M E -6 1 8 3 Ot/tc/oS l1 r{\6fks DtskibuJlOn frn6l Exam AsdEttwrls ,,n"1 da u r^thaf is - fl,rita E,evnay,fMefll"d I rn r.nt? uelhd A t1u$e\ical alal3sts elSinearing for solvhg a ytida varr'ety of Probl4ths. industrr'es. MaJsr atar*lon rh hgirreerirrg SChoo\s and Pesaarcta rir - So\id li€oh8nic5 - Aarcs?^caer$neesl,tg €vgg. - AulowroHt€ - bio- Mech€nics . - 4rvil 4rucf uYaEr$g' 'Ilerual enttg. ' - Ftuid {low '4nabEt3 - Eleciromagrafic s+udies Availablz sq"435 x (ovrrr'rarciallg - ANsys- sui.fabk{ot ganoralpurporc solufion. - A6Ao,uJ - t{onjirrodv ln alvsis - Ffl4UA/cor"UoL - Sr.u4aHe $ot mulll-physicsTrobhrns - loY elzctnca\analgsis l^sc MAR.K &r sohiCs. l--s DYilA- for impact ard damagea''AINs. x flislor5 "l FE/v\'. ioans., 8vP th con{irruunr mqhanlcs. Applted rn6lhern61 Phgsicisl: ?iscq-,^lisa{unctrons lor Engireers" skossos ri. Ln 1q43, an Mahl'ng of ctnh'nr,luwruechanr)cs frrc{rawta skurc{up, (ourant app\ed l,\al+renaheiavr torsional ilsa this for problarnswil4^ -ftr'angr,rlarr€VNns, &rsf inifialcd s'liFlnqssmafrin t,t1416, flt)tner avrd C.lough,At &r {russ , baar ad other 4rr.rctural e\euanB''usin1 elaslcr\ -: ln lq6o, khwram. f'lough: Soluhbn of of kiangutav elpty1.;".yk ayd plarv probl4h4iof ekstr'ci{y gavl'+ha ytar1.e'FEM,. t( Why Fe,vr7 €nhanred Be-nat'Js 4'tlic FF/\ rnclr'tdo thcraasd accuY6c9, d€3i3n ?a{a/'efors, design and bc.ffor insignt rnlo 6ritical ltsltr , {av{ar hardwava ptot"{gpas, a vixttaa( proFolSPing groduetr)vitY' znA loss €xpcnsiv€ dosign cycla , fncraased znJ fncraaSsdYevQhUe. g lvrveslrqa+ibn I Theoye+rcal 6xpevrnrental NuNerical t ,t r-_'-__--__--l v.tv I rrm I I-F[l I lBElrl o Finding Lod- baxir,E 6?aL\\ having a no. ol 6d noh- 11n ifsrtu o R6l€ of c ophbnl, vniftrm 61q'n- <lrflneg drd hole< of p.lalz avbilary shaPas orr'etrfahon. o De+orminafrbnd rI ofa po\lutanLsrh airnospheri6air tritl" Condifions. 4'a'bitaryg1qa. 4luH {low deferminafion fn a chanvrot Neqbe+ alt Ne,enon-uni{ovrnity arA rqducp)ha poblarn {6 arnfla ole. - Sowe cA38S, Apyoxivnations 41a accejfaHc. - lhosl\ho casas, -1h'rs sir"rpli{rtah ovr C uses €(r'ous "f, thSccuracv o\ r{Yong answQ(, , OdrhI : Retairr 6U \he ron- r-rni{orr'nr-fl'Bs , br,rt vafhor ?'rodtlce f-- corrrPu+ins(na(htnet' - approximak solu,h'ovl by x Fini-la Dil-lerancerterhd CtrD^! -kovrdssagoirrt-wr';capproxnrahon16'lltegoverningqn' - I is a diroci n€l&d of - Tho bd3 or hwv soluhbrr. 1fi4 ragion ramainsa spocific poin{s (catled node$ 1lr3\QbcJy' htt wa Sdei'l on +h€ doynainby € - rvleghnefWork . * fini{c F],ovwd /l€{4od I - SrovtJos a eql" wieo aPproxifia+lahto ihe govatn\ng Pl€cQ- solu}iovr'SgstrnnresPoYls€ an indirect rvretl,rl "f ' t'r advance Wi{1rin trrQ dew\ant t! assurueA 'rof crnoln - Tne. solufrbn donrain "f Ltta body does - I{ i: 2- sin4\0,b.dv , rathav rt 4 aplacivrg if v.tili ragions oy vlewer{s. ra 'rnabted ,/ approrimalod inaY'lg s^^l\ iwlarconnecfoisLtb- - elouuer*to elaruvnl scnn€cd^o"rorrly lirougl" nde Cenenls, 5 6 <pnrqrrr .r pVfe . - nol [, Tna boundary. 5iryle ehrtent ; ?rcparfy romar'ns con4enb. o \4ccurrcres Fini{a elevnan,t 6ppyoxrmafi'sn I'r Fp,(, ongina(Pds {apk(ed by a lgebric eql . o hhY FEM Over fDM: 1 v(fl'/,alLftan+ c-omPhxboundaYysh€rJ. "{ be evwQn' -(tn rrrvt amv KinA gelo*ek! ol elerhe\a{ ,aan buf th FDM o\4lg Y1@. 6Yr- accessibV) -r Non-unisr,nitf o{ Eyshrrr qaramaiors, (wifr rh lhe olement proprtr'esrarnain const albatt tt cYntgcslrou eleuenl Io edna*) , ean^'l bz <olved W FD^A' ==: - lzl t2lo) of r en'r. B Alvav*oges 1. -lVw 2. verll . nrefhod wovks evon whsl aU &e rnathds|ail goo'1 in rnal\agi/t3 shaPzs(arbilarg bo'.rrdarv con'Plieaicd \.\ s. Qood rn dea\ing v{ith nan-\rh)lQrmmaleriatfreldProPrh?5. a Drsdvarrllgas.f, rr,lt f" ba no* rir a <rnglc hdy, bu& reatizad is bodyfegiorr aw \. vathar an assamblagod' elemen\s csnnecloAonly at fhz vrade fhen $aF f*tr-|t'o"s 2' fo*ox' {.sPonso fs pra-cla{ermined g*lorpolai\ {uncio'rs). aLrass eleme'rl s. Field {ariablee av usual(J di<contrvtuot"ls s. boundarr'e *frlon tr. xurvrbad' .rariablasat eaeh 'rde ts ovnetr"nresVliEher ' frrrzases) those "t FDr^. (r$ Vari6bla 'hc(aaS€Jequalron 1l can not hardla il4ora'lhan f (has condrlr\trs sing'.rlarrtg. tcinfof si'lubrilg slpulJ haveJ BoundarJ ElcYrranflrtethd CBED - A nunrcrical analysri vuethd. * Boxr-rrdargt'fl+agYalequa{ion tllpfhd. .. Not-wrte\ \isd rh schools6s !'{alt6s i'dLlskr'€s. PopulavrnZlnlg rh lhr acouslrbs. "f Uso (veen3 Theorcm16 feduce the. dr'rnensionahly of the proDle fibld rlolurua grob ------> Surface prob S*tfa* grob ----> l-r?a ProP. 4he rnelfid usa n o connected al6'43 fl,p Wr"nal rndes, orrly r+edes(aeshesl ex{effel boundarlof Fhs region. rzu F Basic Sa'pn S'[ops: "f i.. Dr'scratzahono{ subdivtgionof ! --r2 3 at r + -a- g ---z a 2 3 + 5 6- tha donain/regiot, I vrldy. Ele'tnelt N 0 . ilodado. g, 4 P" 4 4 1, --?Fr'ed"ariabb xt x2 x3 x4 '5 xC irrlernal .r?des boundarl nodes af 'rdes . lalsral 4eneraldtladxi a\awouvYs hr e\eywant Reclangu 1;7 -L)-) \ w l.t & nrdad t ^t^-^r$, llQAAh'(]'Iat e<trtz' A 4-nodrd Mta nQA{61 eleY4eftI. 2. jplec{ron oF (ntargo\ahonFrJ|rcfrors(shaye funchovis) go[nomia\{unctrbrsof ltr'lcr . ovders' - usu6lt5 fn tsvms "t -1n6 llnchbns aYa uqd )h describz lho possible -!ie\d 'JariaHeo{ar a sidr-4<n 'l ?rte bolrdviorlvariahbn 3. >elermine the elevnenl-propartgqua+tbn. *he -'A malrfx€quahon G derived dhfch erPtass sal+lad 566p2 chorac{qnstic o\ lne deman{ vrsing fve [uncdbn. * 3 @foa<lres- opplicationo{ conf}rr diree* ,{yproc-h 199 Dirccl equatioD -R*z rvletrd) Griaftovu\ t(e\'hodlRayteigh -: riros'l wr&! ''tsed' ,{pproaehes ReSflL{a\ - Qalerkns l(e&"d' (-1 tKl iV\ 1r l4 L- *'" "...tu w.da| paramafurs' unknownrtdal tariable I co-etgttent flaktx CKnownJ lt'\atr'x 5M-----' Kr s+lS{Yless P, loJo\ fttea nta\rix t 6t the nodeg' di:ptaeenen Asserrrblingof +p asserrrblingtha matrice .9f demetls. (j-+ eaeh elenen'll - Asarnding oF the lrrdivid'raat elevnq4 - propetb mafrrtes .l.o o}ain 6 sing\e global matrix equahblr, \^{hth hlfll descrfuaItta bahovior oF fta er+ira lomain/region. lrT { u} = {rl 5. lmgosifion .f ,[d+rix€quah'o4. -.' QlobalStrffness ttte giventhglcal <ondr'hbrr< al thu hundery noAes. 6. Nrrvnerical soft,rt'on .f ftr global tr1s'tervr €qvaliD4. silabr_ &v -too ef ) C(ausselfmf'nahbn -:f - &rrfon-RaThsovruelhd. &r tha sgshm rron-hi€aY hriihar neshing ttorahve rnefhodt? used) - frr l. FE/A, "fvon-lal solu-tr'on ,tle.ihd" spciall5 pto&rable. catcqla'hbn4 of infertst. other dr'Fferqni para@ic13 fhan {reU 'tariabbs. 