CIVL 8/ D Boundary Value Problems - Rectangular Elements 1/7

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1 CIVL / -D Boundr Vlu Prolms - Rctngulr Elmnts / RECANGULAR ELEMENS - In som pplictions, it m mor dsirl to us n lmntl rprsnttion of th domin tht hs four sids, ithr rctngulr or qudriltrl in shp. Considr four-nod rctngulr lmnt with prlll sids dfind s: (, ) (, ) (, ) 0 0 RECANGULAR ELEMENS - W will us ilinr distriution of th u ovr th lmnt, thrfor: u (, ) 0 0 RECANGULAR ELEMENS - Whr th function is oth linr in nd nd ilinr in. h four unknowns in th ov prssion m found fitting th curv through th four nodl vlus. u, i,,,or i i i i i i (, ) 0 0 RECANGULAR ELEMENS - Solving th ov four qutions for,,, nd, nd sustituting ths vlus ck into th lmntl distriution, th following st of shp functions m formd: N, N, h pproimtion for u m writtn in mtri form s:,, u un u Nu N i i N N,, RECANGULAR ELEMENS - h lmntl intrpoltions for rctngulr lmnt r: N N RECANGULAR ELEMENS - h shp functions r visull dciving. hr is no curvtur in dirctions prlll to n sid; howvr, thr is twist du to th trm in th lmnt rprsnttion. N N N N N N

2 CIVL / -D Boundr Vlu Prolms - Rctngulr Elmnts / RECANGULAR ELEMENS - On vr importnt dvntg of this tp of intrpoltion is tht th drivtivs r no longr constnt ovr n lmnt. h drivtivs in nd r: u u u u u u u u u u u u Rctngulr shp functions lso suffr from intrlmnt discontinuitis, in prticulr, th vlus of norml drivtivs t n dgs. RECANGULAR ELEMENS - h shp function m writtn in trms of th locl coordints nd : N, N, N, N, h shp functions m writtn in trms of nondimnsionl locl coordints s nd t, whr s = /, nd t = /: s t st N s, t N s, t s t st N s, t N s, t RECANGULAR ELEMENS - h dimnsionlss shp functions r dfind ovr unit squr - s, - t. h vrition of th glol coordints nd m prssd in trms of th shp functions s: N N i i i i RECANGULAR ELEMENS - h lmntl mtrics for th Poisson prolm r: k A N N NN da NN ds Lt s first considr how to hndl intgrtion ovr tpicl lmnt r A. For mpl: F, dd A f h A NfdA Nhds F, dd RECANGULAR ELEMENS - his intgrl m trnsformd into th locl coordint spc using = - 0 nd = - 0 : F, dd, F0, 0 dd, A whr (, )/(,) is th Jcoin of th trnsformtion. Using th trnsformtion = s nd = t th intgrl m writtn s: F, dd A, 0, 0 F s t dsdt st, A RECANGULAR ELEMENS - Evlution of k o convrt th intgrl into th dimnsionlss spc (s, t) th following trnsformtion is usd: s 0 Using th chin rul th drivtivs m writtn s: s t s s t t t 0 whr th Jcoin (,)/(s, t) hs vlu of A /.

