Analysis of von Kármán plates using a BEM formulation

Transcription

1 Boudary Elemets ad Other Mesh Reducto Methods XXIX 213 Aalyss of vo Kármá plates usg a BEM formulato L. Wademam & W. S. Vetur São Carlos School of Egeerg, Uversty of São Paulo, Brazl Abstract Ths work deals wth o-lear geometrcal plates the cotext of vo Kármá theory. The formulato s wrtte a way to requre oly boudary -plae dsplacemet ad deflecto tegral equato for boudary collocatos. At teral pots oly out of plae rotato, curvature ad -plae teral force represetatos are used. The o-lear system of algebrac equatos to be solved s reduced to teral pot collocato relatos. The soluto s solved by usg a Newto scheme for whch a cosstet taget operator was derved. Keywords: bedg plates, geometrcal oleartes. 1 Itroducto The boudary elemet method (BEM) appled to solve plate-bedg problems has bee successfully used may tmes so far. A mportat characterstc of the boudary methods appled to plate bedg s approxmatg all boudary values by the same shape fucto, avodg therefore usg hgher order dervatves of dsplacemet approxmato to compute teral forces. Thus, bedg ad twstg momets ad also shear forces are precsely evaluated. The method has already proved to be eough accurate ad relable for ths kd of applcato. The plate bedg umercal formulato s a very mportat subject egeerg due to be appled to a large umber of complex problems such as arcraft, shp, car, pressure vessel, off shore structures amog others. Usually these complex problems requre accurate plate bedg models as those that take to accout the geometrcal o-lear effects. I ths cotext, several BEM formulatos have already bee proposed so far. Oe of the frst works treatg ths subject s due to Morjara [1]. Kamya ad Sawak [2] have proposed a BEM formulato for elastc plates govered by the Berger equato. The frst BEM WIT Trasactos o Modellg ad Smulato, Vol 44, ISSN X (o-le) do: /be070211

2 214 Boudary Elemets ad Other Mesh Reducto Methods XXIX formulato to aalyze plate-bedg problems wth the cotext of vo Kármá hypothess s due to Ye ad Lu [3], who have used a fcttous loadg dstrbuted over the doma to model the o-lear effects. Vo Kármá hypothess was also adopted by Taaka et al [4] to develop a more elaborated BEM cremetal formulato to deal wth fte deflectos of th elastc plates. Wag et al [5] have also worked o vo Kármá plates troducg the dual recprocty approach based o global radal fuctos to approxmate the correctg tegral term. All works reported above were proposed the cotext of th plates. Several other mportat works have appeared more recetly potg out the effcecy of BEM formulatos to deal wth shear deformable plate based o the Resser- Mdl hypothess: We et al [6], Purbolaksoo ad Alabad [7]. I ths work we came back to the BEM formulato based o the vo Kármá s theory. Emphass s gve to the accurate evaluato of the doma tegrals approxmated by usg cells ad to the soluto techque for whch a taget cosstet operator s proposed. Examples of plate wth fte deflecto s aalysed ad the results compared wth other umercal solutos. 2 Basc equatos Wthout loss of geeralty, let us cosder a sgle th plate rego wth boudary over whch a dstrbuted load q s appled orthogoal to the mddle surface, (drecto x 3 ), as show fgure 1. Ths plate rego ca also be subjected to plae forces (drectos x 1 ad x 2 ) ether dstrbuted over the doma or appled alog the boudary. I order to wrte the feld equatos of ths plate problems followg the hypothess ca be assumed accordg to the vo Kármá theory, for whch the stras are assumed to be eough small ad the fal deflecto of order of the plate thckess h. x 1 g x 2 x 3 Fgure 1: Geeral plate doma. For ay pot defed the followg basc relatoshps are defed: - Equlbrum equatos for the bedg problem: m,+nw,-bw,+m,+q=0 (1) j j j j WIT Trasactos o Modellg ad Smulato, Vol 44, ISSN X (o-le)