19/t2/o) T'I O@ ffi I k2K,3 ur wz F1 LJz r- F2 Elayy'eri !,. hhear{xing o _____o d,o_trT___ o Ut e) uz fi \2 {r= r,a ["^_ (pnry Non-\-vraar t2 F=ka . k, ( \.rr-ui) = \ur-Kirra 7 1l = -(ur +Kuz I In, I 1rtt1 I \ | 'lt?')-<,\ -t? |\ L-h nl \, |' [ -2r f nl l - ) \ |f I r '') l tl\ sr{fnJss I't ui.r,{""" v€ch( makix. NodalUnMot'trt Veclov tr €or elbmerrh 2 . lu- -n-l)u"1= I fi"I L<" ",)l ut) \+fJ + Sfi$rress rnalrlt< Rr ehmpti ! tkl , ,ftt l*' -Kr olI l-(r Lo n, 0 o al .l --1 l-" tv I Lo B KfKz -v2- ol I -wl -l * l{.r-:l I 6r the global S5slunr l. l-\ L" -K \vrKL 'R7 n1{\ \tH\ I^IJ"J=111 --- BorLndarJ ^t'r6;+bns tr, = \,lnh1o!\n 'I C-- n .1 O I I Lkl-=lovz-krl -tl -vl [-] =l'.]'* lnl". dernertt 2 -k2 u'l -t"l K2_ t In'" | -K" vz= -'t ' ? ( U:-? I il L us i g€dqc€d \oi = ( lDl ) ,_> €tobal 5'6q1;"rt Eqo it,\=f:I ? iui) tD J -h tl" - P, 1*-1 L1=L2=l Ur=Ur.O zA, )l E L I r l->? t-- ' - oI i L" ul Ur q l2 6 J u3 - 4eV7 E. ') ,ZA' E r---6-_-l -z t+-x u, \.,l., E t,L EA t1 = E',94 *A= !)i? (u,-rrz; Ll eEA (u,_J2) o - zEL lu.w) .A -'L -EA L'' ,, =!' fu,-fu, - n#u,+f .u =l,l -tf rur ) r Lr = T 1.,,11;itJ"tt 2EAft 'frr e{errenl z FA T L_ _CA L- 'J3 = EA., \t, + T.u?r rr t.2 rz + L lltll =1$1 ;i ;lli:l ={?!} 71.^ Lo v- -r t lLtl:l -z r.tfz ol(oJ 1r=l (nr I Tl',2+t -, llwl=1pl L€ -r 'l I rs t" J I '3ut f J !- az =- ?L @ 6, Ee'= t.(i4i9)/ \r ' Fe 3eh 6.=.E€L' " F. (Ll?dr) L g,T T- /'/f ( ll---Tn \ffi le- L' ------:l o@ d2 0r -r tta& ! . .qJ L '(41-ezl =1 1- gJ -t- T (-Q+o! --Tz .& L T T= aO r: -t ^r >r=9lna 02 F- t-----1 ,2 '1L TI dl 8z FnergJ rGngnvalrbnq\ .ldti -. r^. .-i ^r'- Law ot &= dX _kA Ndel: KA L Nd€ 2: KA -f ' \ t'r+'2./ ? O1 0z f, :lfll "ls'i. _x4 I lr- l2-,/ = Sr? 2tA2/09 vL u3 @x: FL Fe tl,. F, Fihd tnrtt = l lrJ aerl'ar* q , etobat nodes"-,1 Kr+ |(.-+ 19 -f \= "l kl It'l = "l ul -kl :l o [r"] = K2 o - Yt- -KL VL "J ;l "l o lu'J E1\. ti (, o K? -3 'K3 k9 o ;l O fxul = o K4 -Kll il ----> 4X4 rnaf rix -h -' f Ke'l 111+K2_+Kj - R 2 H I,tJ .: tt'z+\+\ - Y9-K3 - I,q Er.o,nlb'. r,l ' Pq t 2 In t' ' ll Eletnart.4 Qldalndes=r IOJ 't 1 t" s ?a' L, ci@ rj .l 5 V = Elash'cnrodu(qr /. t T,, : ts6dy torc€s/Volul44€ Tr. Ty.'Tq - ?oir$ {cvces rr [r'] I Tr -v -Ai -1| xt' d i - diavrre{er' ?t , ?z i"' -'I I f k'] s',rrfac forca /'larqlx ttr1"il1 (3O oo o L't o lI to o 15 o oo o 6o a cO 6 lr,l = m o t. -Artz E o o -ry,|.2 0 o Ur-t o o o _f -o o [r'I = o o o o o o AA t4 -43 -/,ltt ht/vt t- o o o -70 ,X) U7- o Tr,4 I LNI- o o 4! o i-4 L4 = Kr+ K' + Kj+ l,{.r - A'llu u5 Lt L9 lrql Vt o o 0 kq tlrlr , I! -T1- 2 lF l = (L*'"8)'h| g t(+tl. ('t!E i!L') + '2\-< / r s. AaLr .r.!- \ ) 2 1-J \ I je+ a@+I SaqLi. tql-q1| \ ( sArrk . r'.La \2L z rl nl 2-) | o 1l t @t [' 6. [' 6ti \: 6/ Ll + slefllenr I L4 -rf - 3'nptrtfrahbn lo Wluce prevrov<,poHarir. ShaPc Funclon - lnforpolafing f approximahns{uncfion of - They a.r4 useA to ex precs thc variadon / bchauio'r the eAw* 'rvr tor''tts "t lhz {bld ',iatiable witxn inaepnawttvoriables (PoItrorn(aq - Polgnonial funch'ons ava wr&lJ useA' iocautz s\mple and be apptrzd lt I*tpm. eosi\ can kechn\que rh{egration Spoc)al 4-D probkrts RG) = a" -t a'x h'nzar shaPa {'incfttrr hQq - ?6+ dtx I BzyL 2-D &oblens brt(y)' ?.+ arx+ azY h U,9) . '4"1 atx* Gz\ -l 69t-! I a4*L+ 2rgL fl l PYobl2m 3-D Prlx,u,z) = €o * 6rx+ azlJ+ a3z b"V,!,2) # = €.161x{?2Y *asz+ aq*qaryaaa4zL + arxJ+agV?ra9zx Urear shapa{un<*rons in l-D gtobkrtr U> a"+ar x Ltt= Zlo+ arxr I Vt " A.'+ 6tx, . U,- tlr .al \AL Xt '<2_ 4-rLf X1-x, ?o \lt ^- --1r lr{lX2-U2Xl = )e- xt U' UrXz- \'{zxr Xz- Xt - rX"-X l- L -lXL- ll 'r X2-Xpl /v\ x /X-Xr . r' _ .\' _ lL-ltr \., ) ( 4 r- --+ infsrms o{ Yrodal.ld\iabl€ = Ltt()ur + LzG) tlz LrE) ard LaVl la tx) LzB) AYQ /-2-Y- X- xr Yz- *t NEAT shapc-{unciions tr 6enera\ Prqcttrbs Lrt(D= I 6l rro& t 2. LzEr)= o sf noda aL t. Lr(xz). a l.{x,l- I t-rlx)+t-z(x)=t U= LrUr+ Lzaz .X = LrXl -r LzX2 dl-r(t . dr-rtx) -dl JK ..!.\x X2, U7 Xr,ut i=hUr + LzVt $t 'l rzl = 1 1uI tL' I 1''l:.l / tBu v6(. f- tlodalv ar . lil I u1 I L *,ape {unc}r'tn mafrix 04ot/to It \L U : L\(x) tlr t L{*) Vz 2- X. *l 4-- LrG) l.inear shapn{unc{rbn. X2-x t h(x) = x-xl (1-Xt e al coordi42iP Nalu.r l - A n ovmali zed dene* inalp <-ooxd syslarvr is u,rsed widelY ftr .l,.ri vina l s+i{Inesse{. - &(in lhe. coordinatP s5stem is 6f 16a micHle "P tire of LVwt+ @lth, - 'lhis coordinde - fiiis -l 4shtn vaYier Erowr pr{icular coordinato applying an elfictenf !ahi& s <eguird to 1l ) -t < at J X= Xr X= X:_ IJ tL < +a 1.-1 .w I very sui*abk for simple Melkcl oS infog<ahbn' desive lhe z/s/n e4s' 6Yd x-rr " /\ - ( e - x r)lt - L ,r fa sJ<}an s ro a-' - - -,rA- 3=" slat:&rd l-D elenrent. tir'neat r--- t6\ , J,d L________J_,y__:_. .}.E, i 5ol I=o Xt ' )=' X2 x"^xr- )e = r-T N,t3t z Xr-- X h (rl Xz- Xr V6) N,(J) = X- Xr xt- xl '' I 2 U - Lr(x) Vr -r Lr (x)ur = N,(T)u, + N'(T;ti, . t'r,xJ li',] = [r.t]| ul J =n(i1^)_, X-*r = T Cr-r; x : xr.t T 0.--t * =tJ)*, n f$;*. X=NrG)*'tN,$;xz = trr, N^,]{, x, I xt I = Url {r}. q =t^rl{uJ x = trl{xl -Ve ftfmulaio'n in vrhich bolh the coordinah patar4s+qy and thz drsplacerrent S'€ld v{i{-hin ftte dqeft+ cb\} tu ex$es€d rh ]arns o! Ihp Sarne <haY {.,rrrcfionr_ r! Kv10\{nas Tsoyara uei<i < S<rvruIa+rbn tYr FEM t'-- j= -l 4-D J -€- | ''' rluad^rahkelewrent U: 8"+6$ t Azd t3 ---1 F-1" ' = Xz-Xt tJ = Lr(() Ust LzE) \,t1 a 136) % Lp@Lzg,L3(x) aR quedrahl *aPz {unc}r'o' th x. \'l Ft. -_-_--_-'6 .7 j=o 3=-r ,w' l t,l-, z, r=l nt X2 I LrG) 'qtj ) loG) '..t-(! 4ut nls(.3) X') ^ / X-xr r J \X2-Xrl r,rr(1) - i (,-1) N'L3) il, (t+\1 $(T) (r+1) Lr-j) a,5+drJz l= '"n Tkree qtladtahc <haf frl^nclron 1 6" /\-4 :7 N t a13-'na"$ + Ortr) + 6zLtl + O t0. 6".0 'A^. - ol 2L= 11 a\ -- \- Qz- -2^ - V^ 6t = -Y" 'l' t:) = '']. _l *.6;= j/{,'r; u(;t=e3; t-}, 3z J=o J. r glrr;r io$} v 'r{ L Ug 32_ o l'-r \"t re _i L-I - Sl-avrlard 1-o quadraht 4uttenl tL U= Nrtlr + N z_V:_ = f,.r, r.rl {| I, 1 u7) N,/z' -;,\r/ -^ \-E -( I \ N' Gl -lT ) ment fi€ld dt"Ela<e I - ( r.r,r\ A /r..r = ai *l gu' 1 ,,]/ -- 1-dr..rr dr.tz-l( ut ) L-a^ r"l ).1,I -- TBtlytl d\'1 (}X '- fdl.t| --Ti dt rbl =Ld&= dr.r2 dr I --E -ro\ d)( Iu -L- 2 xL- xl Ibl =.B 9-- I ).e dx l ^ L' ' JI il ftr url it - J I, ,€t t-' L Linear L.'o derrnel; 6"- E[B] \Ltt tar Jur 1. L"r \UzI E r ' -1-.L {*y rl l J u=ttt {u} r = tBl{u} lUz\ --> 6= "I \'1 ! , f-r dx.= i;4 . *z--At ) -l clla I \21 | Xz- Xr (I, .Yrramdrlplacenenirnakix) lq2\ ;r5u'?. 'l gDu{<eW t{ = Nrr-lr+N2U2t = [N, N2 Nt \r(\)= QJ? A-9 \u. I , qu€drah( <hEB S..tnchorr N3t{_i iqr 1 luzl t ,rfu t " "i- -12 'c _du - --{x = f ar, 4N4 d>- LZ; d9l fii1 (ur = lsl 1u"\I lu3 ) IF.