRITS :

DNSR6IEI\|C€ TIICOEEM :

( $ . , d o - = ( f n i d F_ J

' t L - E - JJ L r

t .e . ?, r s r 5 g f d & = S f n d rJ L r

NITEEAATION BY PASTS

I ti g d& = - I+ g,iJ-s- + I+a ni dt'-O- .3L f

tezduyar
Rectangle
Page 91: INTRODUCTION TO THE FINITE ELEMENT METHODl. FrNtrE ELETqENT MtrrroDs ( eem) FOR L|NEAR STATIG, ANALYSIS PR,ELI'Y I NAR,IE S I{AIN cONSTITUENTS oF FEM FoE T}I€ 3OLT'TION OF A BOUNDARY-

EE.

i . c .

ltyt)7d-4. = (c ,o,])ca + S+ardrO ) . ^ . t d

= q ' ' - - J r \ v ,

"

pnooF :

J {,.a drL = - j ?,rf csa + 5,*t),;crliL

_fL lL

( e:ooueT BuLE )

= - Ig;fe-a, +, I+a ni Jr3' rr

( Dl\er6€\lc€ ileeg! )

tezduyar
Rectangle
Page 92: INTRODUCTION TO THE FINITE ELEMENT METHODl. FrNtrE ELETqENT MtrrroDs ( eem) FOR L|NEAR STATIG, ANALYSIS PR,ELI'Y I NAR,IE S I{AIN cONSTITUENTS oF FEM FoE T}I€ 3OLT'TION OF A BOUNDARY-

gl .

LIN€AE RSAT CONbUCTTON

NOTATIOI ;

tA : TEMFERATURE

9: r lfEA? FLut VGCIOR ( t )I t

9i r - *ij \j psurreei LAw oFIIEAT eoNOUC,lf 0N

Ii . C . I r - F ! U-

,3 : CoU! UC"IUITY f*lATE|l-

E ,E Syt rrST:rC

liosrLY wF nevF AN lsorEoptc llgDlutf

t^lT l r T N i $ e e I \ t D € N ? l t t i l A " r t r

cr A.i =, ^V PloNecrst D51"o

+ : HgAr SouGtF Pe3 uNfT VOtulrF

tezduyar
Rectangle
Page 93: INTRODUCTION TO THE FINITE ELEMENT METHODl. FrNtrE ELETqENT MtrrroDs ( eem) FOR L|NEAR STATIG, ANALYSIS PR,ELI'Y I NAR,IE S I{AIN cONSTITUENTS oF FEM FoE T}I€ 3OLT'TION OF A BOUNDARY-

10.

DIFF€T-ENTIAL RUATIOTT'

ii,i B +

g ' q B +_

(-r i; %ilrr' = t

g.(-s 'gu) - f

]rEAr EquATf oN

( Polsso\rs tquA't lolf )

tN VEErol Fos|Yl

BourrDAsv c'oNDlTloNS B.c . )

t r= t

o N v TEi"|PEFATU:,S B.c.

( esseNrrAL B.c . )(onrrHt€T B.c . )( p-TYPg B.c . )

ft€AT Ftul B .c.

t xeru?AL 3.c. )( xculrAxN B.g. )( {- TtPg B.e. )

-1 in t- 9 ' 1 1

- l -

- l3 r ' )

tezduyar
Rectangle
Page 94: INTRODUCTION TO THE FINITE ELEMENT METHODl. FrNtrE ELETqENT MtrrroDs ( eem) FOR L|NEAR STATIG, ANALYSIS PR,ELI'Y I NAR,IE S I{AIN cONSTITUENTS oF FEM FoE T}I€ 3OLT'TION OF A BOUNDARY-

9 tLINEAS TEAT CONDUCTION :

\\,EAK FORMUIATION I

l l , L - + = o

( qi,i -f ) dsr-l*$

REITOUAL.sLwEt6TTlN6 FUilC'TION

BY PASTS ..INTE63AT€

(- ! w i g i

SL .sL

+ !*f in idf 'Pt

oN frr

ot\f fr,

I'

!*tin;v(

) wl in;

It

E€M Eri3CQ. oL

FUltcTlON cr'.Als€'ll :

st!$l,At To l-D cAsg

ugA a u- t ox f 'qwe 1f =t w=% o* f.

Yw e\f

dsr- + t

w g;ni dr- I*f*=

o.

dr' - JT

THAT \At =

ginr =

tezduyar
Rectangle
Page 95: INTRODUCTION TO THE FINITE ELEMENT METHODl. FrNtrE ELETqENT MtrrroDs ( eem) FOR L|NEAR STATIG, ANALYSIS PR,ELI'Y I NAR,IE S I{AIN cONSTITUENTS oF FEM FoE T}I€ 3OLT'TION OF A BOUNDARY-

f - ; - - - - r - - - r - . D - - - - - - - - - - - - - - . D - - - J

i - \ w , r j i d s L = ( * l J - s - + [ o [ d r ia J . ! r r

J J i. r L J L 1 . 1 IIL----- - --- ---- --- -- --- - ----- ---- - J

9 Z

THqBt FogC I

- 5.cL

\^r,iqi J.rL -- 5 w{ Ct' - Iwf dsa =e.

CJEAC3ANSI\Ig :

'rHrS lS TtlGi S|€AE "Oe,fYl Vrrelf

s\fBsnr\rnNe FoR, 9i= -r<ijuri

fq

+

f *,, rij urjd$,ilwfdsr

* 5*[cr.sL

VwelfWEAK FOR.M

tezduyar
Rectangle
Page 96: INTRODUCTION TO THE FINITE ELEMENT METHODl. FrNtrE ELETqENT MtrrroDs ( eem) FOR L|NEAR STATIG, ANALYSIS PR,ELI'Y I NAR,IE S I{AIN cONSTITUENTS oF FEM FoE T}I€ 3OLT'TION OF A BOUNDARY-

ct3

NOTATTON :

a (w,u) = I*,

"i j urj Csr-JL

(w, +) = I wf d-sL.(L

(w, 1)., = I w[ dt-t l't

THctc An g ALL syHlt qg!_c . BILINqAR

FORlvls .

wtrlf Tltrs NSTATION lN FrtND, T]f€ w€AE

FOET.IULATION CAN B€ WRITTgN A3

FOLLOWS :

a(w ,u ) = (w , f ) + (w , l . ) rVwel/

t\, '?trlt FcSrrt, Tutg wgAE roarru\A1lo N

LAoES vgEy Slr{lLAP, TO T}fg TWO OnttR

wgAE s|trrs otreruED prcvtor,sty (ttr pp. lz al|rp)

tezduyar
Rectangle
Page 97: INTRODUCTION TO THE FINITE ELEMENT METHODl. FrNtrE ELETqENT MtrrroDs ( eem) FOR L|NEAR STATIG, ANALYSIS PR,ELI'Y I NAR,IE S I{AIN cONSTITUENTS oF FEM FoE T}I€ 3OLT'TION OF A BOUNDARY-

j { .

