'llrey are tlre thin pl ate theory arrd thc" tfrick pl ate theor-v, )e

f inite Ele ent pr.ocedlrr.es us;ed in this arral ysis ar-e preserrted

bel ow.

3.2.5. I IhlrL p*!_at"_e-. f,Ip,-gr:y.i.

I. general , tlre tlricknegs o.f ure we-rr in a gearr. wtrr:r:r is nrLrch

Erfial I er than tlre .uter cria,rnete* 'f ure rrltb o. tfrc, r'i*, Frence rt

wlrul d be app'opr-iaLe to appl y Ute tllirr pl ate Ureo.y r_rf [:errdrrrE ofcircltl ar plates. lhe web strltctr-rr.e iE cliEcretizrlcl into arrrrt.rlar

disc el eore'ts as desc*ibed *ar.r ier'. Trre tr-alrsrver-se di,:ipr acEnrerlt

r.', ts expnesljed as a pol ynoarial f lrnctiorr

w.1 (F) = (4.+;1- f.*a.r n=*a+ Fa)..

Wnef'e 3rt a2,

displ acenrent at

l5 grven try the

a.r and a.+ af 'e

any poirrt h, (F,er)

elipFer3!; i on !

w.-l ( r') cos n6

(3,1H)

corrstalrts , { or the n.*, tr:rnr , I lre

, orr tlre rreutr.a I ft I a e ctf tlre wet:

(3.19)oo

zn=c)

w (r,(t)

dw

Ihe diEpl acefients in the arrd tJ clir-ect iorrri Bre giverr by

( :l . :t(:r )

-Stat ic Anal ysis

)n

0w-z

0s

;1 .:lC)

Ihe el e,nerrt str.ains irlas bel ow.

€F

0r'

I

n

ter'ttls (Jf Llte disll I acemenhs catr br., writLerr

Jt*

-2or.

)r-r-z

es LI+ ---

r

-1

ln

J',

The stra i n-d i qp I acernent

(el- tErl- (q)^

=-z

FEI atiorl is given by

de"

-'t

lDw

de2I'

(3.;ll)

(t.22)

f|^

(tg

The stregs-strain r.el at ions f or., trerrcling ar.e gtvean bv

€" * I ee]

1- 11

E

I tu t €,. ]

ene frot t t+Dl

and the strain-displ acement matr.ix tEl inpnesented in Appendix .3,2.

tfre above le,l at i Dn t5

(3.23)

)is the Foission,s r-Eitic:.

:.1 .?t

J'lrer stress equatiorr is given a:;

(6]- tfil tFl- tq)_

where C6] is the str'ess

for this pr.obl em is given

(:l ";t4)

gt ne':t;g-qt ra i n m..':rtli )i

The pu press i or]

eiven by

vector'. Ift], the

1rt f)lr[!endix 3..l.

{ot' the stnairr elteFgy "jtor.e.j in t5e el emelt

€r.6r +€€6€r + f1,e.r Cngr 1 r- dl- dg dz (3.:tF)

and str aiftc; fronr

eqLrat il]r'l (3.:.1:i) ;rr rcl

IT IEI ttrl tB.l r cln cl€

[KJ.. can now be written as

(:J.?'/)

ug' = : fil'of tlre str-es.ies

respect i vel y irr

y can t)e HFi ttc?r t

egi

,

eng

|-

ItY(tltl= It r o ItlI o o u_))/"-J

al Lr

.23

ene

VA

3.

the

and

st ra

(:

i11

fz(II

)li

By subst i tut ing

equ"rt ions (B.Zl,

neanranging, the

Ehsust

24 ( l-tz)

*tt|l

stiffness rnatr i x

r,r = lll IEt ttrl tBl r' cln de

I -r,r

3.3 THREE-DIMENSIONAL FINITE ELEHENT FORI.IULATION

lhe compr ex grlcjfll€rtry of trre lrrrL'er l:anrrot be exaictr y crer:,cr,1hr.]Lr toa cl ose degr.ee cr+ elccllracy usirrg Lwo_rJinrelrsional f i.i te et ent.=all:

