DEFLECTIONS OF THIN PLATES: INFLUENCE OF THE SLOPE OF THE PLATE IN THE APLICATION OF LINEAR AND NONLINEAR THEORIES

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1 Procdigs of COBEM 5 Coprigh 5 b BCM 8h Iraioal Cogrss of Mchaical Egirig Novmbr 6-, 5, Ouro Pro, MG DEFLECIONS OF HIN PLES: INFLUENCE OF HE SLOPE OF HE PLE IN HE PLICION OF LINER ND NONLINER HEORIES C.. S. Bordó Uivrsias, v. Dr. oio Braga Filho, 687, Iajubá, MG, Brasil bordo@fpi.br W. C. Olivira Uifi, v. B. P. S., 33, Iajubá, MG, Brasil lamir@uifi.du.br P. S. Id Uifi, v. B. P. S., 33, Iajubá, MG, Brasil shigum@uifi.du.br Márcio adu d lmida Uifi, v. B. P. S., 33, Iajubá, MG, Brasil madu@uifi.du.br bsrac. I hi plas, alas ha h dflcio ad h slop o do cd, rspcivl, % of hickss ad -3 rd, h Liar hor (L) ca b apllid. If hs limis of dflcio ad slop ar cdd, h Noliar-hor (NL) of h plas, hich aks io accou h displacm of h midpla of h pla, has o b usd. lo of auhors hav chckd h applicaio limis from h L ad h NL hrough h umrical ad aalical mhods. Hovr, h aalsis ar carrid ou b chckig ol h rlaio b load-dflcio ad vr h rlaio pla load-slop. h objciv of his ork is o sud h rlaio load-slop of h pla usig h L ad h NL ad o sablish h applicaio limis of hs horis. h ork is dvlopd o squar plas, clampd ad simpl suppord, sumid o uiforml disribud load. h fii lm mhod ih isoparamric quadrilaral lms is applid i h umrical soluios. Kords: hi Plas, Larg Dflcios, Slop, Fii Elm Mhod. Iroducio Ma applicaios of girig srucurs us plas as srucural lms, i mchaics, i civil or aviaio, for isac i mal plaforms, i commrcial floor or idusris, gi discs, ak bass, coairs ad aks of diffr sizs ad ha hav o suppor iral ad ral ffors. hs impora applicaios ld o ma rsarchrs, i h bgiig of 8, sudig his problm i ordr o dvlop a pla fudamal hor, o g aalic soluios for simpl cass. From 93, som umric mhods r dvlopd for solvig compl problms. h rsuls from hos aalic orks, from ha im, ar sill usd as bas of compariso ih umric modls. akig Kirchhoff hpohsis for liar hor L (or classic hor, or small displacm hor) for hi plas, isoropic plas, homogous ad lasic plas, rsuls < () 5 θ < 3 rd () h displacm limi from Eq. () is chckd i ma umric ad aalic dvlopm (Chia, 98; Dua ad Mahdra, 3; Zhag ad Chug, 3), hil h slop limi θ from Eq. () did o rciv aalsis bfor.. Fudamal quaios for hi plas subjcd o small dflcio akig a ifiisimal lm dd from a pla ha is udr load prssur across all surfacs, disribud i a uiform a i h ara, p, h applicaio of quilibrium quaios ad Hook La provids h diffr quaio blo

2 + p + D (3) hr 3 D () E ( ) h Eq. (3) is ko b Sophi-Grmai-Lagrag quaio, prsd i 8. h Eq. () rprss h pla flural rigidiis I his quaio, E is h marial modulus of lasici, is h Poisso s raio ad is h hickss of h pla. 3. Formulaio of h fii lm mhod for h problm of hi plas subjcd o small dflcios 3.. Elm propris For solvig b umbrs h problm of hi plas, h rcagular pla is discrizd i quadrilaral lms. h pla lm, placd i pla, is sho i Fig.. Each odal poi has hr dgr of frdom: h vrical displacm,, h roaio aroud ais, θ, ad h roaio aroud ais, θ. h roaios ar rlad ih h slops hrough θ, θ (5) Figur Quadrilaral Elm h vcor of odal displacms of h lm is rprsd b {} δ {, θ, θ,, θ, θ,, θ, θ,, θ, θ } i i i j j j m m m (6) h displacm fucio,,dfi h displacm of a poi of h lm. h odal displacms lik o h displacm fucio hrough h prssio { } [ P]{} δ (7) h mari [P] is a fucio of h poi posiio cosidrd ad i is calld h shap mari. For ach lm, h liar hor for hi plas provids (a) h vcor of dformaio-displacm: ε ε γ,, (8)