LINSAR. HEAT CONDUSTIO N !

6ArgE,E,tN ?OErlUIfrtoN

INi.IEAD OF FUNGTTON €LASIS U AND A

w5 u3C FrNtTe DIHBIXSIONAt AFFTOXIT'IATIoNS

?th A*.D d

TIIAS ll t

h h hr F w e ? l l c \ t w l = O o N q

( o : * b i o ) o

rF tt. d ruGh ot = E

olr Pt

(os *t =a )

JulT L|EE l^|g DID tN Otl€-DlHErrSplrAL

C n l l , h

Assut{E rr{Ar 1f K :T*

AND ^ -h

L|Ft Vs colrE'r EucT JJ 8Y urlal6 v .

tezduyar
Rectangle
Page 98: INTRODUCTION TO THE FINITE ELEMENT METHODl. FrNtrE ELETqENT MtrrroDs ( eem) FOR L|NEAR STATIG, ANALYSIS PR,ELI'Y I NAR,IE S I{AIN cONSTITUENTS oF FEM FoE T}I€ 3OLT'TION OF A BOUNDARY-

1 5 .

pg3!e A ruuernN 2'

(A 6lvGN Fulrcrrou)

sucu TIrAT t^

= ?

o\r fl

( o R 0 ' : A )

Fot' e,vt?Y tn- th

w3trg ,lh = art ,,t h, ?Now c,ttBE, TO SIFF ;F Uh BGtON6S'To -/5

h

o th. ' . d ' =

? O u | 6 , /

E '

Naw wQ,tt€ Tllg wEAs For,t l Fot

*te U'v h e U h

uh e, Ah

tezduyar
Rectangle
Page 99: INTRODUCTION TO THE FINITE ELEMENT METHODl. FrNtrE ELETqENT MtrrroDs ( eem) FOR L|NEAR STATIG, ANALYSIS PR,ELI'Y I NAR,IE S I{AIN cONSTITUENTS oF FEM FoE T}I€ 3OLT'TION OF A BOUNDARY-

q 6 .

a(wru) = (w,f I + (w, 1)n VweV

w G - * h

u+ , *h

.n a(* ] , t ) = (*1f) + (* t r { )" vwLuh

uh r tt* A'

e a(w] .,'*7) = (*1 +l + ("1 {).. v*'euh

a(*1 vt)UNfril|l-f

= ( ",1 +) *(*l {),"-a(*] ilICNOt^f N

:+

( coFtgAlc te THq SALegtslu trc3rrr

ostArNGD Dtfr|teuttY ! nf aI) )

tezduyar
Line
tezduyar
Line
Page 100: INTRODUCTION TO THE FINITE ELEMENT METHODl. FrNtrE ELETqENT MtrrroDs ( eem) FOR L|NEAR STATIG, ANALYSIS PR,ELI'Y I NAR,IE S I{AIN cONSTITUENTS oF FEM FoE T}I€ 3OLT'TION OF A BOUNDARY-

FINITS EL€F|CNT DTS3ESTI?ATION

DrSCI?Ertt€

E\.EtnENT

rooEi

Tuq DonAlN .St trfto

Doprarus -ge Q < e, S nel )

3? .

Drsccell2fTlohl

oF -St AND

r

ltAU€ A1 tg11gT Ttrg€g uoDEl.

rtoD€t5 .

4- NoDti RErr€rsn$ uortuv)

ELE$5NIS WILL

o E , 4 1 3 r , , , ,

( rlrc r^llLL ul3

Page 101: INTRODUCTION TO THE FINITE ELEMENT METHODl. FrNtrE ELETqENT MtrrroDs ( eem) FOR L|NEAR STATIG, ANALYSIS PR,ELI'Y I NAR,IE S I{AIN cONSTITUENTS oF FEM FoE T}I€ 3OLT'TION OF A BOUNDARY-

Horf Do wE geuatg NoDE OBtrEErr6 TO

Ttrrg oCDGS|N6 Oc gquATtorr S 7

AP?Rqllt51q1

oF 3 t t rn

o N n B tt

Ltsgas

NTERFotATtoN

n-- 1 sgr oF oF- 'np J

, '6roBAL noo€S

hnD : NumtG?' oF

NODSS

LET tl =lt

v

I r r 2 r " ' t

g ?

l"

Page 102: INTRODUCTION TO THE FINITE ELEMENT METHODl. FrNtrE ELETqENT MtrrroDs ( eem) FOR L|NEAR STATIG, ANALYSIS PR,ELI'Y I NAR,IE S I{AIN cONSTITUENTS oF FEM FoE T}I€ 3OLT'TION OF A BOUNDARY-

e1LET tla Bs TlfE srr oF NoDtEs AT wlftclf

h r l x* " = ? . t ' A - N o D € s ' .

rHE csretGrrcrr oF n(O rn nt ( i... T)rrr

gtr^Atlltr3 r.nuut ) t3 THE sFr oF

NoDGlS AT wtllclf Uh ls uuruotrtf '

wrlA't t3 Turg NumBGE oF gQvA'tloxr .?

= uurotFE oF \lCDlEls I N Tl.lg

sq" \-rb

= ..1 .

Tt{tx, JFclt AtL Wh C, tf b

tr.|lF cA\l uretrei

wh( l t= Z NA(I , )crAcl"lt

trlt{nl,E NA1I) ts rrr€ sFAp€ ?tlucTrorf

AStOCIATIID WrTlt NODS A.

Page 103: INTRODUCTION TO THE FINITE ELEMENT METHODl. FrNtrE ELETqENT MtrrroDs ( eem) FOR L|NEAR STATIG, ANALYSIS PR,ELI'Y I NAR,IE S I{AIN cONSTITUENTS oF FEM FoE T}I€ 3OLT'TION OF A BOUNDARY-

ts wh=o oN f-. ?tl-

wlfY ?

loct.

NODA L VALUS!