'net.lods ' In pa.t icr-r I ar', c.rnvec, werr.; ;:llrcr t.1per. rJeb:.i car'rc.rt beideal izrd erxactl y using re twu_tlinrerrsrorral lle lorl . ]rourever.,th.ee-dimensional f irrite ele'lert flreLrro.J can bc, r-rse,cr irr :ir.rcr.lca!5es to advan.t:erge. The geomelr"y antj tlte dtater.ial ,rr.opet.Lies oft.e wheer a.e the gar'e at ar l r;icJial c.'ss sect inrrE, burt .t:rr*I.ar,ing is asynroretnic. Herrce a _.;elni**r'al ytical { ittite el emerrt:niethod can be Lr:sed . A corritr i rra.l- i un of pol ylromi al + arrd ltar.,rnor.r i r:functions af'e Lrr:ed to 'epresert tlre aii;p larcerrrerrt. r.h., rrar.arorricf LrnctionB satisf y tlre bc.runtJa*y coar.liti.,rs:i irr

're tJi.er:t i.'a i.rr..g

lvlrich the nateFial pnoperLir::: arrrl tlre,: eennreLr.y i,rrcr ttrp- ,rjir:l,nE -Finite ele'nent cri:icr'etizati.,. .f tre wrree*r c.o:;r:--gectiolr iriLrJBnnLtl ar- tr.iangr-rlar elemerrtri is rilrowrr irl Fiq.3.6-

lhe stnlrctune

displ acenrent in

is described by pol i r..the Cro-cjirection is

co-or''d i ltater; r. rz ,(,.!t i ve by

I hr+

LI

w

2 u", cusrr€r

) v.. einn€r

2 w.. cosne(:t,:t9)

Stat ic Ar.lal yEi i i!:J . 3;l

t-"l''l13i,,l"lr0

9

E

5

5

a

3

?61

260

zs9

25€

755

ZsL

f6 3f t6 6t 76 ,t 106 l2l

Fig.3.6 Typicot discretizotion of a wheel ross section for

r9l 208 223

three - di mensional onolysi s.

the nodal displ aceftent vEctor- of

is given by

Llre el enrerrt fBn tlr(" n Qr'' I t;ir tlctnic

't'

(ql-

u,rner€t

i=l

aL the

I ct1 Vl ]fr Ll2 v2 tra Lle Vre wtr f^

r"r,v,br ane the displclceflterrLq; at arly

to 3 are suf { ixes cor.r.et:lJotrLlin!, tLl

thFee nodes .

(:r.4c))

pc)1nt f tlrc. el e,llenL ilrrd

the rroda I dir;pl ace.lterrl.$

inear f ttrrc t icrrr

r3.41)

The displ ace,ner.tt:; at any point ar.et elrFf ,cr:;riecl a:; a I

of the f or.m

Lr Lte * L2 Ll2 * l-a g.r

whtre Le rLe and Lrr ar.e nAtLrf.al co_(rr-cJlnates sltclt LltAtLr + Lz + Lc = 1.

Rearnangingi the above c"xpr.ossions, the displ acrlerrte crrnr4Fittrn as

Lr (ur-t-ro) + Lz (uz"-Ltg) + rta

tre

'l he

the

Lr (vr-vrr) + La

Ll (wr-wa) + L.,

( vro--vr,r ) F vlr

(w5!-wrr) + w.i

el eotFnt str.a i ns aFe

el eaneJrt displ acemerrt:;

otjtairlr.-'d in te'r.nr:;

w i t lr r-ellpect to

(3.4.J)

tlre drlr'.i va t ive$ ofI ucal co-orciirratel;.'

of

the

Stat ic Anal ysis:r -:J4

AB t^e el ernelrt stnains ar-e cref irrecr irr ter-ns of rrre rrat:r.r'ar co--oFcJinates t F arrd z ar.e tu be expr"essed ilr t elng................; of laLLtr.a I c.'_ordinateg with I inean shaJre {lrnctions.

tr = Lr (t.r-r.c) +

z = Lr (z r-r:e) +.