3 Procdigs of COBEM 5 Coprigh 5 b BCM 8h Iraioal Cogrss of Mchaical Egirig Novmbr 6-, 5, Ouro Pro, MG hich rsuls i {} ε [ B] {} δ (9) (b) h rlaio of srss-displacm: τ Ez ( )/ {} ε () (c) h moms accordig h srsss: M / M dz () / M { } M z{ } Usig h Eq. (), h soluio of Eq. () provids { M} [ D] { ε} hr [ ] () 3 E ( ) ( )/ D (3) is h lasici mari of h pla lm. 3.. Poial rg pricipl For drmiig h quaios hich ar usd i h fii lm mhod of h hi plas, i is applid h pricipl of h miimum poial rg (Pilk ad Wudrlich, 99). h variaio of h poial rg, Π, of a pla is Π ( M + M ε + M ε ) dd ( p ) dd ε () hr is h umbr of lms hich cosiss h pla ad is h surfac ara of h lm. h Eq. () ca b rri i his form ({ } { M} p ) dd ε (5) Wih h rplacms of h Eq. (7), (9) ad () i h Eq. (5), i is obaid { } ([]{} k δ {} Q ) δ (6)

4 I h Eq. (6), h siffss mari of h lm, [k] ad h vcor of odal forcs of h lm, {Q}, ar, giv, rspcivl, b [] [ B][ D][ B] k dd (7) { } [ P] Q pdd (8) s h variaios i {δ} ar idpd ad arbirar, h Eq. (6) lads o h folloig prssio o h quilibrium of odal forc of h lm []{} k {} Q δ (9) Cosidrig, o, h hol pla, h Eq. (6) provids h folloig quaio ha is valid for a variaio of displacm { δ} { } ([ K]{} δ {} Q ) δ () hos dvlopm rsuls i [ K ]{} {} Q δ () h siffss mari of h pla, [K] ad h pla odal vcor forcs, {Q} ar obaid b suprposiio of h siffss mari ad odal forcs of h lms, lik ad [ K] [ k] { Q} { Q} () (3) h gral procdur for solvig problms of hi plas subjcd o small dformaios ca b summarizd io h folloig sps (Ugural, 98):. Drmi [k] hrough Eq. (7) i rms of lm propr. Gra [K] hrough Eq. ().. Drmi {Q} hrough Eq. (8) i rms of h load applicad. Gra {Q} hrough Eq. (3). 3. Drmi h odal displacm hrough Eq. (), saisfig h boudar codiios.. Drmi h srsss ad h moms i h lm hrough h Eqs. () ad (), rspcivl Isoparamric quadrilaral lm h displacm fucio is prssd b a hird ordr polomial, lik a + a3 + a + a5 + a6 + a7 + a8 + a9 + a + a a () a + ha dfis h displacm of a poi of h lm ijm. h umbr of h rms of his fucio is qual o h umbr of dgrs of frdom of h lm. h rplacm of h Eqs. (5) ad () i h Eq. (7) rsuls i