-'l L-J f ,, 1 dNl I L Ti A - .?, *I z lt^ dNz ql dT tl't+I) .r 4e lrr-l l\ \ z, @$k Qi1) d'lr -+t dxj ^"t '/an \-zs) ] C"H) dir -r; tr:Jlu] o\- f" Lt1-,, G*h) crl)l 1'ri o\/ot/ ro n FEM Wj rroa+rlr B4r1$t gropcr*g by (a) girBet mefhd ol melhd Variottonat Lc) Residual l'!t@{had L-D WareWALrt h tt (i-\ / p:T:" trll l,L- xl lZ r- ' 7't i<-_-- t-->l rk"'l=1rf' Le L-l I [r1{u}={ ri II rnei?rd Variatrbnal Variatiorral Aethd gslenlia\ eneql t. lv\ivrirrrr,trtr z. ?agleigh-Rilz rdnod. aVptoae\ -> 5o1idrnechanic<' t: l,lrhlnu,.r,r pui€nlal energy $e.Nhd -l-ctal ? 2ulen{ial €'4er{} = ri,g1 in&rnalerrcrgy 6lorrd bj lhe bdy due to dfirrnahon = TI Slrain enzrg.l U + Tr Workdona 0n tte bad! by he exhrnal toadr. WorK Pui"rfihl (wp) u- v'{P 6rr t =0 5 tu- blpJ Vafr'atronalfaft YdelLr iP ir rpri'q = Fnerfl <1orad T-T__ I llr =| *@rcain S7riq)xdisT\acenervl + Av 4K =\_ ^ K,\.txu I -7-^-7-T =f, ru2 \{P= Pu Tt = !w" -?u -:) * - f,.zuu-r ) (ut-P ,-.------------1 = + ff=tcr,ti + L-- ku=P U=+[<redv ,^r vtP= t'- Jl; udv = | ur -q"av ,tl tt= \ra =trr lE Y Wp-= |^ r ''L u ras J T- {rac{ion "5a.ft.tn J = lcev. wp^= Lqvt *,= + [{ultrar* I urp, 'fir1'Rdv ' | lr t is -r.T^ 'tl | ^- )' t. .tel , z- ,' =+' 2 - .t{ct' { rTdvl{uJ f"au_I juJ'n,,_ Jl ..vi F Tr - f. J-,--lf. =JT . -J u'.f"eocx ? Ju rdxl<' € Aedt Z , t.,D, e2e ? n, = ?ljo'. n 'l u' d\ - + ai P; nnax-?f,rt.l"e.a^-l-J 12 ,, wQ 'i I f - ) ste Aadx Xp 4,4t - NodaldrrPlace*rent Nr4r+ Nz9z ShaF fr;ncHort Nr Na- tNJ{q} I .l - rbl\q\ {(1 -- E rBt{q} 3v . Tu1= E-u.a: d^ dT LdY JTI| --r Lt )- ^::- o r, )e JY, rl h) l>it l-l 4eL ll ) N t =- * :-, .' N z ' - Tl , j ' I tL tr , ^J, 1r (rrai{aU ( trj {q}J +" dx ue= r. tl t' )<l =11 :>aT'#*,.ar=fra Xr- Aaax Cv7 r \al rc {qi' YI Ve .d*-$J\ -z {ct'(!n AeL6rrr6t 4 a\) Iq\ -l ur i r; {11 \^f(ry'f H";' -r \ 1.*_--/ -z _l :J o V\2 ylPr dI) 1ct 1,1'Q +[llillir){ql -ll) (T' \qr' I-l r'i -^'T i rl ^ /X-7t r ''5;;,J-l -z 101' rnt q1. f ----= 40,sing\efur e\er'enl _I J J(t^l{qfr"o. + "J 1'qi'{"Ae dx lT -l ,i 1 \ / 1f T {q I Aq ls fo ---;- tNl' d] I I ( tlr1r\lzi-r nele { / {o-t- \-- _l I neln ( ) $" '--'7I I .l! o\ \t -t ) rllIE Wn-- 1"1) ( {,}' WorK donc )la+!"ih tlo A wp"= on the bdg lon.{ W kn bodY 10rc€. I t't'T dx, tt I = [rqf Ln{]tT J 4! < r ! I ,T = '{qt., .Jolrdj le -l J{z d3 \) . {qlt ! t7- t I I rKtlqt 1cI ,ra T''e=i rl ---> dis+ribv+?d qltally at {'oro''rdes' I c ^or / ft Aele 1YI \ -=- {t} \- {.neln,11n +h - - {cf aTe = j{e1'rr:{q} i--'-T \ {r,tule --L =I lo\'c"t J - .lr1 *.- +\Lre+ b" I bar eluna'rt (^a LKl lqf = . L - \qi lr| Spring elanant Ir = +ulklu a, )- \q{ ITt? t r l l \21'r,/ fAts ft -:ll^;,\ "tll {{} 1,1 Ndanunbar. _' o91o tno B, t14a1 -nducfion .quation L3-il -(4a--4*+F)*e= (( -?r .d1\T**-xj,az,/,i-..'in\ /wlutx mfatnal heal greralionf avsorbfto" qr' - K+ =-rE d n 4v'-k* qz . _K +' A /,,Trl 6 r^7;/ (q^l + 3Z +Q = (c /?t +$ t^F,)*& C-tl Ir o o-l !)%" I * "i\>2" I J" -1. = 10,f o (J latn'\ \sotro?ic^^lprial Vr) o'I ! q., =-1 l-o* o Ky { f lr") 1", t-tl (,'1 lo' ( j l{z L' l- K,, I I o, L *r' .1J arAx\ I 11u{qria [xlhot1e7\c u(r, (_o \\ *rlf o'rt.J Yrz <u lTAr ( I \Itry . \( *'\\ Kzz Kt1 Kte'Kz\ xzt) ,n, .t41: Y-ti' -., ,Anigo\xofl'e 1- K1y. Vzg l 1 Ytoat Cnndugtiorlqqa \erP< al tr l-o ?ro{)+&,=o - i( ox/ a-1-=-. - g--, ,-- * | ---------+) | X.b CasaT: T rs 9iven, (asafi . Ta, -Tt tred {lur ls qiven 1^'9, cass.11: (9c; Convecdsnhurdar5 eordi{ion. d dT\ 6xJ a )= (- 2? ..i- !l dx ^ l. ,L righi tov tov letF end -ILHA ( a ll r- Id-E L'u , ) evtl ,.lf A tlo = HAltq- Hi = t tr-t*) #l -,. \g 6rall kins nrethd la{(ki Dt&s. - An apptoximaio ^u11*t6t'lral nre'fhdSr <olving - a resrdual nethol ' ltr.retgnka resfdual vnetlid) fi 6nides a ' d'Y ---_-i' t d*- O.o's--) ganoral P€\ ' jI + {(^l 'l = F or*t I \'/ s]\ SFapU., la+t-v1 n hvtat+ or g'(a) =L, ? ,t L h.,etb) t qz c'(b, =rL I { (egdue, € G<)= LHs ol DFa - RHs of DEd I KesidLla\e\ror iuncfton = +**hk) Adx+qt*)Y-F g,rrbslitu&a €(x) .o \r.rhaneract e(s /" ' .{ov approrirnato Eol*Hon 1bpz: €Jtuiliry an soludovri5 n .,rxorir\att -rl so\qtto"r vfu) =Q AQ) + cz{x(t)- g{tlx)''' - = 2 q" #it*) Q - arb\{arg cons{an}g OiG) - baea {unc{ions' - vGi shouH eaHe{.J E.c a 4iCx) 'dsz dnear\y ;rde2ondonl fqnctioas' di{"cHJ '.rith 9 . boundarj. g6Yd+onr gorqning €q4 and ^rN16y1 andvanlage . eq[ Salarkrn's J z (4 wior)dx = o Nr'(x) - r^lerlhtadttrnctr'on, ivrthts <ace, t_ '^iiU, . $ (x; 'la- ri X2- ?(I eawat €{uatron, e"u *C-+)n e(^)--&cKs) o q vtx) = c,Q,6) + QVzlb+ GQ"tlt--= cr qcx) + 94-tx) z noJer' as ltp'z V ontT = Nr(I)Tr+ Nzk)fz =[rtx) .nr,t-rl l],] = rNllrl YL (ahrkn! pql J X1 - 6(xrutiGl 'Jr = o 4qnchb N, r{, -unaarldcrptiahng (shapdrrcrbnsJ le. n ( ) NiG) leG) J dx -o *1 XL + l{* =) cX9+o'} NiGrdr'o o t\ Ni(x)dx.= [*l I l.r ^ 4, )* -, J*, 1 F't'' a* xL Y, & Jx * dx riLx, ,<]Lt ["u,,..1 f [ '''*'' r 6K' + dr J o ) XL {"t^" r#' ( -t, "-l ro + rui(x) \ ft I 'xt r L *, ( f > d. r \\,\ I i r.a _ 0\, 9(x('T-:r) d t ,* J)tX \ x r J )6-dr. +lNi 8) xL Xr- l,r,rn*S.1-- f $! r-?r lv .rr+ [ r.riGl -' dx o J L''*'''--&-Jx,' { 1 J & *t Xr X2_ f" J = klu,o,*]*'* ? * [*$ *.0* u,. JO* (x{.) adx I /l *1 t1 -{r {\ K *)( J ,, LNitx) = r/l K Llr4 j +" Grl\ t'=| Lx"- Ny("'l $^ /- =-k*lx, - u,l;*^*ol - [-.r4,*; (-") ' k +0, du )t -& = T-n \rNtlriJ .., = ldr!-l{T} I I' LdxJ ; dNr dlrl t x 1 = Lzx dxJl tLt. /y -7 [ a'.ri J "R' [$l {r1.a" Xr Xl I a*, lr] -U I-4r t-rqla* dj( l lar' J Yg tt d* tJ ._f" t+ {j}l ?'} dxr :; lot l1 f(dur I : \?Tif+ rl I r Jla*. 1xr I Ex- /,re, dx , sr't flax] o* lr'l Rowr eqn A) ii++ j--^ xL ax l{,ar,t"r I r,.r'r ,.r1 (-**'l\-dr {.1,-si 1r- ]r" \ =o{ a,,, ="-l ! i" *t'r I lJl'o*t\ L*+, * +Jo. *\ u,(1)-_ -2-r-t / x- xl L nr,(1)= T ) ' u\lfx,fa 2- & xL-xt 2 \2, ff4lr \1" Yla,q ,LT5 :J I x rt d.s dMdx Ill I'o 4Nr dT A"9 dil dJ Lc1- Vu"l Jx I Ar" d3 dxl (X? at1,;l' l5 t d^: I b dJ t(l IT") -vr.-1t?" dJ ( l'"' 44- dx ' I r. t 4l< 1i dX J l-Yu'4 -l ), l-ro- &Nz .-1- -1n"1 l+ js.4 I dt dt' d t dT dx Vu- ) _,J I t'1 ITz\. [-" =K l1; l l-| *r-u1* ( (lte/^ '' l ) 7 lq! l *txt J I I J & d3 o r.1(1;. -l ]=r I r sn,tt)B.d 5 J - Elotutrt\ *,n, \ a.t4- ) slffnnrs ?rop€r-|3€qa v..li$ intcrnal apptrd heal +or {nr !.0 4itgle ebnul genardion' gloFa\ shffrry "41 "noltr ringle& au K2 LL K.Y" -+:Ll 44 -K2 l2 o 0rc - rK z - K z . k-iz -r\1 - ! -- o - o A *1' l\ I LL -k, Tt K3.ulq.-tr -., o L{l L3 Lq I Xa - kt (rlobal stffness me*rix €q4 \r kt 2- Kt3 (zz Kr? Klq Kzl . -- Rtn n Kzyl l 7 K9t K32 K?) Kjq "t-. (Al I dr'/ { I F.nr Khi krt vn2' hr Vzt K3t KyL knq Vn$ q3 Y.:29 Y13 --- ktq - - - Ktn h4 - - ' k34 ---K?n Tr knn ' Kzt\ I o+ &la T? =K o+ A\0 : Knr \nL Kn3 lGq - -' Krq^ ql.o -KdJ6D.F d{ lTr- : -t fd J <t*"t '/ 11 qF Ktr + 4z 1+tt ?w- Ylt Y4^ l<2-9 - - v33 - i Knr K4, t- hs--.