3 CIVL / -D Boundr Vlu Prolms - Rctngulr Elmnts / RECANGULAR ELEMENS - Evlution of k In mtri form th ov prssion m writtn s: s s 0 J J s 0 t t h mtri form of th trnsformtion m invrtd. J J s s h mtri form of th trnsformtion m invrtd. J J s s whr J nd J r th first nd scond rows of J - RECANGULAR ELEMENS - Evlution of k hrfor, th prtil drivtivs of th shp functions m writtn s: N N N N J J J J s s Sustituting ll th pics of th trnsformtion in th k trms givs: A k JJ JJ ds dt JJ ds dt whr JJ = (J J + J J )A /. RECANGULAR ELEMENS - Evlution of k h vlus of th mtri nd JJ m computd s: t t t t 0 JJ s s s s 0 h rsulting lmntl stiffnss mtri k is tht is ddd to th glol sstm stiffnss mtri K t th corrsponding nodl loctions. k RECANGULAR ELEMENS - Evlution of k If th lmnt is squr, thn =, thn k coms: k k RECANGULAR ELEMENS - Evlution of f In gnrl, th intgrl is: f For gnrl function f(, ) w will ssum th f vris linrl ovr th lmnt, f(, ) = N f, whr th vctor f contins vlus of th function f t th nod points. With this ssumption th intgrl coms: A NfdA f NN fda da NN f A A RECANGULAR ELEMENS - Evlution of f rnsforming th intgrl into th non-dimnsionl coordints (s, t) ilds: A f NN ds dt f f f f f A f f f f f f f f f f f f f h rsulting lmntl lod vctor contriuts to th glol sstm qutions t thos loctions corrsponding to th four nods dfining th lmnt.

4 CIVL / -D Boundr Vlu Prolms - Rctngulr Elmnts / RECANGULAR ELEMENS - Evlution of h Considr th intgrl: h Nhds whr th intgrtion is long oundr sgmnt of th lmnt. Sinc, th intgrtion is computd long singl sid of th rctngulr lmnt, th originl shp functions rduc to: j k h k s 0 i l h l l s l h Nk hn k k hn l lds Nl s s Nk Nl l l RECANGULAR ELEMENS - Evlution of h Evluting th intgrls, th h trms rduc to: h l h k hl hk hl h rsulting lmntl lod vctor contriuts to th glol sstm qutions if th lmnt hs sid s prt of th oundr. RECANGULAR ELEMENS - Evlution of h vlution of th is vr similr to h cpt tht thr is n tr N in th intgrnd. h vrition of th function (s) will pproimtd s k N k + l N l. Considr th intgrl : NN ds l k l k l k l k l RECANGULAR ELEMENS Rcll, th glol sstm qutions r composd from th following summtions: K k F f h G G G G G G h rsulting sstm qutions r, in mtri form, givn s: Ku F G G G h rsulting stiffnss mtri contriuts to th glol sstm qutions whn th lmnt hs sid s prt of th oundr. RECANGULAR ELEMENS PROBLEM #9 - Vrif th componnts of th k, f,, nd h mtrics conform to th rsults dvlopd in th clss nots. In othr words, show ll th dtils of th drivtion of th stiffnss mtri nd th loding vctors. RECANGULAR ELEMENS Empl - Considr th non-dimnsionl torsion prolm: with, 0 in 0 on G h strsss nd torqu for th Prndlt strss function r: z G z G G dd

CIVL / -D Boundr Vlu Prolms - Rctngulr Elmnts / RECANGULAR ELEMENS Empl Discrtiztion - h simplst modl for torsion

Lins of Smmtr 0 (,) (0,) 0 0 n (0,0) (,0) cross-sctionl r 0 n RECANGULAR ELEMENS Empl - Elmntl Formultion - Using

f f f f f f f f f f f f RECANGULAR ELEMENS Empl - Assml - Sinc thr is onl on lmnt in th modl th ssml is simpl:

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 RECANGULAR ELEMENS Empl - Solution - In this cs, th solution is quit simpl: G

g RECANGULAR ELEMENS Empl - Solution - In this cs, th solution is quit simpl: G (0,0) (0,) A (,0) (,) 0 (,) (0,)