OF \A

SiHILARLY !

vhcr )= f NA( r )dA- AQlne t- uNENo.^,l,

dA : Tn€ uNrrbNN (ro Eq FrrEEJlrrao )A'r uoDt A .

i.c , TtF}rPglA'tUtE At NODE A.

lf,lTEEPoLATlorr oF ol t r ) oN lta s 3 , c

A'rt)= I NAt!) ?*Ael

w lrEp,rE fle

s A (L^)

Page 104: INTRODUCTION TO THE FINITE ELEMENT METHODl. FrNtrE ELETqENT MtrrroDs ( eem) FOR L|NEAR STATIG, ANALYSIS PR,ELI'Y I NAR,IE S I{AIN cONSTITUENTS oF FEM FoE T}I€ 3OLT'TION OF A BOUNDARY-

sulrrtrrr€ lFr. *b, V

6ALERr3tlf FoR t'l I

f-"I S t / u r l

h hAHD

0

lo t

rN rnC

: a(NA,t lc) dE = (N ̂ ,+) t (Nert ) r8e1-1'

-f a( gArNs) ?sBelt

Va e?t n7rlr€t€ AEF h-- tFqgATlg;ll ttggg--l -

Fos tlal \l!EW

wtrCgt h..t lt TlrF NUFrBge- oF

NGDE: rN Tlre sqt 1tb.

To DgrgEHlUg Tt*lF STIFFN|F�SS FrA.tPlI,

AhlD rnc hP3g uRTo:,, EQUATIOU lJurlB€II

t.tulT BF ESTABLISIIED.

Page 105: INTRODUCTION TO THE FINITE ELEMENT METHODl. FrNtrE ELETqENT MtrrroDs ( eem) FOR L|NEAR STATIG, ANALYSIS PR,ELI'Y I NAR,IE S I{AIN cONSTITUENTS oF FEM FoE T}I€ 3OLT'TION OF A BOUNDARY-

frDft g\CC,Y NoDC tN ,tna

AN FqUATION NUIYIBCE.

Ft rne NoDet rN ltg

NunnIEg lt AitlSuGt . (

sDe,rFlE AT r)fAT NoDg )

,

Dg$.TTNATION AEEAY ; Ib

lol,

urtF ASSfeN

NO QUfrTIOTTI

hEEC,AUIE U, IS

: NUMi€P Otr NODALDot lrTl

: N01{EE3 oF EqJATnrt

Asst6NrlENt OF

uoDG tS ACH

A R R A v ! D .

I D ( A ) =+I

-teLoBAU troOE

tSorrA'Ttorrl rt t'lGEPS To

tsfEr) 0) Dtsnpqr|crl

e f oLoBAL EluAllss .8.

i F ,F Ae rl-rfl

\ o r F A e " ( l )

NumgeR.

NGTT' THfiT

t < A ( h " P

l < P S 6 c l

nnp

h=t

Page 106: INTRODUCTION TO THE FINITE ELEMENT METHODl. FrNtrE ELETqENT MtrrroDs ( eem) FOR L|NEAR STATIG, ANALYSIS PR,ELI'Y I NAR,IE S I{AIN cONSTITUENTS oF FEM FoE T}I€ 3OLT'TION OF A BOUNDARY-

lo3

wrlAT tt Tfr€ trlHEfi$ox eF rxE

:E,D ARRAY ? . hnp ,

MATR,IX SUATlorf I

K d = F- r - i ,

K"u = a (NarNg)

P = fD (A )q = I D ( B )

F p = ( N e , { ) t ( N e r { ) r

Z a(NArnr.) eeec1?

Page 107: INTRODUCTION TO THE FINITE ELEMENT METHODl. FrNtrE ELETqENT MtrrroDs ( eem) FOR L|NEAR STATIG, ANALYSIS PR,ELI'Y I NAR,IE S I{AIN cONSTITUENTS oF FEM FoE T}I€ 3OLT'TION OF A BOUNDARY-

utuee*. neff cOttDUCnON :

ELEFT€NT SIIFFNESS MATRU A}ID FOEC€

vEcaoR

ELC,;feN" STIFF NCSS t'lATBlI

ELCF|CNT FGl, \,ggToR,

tfLCI.I€NT

iloD€ xunaB€ R

n

f,lutrrtB€B OF

EIEF,I€NT NODES

S a r b ne

L

k'= L klr]

f"= tf: l

*;i*u,j d&-tge'

o-l(N o r

k : = Nor.a ( No, Ltr)' -

4t r-t c r - ( No,f )e t

Tg N " ) * ( g N u ) d s u

nr

i),.-F: klufi

tezduyar
Rectangle
Page 108: INTRODUCTION TO THE FINITE ELEMENT METHODl. FrNtrE ELETqENT MtrrroDs ( eem) FOR L|NEAR STATIG, ANALYSIS PR,ELI'Y I NAR,IE S I{AIN cONSTITUENTS oF FEM FoE T}I€ 3OLT'TION OF A BOUNDARY-

t06

NcFr€ TllAa :

- c - |t [ = t ( IL) te b rs A ?-tuoDc

e O l F b l t N o t A ! l - N s l C ,- o

A COiI\,GNIGT|T FORT'I OF WR,TTIilO B. :

. G c , Tn . u = I ( g N . ) x ( g N b ) J - s

ge

DCFlrrC t

D= * ( corlutrlultt mattlr)

E.= Y N"

E = [E ' Err . . . , E. . r r lTHCII :-

k iuE I B :pEbJ-o .gc

B" =a[ g'P E J-sr

tezduyar
Rectangle
Page 109: INTRODUCTION TO THE FINITE ELEMENT METHODl. FrNtrE ELETqENT MtrrroDs ( eem) FOR L|NEAR STATIG, ANALYSIS PR,ELI'Y I NAR,IE S I{AIN cONSTITUENTS oF FEM FoE T}I€ 3OLT'TION OF A BOUNDARY-

lo6

coFrgrraATtoll oF HEAI FLUI U€C"OI,

LEr d-'= [11"1wH€rE d: 3 uh( I: )

rlr€N q = . D B d C- f l e n

3 , - ' P r g o d :i -

a 3 t

tezduyar
Rectangle
Page 110: INTRODUCTION TO THE FINITE ELEMENT METHODl. FrNtrE ELETqENT MtrrroDs ( eem) FOR L|NEAR STATIG, ANALYSIS PR,ELI'Y I NAR,IE S I{AIN cONSTITUENTS oF FEM FoE T}I€ 3OLT'TION OF A BOUNDARY-

fo?LINEAL }I€AT C.oNDTTTIOil :

DAaA PcocCSslNG ABEATS r grI9, !!!