3 Boudary Elemets ad Other Mesh Reducto Methods XXIX 215 where m j are bedg ad twstg momets, N j s the -plae teral forces, b represets the -plae doma loads appled ad m s the appled momet over the plate doma; the subscrpts are the rage,j={1, 2}. - The -plae equlbrum equato: N + b = 0 (2) j, j Assumg lear elastc materal eqs (1) ad (2) ca be wrtte terms of -plae ad out of plae dsplacemets as follows: Dw, = g + N w, jj j j (, j = 1,2) (3) 1 1 ( u, + w, w, ) + ( u, + w, w, ) + b /( Eh) (4) j j j j jj jj 21-ν ( ) 21+ν ( ) 3 2 where D= Eh /( 1-ν ) s the flexural rgdty, E ad ν are the materal Youg modulus ad Posso s rato. The problem defto s the completed by assumg the followg boudary codtos over : u = u o (geeralsed dsplacemets, plae 1 dsplacemets, deflectos ad rotatos) ad p = p o (geeralsed 2 tractos, -plae tractos ormal bedg momet ad effectve shear forces), where = Itegral represetatos I ths secto, we are gog to derve the tegral equatos of the plate bedg ad stretchg problems cosderg geometrcal o-leartes wth the cotext prevously defed. To obta the tegral equatos of both problems oe ca apply the Bett s recprocty to the lear parts of the stress ad stra felds. Thus, for the bedg problem the geeral recprocty relato wrtte for the 3D case ca be tegrated across the plate thckess to gve: w m d mw d (5) m, =, j j j j m * where w ad m * are the well-kow fudametal solutos of the plate j problem gve terms of deflecto ad teral momets. These fudametal values ad the other resultg requred values are gve the specalzed lterature [8]. By tegratg eq (5) by parts twce ad replacg the secod dervatve of the teral, m,, accordg to eq (1) oe has: j j WIT Trasactos o Modellg ad Smulato, Vol 44, ISSN X (o-le)

4 216 Boudary Elemets ad Other Mesh Reducto Methods XXIX * (,,, ) ( ) c (, ) c( ) ( ) ( ) ( ) ( ) ( ) ( ) = 1 * ( V( P) w ( S, P) M( P) w, ( S, P) ) d ( P) Rc ( P) wc ( S, P) ( ) (, ) ( ) ( ) ( ) (, ) g p w S p d p + Nj p w, j p w S p d g = 1 C S w S + V S P w P M S P w P d P + R S P w P = + + (6) where V ad boudary, respectvely, M are effectve shear forces, ad momets appled alog the * R c represets the corer reactos, V, M ad R c the correspodg values obtaed from the fudametal soluto w * accordg to ther defto. For ay teral collocato s oe ca dfferetate eq (6) to obta the tegral represetatos of rotatos ad curvatures, as follows: { } ( ) * ( ) ( ) ( ) ( ) ( ) ( ) ( ) w, s = V, s, P w P M, s, P w, P d P R, s, P w P ck ck k=1 * { V( P) w, ( q, P) M ( P) w, ( s, P)} d ( P) Rck ( P) wck, ( s, P) k=1 ( ), (, ) ( ) jk ( ), jk ( ), (, ) ( ) g p w s p d p + N p w p w s p d p g (7) { } ( ) * ( ) ( ) ( ) ( ) ( ) ( ) ( ) w, s = V, s, P w P M, s, P w, P d P R, s, P w P j j j c j c k = 1 * { V( P) w, j ( s, P) M( P) w, j ( s, P)} d ( P) Rc( P) wc, j ( s, P) g k = 1 ( ), j (, ) ( ) km ( ), km ( ), j (, ) ( ) + g p w s p d p + N p w p w s p d p (8) Aalogously, the Bett s recprocty relato for ca be appled to the lear parts of the 2D stretchg problem to gve: N ε d= N ε d (9) j j j j where N j represets the lear parts of the stretchg problem teral forces, therefore gve by: Eh 1 N u, ν = ν δ + ( u, + u, ) j 2 (1 ν) k k j 2 j j (10) After replacg N j eq (13) ad tegratg t by parts the followg tegral represetato s obtaed for the -plae boudary dsplacemets: WIT Trasactos o Modellg ad Smulato, Vol 44, ISSN X (o-le)