Jac

a

dt-,

__a__

dlo

LJI

(ra--t':o) + r.,a

( z *-z:r ) {. z.n

L2

L?

j

,l

)I

( _d__ )la',, IItla- |[a,' )

an tnansfoFmation of Llte

f D" d= -r rt---------tl_ lu" u'. I I- I )r a'l {t---- ---- | I

I 0r. Dr_- I Ir.t\

'f:(}t.rr

?_Dr

_.?_dz

(.1 .43)

('J .44 )

(3,4rj)

( :.1 . 46 )

the rratunal

rJ Lir. i vA t :i vr-.',.; .

be obtained

I Dwing w..ry !

rI t"r-,',', .=.-r.',-l

= Lt^,,-.., ,=.-".rJ

[ -u-- J

tiJwhene f Jlco-or.d i atellre I ocal

thnough the

the Jacob i an oper.ator r i s t_rsc,cl to rel .rteder.ivativeg .Lo Llre tocal co_crr.rJinate

deni vat i ves w i tlr r.espect to n antj z canirlveFse rJf the Jar:otj r.rrr rnaLri:l irr the .f r:l

Stat ic Anal ysi s

'lI

( s--

lr"loL;;_

wnere ar r

Jac ob i arl

tJl

rdraraar and aa:r

matnix.

-4._DL,

-4._A'

atl dra

Aa r ?\az

r

ilrlt

-"4_0L.

-.0.-.

dL,o

(3.471

(3,49)

Du D r-t

ane the e I emerr l-s of

= tJl

Llre inverse

EqLration (3.44)

nent vecton (q)

dL.-

c:rn be r-ewr i tterl

(3.4[])

in terrng of the riod:r I displ ace-

Dn

DuTJ]

DL.

E r-r

-'I f"'-"- JI

Lr?-Lt.r JI0z

1,,-l

(,(:r

0

Ir-r I

L"

br-r

Dr

Du

bt

tJl

(_)

Substitr-rtins for tJl-r f nom eqlrat i r:n <3.41). r we qcrt

1

I

0

(t

L' -l

r:r

o

Sfat ic Anal vsi s 3.36

" "-l(:) (j I

Ct Et r:a (-) (,)

(:l Etlir* C) 0{iill:(j

Ct

(*il1s-€r12)

( --;t:* r -Aa:r )

(ql (lt.1i{r)

The expnessions f (]r- str-airr

nates is as f ol I ot4s i

Or-t

;;

Dv

Dz

_:_r

(€r^

cyl indlicirl co-or"rJi--

{w

(ll .';:l )

l'r.l!\r!i al"rf, nrul L:i1.r I ied by co:; tr

El-r':l i n-d i spl ;rcement r'el ert iori

€n

€z

e,"et

f=e

bur

bn

Dr,r

b.

cclt pcrnrsnE|t g i n

bw

de

I dt-r

"04ldv

F. 0e

be

I n tlre atrove

and the I ast

is now giverr

nratr i x r

two j-ows

'f ir.g;L f olrr.

sili r. Ihe

'tlre

tly

brr

3z

_1_r

bv

bn

$tat ic Anal ysi:;ll .:i /

e-1I o

z

"-t

a2r

L1 o

"22

(-ar a-aa, ) o

(-arr-ara ) o

tll^"1_le l-tltl

r"2t

'l nL.rIr rt'-r voI

J

a2r "ll o 422"r.2 (-arr-arr) (-arr-arr)o

r"..-!r( arz- . -L^(:"t:.-"re) t'

"2t 1Z-t,,43

GtLzi^zz

Ill'.?

r

o

Lz\Tr '\9

Y

z

ur

ur

*l

u.

*3

The el ement

of f r.eed crm is

il

matr.i x c Llr'r.espclrrct i ng

f rom tlre v;rr,i.rt iorral

to the I ocarl degr.ees

prrr'lciples.

stif{rress

ofltainced

tlrl tBl dv

the above equat i on t t. .l i s

rnaterial pnopenty marr.r x .