ELEFTENT t|Ob€: ^*31" : lEN

A = X E N ( c . r e )t

' f-c.I.-EL€r'r€NT uurrB€R1 t-Loc.a1 (ererrGrr) rsOS {f

6tOBAL NOD€ {3

D€s'lrlNATloN lglfY I ID

P I rD ( t]- sroBAL NoD€ surrB€?f

euoStL Eq,r|rnON ll$}|t€R

LocAT\o$ gAfetl : LM

P = L M ( q r e )

t: Er€}tcr|r NrlFrBgrEL€FrErn rloDE Nr,P|B€P6r.o8AL €QITAT|oN NUF|S€:

tezduyar
Line
Page 111: INTRODUCTION TO THE FINITE ELEMENT METHODl. FrNtrE ELETqENT MtrrroDs ( eem) FOR L|NEAR STATIG, ANALYSIS PR,ELI'Y I NAR,IE S I{AIN cONSTITUENTS oF FEM FoE T}I€ 3OLT'TION OF A BOUNDARY-

S ,sNg

AND

6s"s uP I!:NTHCN OT}TAINS

l € .

ABI,ATt

?OLtotrll :

ArtD gD

LM AS

LM ( q , re ) = ID ( f EN (ore) )

Page 112: INTRODUCTION TO THE FINITE ELEMENT METHODl. FrNtrE ELETqENT MtrrroDs ( eem) FOR L|NEAR STATIG, ANALYSIS PR,ELI'Y I NAR,IE S I{AIN cONSTITUENTS oF FEM FoE T}I€ 3OLT'TION OF A BOUNDARY-

E"IO AEQAV ; 6LoB,qL rloD€ t

I z a 4 s a 7 t t

rosExAlrtrL€ 1

E33t}JtTAL

B.c. DA?A

SPGCI?lco

( i .e.

€TCFI€NT NODCNul'rtCElNG

l = t 1'h= l' l

\-11-- I I

1'10 HAt NoDel

. ' . UNE,f\FWilt

EQVATIOU t

hntr=

7 -Nosct

lx .

l_xf

A :

2

2

P = E D ( A) 6LogAL SrATloll il .

q.l =

Page 113: INTRODUCTION TO THE FINITE ELEMENT METHODl. FrNtrE ELETqENT MtrrroDs ( eem) FOR L|NEAR STATIG, ANALYSIS PR,ELI'Y I NAR,IE S I{AIN cONSTITUENTS oF FEM FoE T}I€ 3OLT'TION OF A BOUNDARY-

i loElAr'rgL€ 1

cS3tlrTtAL

B.c. bA?A

ggtFcl?lcD

( i .e.

. o. , ,2!

r o l

t t = [ t

t l l = l

, L r . . . o

t r { ,7 t

1-1t= [t't t 3'orl,rrtrlt

-Noscs

lx.

l_xl

ETCHCNT }TIDCNUTiSCEINcr

ID lrl,QAv r cuoBAL mo€ |l

S 6 . T t t l o l l l

ll-ln n.r E NoDel. a ,

.'. A uNE,Nowsl

A EeuATlot s

l.lntr= 12.

q . l = 8

E.A :

P = E D ( A) gLogAt SfATlOll * .

Page 114: INTRODUCTION TO THE FINITE ELEMENT METHODl. FrNtrE ELETqENT MtrrroDs ( eem) FOR L|NEAR STATIG, ANALYSIS PR,ELI'Y I NAR,IE S I{AIN cONSTITUENTS oF FEM FoE T}I€ 3OLT'TION OF A BOUNDARY-

E(€HCN.r

t q 5

i l Ir€$ AER.AY :

LocagNODC

cr.|lutqttt

hen =

LM ACT.AY :

t\g,nE

\ruFl Et t

3

As fEn (cre )t atoBAL NooG

€LEFf ElfT Nut\. Cte

I Z t g B 6.csnL ( t

Irroot o I .

uu,rqtl l !I\.t

hel 3

usr.aB€R

h c l =

= ED(fer.rlerc))

Qvnnot.' uumB€R

P= Lt*1 (ore)

t'- Gtog4g

Page 115: INTRODUCTION TO THE FINITE ELEMENT METHODl. FrNtrE ELETqENT MtrrroDs ( eem) FOR L|NEAR STATIG, ANALYSIS PR,ELI'Y I NAR,IE S I{AIN cONSTITUENTS oF FEM FoE T}I€ 3OLT'TION OF A BOUNDARY-

C\€t{CN.rT€N ASR,AY :

LocaluoDC

q.luuiet

h36 = '4

LM AC:,AY ..

hen= 4

As fEtr (qre )t eroBAL NgOG N$r,rt€lR.

Nut\i Ctt

hc l = 6

I ta

\Jut teC,l,

h e l = 6

= tb (f er.rlarc))

QVerrOu uUr.rB€R

r.ccAL ( tr.rooG " J

.tttl,ttCl

| !

\. '

3

6

P= LM (erc)

t'- €tog41

I L + 5 -l I2 3 5 6 g 35 6 E 7 f l l!

1 5 7 I l0 ff

€LCr{E}I?

o I o 3 o 5

I z 3 I 5 g

3 I 5 b 'l Io 3 o , o 7

Page 116: INTRODUCTION TO THE FINITE ELEMENT METHODl. FrNtrE ELETqENT MtrrroDs ( eem) FOR L|NEAR STATIG, ANALYSIS PR,ELI'Y I NAR,IE S I{AIN cONSTITUENTS oF FEM FoE T}I€ 3OLT'TION OF A BOUNDARY-

EXAMPLE 2

COtlSlDEn, THE

i'|ATRTI ANb

f

FOLLOWIilg E,LEI{ENT

FOnCe v€croR :

l t3

STIFFTIESS

{.'

r lq '\ l

k t=-

o

{.:?:i

k;

k;

6f

$.

iit)

f,

ki

t 5

f==k!,

@. , - ^ la r -

- - ' r

ik i j tkir l\- -- l \ oJ

9

ASSUr.r E

E LEFTE N"

TIIAT TIIS LFT

15 6l\tc,N At

ARTAY FoQ, Tlll t

L]*1 ( t, €)

L l r (2 , €)

LFr (9, e)

LH (rrc)