-- Vnz - ki: + O'tal. atx - kzn d.l.€. K3^ ,ek HTa+0Q, ta/o(ro U3 t{r --t--------V-- tr d2 -----> L x. I x3 X1 r.-t I-- J 1,1. inea<'&metl t EA I' -rTtvrl 5Fr'l -z]rr rl1u.J=1r"I l ils9r N1Q, + N2t, 1 = tfliqf =-u-3u\) r,l1 z t{3 J " '(**)-, = | ] (r+i; dx=4lt -t' = Q'? t-''i) | 4l tBl1ql - J<1 - F tBl{q} \ tt P .)t ,-v - - d[-] J-dNr qa ' 4r dx \ lex, Lf| - \-2Y ------: t.r-2T 122 ^'!-.LT 'I { {<r'12}Adx J t j l-J -2 I TF t.f _J t\\r? ''r u/' 2 l\u\ I,e dx - f {uJrrdx- 2uifr, J' T. (r tal{q})'frsr.{q}JA aQ/"d1 \-L-ll|'1,/\L-JE''/). 'l .t- .-!c 2-71 q{(ry \lut'tst.rtI{qJ. ''I L'IJ' J -l . a+ \e{ tnl{io? Eovrre# stif?ness. f Let = E* .J tfsl're) ai -t I r ", Fi{€ ) \-r-ri 2 --\s/tll'* 2_ I <' \( I FzT '+' . -2! ldr . I lt+z] tr lLa _l\ . / l z l,rr\ ft l r -tre ^€ jr --:- JU t 7 -8 -l ' -el^ tt I 3 l-e. -"3 Fz=og,t\ Jl"f* ^t ir d l _ F"Al{ \'{ ?l I {ul' r0fiu_ ({ ,)l Itt" NrdI I t tt'd5 | | I ,fs{3 rJ J -,-' ce' f,; .---- SodJ &t F"AIs = \^r1 tk 1 to = ?Y. I lg"tl \q1sJ Vt I C "? t] --> rnaffr\ )Xb _______-___+\ F__ L Sdrctfle structure cqnsidering2 {urad Fts. A1 Vz \trs t1q th -T '-l i- . . r o .11 qNB | '2 i LNB I e) "z 2 + 5 bar LND L local r'toJe 'nrrnrber i't3 ' tr l*')=Ed I, lz r r.a-t Lrr= -sl -tllz '8 d- -8 I /l e 2- teI l-= r.-r-slt6 l1+ l, -8 lt EA 7 = lVr-l L iJ z !-J I _s\2 "lL) s I 141 o EA (J\ -8 1{ L -$' o o -s o o L It 5Lr *3L -c l6 Ut I I Uh tu5 . :9 {+l}= rnr.(7' JI; )23 1n +/,I | {n'} t7. = FAlt \vol 1vr) I 2 z q lll6l \/ 3 q tt Err 4l ol (uz) , rb -8 -8 1+ 1"1= - l-- r t*l -*l(u, l=1b I t ll f ^1 I lD L re-Jlut'l -8 0 (or q\obel q^1 t' u - l { q 5 r'l' , Y/"o | - t 1:ll -' @ (t ,^'nN @ ..1 tl h- d '7 I \Az,ru,r\.?. ) I=-t 1.o 1 -+4 I E t;l=lil -ll ( Er{ [r tr L' 'J1 Al = u1Ca9E r- qr"q r.l/ - U24osts+,zii'€ "!= eOSv y1. gin€ Xz-Xr. = -- G- I 4Q v.-q, pIJ''*rr'-r)" ------1- ; lo (urr? fi r o 1u;l= l_" o r t u')1 ['i \u/-J LD .\\:l ";l ul Y'4 {u't = tRr{q}. o 4z :'l .4 ,\ )' t llr t nliql D ,r, = i 1,litr"dr{i} = |' (re:{oi}.trlt I*l { 1 = l.z { q1T | .,'rr T#l ltt \LFr IR] 1q1 = .1 ioit tkt{qt. -'l ' [' f rl Lo L-l L I l.^ I = tr11 -T: lll o d I t L€ - lrh €^l -; tr- tn L lr^ n I| -[n ,,1- L L-* -* .L rrl ')- zx\ (\n ) trrt ,a M, -lnt -lm _* "{ -r'f - Itrl tL- (n lfil n\ rL 1'1- !m ttt\ f|l ,, t'x1 -tJ\ ')D1 -lvtt 1rl Fyl 62' Fxt R. o lSlor lto ( ,i 1 1',I - ftlr 1 ra o "i o ,l ,J 2qi lrl =[: lvz I =ffltu'L [€ = le1rgq6R] =#t 4 [,,'lJ [+Kq L u("' )-:!--, F,< , (xr- Yz) " .,-_+ Fx, Cx,9r) L= xz-xt le m= ve" f, L La = I tx,- {,t t- 6= E€ {.L-*r.f - t? - I ( t!'1 v[-r'tt E = r-l l _le -,tt ,tl )_ f' 'l \ui I. 'l "l r[ t Ti l-r lz E i .1. t? [-r U IRt{u1 Iur ') " "1J",\ nl- \u, { 'l L,,J r -' Iu' "J1"' ) l''{'l teL =:- E -m l-L ,,r] It ull . J I U2. Z-D (xr Y,?D (" v"zt ul r {ur + mut FnNl Vl = [W+ qu;-t nWl L. @s(xlx).n, cc6[/,y) Hz-Vr Le Xl- K n. cosfx,z; = .Za_Zt --I;- Le = f cx,-r,r' + (s-ct- + (2,+l'f J 1l'1' t: rFl4O oot = [R_]{u} o ol u (ur l )'rr I , 'l 11;) lu, I l.t{J g (ul = [p]rrr;l Ltn LN 1l- -(m )J1 tlg L4 LA m mn tnn t2: -Lwr -ln - {lvl 4 _m_ _nn n - ln -/rn -nL -l'4 -1n IL tm l4 -n( -w\n Ul tt,- t|n -nL 'l.n fin nt' 4- -mn lxl \rl I kel LR'] rv, \'lr uL Vr ll ryl F91 Fzt dt$ness rqatrix €q\4ahtn far Eleraaent in s-D +ace. a *r vtss elevrrg* If rlabrial gropq(ly. F Area *4 I ,t. {DCos46" ra= ,$h95'= tm -Lx' fi2- 'Ln LL qh -ItA -rnl -I$ J t,. F, .,L r,- Lkl 21 -1, -y2_ [+ -,. l-? v l1 l- l-y" + l. -Yz UY" .-lrr^I -q,l rrh Lltrrr rh" -tzl / -bt , _l,z_ -L ,z Y2- t. ,2 v- -4 ,l/ ,t) - v2- I E/l \)" I J I . l:0oslg5=-h 14=.t4!|35 = + I J. €A -f)- :- t22, ,2, I T,- Trrl _t2- Il+ fkcl = I .i >,'I -v. ) I-L " i-n' -/z- L -\ o lo I o 4 lo t7, o ! 1. -,L 1t .! z I ' 2l ) L -/Lt . IJ ,l--" axi9 Cteukat \t &oss^Secfion o\ beavYt \r- --T- ,.4 E ,Q- dvM -i-" : -=:EJ U . tefbelion garane{er. r: Fxgression{or bar ot inlarn al 6l-orad €nergY due fo bending of c lengthdx. rf du = b lc-edeax = J- [s"4pdx 'tF = * l) ( r"s'r dnax J E \ I"/ tfnL =+\L.-_vianax =J' r r.-, dtu ? tdA dx ' { -ttz J l" ?x"J , ){vlL e ^ = + ELT') Jc'rr& A . 'L t1- = !' t - E1 ( o,r, )dx \ dXLt Total po{ential€nergJ +r fia fuam F. disl-ribqfed lod. n = U_v,Jp - L ^' : +- 1., (*-)"d*- \ bua"J J"'td*"'' Oo lood al x: x,n, Fn-^Concentrated lt< = Monerd --2 t kr,{z- _l )- "l fte cougleaf 1=ao Fr,l Urku -f:v. I P"ur - IMI'* m tr tl ,. I .d o------:-;-- t / 2=1)_ ^ \q [=--fFF-1 ^ ) ___*rv2_ I \ ur p, '' c.r -;. 4xq v2ft- 'xz M2 x, '"1 'e, )-/l y- de!(ecltovr af nde I 6 = jY rotatrbn 4+ nde dF r. | | ,2ffI,. - - Nochl {c*ce Nodal nrotnent. Hertni+ lfilaryo\a+i$t/ Shape {unciirn H;(f] = A+h'T +cti'+di]' llr '. 6r-tbr3 + Q 1o + 4i1 ft2 = 2' tb =As+b?] t{q +b"\ = 6q+bq\ -L +o\ ' -3 rozj , +st. -L + arJ, +qr1 a- ,J +q4! I -L A t"l g1oPe.4 'd5 T! II s"l - __J* 2 .,{\ hpetttes d' Hermi{ $op {unctior coordina{Ps3s}am' f, Four <frpe Suncftws &( fJ ?raTuraI baavne\ewn+ r+r$)=f ('-r)"tzt9 Hs(T)=f, Lr+1)(+9 HrG)= | l*t)(j*9 tt+(I) -; (r+91'z(1-! "9= cuht vaxiatrbn oF the Pata'aeloc (tlermit {uncfionsJ . .lro- rr,", r rrr(${)"i HEuz+ H,r(* )". ,= 6rr{r} tz/ot lto | r /e'7 2- \Yez Vva q tc rlL +" \ \,^ 1lr9r \'l' r.z Mr- F- l.-l 's noda\ deRbchonz' gign convenhbn beam derv€r\+ . tigl l-D - 6hay 'funchbns. | (r-])'(r* r) Hs= I trttf (e-:) u ' | l- - lL { tr 11r1i+r; Hq + Crtf 1"1-9 L , -Ue= (, ,.v .2 d" ; tr'JQs,.) + 1r{ ik"l{c} o vG) = ttrv,* *{j1nn), * r J= ^/'_ x-X x:--xl ,2( -- o5 ,Jv I \d3/l dv -\ - f vL d^ HaVr t I -l ar d,j =d' ?x' 11 /dv. Hq["'g)z 1/- = 9r g' rdv\ - \ot-/2 -" -'O't \r(i) =+\ul+FL .lel^e, + H3YA+ HLI f =lHr Ie/"A" L' '-1 Hz azn,, ) Lu, ' lrJ -.T 8al i l,t - l Lo"J = fHl{q} - Av _l -dx l dt tHt1{l = d n a d"{/ -- J- It J X ! \2 att Lq _{!* {r1 l, ivrt '-aZ, dj --rt tq\ t , 2 'qdH 1cl. ld\ l€ ),--4 , ,t- -L { -]; , dH ,r. tI - -u. . : - .dH .'...- - (^r\_-2 .41t JI" \l"" 4l- f,i-J - J lt, tv,r, L- 2' 4) t-t: i ^ -' r fzE I I ( ,, \ jl- J Q r4 -L \t 9,,11) nj ,^drduHrT ltll ll L'r \ ) r- zt J t,r.(*,"it*rtt+ rr' -- -7,j ,ry ,rt la./ .-l+9! z< dY f| ----a -:'(- t'\/ t- ^q,tt-\tl f*I rql -1 ' T ,^T + Er lqt 33 )BE:l | -; --,a ' *t \ La -lrJt -;-, =' - ru JL \aL[3T l-; L- ,141 '.ail z-J 4 Iq tL . -\+g"t - GT 7 lu12, ' r+}t lJ?'f lqf ";) ) I l t. : 'l3sl^ I 'r?H''-\'t?' =+{qrr?r,, Lg [l e-'ry i 'nr. \ Lp-ll I ( ^, 3 lt ,i L t +ry) 1r \- *xt -- + 11.{f lK€/ [r"-] = {q} f' ( \ JI gEL Jr LQ ldt r ]"-'t lx t d+) . - sEr t [-rr"rr.'1 Jrix4 tt.l E! .'_^. 1p2 - \ | I\',,/ f'' -t2 61.