5 Procdigs of COBEM 5 Coprigh 5 b BCM 8h Iraioal Cogrss of Mchaical Egirig Novmbr 6-, 5, Ouro Pro, MG hr {} [ C]{} a δ (5) Ev so, {} a [ C] {} δ (6) I h Eq. (6), h mari [C] ol dpds of h odal coordia lm. Hovr, h Eq. (7) is rsricd lik { } [ L] { a} (7) [L] [,,,,,, 3,,, 3, 3, 3 ] Usig h Eqs. (9), (5), (6) ad (7), h vcor displacm ca b prssd lik {} [ B][ C]{} a [ H ]{} a ε (8) h dformaio-displacm mari is obaid rplacig h Eq. (6) i h Eq. (8). h rsulig quaio supplis ha [B][H][C] -. h rplacm of his quaio i Eq. (7) rsuls i h quaio of h siffss mari of h lm lik [] k [ C] [ ] [ H ] [ D][ H ] dd [ ] C (9). Fudamal quaio o dflcio of hi plas subjcd o larg dflcios Wh h dformaios ar larg, h pla lm mus b cosidrd i is dformd codiio. h applicaio of Hook s La ad h quaios of quilibrium of plas, i his cas, rsuls i φ φ + φ + E (3) + + D p φ φ φ + + (3) h Eqs. (3) ad (3) ar h quaios dvlopd ad prsd i 9 b Vo Kármá. I hs quaios, φφ(,) is h srss fucio. 5. Formulaio of h fii lm mhod o h problm of h hi plas subjcd o larg dflcios 5.. Elm propris h formulaios of plas subjcd o small dflcios is usd i h cas of larg dflcios, ha icluds h ffc of dformaios i h midpla of h pla ad is rspcd srss. For his, i is cosidrd a pla subjcd a firs o h applicad forcs i h midpla, hich sad cosa durig h bdig. I his cas, h dformaios ad h srss io a midpla of h lm ca b rprsd b (Ugural, 98),

6 {},, ε (3) N N N τ or { } { } N (33) h dformaios o h bdig ad moms ar giv accordig o h Eqs. (8) ad (). h rsulig srsss iclud h srss provocad b dircd forcs ad b flural moms. h srss ad dformaios du o bdigs mi hrough h Eq. (). h srss ad dformaios i h pla ar rlad b {} Ez ε τ )/ ( (3) 5.. Poial rg pricipl Cosidrig ha h dformaios du o dircd forc ad du o h bdig ar idpd, h prssio o h poial rg of h pla is giv b {} { } [ ] ( ) + Π dd p dd dd M ε (35) alhough [ ] τ τ (36) i is h mmbra srsss mari i h midpla of h pla. Wih h displacm fucio giv b Eq. (), h slops i h lm, θ θ, ar rprsd b {} [ ]{} a S θ θ θ (37) Rplacig h Eq. (6) i h Eq. (37), i is obaid {} [ ][ ] {} [ ]{ } G C S δ δ θ (38) hr [G] is ol fucio of h coordia of h odal pois of h lm. Rplacig h Eq. (37) i h Eq. (35), i has