5oo9

Page 117: INTRODUCTION TO THE FINITE ELEMENT METHODl. FrNtrE ELETqENT MtrrroDs ( eem) FOR L|NEAR STATIG, ANALYSIS PR,ELI'Y I NAR,IE S I{AIN cONSTITUENTS oF FEM FoE T}I€ 3OLT'TION OF A BOUNDARY-

l |4\ rnAT 13 Tll|F

ELCt.t€ilT TO

*tu

K t t

K g r

collrr.lBuTloN

K A N D F ?i r - - '

n E

Ktp

K r,,

Fs

Fl

OF THTS

+

@@@

@@

+

+

<-

(

f,e = - ti,u:-[&;+ ; G

, - --'o" - iuiint\- - -

-$itt

Page 118: INTRODUCTION TO THE FINITE ELEMENT METHODl. FrNtrE ELETqENT MtrrroDs ( eem) FOR L|NEAR STATIG, ANALYSIS PR,ELI'Y I NAR,IE S I{AIN cONSTITUENTS oF FEM FoE T}I€ 3OLT'TION OF A BOUNDARY-

f t5

BILINEAR, quADf,lLATERr\L €LEM€ NT

(-tr-t) (tr- l)

ll ttPARENT DOMAIN

:;ll(r;, g: IFt

(x;f)

(xi, g: )Dor{A111

lfPAR.€t|T DOTqAN ,

nIIATUR,AL CtrORDINATT1S

!=[;] =[;Bl1 I

5= [ ; ]

!=[;] =t;l

(ri, ei)

r ) I 1r,r1 3

I 2q

Page 119: INTRODUCTION TO THE FINITE ELEMENT METHODl. FrNtrE ELETqENT MtrrroDs ( eem) FOR L|NEAR STATIG, ANALYSIS PR,ELI'Y I NAR,IE S I{AIN cONSTITUENTS oF FEM FoE T}I€ 3OLT'TION OF A BOUNDARY-

rf tr tl6TH€

--NIVSTCAL ceo:,Dlurffis oF A DotlT

rN .$l.,e :

x -- I xl -- I x(lrl )lR' L :J tU( l , l ) lAe€ 6luQ,N tY THC FottrOuftilo MAPplt{Gl :

r (i,1) =4z

C,=l

eN" ( 1,1) X.

e (q,1) =+I(l--l

No(t,l) g:

u'(q,1) =1z

lNo(\rtl ) d:

oEr tN v€cToR. FoBH I

+I ( I) = I N.(!) I :

C=t

|$TECDOT.SnON G A Fur|cTrct| HAS TTg

sAtlE Foc,ttl :

(OORDINATT

TRANSFoRItATtoN

I}ITIRPOLATION

Page 120: INTRODUCTION TO THE FINITE ELEMENT METHODl. FrNtrE ELETqENT MtrrroDs ( eem) FOR L|NEAR STATIG, ANALYSIS PR,ELI'Y I NAR,IE S I{AIN cONSTITUENTS oF FEM FoE T}I€ 3OLT'TION OF A BOUNDARY-

ll?SHAPE FONCTIOT'IS :

ONE_DIH€NSION : ( R€,r,flNDeR)

5,=- |€ . = + t

N , ( t ) B * t t - q )

Nr(h)= t ( t * i )

ruro- DtH€NSloNS :

N.(t , r t ) = *( , *h. tXt+r lo1)

t ,= - f " l l= - lN,(i,'t)=*(rtx,-l ) t= *, 1tr = _,Nr(i,r)=t(rq) 0-q) l'=

*' t3 = +rl q 3 - l r l . t = + l

ur(t'l)= +(q)(r+q)^l(i,l) = *(t-rx,+q)

N,(h) = * t t+ i . t ):t

Page 121: INTRODUCTION TO THE FINITE ELEMENT METHODl. FrNtrE ELETqENT MtrrroDs ( eem) FOR L|NEAR STATIG, ANALYSIS PR,ELI'Y I NAR,IE S I{AIN cONSTITUENTS oF FEM FoE T}I€ 3OLT'TION OF A BOUNDARY-

TTrs ts nouf A snAtrE ruucTfoNI rg

LOors ur,C :

NA

t r lA T€NT .Ftort NoDe

rf ,l

€F Tlfg .r€rn Eulttlrgst, ll

Por€Tlfg

A

Page 122: INTRODUCTION TO THE FINITE ELEMENT METHODl. FrNtrE ELETqENT MtrrroDs ( eem) FOR L|NEAR STATIG, ANALYSIS PR,ELI'Y I NAR,IE S I{AIN cONSTITUENTS oF FEM FoE T}I€ 3OLT'TION OF A BOUNDARY-

nlutueaR TRtAtrlguLAn €L€FI€NT :

ItAN EXANiPT€ O? Dg6gN€EATIOil

-t LIN€AG, TR|ANGUIAR ELeH€NT

<^AN BC D€BU€D FEOT}i

+ BIL|N€AR QUADR[ATEIAL gL€T'r€rtT

t.| c,oAtFscttlg soDES 3 Arro { .

{

r N K = t N q I : r . , s s e r I l = I ; .C--f

TllErt :

! = 1,, I:r N.X: +(Nr+Nr)I:

= ui li *NiX: {r N:, I:

X B-

3 r -rlD - -

) N - XI

- s + ( l

o= l

COORDINATS

TP,AilIfrBT{ATKIX

Page 123: INTRODUCTION TO THE FINITE ELEMENT METHODl. FrNtrE ELETqENT MtrrroDs ( eem) FOR L|NEAR STATIG, ANALYSIS PR,ELI'Y I NAR,IE S I{AIN cONSTITUENTS oF FEM FoE T}I€ 3OLT'TION OF A BOUNDARY-

= N r = ( r - t ) ( r -q )

s N l B

= NrtNr =

( t+t)(r-q )

(l +'1 1

I1

!4

I

z

,

Nf

lNr

lN3

taowHcrr Tllg srnD€ FulfsTloNs AtE:

t TTl3 wAY ) htll€TlfgQ trr€ lfAU€ A

quADEl LAT€e.AL €tgr.f€llT OP A

TR|ANGULAR €t€Frl€Nr Ul€ CaN

u5€ TUE SAH€ PAPtrlT DO]"IAIN.