< +6 ""-* l'o ^ ate -', | L* C[€ -l ": zte" \ -ak I "t u*"J 1L - 6lz !"l,eL L a' rr4 \r] \{ lql a-r U,, J- o ,t Lrnlr l -/' c,f ul ''-;- q} . , 1 ' = I' 1a I 6IA 1 t. 6\a '{& I nr] (( u, *' ) -1*' \ \ u' \- - br? :\ L (,fu'l \r,' \ lts'J ) L*, ,{ lo &sl(ihrtc tod r! gresen E wr,' .L r1v)rdx J ot *,"or = r (rutlol)' J -t l_ =\cf (+ f [n]'a'l) \-". I L€Hz, fr = lqtr ( Ple Lnr' z * I r'' -t , fl3' .T SHrJ \ dT - {r?' , | I't(',-rl'tr?) I Pl4 .L 'o* i' O*i't'-')'tr*'lar ! \ , \'u*rr"(r-T)'dl I \ Prl DtL -t\ '('+ill(')-r)dl t Bi Jqx, - heo ^l = loitl itl = FI {d dv dx" lr Ie" = t 11 Lt"== -r rI td- 7d"H\ Vt'l lx4{qi4xl 'Yr-'Y I d'! l') l7 L' l fr' I qt - - 72t r+3.! t"t -, l tal l nil \tl- \ g," J - dM -'1--- dX tr cr *"'T- 0 T- vq'a ) d I d? le- L- I ut, ,t\* 'r =bE 3l t = -1, , ?tn1 -{) L l€e le 1+e1 -;-'-* f9 q le r+ejl/ (', I 1u'I lu'l L('.j _11 Er,/: Na{at de{lecHon Sheat ftrce - 6tih ends lixed |earn Reactton 116rxPn'l= ? +x+ - t-ou\ s\i\t"ess mafxix 6Kb- Q\oh\ rIl I u"l s\i{hess vna}<i< \- tr- Te \u' I-'o -5L a^\ €T I'o -l-, -tbler- L otde[befion Sbpz JdeQkohbn IL qr--.l -6L L6L Ik"1" o',I "'- II -eu 6L 6L -12- rrLry -bL - '6L IL gY -61 =9 ^rl r-r: I _nt '.ir l\ Jl lra Lr" a-) \-k4l=F.*1,+ [K.J'6L IL 2( o ,tl o -lz btl o EL -6L' .t 2- 6L l- lz -6L 2L- -€,L o 6L t24 l LO --' aav -b 4f ;f th1={;i1 rlf:' C:.;cci,t:t, . -- , cc.Vi Pc-,il y(d Fl .9r /'rr\ V1 -P D2 ll W" F2 bl" My -- 2 5 / O tl t o t^ ('tnb"+*n 'e9oF n . Paravrrelers) (Load'lrg pararoreh$ I{p Equivabntfid6t {trces and rnortanft dqe 6 dislriLL{l€dlmding I rr {u}= {o} Ff l2 6le -10 Ele Gl€ {tA - 6le 2\t -12 -6lz tz ^'"I lf-\ -{le \ -G\e di:\tit'iC \"ril"t1, Arf l ( ?te1- Fur lll 1\ E" l{z r, \V" \+vtE" l_P,1 Llz tr $Ue lite z-eleraPntrnsdel) &.art, Prtbbnr b7 Pg^l ('faxe a 1 s tarrent r lt2 [K"'] = trrl t3 -l -ilz 6L stl \ I -!q I 6L 4L I -t2 -61 la -"1 ot$ -6L qL- I I & = lkll ') + [k[l -to a' ,1, xl 6L ,l^l |'t ;I-l L L- ?4 IL o 6L -5L a9 2L il I"J \* -(L -6L | Fi, I 1-pte /Ar I i,pti dl \- y'l+<"" 't#) l'.i, -'" i l ' lu"o\ ) L&,, n $t,,a +he bednt sugporfed b/ Et ing.Teke lt1" be'am as 6irrgle eleMent' t-. KVz (as l. l5L l-r' I L6L I l, a "pv,rarJ; CL ET f'' ft tl- -n Systen'r Cttsu. 1 '" -t, 4* -bL - bL l2 2g 2xz fi - 6L - aLI ( r,z1 1-r_F' ) o { lrr \3q" - ldAt (-*"'-B ' ,'/ o [0,- ur]1 lg ^- \-q ?, f'rz1K -6LlrvL\ L' qr'l\e'l l.gl L-,, rl Jl &nErder c deY'e"\\. e\evnenfy' anothersPrtng' 6L IL Lk'l = . -bl- 4g LL. _6L -la 6L r. 11 LK l = qf -".] [R L-K {*} kJ \q ) -6L r t2tK D +X,L- , - tzl- - o o -6L e{ -K 6L o 2f o -il -k LrJ2! E-l K ;1 1 i;i= +l []:,. j tl fv Xrrle elenenf par rnodeJ ; 6x6 nakir (e-ote !@2 6"---^ '1 _,2U2 111 r, {\ t [__-,------rJ?l 4 \- /l \1,/ \:// tr 0; ;l / / A = J ' !- - h'I,A 3 -r 4 \v E------Tl-r--x \l_/ J- !,lr tJr €q tl2 Vz cz Rafie €l€Menf. @upe<iosiibn) _ \,{r IEA rJ1 _ EA I -T Lk*.] = i'I -CA I \1 11r q1 qq EA I l-i-i-l LLI- 2 356- &aht tl1 81 {r, \rl v1 J+ oJ+ o o - L OL 4o o "l "l[u'[u' vl I lt' '+ '+ '+ -:+ f, :+ ry +'-t(#\ .#l\q o 6F-L 14 L CEA o T -!Lt rnl \ o V - l-Lrl E-I L o Jq y. O I ol I lu' I 10" -' -YE"g,o '4 -FJ I o Ir L- 6E {1 I Ir o -6tf fltItr1 o Xz fY,lL w I 6xA 5C1l VLLYTJ \oad Aitfnbt^W lr- |ransversn 1tad. disl{ibuk'l q- axial * $i.Snes wralnV &r sirnglr {ratr'tp nh^efr' 3dor/to tr g,1 ,A'L 16-Z----# -__-___-+V2 -----+Ut A Irarrp ebuent lvz lq (e \ \.? u2 . dof - beam dewre.* -- l)---- matrorr4 rlilh nodal axial 4nfov -------- r6d t-I l1n ( ,142_ rir iY' | \ 2 ^ _-____? t-v^ ,x\ axialrca - ,,"t*, /lrV lding garanrehrr' lr; * oo#. o o vyo v veo V o )e * - t-l' Dl d7 o 60* o vvo d w90 {kom bar ebr,er* {ro rln beavnelener* v V v v, VL rrt tr v, r-lr'= rlr Cos0 + rl1S1h 0 u/ -- - r.llsirrO+ urCos$ I I, x 4eonre*rg ol {va '" e eler*ev* -dr- lv" (u lri Lqs lc 4 c,, s- ---- '"' th: ) ' ; lof lret \r't - 1oit+t € vz w2 fr- - ulCcs0 q5in0 " r.!$in0 + vr CosE -0" ulihx-axis r'l a,'sg- t U 1 / = l r . - 1 11 r 1 r . 9 1 Srh€ = Yrt rll = -mrlr + lur $l =y62a mv, Vi ' -rn\iz+ l\tz ut Lll iql - v,' !a'\ ll- d,' Vt 0r lJ2- w2 VL u 0z ui t D,, -m I 0r' o o o w1 \t; g,' tl t o o o 0 o a o o o ,\ 0 o o o o (l o - Yl1 o o o c) L - l*lex, 1qi,^, lo'1r_, rnakix ft66ftrmado'1 coo<rlineta ti i ' t ,,7 = Us { lq'f 1r"I { r'} 'f . r ^\ {q})' [#l uer117 = + 1q1'[el ' [rJ] LRI'fqt -I(nt\ - -2 U1 U1 0{ rl2 vL 02 Ia \r-}'[re:'rkirrc {cl =* {qi' fu4\ql 6Kc + io'l'{r.} = (lnt {t})' { t" ,} = {qJ'(tt# frl),-, - \qlr {te\6K, fe= fl 0 W 1Fa?= II ' IPJ r*'v pil" o .PV" -?(r', d.' [r"] = [R.-]rlkd] LRI 5oo{1?r t] -f Jocc lb E = :o xro6p;i I I = Al inq s 4 . 6,8 \'nz ,n'-) Fo< e)evqerl' L. |/t' n1=0. t- --R.z a o F+" FJr \\\ o I Inl = o o to{ €bn€rh t- 2. J o o o o o o o -l= 0 , 1 o o 1_ oo o to o 6L o o OO I4t.'1. tr^ oz [R]-'- = Ol] H t 11il oo DO oD tr = - 300a , r*t'{o;1 {*"t -?2000 0 - JOo0 7200 Boundaqcorrdiio n. Fy,= 3oo o l3 ="\ :OJ rJr: u -$ AAI t{q. o R2=o u4S\' Fgr= o * lNz = o I' 0r/ o zlto L.l p=arrd nl € = 2 o xlb3 lt/ilr\L (a) ftb1d Find the 1dr:Placemer* \ sttesst'tu \ - I l.i"reaselerrcnts uc\ng (D quad<ahLeleqeflfs \eactloY15 J I s"PPoot cons'(de( ba{ denrent. (,a -1-1 EA fl LKr=T L-, rl t42 '+i' " F' EA ft I ksl ll'l -l o u. I L 2 l-t I Lu 1,r3=fg -l -r I htm ,{'o -rl Ju'l \)<1 l< 7 - - l-' e* f z 'l\ul\ -/ , t12 ttttlt \.42 = 1,5 nrn x. 'c = - !O XN {) Isc i V3 \o'I <)41 \rrl ( (,? _L <t |LlzI zoo ttlrirk a EX fl - l , i-l \ | - \o rt/mnnL 5upporlr€6c+ion. d = = wt- L --i- _5O KN \Ur lvs 06.02.to If I L t 992 yA F--L lll FN uL- L54 A z €\ = 6,'€ xrdoN/m Er - ro'? 11Jo * l"t L3 = 3'59 m rt/nf e, = ro 'a xtJ! \= 2'54n A1:> $2'! Az- 3s'A em\ 43 = 25. fi, crnL itt€ \hras na^ba( iY\lss- FrnJ sr - Nodat d€tlecnurI - MernbQr - SftqsJ (€a(h'ors ' va'- yr^oor l - F a. ,J I: t: L; fk"l - nI 915 lo l. l-r I a, -l o IK'] = 1;.7.7 r+ I -l o o o I t 4: 9 qu I -l ,^ r l CtttL 9s 9q 4t Qz- 4 I , 5 I l K'l _- -I: n".r -t { l L; -l 1 I il 1t I -,1q" -;lI 2 q- t -t -l 6 ut zzz'<N q IIt KN ur- o as trxed. Sr,rpporfre6clron Vj tlj 7\5+74.c - f +,t -b\5 o o -7't.U h.+ o *715 o A' L<'J = o o -74'Lt ?4.\ l%1\ ?ri.+ L- -74.q o -z+.+l \Q'l 1+.+ r+u I -74.4 | 0 a7.7 e 1\.q -?q.q - d 671 l t ?+'tt :+h.V "l -37.? I -7'f'ttl 81,71 +"r{.!J (2l1. = ,,\ '+r-i 1 1-','I {r. 4L' tr -rheot! oq <bsh'citJ U 4c l * I sv" s** r t'- n,ffi- ------2x Qt {.} ' = \u) scy caz zyJ tzz zxy uxy L.ta 6Ja 9-. 6zx €q\ equiltibrir'tttt 4 S b (x* b(xy -r- bfzx -:"p":o bx'dg ?6yy -Dy- n Ddzz __:_ A7 Z 'dx bY{ [*- a; I bz .*- P1=' o - )Se b? d(z:/ * ?, -- O . -PSz -. -T7 - A_t - r-'<-w f6u, - v 6xi 'c. -6 tdxg o bodvin L-" LQ* - tcn^ 1------:fouard whal Lt axi. porp.'l'ot,l. 6xr 'z t r lsr.,- cyu- 6al - crv\ er. =-[ [6rz- e7+ .V,.