5 CIVL / -D Boundr Vlu Prolms - Rctngulr Elmnts / RECANGULAR ELEMENS Empl Discrtiztion - h simplst modl for torsion of squr r, utilizing smmtr is singl rctngulr lmnt. h gnrl prolm domin nd th FEM msh r shown low. Lins of Smmtr 0 (,) (0,) 0 0 n (0,0) (,0) cross-sctionl r 0 n RECANGULAR ELEMENS Empl - Elmntl Formultion - Using linr rctngulr lmnt th lmntl stiffnss mtri componnts r: k A loding function of f = givs lod vctor of: f f f f A f f f f f f f f f f f f f RECANGULAR ELEMENS Empl - Assml - Sinc thr is onl on lmnt in th modl th ssml is simpl: Constrints - For this modl, = 0 on th oundr, thrfor,, nd = RECANGULAR ELEMENS Empl - Solution - In this cs, th solution is quit simpl: G Computtion of Drivd Vrils - h prtil drivtivs with rspct to nd tht dfin th strss componnts r: G A G dd G.G G ct.g RECANGULAR ELEMENS Empl - Solution - In this cs, th solution is quit simpl: G (0,0) (0,) A (,0) (,) 0 (,) (0,) 0 0 n (0,0) (,0) 0 n RECANGULAR ELEMENS Empl Rpt th prvious non-dimnsionl torsion prolm using four rctngulr lmnts:, 0 in 0 on with G h strsss nd torqu for th Prndlt strss function r: z G z G G dd

6 CIVL / -D Boundr Vlu Prolms - Rctngulr Elmnts / RECANGULAR ELEMENS Empl Discrtiztion - Discrtiztion - h simplst modl for torsion of squr r, utilizing smmtr nd using four rctngulr lmnts is shown low. Lins of Smmtr 0 9 (0,) (,) 0 0 n (0,0) (,0) cross-sctionl r 0 n RECANGULAR ELEMENS Empl - Elmntl Formultion - Using linr rctngulr lmnt th lmntl stiffnss mtri componnts r: k For lmnt & : k k RECANGULAR ELEMENS Empl - Elmntl Formultion - Using linr rctngulr lmnt th lmntl stiffnss mtri componnts r: k For lmnt & : k 9 k 9 RECANGULAR ELEMENS Empl - Elmntl Formultion - A loding function of f = givs lod vctor of: f f f f A f f f f f f f f f f f f f f f f f 9 RECANGULAR ELEMENS Empl - Assml - Compiling ch lmntl mtri into th sstm qutions ilds: RECANGULAR ELEMENS Empl - Constrints - For this modl, = 0 on th oundr, thrfor,,,,, nd 9 =

CIVL / -D Boundr Vlu Prolms - Rctngulr Elmnts / RECANGULAR ELEMENS Empl - Solution - In this cs, th

h prtil drivtivs with rspct to nd tht dfin th strss componnts r: G dd ct,9g,0 G A.0G.

strss componnts r: (0,) 9 (,) (0,0) (,0) 0 n (0,) 0 9 (0,0) (,0) 0 n (,) 0 RECANGULAR ELEMENS PROBLEM #0 -

7 CIVL / -D Boundr Vlu Prolms - Rctngulr Elmnts / RECANGULAR ELEMENS Empl - Solution - In this cs, th solution is quit simpl: G G 0G 0G G RECANGULAR ELEMENS Empl - Computtion of Drivd Vrils - h prtil drivtivs with rspct to nd tht dfin th strss componnts r: G dd ct,9g,0 G A.0G.G A RECANGULAR ELEMENS Empl - Computtion of Drivd Vrils - h prtil drivtivs with rspct to nd tht dfin th strss componnts r: (0,) 9 (,) (0,0) (,0) 0 n (0,) 0 9 (0,0) (,0) 0 n (,) 0 RECANGULAR ELEMENS PROBLEM #0 - Considr th non-dimnsionl torsion prolm:, 0 in 0 on G Lins of Smmtr cross-sctionl r 0 n (0,) 0 9 (,) (0,0) (,0) 0 n 0 End of Rctngulr Elmnts

INTEGRALS. Chapter 7. d dx. 7.1 Overview Let d dx F (x) = f (x). Then, we write f ( x)

INTEGRALS. Chapter 7. d dx. 7.1 Overview Let d dx F (x) = f (x). Then, we write f ( x) Chptr 7 INTEGRALS 7. Ovrviw 7.. Lt d d F () f (). Thn, w writ f ( ) d F () + C. Ths intgrls r clld indfinit intgrls or gnrl intgrls, C is clld constnt of intgrtion. All ths intgrls diffr y constnt. 7..