-) rLElncr$r INT€RDorATtoN nA: Tlfg SAHE

tuElr AS COOEDINATE TEA[lSFoEttATlON. t.e.'

a^h = i Nld:Csl

N; d: + Nld;0 .e '

+ Nsd,

Page 124: INTRODUCTION TO THE FINITE ELEMENT METHODl. FrNtrE ELETqENT MtrrroDs ( eem) FOR L|NEAR STATIG, ANALYSIS PR,ELI'Y I NAR,IE S I{AIN cONSTITUENTS oF FEM FoE T}I€ 3OLT'TION OF A BOUNDARY-

r! llOw STIAPC FUNCTION

lal

Loors 1rt€ :A

nri

ELEHTNT STAD€

FrrreTtOil

Pt?Atf lDII

EtoBAL $TAD€

FUTCTIOI'

t'i/ l

}f, 'eer-)

Page 125: INTRODUCTION TO THE FINITE ELEMENT METHODl. FrNtrE ELETqENT MtrrroDs ( eem) FOR L|NEAR STATIG, ANALYSIS PR,ELI'Y I NAR,IE S I{AIN cONSTITUENTS oF FEM FoE T}I€ 3OLT'TION OF A BOUNDARY-

la2-

T}IR€,E. DITIEN SIONAL SIIAPIE FUNCTIOIIS "

TSILIN€A:, II€IA TITDPAL EL€}I€NT

tfH€lAlfEDSO N , BRI(,B

5= q ([)

TIf t tPAr'C i{r Do}lAlN

PT$|CAL Domerl

X = X ( I )r l

DArS\r D$rAtN , tlAlUSAL COOIbINATS: :

t q lq B I n l' - t l l

- t - o .

Page 126: INTRODUCTION TO THE FINITE ELEMENT METHODl. FrNtrE ELETqENT MtrrroDs ( eem) FOR L|NEAR STATIG, ANALYSIS PR,ELI'Y I NAR,IE S I{AIN cONSTITUENTS oF FEM FoE T}I€ 3OLT'TION OF A BOUNDARY-

rz3rt tl

ru€ Pfl.rsKAL csoEDlhrA.ttl oF A DOINr

f N - Q e s

l r ? t r ( t r t r B ) Ix = { : t = 1 v ( \ , ' r , i ) t- r \ u ( l , I ) S ) I

Ar,t 6lulEN Br TH6' Fo,.tohfltf. '{APPIN6S :

x ( l " t r5 ) B6z

C3,

Irq 3 f

aI4 = l

No(q,l,q ) x:

9(t t,5) = ru.(t tI ,5) yi

z( \ r1 ,5) s No(\, t l r5) " :

a2 t frf vEsToS FO3M :

ax(1)= Z \ t :) [ l

e3l

(ou:ilNAr€

TRNININXOiI

Page 127: INTRODUCTION TO THE FINITE ELEMENT METHODl. FrNtrE ELETqENT MtrrroDs ( eem) FOR L|NEAR STATIG, ANALYSIS PR,ELI'Y I NAR,IE S I{AIN cONSTITUENTS oF FEM FoE T}I€ 3OLT'TION OF A BOUNDARY-

Er€r.r€rlr nrERxttATlorfAS 'rhtc cooBDtNATE

12"4

llAS rt.;" SA]f€ Fo:t{

TEANSFoBt T ATION :

uh(trntr S ) =E

T No(\,t tr5) d:Cr3f

EtEruNT NTERTOIATNN

S}IAPE FUNCTIONS :

No(l ,I,9) = t(r+h.lxt *1"1 Xr+5"q)

v.lff gt€q, H. 1o 5oI - l - f - f

z + l - f - f

3 + l + f - f

q s f + l - f

5 - f - l + f

6 f - f - l + f'l r l + l + l

I - l { - , + t

Page 128: INTRODUCTION TO THE FINITE ELEMENT METHODl. FrNtrE ELETqENT MtrrroDs ( eem) FOR L|NEAR STATIG, ANALYSIS PR,ELI'Y I NAR,IE S I{AIN cONSTITUENTS oF FEM FoE T}I€ 3OLT'TION OF A BOUNDARY-

t , l f

DE6ENEfI.ATION OF B?TCT. TOrr rl

weDCt,fz3

eoALESC'€ ItoD€5 AND +Ah|D

Axb E7

1s$ll:

I ( ! ) =

trf tfEF€

tNa ( : )

? ,+ Z 'N . .

creS

?.= t e2- Nq IoCl,3l

el .

Inutr)= l"'(:)t*+

(!l

, Q-lrlr5r6

tNr q) ,tNa(5) ,7

a=3

q = 7

jO O E O

- e - - o - -

Page 129: INTRODUCTION TO THE FINITE ELEMENT METHODl. FrNtrE ELETqENT MtrrroDs ( eem) FOR L|NEAR STATIG, ANALYSIS PR,ELI'Y I NAR,IE S I{AIN cONSTITUENTS oF FEM FoE T}I€ 3OLT'TION OF A BOUNDARY-

n e

DGo€N€:ATlo$ oF t wEDo€tt To TgrEAn€DRort

cort.EseC ;1obEt 5 1 6 A N D 7

Tl€lf :

x( ! ) =

$HCR€ :

,,

N.(l) =

t ,

I:, + Ns(!li N:(r)a3l - x

t-

[* l ( 'D[ " ;

( : ) *

,

,

N. (I)

4 = l r a r 3

+ N'"(I) q,= 5

- r . G I - e ' - 1 - e O - - ' -

Page 130: INTRODUCTION TO THE FINITE ELEMENT METHODl. FrNtrE ELETqENT MtrrroDs ( eem) FOR L|NEAR STATIG, ANALYSIS PR,ELI'Y I NAR,IE S I{AIN cONSTITUENTS oF FEM FoE T}I€ 3OLT'TION OF A BOUNDARY-

l2-?

lll6uER - oIDCR C\CHerlTS

IN ON€-DIHEN:ION "

Na

N , ( l )

N r ( l )

N3 ( t )

= * t ( t - r )

3 (r-q)= ( r+txr- t )

e.sralrl'E, :

h c n = 3

@

@

= tt ( f+ , )

$16$€l-oRD€l t\€MErlts lN

Trdo oR T$tsg Drn€NlloN:

C.Ar| BE OBTAINCD IvoCSUrsT BUU€

Page 131: INTRODUCTION TO THE FINITE ELEMENT METHODl. FrNtrE ELETqENT MtrrroDs ( eem) FOR L|NEAR STATIG, ANALYSIS PR,ELI'Y I NAR,IE S I{AIN cONSTITUENTS oF FEM FoE T}I€ 3OLT'TION OF A BOUNDARY-

rrETlfls ts florr fi wotrs .'

i l(1+r;

@ ( l-rt.)

ETA'iPLS ..