= - oYiJ Lsex_ 6'q i z.a{ = -Ti -rx cry = -T 2r* = -]a. , = -5! b\r l\, zxy {- Ti 6q, ,- _- XU i\,1 r-E 5t a.*=t* *# ( .i? luJ' 1"1 lwl aPcoach {or tsasic r-0. as ,assqllad bodJ 3-D E - Plan€ s'|r?ss (zo) *:9 groblern g^} 6*.€v,s;y - ?tah€ st\ 6tn Vo*atl. (zD) - urfur y'ane dro,s alSurn$ott I {nr-1""' tl:'\t l<"v t vJhen one dinrcrrsiont! for <tort. bff* S.*r.,.'"" ?(yy + )(ry I gy: o Fbx ('"1 l* "l {ul ) €,vrl ^ *, t\ri \t 1 l" l6'l X--l r? {"} L& t' \-v t+ C e<**- elr) --- Strain pery.{o the plane. Plane strain problern - lrlhen a ' -lhal bd) {s 16l lorrg sbiect. 9Y. can be assqmeAat 2-n PrassurfTedPiPe' '€x:,, €yv s-*, (lr >tr_. '-:}l )- #"xK.., + F =o d'l dx Dsry ?) ' bf tx -+ Fv = o v -'1 \,<- oez? E i_ F3 :O +-6rr)' 62, = if<xx F exy , siv. <>z :lbtu tf ?la.rp F{agh'c Roblens a- -R [rea< io6rithrdernenk d'a diecsetizahon u,l rl- 11t "t) L4o \- lqu l4c r7l [:' R,. \ ) :E' . . lr' txL 4, lt,Fx7 \4c lFx" \u [.t' 1 3 '^^)tc-:"/u fi clocKwts€ r iflq trutnrbo< \,{ = N,dr ; N2ll2+N3\i13 Ll - -t N1v3 : N'u, + Nz-Va s\4e + Ikl tul = {r} \:, \qu \-r FE: usin€ 4<iarrgular bdY Ir ul url_ U, I \,{q I lq, q, frl .n, 2o fi'trtc'tions' preterabte. Nr4r+ NrQl+ NgQs 'NrQz-F Nl9 t{ + }'la9t q It aq, I " I (ul f"l, -l -[o 1"1 OtlzON3 Nr dZ o o "l \ Nel lo, q,{ \ I \qr i \ O. a {uJ=trl{ J Tteee Lnea.c shapa {un6{ionsh naturalcoordinat< syslen(Nc9 Can ba taken at \tr,3 Nr, Nl. N1t,t'L+N)= I P<operi'.es: Nt,o nod'. t dl- node 2l't "tnoda 2 Nr" I *l- o r.l1= af node 1t'g Ne"{ af N3:o I f a '1-] rl nr " l-1-"1 Nr' t - el qod'c 3 vPle 2'l !'t4<l ), l - -TL _-_)- NrtNz+ tl: . o 5 lineac 4"ria'ngrrlatebrneni [r-rr) bnstavt stcatn tr i angle N1Q1 1Nz43t .^ -4-\ 'b)1 .t \'1r 1\ , 4. , - Q. r.J\ - in -,r ar-g Q'-ua) ] t (,ar-ur)? +,t: Nr9r t N2qqr 5qlid mechanics) N:9s T4, + nq. r- (r- !-n), .l t)n5 I t, , \" U= t\n N3gb r(r_T-,,1),.rS J ur+ t Ltz = 342+'l(++(r1-1yu = J Lq,_qr)*rt ll+-1e+tt :Qe-ur)j t 1f,r,-u3;.'v, U --.Qtr-uf +Lr.{z-ul)l+ \.t3= urr} + rtul _r_ t(t J u - Lur-tt,\ + Lu,.-rf) 1+ g -- vral _ruzs!+ vc fo\ ' iso-paranretric{ormulahon x = Lxr-x3)T + (xr- x3)'l t *3 - xtal *xz3l1y3 Y = Y,r\ -t lzr'l + Yj ts61din6l-a gtotzl c 'nafr,l<al sys{awr<elahbrtshi?' du dY 2u -r: )x -fr irl ,I _-i:-: #.,," -r, \q tx dYI I ?'j =- o.1 oJ ) iw . -!L,t I- ., .lr3 ' ix -.br -tT' )v ;i ?1 " S*'n, n-$'$., t )u I r (bu',) Ll J sr y i\ { [*rc 1T{} = I jn l!-\ szai I br^ I .J t\ | "1 I bt't > l .\ | j"r] I I!.r fr 'l l. 4ransitrnafr L **r,un vnalrix s{ coordinafp - +{ br.l aul OX oE( - 0u I ITI al b.t dvl : 5rl I :- I bu I ur) 1.. @'I lTl def[r] aLl I I -$l U-^ I I -x.| 'trrother relation {ar ldei [J] | = lr Aarea __tL xr3_l 6eomeky "l -r',1{#'l e", f L:"r, -''l1fiI --tr) qY'l a1 |' tu I -l !alrila<)Y] AI J -Friangle Au*a. der lT I Ar+ /q-+ 43 = A o'. ** * ** ='t ) N r-'l- Nz + Nl3-' 1 ., \: ,$r i-, -r=F , ^a,* t1la2/t" f-] U : \lrUl + Nztlz + Nsv: N,91+ Nzua+ Nov.3 U: Nq{,f Nz43+r{S9r E.,X) rJ = Nffr -t Nu4rt + Ns9{ l-inear{"rian6,,lar elenrent =l: --l- o N2_ {.,1 ur-\ Nz o t{1 43 lqqe | [0, - e' .\o]ro = [Nlr*. 1vt ' 2_Xr tlr. Nz. vl J , -aq I I *l = [r]-'JLu ) [ I bu I b€tlrl 6l I t : 'f o" I| -7o L-' bd- LJ1 = xls y23 - t+ [^' L xz3 1 IT LHI I ''l tl3 . l-]- z 9rs^4 ill bu E iu q I -_^ s,sl I r..r"". |'-- Del{Tl L:}L? _!,"-l xrg _l - 1n'* l- &, 6-u3 uettrl rJ.- -Irl 5u " t, **,a *' l 5irr i Iarly, lzv b! 6X -;r LTI bv ^- 0J - \iT [235] l 1l0v detkl 1"1 \"\ r*, = (€"1 {tvr l - l.*,) l-ro\ \F 1bv \aJ I | 0u 0) 'n 'l - I Fr-a * -,rfiJ I bt\ U29 -18 -l l-*" -19 ,,r+l dv 5i 6U 5T -u* -r-n: # $ -*. + +xnfi *v.rS-v,rfi u: Qr-rdl +Lql-qtl t c5 (ez-eu)5 +Q+-9e)lt lr - crslag-l4 ' -F-61 IYBLq''qt 1.1= + x1e(4t-4u) {-rry.-r,l q,l Yzr(%-qoJ-vptt+ !1,-cr) 1 xrI (le-4sJ \-xrr gs {, - urais- Qzr-!ra) t9z: g, d"tFf l "urtxrlqq + Q<"r-*rr) II >tzrlt [- I I | 9zr4zt xrzg:- yrg9v+(Xz:-*'4r5- (ve3-vr3) 9t J (9zrgr- cragr* lrr'Qo ,l t l I e6. l+ ttt 1-xugr-Fxr3{q+ x621) ll I vzsq"+ Xtlgr -Jr3tq txzr{,s+ v119c J f"234rt I 9rc1L | 921171 Ysls xtt4z * xralq + xzlQ6 )d trt ( +YrrYe I xSzlr + Y2j{1 }x13i1+ 9j1 {q +Xztls o ,n3 _1 \?z o U=,tSf vt92 - tl - \4" lA\ '') L lar U4 *r) -u osizo x'5 o xzr '14 !a AL., 9rt q- {'1 7xL L j Axt l; o Yrt *7L o rlrt *r3 YIL O o I xr, I v?r xlr Xa 9r d6r \J_J q rolal pobrhal sng1g1 of ,a t' = t the bodv fr n rT nlv ieldv_ itut liel -\ J' T z lul'TdA\l, _,/ {ui} r,. lLt ' PoinN diskibu+ed body ' -J{u}r rraeI {u1 r+aL- i,{nrfli \*=t'r =+ (nt'Tt'tt'^ J )' t \ " 1I 15 = = i""\ I s,ntJ ['*l t1t [u*I ;J 1"\ \€-'l v! \- va plarFsh€ssawtu<. oq - Lol\n\ t\ q"Iq f'-" E lrs : (u- [ r"J6o I Ld I"') = lo]lnt t'r,,, \t L-ls rl 6 Plarp Slca\n ; 'Yy l a?rox. 6^y F - --z :- ^ l- rJ" U'>.-:-* ..) T r- + liet Lrl lef IT e T dtr - e tut {ol rtr' u" =*- J lgrltal)t -* r /'t I / 1 --? ' \ Lt1 1-tr1 tDlT rsl lq\ {u} rta* Jl*l"t'L' +an {dr -- l2 - qaY-1 . -4. 2 1ufI rt =i lol' ( Jtro'u'trsl da) I'J = -l- JqlT 2 \'l - rt \( L2 \K"l*e = I t,ir'r"toJotl't q1. (tn"I*1trn'rer)h! , \ql tr"l tql tAe ["tt to-ttttr "T { Ndl€ numb6ing 3nould bo ecw. 2olo2l1p rl [re] = +A" tBlr tDlrtpl ?r_ Di l=l els+xets P/ skaln ne=.| lants1 # t5dal share al el<msi bodv ftrce (ft) J{u}r'1 av tf {li ru J e levrrsrt bcd: = {Q'o'. J flJ t an + (r'qr* *tqu1xts9e) r,r-o^ .r No.9cl {x 1,(t'r' {nan)+ $1{ ae Jx, . +dA=J(,.4&rurv)+dn o'f*9 1r(J+"t'a9* n u\I+^rt-dt }+rt1iy do) + q,"(tNefldt r/olumeof tte tek hedron :! x ba€aroaxheight. = -lAexr = + Ae r lx fo, 1t 4y iq gs Nr',A Jr CJ ' -__t i N 1d lfx,ll Nt7_, 4t +frI l,N,_dIA t{ i \f' d l N2i s' r'rraA I fr \ fn ^''11 ^ :'LY\ \t ^ 11 = 5q1r 1c"'1 . ' I.) ^- a.1 4k lS -ta]i c'9 ndet eorrngonents 1-,n bodYtorce' [\ r{{""}=ry lr," rr" * 'Tx ' n'l {r' fu--, f"o t' Pr' huHon ftrce'. Ndal share sf Elemenf +rdc'tion{orce _f -j fr t.{ _f -J r)( t { ,tt r Jr e -t..,:-l ,l dI -'(t l .I Lr- I J I $, YD TY x Totd as al 1 t' force dqljd4 \nto 4 patls or cofiPemenls U^ r..lr9r1' r,h9r lx = Nl lxr +NZTXL l, = tlr Qr t t'lr9q -! = {1Ty,+ Na\, tocdon tzrc€ = \IuJl 5o. 'e.terrrent I zl9,,t',% t: l* /- - l:l 1 L'YJ td( l_L - l(ur*+ -J- uT:v)tdl = [1 nr,r,tu.t)(or'\o*,r*i+Q\qu+r2lf $,1,r JLtr-z = f '1, u,nro t, + N,N2 q,r { xl1N2Tx,9:t llrr,.1, ^,lrrlzTt, 9! + Ni TvLl q) t dL t ; N3rn 9z t N,Nb1'z'9r Jwar r-rn J ? n '$,*. d,= * ar = -=-l-rr [,'r/ J- Lro 91 .1L Nr tx 1r, I ,1 + ,-l N r N t. t >aZ x N rN z ll - Ta .1 '1L c4 I '-F ry" ,"1 +NL 1T^ 1 1 Tx2 x2- f Lrr L'l I Nr' i ! xXl- N 1ll1T1t t(1 NxTy, a, Vz I ( 2 Ty, +rTTYt \Yz J*2 Txr 1 - 1J 1-T*: Tv^ Tyz + 22-'l \r n.T tql r -o 1 lrl L'l4, + Txz wherB, 1rnl- tli! 2Tv,4. \ll t /:-\ LDltBjll+ApLbl -6X6 " 9r 4z 'Tv1 z rxa f{ r-x Fxl a, "t q.,) q/ s- Elbl{at -t. fDl Fyr tAe t 2-fy,+ T*. 