7 Procdigs of COBEM 5 Coprigh 5 b BCM 8h Iraioal Cogrss of Mchaical Egirig Novmbr 6-, 5, Ouro Pro, MG Π {}{ ε } {} [ ] [ ][ ] M dd + δ G G dd {} δ ( p) dd (39) h applicaio h miimum poial rg pricipl supplis a modifid prssio o h quilibrium of h odal forcs of h lm, lik { Q} [ k]{} δ + [ k G ]{} δ [ k ]{} δ () I h Eq. (), h rm [k G ] i is calld iiial srss mari or gomric srss mari, hich ca b obaid hrough [ ] [ ] [ ][ ] [ ] [ ] [ ] k C S S dd C G () h mari [k ] of Eq. () is calld oal siffss mari of h lm. h gral procdur for solvig pla problms of hi plas subjcd o larg dflcios ca b summarizd i h folloig sps:. Cosidr h srss of Eq. (3) du o forcs i h pla, a firs qual o zro. ppl h procdurs (sps o 3) of h scio 3. o obai h soluio of odal displacm o small dflcios.. Drmi h slop io h croid of ach lm hrough Eq. (38). 3. Drmi h srais of h midpla hrough Eq. (3).. Drmi h srsss o h midpla hrough Eq. (3). 5. Drmi h gomric siffss mari hrough Eq. (). 6. Drmi h oal siffss mari of h lm. 7. Rpa h sps o, uil hr is a saisfid covrgc o h srss i h midpla of h lms of h pla. 6. Numrical ampls Wih h prsd formulaio abov, a compuaioal program as laborad o drmi dflcios ad slops of hi plas ih oliar bhavior. h obaid rsuls r compard o h umrical ad aalic rsuls obaid b ohr auhors (Chia, 98; Pica ad Wood, 979; Sigh ad Elaghabash, 3). h rsuls ar prssd io h odimsioal form o h load, Q pa /E, ad o h dflcio, W /. I as aalzd a squar pla hich sid a m, hickss 8 cm, E GPa,.36. h Fig. ad Fig. 3 prs h dflcio i h middl of h pla compll clampd ad simpl suppord, rspcivl. odimsioal dflcio W,,,,8,6,,, papr Chia Pica Sigh odimsioal load Q odimsioal dflcio W,,9,8,7,6,5,,3,,, papr Pica Sigh odimsioal load Q Figur. Dflcio i h cr clampd pla Figur 3. Dflcio i h cr simpl suppord pla h graphics of maimum slop du o h loadig for plas ih ih clampd ad simpl suppord boudaris ar sho, rspcivl, i h Figs. 5.

8 ,5,,, maimum slop,5, maimum slop,8,6,,5, Liar hor Noliar hor odimsioal load Q,, Liar hor Noliar hor odimsioal load Q Figur. Maimum slop for clampd pla Figur 5. Maimum slop for simpl suppord pla 7. Coclusios ccordig o h Fig. ad Fig. 3 shod, o h h squar clampd pla i all h sids, h maimum diffrc b h rsuls i his ork ad h obaid rsuls b Chia, Pica ad Sig is smallr ha 5%. For squar simpl suppord plas i all h sids, h maimum diffrc is smallr ha 7%. I h cas of clampd plas, h Fig. shos ha v h agl of.5 rd h L shos saisfid rsuls, or ihr, ih a 5% limi ovr h ormal usd of. rd. For simpl suppord plas, h Fig. 5 shos ha his limi is of.6 rd, or ihr, 6% ovr h usd limi. h dvlopd modl do o covrg o valus of odimsioal loads ovr 5 i h cas of clampd plas, ad of 3 o suppord plas. hs valus ar lor o h os obaid b ohr auhors, ho us odimsioal loads qual o. Nvrhlss, i h cas of hi plas, isoropic, homogous ad lasic, ormall usd i h srucural applicaios i h diffr aras of girig, odimsioal loads ovr 5 lad h srss valus alrad, i h pla, ovr h lasic limi. 8. Rfrcs Chia, C.Y., 98, Noliar alsis of Plas, McGra-Hill, Uid Sas of mrica, p. Dua, M. ad Mahdra, M., 3, Larg Dflcio alsis of Sk Plas Usig Hbrid/Mid Fii Elm Mhod, Compur ad Srucurs, 8, pp Pica,. ad Wood, R.D., 979, Fii Elm alsis of Gomricall Noliar Pla Bhavoir Usig a Midli Formulaio, Compur ad Srucurs,, pp. 5-. Pilk, W.D. ad Wudrlich, W., 99, Mchaics of Srucurs Variaioal ad Compuaioal Mhods,CRC Prss, Uid Sas of mrica. Sigh,.V. ad Elaghabash, Y., 3, O h Displacm alsis of Quadragular Plas, Iraioal Joural of No-Liar Mchaics, 38, pp Ugural,.C., 98, Srsss i Plas ad Shlls, McGra-Hill, Uid Sas of mrica, 37 p. Zhag, Y.X. ad Chug, Y.K., 3, Gomric Noliar alsis of hi Plas b a rfid Noliar No- Coformig riagular Pla Elm, hi-walld Srucurs,, pp Rsposibili oic h auhors ar h ol rsposibl for h prid marial icludd i his papr.