(3.1::;3)

gLif f rress fl'atrix ancJ tl-tl

i:: writte-'rr "rs bel cw:

= 1fi,,iIn

tstlre e I ernent

l'lat r- i >r t Lr l

tttl =

where

[r.I

l"lt=foI

foI

lcr

E5!

at

Er.

0

(,

C'

E2

E2

cr

C]

(,

o

C,

o

E.r

o

{,

0

C)

o

o

E:|

{r

o

o

o

(,

(:,

E,e

Er.

Ea

e rr-i I

(l+t) tr-zirc.Y

c+ i r (r-2r)

(3,114)

(3.:iil)2(l+ i 1

Stat ic Anal ysi s3.3?

clv

Tfle e l

L.. and

where r'

Jac ob i an

equat i on

eflleftLs o+ tF.l ar.E .f LrnL:ti()llri r){: nat:Llf-;il co-u}.dinateij l_.r r1_'.,

r. lrle can wr-ite dv in etluaLion (J.FJ) pe]. urri l- r,ad ilrr "r:;

det cln {:1 , r_i6, )

is the, celrtr.oid.rl r.rd:ir.rl;, rleLJ isoperaton ancl dv i s tl lr: r,ot at i onii I

(3.56) irr (3.F3)

det e'r.rn i narr L o{ Llrei

Ltnre. SLrbst itLrt irrg

(:t.:i'l)

the

vol

tk l

An expl icit evalLlitt ion of te

general I y rrot possibl e and hc.rrce

used. Tlre stiffneEs fltatr.ix carl

J r"rtro, [Lr] n det J rJ. dz

i rr t egr'a l in equatiorr (3,87)

nLrrnerical integr.at ion i s tr:be n(]w wni tten asi

l5

be

tKl

wheFe tFl = (Blr'l_Ltl tEl .r".det.I and Llre in.L€tsr.at iotrlll the natLtFel co-cJndinate system o+ tlle el emerrt.of tFl depend orr Lr ,L:r and L1 . lJsing tlle nunlenicalthe Eti{fne:is natr.i}i ts rtrrttr,:n as

Ik.l=)d.-r,.F..r*i jlr

J ,,' (l!5dr ( 3 -::iA )

1g pet.{:of.otr,td

Jfre el enrerrts;

i n Legr.eat i ori ,

Stat ic Anal

whtre tF'l iri the nraLrix evaluated at poinLs L-r. .L, ancl [...i,

oQs* af.e grven constants wltit:lt depend otr tlte valLle,: L, ,L.- a d

La. 1-hese constants ane cal I etJ weights. Ifre val rres o{ Ures,e

congtants for tniarrgnl an el ement are givern irr 'lable 3.5. 'llre

sampl inq p.1inLs l-1rL,r and L-5 "1nd the cr-rrresporrclirrg weigllLin.J

{actt:ns are chosren to obtairr slr{f icient accll}-acy irr re irrtergr.a-.

tion. Ltoubl e pr"ecision is Lrsr"rl r'.o that ttre r.e*;r-r I Ls nray be acclr-

Fate. Calcr-rl atiort of 6tnes$es and dilrr.ll aceflletlts { l lcrr,rs3 tfre Etrrrre'

procedLtre il I t-rstr-ated for tlre Lw{:}-diflrenE;ion;\l or.c}blHansi.

]'AELE 3 . 15 I^'E I GHT I N6 F ACl OTiS TOR 'I'II I AI{6UL/rII E.LE:I'IEI,IT S

ELEI'IENl' I NTEGITAl'I ONPT] I NTS

hIAI URALCO-ORIJ I NA'IES

WEI6HTSod ..r *

{t . r:i

cr.c) rl . t_i

(l .ct

(r.ct

o.5

() . ';;

t-t . 7 666'7

Q.r66h7

Ct .7666 /

Static Ftnal ysis 3.4t