1Y + +L, -t- '16 11 ?*g 11" 2'ftr+ Tx2 {1 1Y . T4 +2\z fyi t zTyz o o -_ elactl' Progar\ l-- lEr^rTq*rl9t.*, &orn coordinzte Lrl Y1 I I Yrt I q- I x U= + Nsus + N6u6 N r U l + N2Uu+ N9u?+ AtqLlq \J r l{r lrt + Nr-vr* N3\rl + r'l1vq -l- NE\'< t N6vt ' '.l.- r.l1{1* xl2! 3 * Nsqs + Nq471d54c + Ne4U V= Nr4r + tb4rr tNggg -f Nqgg+ r'ls9roa Nc91z ('..{1 1 1 ,I .Nt - I nt, o\tzoNsor{qor'lso I Lo Nr Ne o n r'lc o N4 o Ns uool o*l 2](tL 1ur= [nt1r*," lalrr, U.)" U't* +bs rY i bg! r t +bz-x h bt tbSr+ 6** = * €vv' y €t]y = { j l2t b, Y- b', Y' brrxY* I btoxl bg'i + = ht%qx+bs9 = $+ bslx-r?brrY b5 + ['x \ ] lb1qx 4 b r r y +bg rb6Y 1 1 8.ud€)rc shaPn *.rnchbn. ., S{rains ?rs lr'n€ar. Stress 6' 6frain va{r)e< [.r"neor\y. S Shap {uvrcftonsin Nr = r.lstuxal cocrdinalzsgslen: 5 (r1- r; Aiz .l(E-,) N? e Qvr) N4 triargle) l tt,ear strui" , ut:1 Ny ' \ 1\P tl6 " 4 Y\ \o ^ -.fv ) it' 1l 6,\) Lkl . JJ + tb-l-rtDl t{ & t Cel+rol tsl drdY JJ vi "lJ tD1Le.) d€ttlldldS dxdy= &t ld Jt d1 . - q,'l,:'ol'^ -_lv-rIv tl gr\0 linea<ty .ratgrng ?regglt(€ t=40 4h. M?A 7- $m,eo) Fi'd th" 0 41,)_ co01P0nen1idf Yt -T -L L*,-*r)' + (q- Hif] c a-= ^12' -Lr 1*t -kacttbn force ?t flode ?,3, 9. - o-bl t = -_ LfL =2f Xr- Xz = 0,8 lrt ll1 _T-. lYz - =-5f1 = - z E >l-z ^' IJ t_- t e (--=, ( \/ ti 4r.bnNs 2r- I L*.1 !i \, -T_ lgx = - 0.8 Tlv = - o'6 = -1.6 Ttv= -t.z | [te- la-e! "-7- lTxr t- Tx:_ 2Ty,+ TYe Txt t 2tr:_ \ + zt-r" ,l {000lb tr ' r,t\ .f ^ n,q A E = 3oxto" psi lileet) \ ; a.rd ehrrent t rnd ihe nodal dirPlaceTnenr: -l _ Q,2T SkeSSes. ,* (aye 4 lriangulat elev(en\. Vsa ill plane stf€ss ximahbn, aPP{o ,,i,+ El FE sDluirhn o{ z-D elaslic Pr0bl€l?u USing l-rnear quadrilaleral Q@meil(. rh (iz vz] qs Li Ur ' (xr Y) (tq Yq) N1U11 (2Ur + N1r31{ N2V1 + N1u, -r Msvlg+ tlqUq N r ,D , , 0\ U=N,o'rtl V: Nr9t t NrQr -r Nr94 t tt39p + Nq9s r- \ut <l t - l l- lul Nz9s+ \9r lN, I o Ld N1 N3 o Nz o 41 Nuo xlq N3 o. :J Qy YLI \ u]r L-r 3 IN] x\ :l ;t) IT 1 2- (l ,) l tt 9o Iq1 2Y\g l '$xl ,(l -t ; | z-\ _ J Lt _) rl -4 I - t r zl' \Mas - L i.l I t,h- 0 o o I \ 0 cl N3 Nqlo r'r,"efr_j1 qu- Nr .lt N) Nl1 q 0 o L o t r(r,-r; 2 (\rt) (r1.) t -- cxq Nt= o l ,r) + c= \74 1) t'-9 r+1)(\-1) =i( r+]) [tt ! .+( t-1)(t+11 x = NrXrt tlr- Xr + NgX: + N,rxq 7 - NrYrt Nrh + N3Y3 + *q )rr lil =F'Jl: I; - tlo-pataruetvic &rrvrtl\ation shapr{r-,rnction5. - ri =-1, (rn ? ir) (r+ ,t 1 li) whqe 5i, U -- Nigr t }f d3 E N.t1+ N:9? * N g 9 S t N g g ( + Nq98 oF -a-y Zl u )v dT I coor&hotpa{ nodcs. =* &.s + ^)f - bf ?4 are the 1; NrqL + N:-1,r + \t - t; |,2, 3,t+ }L frFl \tri. t-) \df I lSnl LI- )X \l b3 E . -!l t'{ r b+] l" I <( lbt D4 I I L" t s-l dxl gr 'tx "l tf o't ) ?t\ OF _-l I = LJ] ;.1 .) EE ) l+ b'l 1. -(\,.l);] _0r = (r-1)t+(r-1)xz r.(rr1Jx3 + LbT ) - i'*'i;':lr',1"\1t1)$ -tL-$!r *lr-r)Y1 z8 ^ 1 f J2X:9 ix q tr t Ir!] 'i')q 1 * * t -- L- t, g){r (r*3)v, (-,' T-. ag --! i 5-1 = .L- (,-a)'Jt- (r15)g_+!r+1) 0..,u-)\l - Tr, JSl _fr,, ",,rJ 5; l lgI-1", '"1\E 01- I €r ttTl wJ r {LJl f* u"" "11"5) It - JrsJ L--*r2l ( 1q 16I hK = iz? lto,{ 1 lv t€o J \ "drl . 1 \ 9 +-\ I 58 t f , * f bu') \5x t <r lbu .- I J [57 aul 5iI nrf ryJ :-- I II ',z. dsLf,JL-q, I ftL de+ It t"r -'l'i' u") dl\ Trr ql Ivl dal I Trr I = .*1 IDu )* = 1tu {o} l5Yl0q ,T tul -) [vl EI b\r t'' xzzzT zT :\\,{ +zrfr+4, $ og - L ) -Jzr\u + .-rtu , "'za t'I J.,tt t Tz'\ '\r7= -"-l - lit rzt -521 o {zr dr Tzt "u \ .r' -"" I J 4rI VN\ 5; 0u, 5; -l 7\ ' [p,] fa aq hJ b,1 4xt {,1 8(l (r r,lL ' ' \r1- tRl qe ?xs Lal . [n] [q] '1rt t7 A! q. IT 1e ft\) -- lol } {u \,1 , f>-lLAItcl { KflONN 4) 'l covd .t I ' r t ve--2 +lj6f ^T (.", 1rl+e d A }\ A (t, .r., , ^rT..-(^l dxdl jqj -_ r+-l(l-br Lq _ L _ , f, b-llq])' I r/ L_. 1|\ Jl_ : _.- =\ Car-T =l+,1[qlt -2iJJl fBlrtdfr,,l !q1 dxJr tL"r h '( E_ r )) '(lu ral [Bl defTrl d? -- iqJ [s-]r JJ ,, I \ql =det-Fl* d? dxdy qt e1r tr..l { t r( =>1 '1, rl ro ti =6J x"1 i i -{ -l lJ dl ant fsl [B]' LDILB-I r -^ 8r8 of rhtet'Po\ahon' nrethd $ Garrss \ira\radure r-[qutdx -J -l 1-: o1{(x')+ cof txrl+crfu4 t -- +Cltr9-Trapezoida - - = vlrt8i 1*1f.t*z) + w?+k3)+ n +wnfSd -. , = Lu.li ttxi) | 1i= \Nr = \laightinq €eeFot Yi -- qawo PointsGP) {kr Ut \l^lr'\ t\-J €rx 0 = ?"t unknOV.]n . I Error ' I \fl.odx fa dx -Wr 6:v):o J -t Ira 1 i 'x) Ax [ (a"*t J) ']'' lr- ' v.',(a;1a1t) O --l 3{' I > lax+ L zJ -l 2a" -W1 !" + r - wt?o-ulrdrxr :o - w 1 3 1 X 1- - o a.[2-wr,ts (- a,w,x4_ru 3" ( z-wJ= o d qW , x 1: ?w--z > Xl :o + 2- point r.f= ' I f(xt dx "l J -l = wrf(xr)fw2fLx2) --+ l+ unKnown cubic -, po\ynomioJ, f(xl = a"+ rx -f 4zx2+ 2y.3 I -( Lrro( = I t(x)dr J - 1^11-l (Xr)- Wzt (xrl = O -l ., e I - ,wrT(x, -\^/2t U, _-o dx alx ?zxl I ) lf G+orx+ t J -l :) fr-* * tl:o * Ir ?"= ' u | rvr I +"* | t f /. . . ", _ u'.r / +,.'./ ( x , \ _ h l 2 T t x z - ,=) 0 --l ) 2a.n tu" - w, (20-f 31x11 arxf+ ajxl)-uA-(a"+ 2sr+a,t{+a3f) t-*r, =2 2" (2er'-wrJ- a (w1 I ) - 2t (w,x1+wrt]) -o r rr rrnrzxd + 6'( 3 -r^r'N V,l1it{1' Q \ N 1 x11W 2 X z = o r^lrxr!- wuxI= 2lt W,121h12X31 -n = 1- W1=W2 .n. t : - - I : -0 5?7 \t Xr - +!,t3 ' O,Ei4 t" f(-tr)*f(+,) # -t -l J3 * Tor higher accl4(acyS-poin{ .'l I= J{Cxl{x -l = \^,tf[x) +NLfkL]+ N3ff, i ' l\ \o' ' -\-or7 - \ 0'tt5 o.8t I \ \ u'?? o.r, rt r! i'bw,r= [ f {G,0d]1 JJ EoM t o r t3 used' r l, Il l Z/s - J \.=r w, t(1'. ) d,1 \ ' r - 4l r/./ | I I I ui- (Fr"rif (q, -"' .'r;l) 57 ,:'r : \, crj 22_ f ;l- t t- (r-9 \^Jiv,{; ' ., €(r.. "-"'];) )-r vtl = |',lr"f (j,,1,) + v,l'Nr.{(11,'1,)rwrvlz'| tl-, -. \rt., /r/ tWzte,1z) r ={(}r,-g;* I G,-St.-r bfi,h,)rr[},,#) 06,03.10 a ptisvnaficbar under torslon E fiess. analysu a r-tnitorrvrba 4 any .hntslinS6or,rra,t u r,rsuat\ lakenas Ittp disglaceuenlf'eu rf Sechonunder tJ= - BZ$ \) = 07x u1 = 0VG,1) r,rnlt lengtt f+e bar +ilisl W{ argle 0"f {^'Filg '\nl '{orsion on'. f rrncl-t funcliorr'o\ Y- €xx )rl = -:= -/\ "xg AY €us = 9Uz -/\ -:: o\l -.DEt - 'Jxx )bl _-, = U -& * S =t#-no = s(* -9 6vz sZZ =v SVt = (J b ( X e =" !6N ^ -'v"'--6, Ax d"gl 't. ?szz --i=- qoLH -9 , :\J "xv =, usxx - lxz -^ -\J . 014 = n/) qL_/+ 6^ )\l tEE irg -6U " z-N fi --E oJ €zz il, ----rt\ <e(fi+x) \'r b6x! -[o6{z _ ,..,_ - -C') r 5;f 6(xz + 'D* +-0-6ia-o- -Q'J b{ 't DCxz DdVz= -trr 't -;J" -- .to sdve 4over"rlng Eqr -torsionalprobienr. ( b( 'bx n / )\u tEb\ii .? -Vlf + tGo-a"J , c ) -E! -1- ff' * *" =t & 1e"($j "xl|=o (4e )Bur]= o --------> hpla"ee{- Boundari qondition ':\r) ? / v' -V) r-a( -"\ bx r drr€eHonoo$ne. n-- Z=gr.4x+Cg2.ns=o **)nv =o r\y+'40 ($ Yl1= CDS0. ng = 6in0 u(*-r)+-ffi*l This has to be sals$ied of t'he x- 6ec*ton. dx ds bu A ds. --d-rds _N r. rl tte bounatng Sufrcce ,E=-5e -' DY ?fi 6;, =# 1 ur='-# I skegs fqrr+'on ft6 lppsienal problems' 66'J) v $xs= Qdt#-U >ct 6lz =eS +x)e. =N - / ,,n 6! H *&rn-= - se #r-qq+ eeffi-ae = - 2-q6 {fi b(r- >v -+ -:__t_ r_ zaa,a \-,t v t -rr -v €overning eq! ip terrxs Skess funch ot' d6,tt 6iz.n1 +\2'rty'g + dxr =' bd '.