@ ! I (1-')

* t (t-n) ( r-ht) t t(t+r)6) @

Ne(f,r ) = * t (l-t) (,-rl')

Page 132: INTRODUCTION TO THE FINITE ELEMENT METHODl. FrNtrE ELETqENT MtrrroDs ( eem) FOR L|NEAR STATIG, ANALYSIS PR,ELI'Y I NAR,IE S I{AIN cONSTITUENTS oF FEM FoE T}I€ 3OLT'TION OF A BOUNDARY-

lz?

NU'TIERICAL INTI6SATION

TO CAl,cuLArg EL$Fr€NT STIFFN€33iiATntI' AND Fotc€ U€CTOR rrr€

N€€D TO , tor|gHohr, t\lT€5EA'rE

TCRltrl Lftg Tllls ..

,flf t1)ds -- ?

AAr{OL€ : FoR' tlgAf CONDUGTIOr|

?rlr3 COUID B€ AS ?ot1'ouf I :

ft l)= (gNjs (gNu)rscN, Tlr|E t\lTg6BATtoN

6tugs 'rrrE a- b cor'reoilg}lr

oF TtrE el€rrertT S{IFFNESI

l.rATEl |, .

PoR Flsd=l f j ( lnco;r^ i l )

| - / - . - t t

_\ f t*rdx = \ t t r ( I ) ) *r \ ( t ) d\d J - , J

Page 133: INTRODUCTION TO THE FINITE ELEMENT METHODl. FrNtrE ELETqENT MtrrroDs ( eem) FOR L|NEAR STATIG, ANALYSIS PR,ELI'Y I NAR,IE S I{AIN cONSTITUENTS oF FEM FoE T}I€ 3OLT'TION OF A BOUNDARY-

130

Fot' n3g =L

^"lf(.,g)dgJr_J dfl?i'fh(rr\,rr),e(\r)) j(1'r )dt rl

O | - f t

det

t--fr..:

JA(oE|AN , j =

Jlar

H

f

(A) = detgan3-r?1

Foe nsd = 3

t

Jf(\sp)j*- =+ drdgde

+l r l r l

.!.[F(xm,5 ), 9(r,T,5 ) ra (t,?,b ) ) j(t,?,5)- t r r r f

{ q { d l

j ,B dct

?.I"5u-"5??

-2"3

?r?t?.:l2 t

a!"q

?r"teq

J '7f

??fra-a1

Page 134: INTRODUCTION TO THE FINITE ELEMENT METHODl. FrNtrE ELETqENT MtrrroDs ( eem) FOR L|NEAR STATIG, ANALYSIS PR,ELI'Y I NAR,IE S I{AIN cONSTITUENTS oF FEM FoE T}I€ 3OLT'TION OF A BOUNDARY-

13ltN CAss Cel€ , TH€

TAis ts :

INT€6:ATION

-l ts(h) rt = ,

e o g t \ - t = L-lE

l : f i

]tlt ?(t,r) d1\ = ?FoR nSd =, $

= ?

,WC D6Tr WANT 10 TPY TO TTIVC

TH€sI INTSBATTONS €IAgTIY.

Tlfcc€eonc, w€ AgPeorlMATE ?lltM

BY rlut{tF9rtAL rNTE6lATto}l

-5 fr(r ,1t5 ) J1 J1 J5+ l

Page 135: INTRODUCTION TO THE FINITE ELEMENT METHODl. FrNtrE ELETqENT MtrrroDs ( eem) FOR L|NEAR STATIG, ANALYSIS PR,ELI'Y I NAR,IE S I{AIN cONSTITUENTS oF FEM FoE T}I€ 3OLT'TION OF A BOUNDARY-

wk

tr int

t(E)dl * Z ?(qr.)k= t

lltf3z

NrrtrrB€9 oF lNle6larloN 0otUtSn irtt^,

H c

wL

kt*

at

ao

1 CoOBDTNATE

INTIEGIA-IION

WCTI6HT OF

Forrrr

OF TIIE

DOTTIT

rHE TNT€6}ATIOI

w, wa

Acuplr6

OATTOMIATS CTACTIY

Tllk

btAr{DL€ : TEAP€?OIDAI, IUU€

nir* =1av

l , = - fq " a * lw r 3 I

w t = I 2*D oRDGt-

loogt€g

: f s , + lr-1-rrlta/\-r,

trrT€crATgs

Page 136: INTRODUCTION TO THE FINITE ELEMENT METHODl. FrNtrE ELETqENT MtrrroDs ( eem) FOR L|NEAR STATIG, ANALYSIS PR,ELI'Y I NAR,IE S I{AIN cONSTITUENTS oF FEM FoE T}I€ 3OLT'TION OF A BOUNDARY-

e,IAr"rpt€ : SfltpSoXi R$LGra3

W3

nint^,q ,^,q.d

I g

|NTE6e56=

EI lc tLT. rT

Ws = t l l

wr - 4lI

w l = t l t

RD3f 3

= 3e - l

= O

t r + fq

wa hft

ggDE:- DolvrroirllA Ls

4t* oEbER. AGcuGATE .

RCl.fAB:3

l .

z .

FEh|€3

cstT

IH€P,€

ACr|lEri

FCI^I€C,

INTE6:ATION polrlrs ]rgltN tgSS

ARe trnggHTlox ILULE! wntctl

Tlr€ sAlrc AsB$tec v wlTtl

r1n.r€ql5tt6N fotrrl

Page 137: INTRODUCTION TO THE FINITE ELEMENT METHODl. FrNtrE ELETqENT MtrrroDs ( eem) FOR L|NEAR STATIG, ANALYSIS PR,ELI'Y I NAR,IE S I{AIN cONSTITUENTS oF FEM FoE T}I€ 3OLT'TION OF A BOUNDARY-

t34

6AUSSTAN QuADrAtul€ RULCt :

u TH€r ae€ oFllifAt lN ortc DtFrENltor

ii) \rrrlr ( nint) FotNTt/ TnGt Aclflcug

(z n int) oe,D€" oF AccutAcv AND

nn€orfiG (Z nirrt- l) orDEB FotYxoFrlntt

EXACILY.

l . hint = |^,

{ , = O , \ r J f = Z

Z*o OEbtS acaUpATG

I ET oc.Dtl DotYltoFulrr lllr€lb?ATgD et5tY

Z , n i n t - 2 -

aE|

l , = - t / { E , W . = lx ^ - -f - = 1 t l { 3 tr J

4"* oet€e eceu:nrG

3LD oRDeC. FOLrNoFilAtr n$Q3qaau EtrAsTtY

W . * f

Page 138: INTRODUCTION TO THE FINITE ELEMENT METHODl. FrNtrE ELETqENT MtrrroDs ( eem) FOR L|NEAR STATIG, ANALYSIS PR,ELI'Y I NAR,IE S I{AIN cONSTITUENTS oF FEM FoE T}I€ 3OLT'TION OF A BOUNDARY-

133

3 . n i n t = 3rylq , = - \ / F , w f = s / 1-

T.= o , \= a/ l

i =*{1F ' W3 = 5/?