-d6 d,l + ., -6;J'\: i9r/ -ds \J Ty +ff t+ dF E3 oP ?x <IX ds -" - consfonf along the boulndary. f=o tr No|^/, - q' ' axae ' dxdvx vl Lq" J .J =JJL-g - u-!gl " bsJ -= - "JJ dxdg flp,{axtt 6 (xzvz) '2- g=Nrdr+M/r+Nl/, - (s'l l-" N1 *l 1t"l \Qg) -- I-r'11 u'l ?g d\ {r}' oF lx .,__-Q!.:: 0* 11 ry bj. zF -fiMtx b1 E- bJ -,bFT-u'E. + \ !r q )rl I osl lbx l5-- bF -rT l" I l ttx x= NrX+ Nr(2 +Ngxj v^ TrJ 1w1 -l td n \rx t'tgC-t-'? f-{l ^- t ,,-, -"\tr \ =lrj I''ax 59 Lu d€llfl 0v 5? 1 [t J I= lr" IIJ fu L-"tg Ttu ddD't"r, ( af \ \5TJ dp \ar v,r-l Lxzs !r4 t d61 Iar @ rlcfr{, ttz"tl t lrl =f*', - [r)] ap [ tlg IAJ Itr J L- -[ - (dp \ dF xl vr + N2U2.+N3V3 N1 b6 1 q EIJ*I tv l\tK I -r--l xtc) fFrl . -'l lr.l -r I - r'ul f' I 01 lr, l\ L4 ,zr) ')-i2 xsJ -,-.[4) d31 -r'o =Lo 1 -'Jlid,\ p,l ---/ ) larl (ql ( p/'l coo -\3 Q+zs*YIf .T ) 0,, [ _ -]_ - dui6l llrr\ l-xzt xle ( -*t) 1as lxze _l \ fu ) l-rs' . : -lfvzr +$st defJ l ru, xra = rst {sl fbl . s,rl \ql ."\\k\ I &J fua L Nsr xta ,nl lzr J "2 ure ttEfhod. €alarhn Qowtnivrg E+ tF --tE n,oo -'" Dgx bxx e(x,\) = Db*tg ' bJx +288 bx2 Ief ug aonsl"derkiat sduh'on: vlx, vJ Qalarkinr equdtlDn 'o fr €tx,uJvltx,9 dxdS ll JJ =o -'S, -,-'onl dxdu ==) J! "L*" A -!- f ,'nlFr bx \- bx/ ," fls "6 c No\r.t, u!? bg! = 't*$o^ # =Eb to gu*) =i; t f\ rr lEt ?g J \F ),,, -ix i,'' bv od bU AC Y'J"* rf [* t,+rt- H # +&tY,)r'e' JJ A -o drdv .. A+ zJJvqo d*d * rzJJne # e#). {i!,^u, t fr * Jar "JJl$ 1jE t A fk ^-\ . --. 1 l^frr," rr, 1,r " --.r lc lo l-r"nefrrfegrat' Sutr8acB o Qreens -tt€otzn( I L"rre s ' tr ft'r z a@a A' blundd bv bourdrhg r1ll0 5rtrtobl5srnoofh {uncibns, 48'v; ad Fg, Y; (rr arG,$ 6(x,sds. a^ar= h rs,9d^1 t \f 591)_ I I J ax bY 'o ' t4+egr6, (.5urtbce \\L JU ; ( Lhe rht€gral) \ -bx r\ -,0 - -DE >\r -_l ll I lt vJ | li r \cJ\ I u I i" \g \.1dxdU= -lZ- lrO vPl I I! - br \"t;e ^ t t-f -ni 'J \" lcJj | A dx .gH_l.E J tr- vI E Jt- i6 nv l & =Q ' eoflditbn boun)tdar3' lat aclrrHo n <hpubl saHsNg "JJ fau L5r' ?F ' tl #.H#l,o " ln Ibu -:- + -!a TE 0x by L0x \6 \U li--. =,JIt* ' _( f'0" aF 2l\ u Qe dxdY dJ dA. "JJ" ana*av |u l{ D X .-oqll -l rn trQ€ Axdy--e nJl bp OJ :' JJct{*1J' ( rsl\v . t)dA 'J =z\ cn I J J q e fur{v} J"JY {v}-qt"uat v'irtual skegsfurcHon 'JI =ncteJJt-fr-f"t u=g'rii v] ta:tpi'ao tarr lv1' =r {nJ' {'pittnt ,t = {rr (ee lpt (rur.ru: iliii} il,) .i^ fr NidA= +4o.r =A VJ ll +LkeJ 1e1' U1=lvi' i a1 t =AeIsl L0l tK€l ,qeo, \t,\ { -'r rel= { vJ' (r*l ls t - lftij =o ----;- <'l l') ^'? . ^ " \ Elanenf sfrtfness equ atrbn' r 1 _ = 1";rl !cy.l ' f rol \il{ =rnr{r} i;;| ut Iiv) L,I %- probbng " # t#).{- 2q€ ,o (rorsio-n. #(*l I = skess €unchbnfin torsional analgsi:. *(^41)* (l^ \ 0tr *(*]rt 0X\ tf _/ T. rewr. + e =o e(&.*d!)--e \ ?xbs'J Eleckicat Problenn 4 - etedric Aolanfral ivit-g of l\e 6 - Per,rr.l+f contluctor ' e - Volurire ohargedensify 'F[+(])*L/lg)l=o au \0y/J L0K\dx./ u- rnqnehL Potential 4 t{te nal'arial P - perrrreaH.L,lg rt 46 I at2, z- cz ft, vd k',D Ql .o) 8t"D C",-D (o,-, Staidard qudtatrc quadrilafaral e\evnenf {qil6xl tul=r': r-xl6 2xI I o 'l^\ Nr oNro LONr o rl1 . O N,t N9 o q,l 't qrl oNloN6 ,lT ' o ika Al4oNbo oNi o$l :i 46 U: Nr4r tNr4st '10 - tJr4r t r.lr-4+ lsopcarndrtc Srmulaf(ovt r.rlgg + Nq gr t \4q +,t'r'5(n t r(71rs1 Nr 4rS ' -l 2 r.tiX,' = N1X1* Nr,{l+ L tt,Ui = Nrgr + Nz.Jl+ - _^ \, l.^; or ,! ,b Ngxt + NrUB Nr' I = - -\' N.^ -4L | -t LrnSY)tf*$i) (t-3:;-4tr) Nq=-L Nat- -h (r-l)("1)(,, i-0 Ns=)- \ - / (r-1') ('-1) Nt" t r,l''r Ng * tr+j) (r-1) = t(t-3') (r+1) " t t'-9t'-9 IN tr Thr€e djnrensionolprob\ems r 1 t 3 9o\E q- 4. (q-ndu'd; Sotrt elen€nf. rtslrahedral ' n11"sh5nle5 t /r\ 'I t //(Qo, J i,lasfer elerrrer'rlfiY lhear 4uetrilah2ftl elem€nt {'}rr, = [u]r","1'4t, ql qL = [u' lo I o o Nr o o Lo Nr Nz o o r.lr o o Nl -l o o.ilqo'l o 0 o'ilr NL oor.l3ooN! 9V o N1 ol I alllr+ X= > Nixi -1 Lr{iY.' t:_ t = N1X1 i l!-Xrt NgYa-t n\rtx4 4 . r..l,Y,+ }.laVr+Nsq3i- r.l1St, -', 7= ? NiZi tat Shap {uncfiuns ll. - ? .J r.t". I N3rV Na --t-T.-n-!, tl iFCB] + {4') lD5 I <l Et\lt I a; l()T I )Y Ki \# t "l t" l. \?ct: \q l0x l'l II lnu }I lax ?9 lq i - f--J | ,ltl \r; -l LV 2v1 av\ I wl I lbv I IDTJ =[r] {$\ \;.^\ \?-'J [H .J.,Q t $.t' l I -1 l/t l L:J- lTl LJJ xrj -- xi- x'l \ -4\ I 7e+) a't I al-tI I r,r1 lq \ =lAl )!s 7 0'^l ) irl \"1 I lrl \ul IbY J ]=" t.l 'rl as lx A.\ 05 *t\ :"\ z2l' ) 0v bu. b$ 5qr br bvl +T ?g . ) 4 -D< lb,^tI |.6T,J o ,# * C. l5i I \* t J rt I Iau I {+}=t^r1H lH ll \^ I lDt 1 Lv l\ tet ''l I bu \i"- [ IdZ tl t5i I t ;.,1 J . azl fut l a9 fal llr il tf, u= Nt9r+ Nz94 tlrQ- t N3gB 1 N49rr Nr4r- + Nrqi t ilr ,ta+ N2q6 + Nqlro -r N3qq 1 r.t4?r- - 1il6*,,' lol t 'r lzxl {ti* = fp-l - ' i(l , | 6xb ext .|il - 'L*tc- \ oq '1 ) { 'rlzx I tDl = OO * o o o o q-q o ^E ' q? -- I | { .ls}r 1e1dv )' V , .J, \ { t(r't tsl lqll ( rDl{q]}v JJJ\u-^n =L L ^1 'iqi tslJq1dv. [D]'tgl lf l JJJ T T , . z(r+lt a )) -t --T 0 b * = {(tr,}") lr .O *q o U r O. ,L * It o O + =t^. llr llll ' {ql' L[;ir,t' f '!{r} b)L J ) ) = t{q}' 'LL ' Ju (^ r ) ( ta'l'tul'IBlv.J 11 | \ 't {o\-tri {e} ftl 1 , r(€J vlq r--1 : ve . Lhl -'f lr^-'rlf v"=t I d€+ trl I .--l LDI LDl =JJJ{ufr'a" dV ' llf(t-'lq])'n -jffiof r^.,1,, al d\ to = lql'\ Ur -r,\ (f( -/ t6llltNl ov/ \ '.rJ =101',F 1:l 'I ry I I {ul'{r" = I 11,]'r an Np an = [f(r*r{q"f)rr. tl\--r,,f JJ ( ffl^riIr.d, . \q]. \ ., \.r,,, =1q\'t! T.*Tl, .Tz, r*,T., .T7, Tr,! \t \ l- 1 =?l{'f [r lK t2q It q r lrlt r) , Tz , o. o, , "1 \Pl I I '( J+ I \tr It ,i' a^ q't ,{o;oO' L1r t) (1,v'z'; -t 13 a+l -t 4 Yl_1rt - | z 9 4 +l 17 {re + n'tr Ns116 1zz 1 u=rlrgr*Nr9rr + N,11,. rNgg+t 1N5q13 1J: f{1qltNz95a N3{9+ r|\11 9rr t nlsqh + Ns 1r+ uH^ t ^ls129 ,.1+9a + lr 12,t 1^1 . N11] + r.19161 ilgqq+ N4lrr1ilg{1y + ila9rr + !t /\fr = lul | Nl l:.. [nJ. lxn\ o Il .o'| 'r 2l{xl Nr"'l slr 8 x -- : Nixi t:1 s Y .. 'l |tr'vi t.t 8 .,.-. 7 -- a N'4 tir elenPd "hexahedtal sotld Shapfuvrehorts Ni=f(rt3j,)tr+l '1i,)'t I+ v vc) Q= 't n v" l( -' ?(e '' t't' - ' *\ of tlt<- coo<dinates eac^ vrde ln 116tur6l Jrv ' co-o<dinato9$shei\ r tl trrdv =I t ld€fLrJd1d1d{ l\ul'frt tgl tq =r"I JJ J" . JS' JJJ It Il{a', JJJaxrsr 1.. fi4 1, dl/ dr4' dY l'+ t-r'l u<, LJJ