6T'r o3!GB fcuBAT€

STtf o:,o€t Do\yr.oFtfAts r$rc6PATtD tl GTty

Nulrc3lcal |rlTElSt A.Trc|| t\l Tw0-Dtwlsltlout ..

nn1 :I6(i,fr) wrL= l

rrr€clArrorr PorNrt ({.rtr) ArrD

TlEll:, (OSg|ESDONDInG l^rGlCTT$ wk CAu

8e oBTAINgD Florr oNQdt|rNsloNlt

6hfssr0rr q\ADcAa\nE l,ur,5 Bv PSotucteuLc

Page 139: INTRODUCTION TO THE FINITE ELEMENT METHODl. FrNtrE ELETqENT MtrrroDs ( eem) FOR L|NEAR STATIG, ANALYSIS PR,ELI'Y I NAR,IE S I{AIN cONSTITUENTS oF FEM FoE T}I€ 3OLT'TION OF A BOUNDARY-

B6

Htlrg ls ttow lr woPtss '.

Ef lraD\t !-a-

2.x 2- r\rT€6rNrlox :

(2)| w ,n*i l l turlil I

^r(3) .

tr e - tllj*fit I

t:-r{Ibrl? |

1

t

(r)g = {

,.ut+ t/(ir j -

"l{

t

FrtIIItI- - -.rn- -a;-i t(d,{.): wltvt'i

:

t t '. - -€€.- t - - -

ti.ltl?iru!'r^rG ) t' ' r ' --?

:I

i(,il i:)l/ "Tw,"'a \

- - . j f O o - o . . .

i

Page 140: INTRODUCTION TO THE FINITE ELEMENT METHODl. FrNtrE ELETqENT MtrrroDs ( eem) FOR L|NEAR STATIG, ANALYSIS PR,ELI'Y I NAR,IE S I{AIN cONSTITUENTS oF FEM FoE T}I€ 3OLT'TION OF A BOUNDARY-

t3?

fieSumnrlAtv oF 2;tL c,uLc:

Ir int = 4

(q, 1"., ) = (-i r-ft )(d,{.)(i,{ ')(s* ,i.)

NOTS THAT :

hi.t

l-

L=l

= (-t )-t)(*fr 2*+)t t , * t )

, wf = I

w a =

w r 3

w r c

( ArGr oF Tlf€

DAg€rIr Dometrr)t\OtE: :

stlrltAgtY , oNg cAN APDtV TlfE PlODrr?

o$\€ To TltrcE'Dlnslltp$AL rllFBEPffilorf.

Page 141: INTRODUCTION TO THE FINITE ELEMENT METHODl. FrNtrE ELETqENT MtrrroDs ( eem) FOR L|NEAR STATIG, ANALYSIS PR,ELI'Y I NAR,IE S I{AIN cONSTITUENTS oF FEM FoE T}I€ 3OLT'TION OF A BOUNDARY-

|3A

C-ALCULAIIGN OF ICBIUATIUEI OF

'tlt€ 6lrAP€ FUNCTrcNI :

wE N€€:D -ro gALGU1AT€ trrc DEPTVATTIF'

oF Tlr€ sl|A9{g Fn4alort wfTtf BBDrEtr.TC GiloBAL CoogDrrr ATE!5.

Wg HAVE Ttig IIflVATt\llHl W'9;r' \ocAL

Cq;g;ROtXAlF.

h s d = I

Nor, = Norl l, * ( eest.l;l-.r-../ t

V

w€ neuc 2-TPt r he

hS{ = 2-

Norl = Nort €rr + Norl ltrr

ntorb s Nari qr? {- No,l nr7

Page 142: INTRODUCTION TO THE FINITE ELEMENT METHODl. FrNtrE ELETqENT MtrrroDs ( eem) FOR L|NEAR STATIG, ANALYSIS PR,ELI'Y I NAR,IE S I{AIN cONSTITUENTS oF FEM FoE T}I€ 3OLT'TION OF A BOUNDARY-

r31

llr Frtrttll FonFl :

Ir,*( *o r t t Nor ; ) = ( r t . r t rN . [ )

|

H Ln,,t 'u]

na J'TtrGtE

-

?a

[ ; ; ]

xrrt

?nH-l; ]

, lqn= f IFU,r

= 3 37x

_

f z x f l- . f -- l -

\ ?1 )

- l

Page 143: INTRODUCTION TO THE FINITE ELEMENT METHODl. FrNtrE ELETqENT MtrrroDs ( eem) FOR L|NEAR STATIG, ANALYSIS PR,ELI'Y I NAR,IE S I{AIN cONSTITUENTS oF FEM FoE T}I€ 3OLT'TION OF A BOUNDARY-

h s d - 3

T(esf )'-

Jcof

4-:-

J

140 .

?N.-r

?t-

?N.-- =

? x-,

a+.€-

? x-

?x.t" t

-

?N.E

-

-- ?Nr (Ef t-EE , l

coF rr =

cdrL =

Cof rl =

gofzr =

Cofet =,

cofa3 =

9rtl zr5 -

9r\ zrt

9r t? , I D

Z\ t r t i r '

?rs x4 '�-'

Zrt Xrtl rD

gr5 a,?

9r; ?r5

grl arq

3,i x,l

7t\ Xrs

zq x' l

Page 144: INTRODUCTION TO THE FINITE ELEMENT METHODl. FrNtrE ELETqENT MtrrroDs ( eem) FOR L|NEAR STATIG, ANALYSIS PR,ELI'Y I NAR,IE S I{AIN cONSTITUENTS oF FEM FoE T}I€ 3OLT'TION OF A BOUNDARY-

14 l

d., = Xrl 9r5 - ag 9,rl

t o { r . 3 t S 9 q - x r + 9 r s

tof sr a Xrl gl '�- *rT 9t+

j = Y,i 4o$,r r xr1 cofru + Xf GFlr

\, = ( Nort 6f,, * Nc,, c4re +Nqri C"f ,r)/ j

"o,b B ({rt @far * uo{ Cofar + Nory coFzr)/i

Nqr? a (uo4 Cd' + N.rt (ofrr + Uerl Cfl )/i

r - - : l -