5 Boudary Elemets ad Other Mesh Reducto Methods XXIX 217 ( ) ( ) (, ) ( ) (, ) ( ) C S u S = P S P u P d+ u S P p P d+ j j j j j j * 1 * + uj ( S, p) bj ( p) d Njk ( S, p) w, j ( p) w, k ( p) d 2 (11) By dfferetatg eq (11) ad the applyg the Hooke s law at the collocato pot s accordgly, oe has: ( ) ( ) ( δ ) Nj s = D jk (s,p)p k (P)d S jk (s,p)u k (P)d + D jk(s,p)b k(p)d (12) 1 Gh T jkl (s,p)w,k (p)w,l(p)d + 2w, (s)w, j (s) + w,k (s)w,k (s) j υ where ν = ν/(1+ν) s used to smulate the plae stress codtos. 4 Algebrac equatos Before trasformg the tegral represetatos derved the prevous secto to algebrac equatos let us replace the rate values by the correspodg cremets. Let t = t+ 1 t be a typcal tme-step the tme dscretzato. Ay rate quatty x tegrated alog the tme terval t becomes x = x+1 x that wll replace x all tegral represetatos already derved. As usual for ay BEM formulato, the tegral represetatos (6), (7), (8), (11) ad (12) have to be trasformed to algebrac expressos after dscretzg the boudary ad the doma. The plate boudary s the dscretzed to elemets, s, alog whch geeralzed dsplacemets ad tractos are approxmated usg cotuous ad dscotuous lear boudary elemets. I plate stretchg-bedg problems dscotutes are always preset, partcularly at corers ad tracto jumps. The dscotuty s always troduced by defg the collocato alog the elemet or at ay outsde pot ear the boudary. Before trasformg the tegral represetatos, we decded replacg the desty of the doma tegrals to sgle values. The doma value Nj ( p) w, j ( p ) at a feld pot p wll be replaced by a scalar value T( p ). Smlarly, for the stretchg problem, we replaced the doma value w, j ( p) w, k ( p ) by doma tesor Wjk ( p ). The approxmato over the cells are ow appled to these ew doma values T( p ) ad Wjk ( p ). Tragular teral cells wth lear shape fuctos are used wth odes always defed at teral pots. Thus, dscotuous cells are requred for cells adjacet to the boudary. The cell tegrals are frst trasformed to tegral over ther sdes ad carred out as boudary elemets usg ether aalytcal or approprate umercal tegrato scheme wth sub-elemetato [9]. Carryg out the boudary ad cell tegrals eq (6) wrtte for all boudary odes leads to the followg cremetal matrx equato: WIT Trasactos o Modellg ad Smulato, Vol 44, ISSN X (o-le)

6 218 Boudary Elemets ad Other Mesh Reducto Methods XXIX w b w b w χ w b = b + b N + b H U G P S T B (13) b b where U ad P cotas all boudary dsplacemet ad tracto odal values of the bedg problem plus the addtoal corer dsplacemets ad χ reactos, T N represets the codesate summato of the -plae forces multpled by curvatures, S b w s the correspodg matrces obtaed by w tegratg all doma cells ad B b gves the doma load effects. Smlarly oe ca also trasform the -plae dsplacemet tegral represetato, equato (7), to ts cremetal matrx form: H U =G P + S W + B (14) u s u s u θ u s s s θ s s where U ad tracto odal values of the stretchg problem, of the rotato product defed at each doma ode, matrces obtaed by tegratg all doma cells ad s P cotas all cremetal boudary dsplacemet ad θ W θ represets the cremet u S s s the correspodg u B s gves the -plae doma cremetal load effects. To complete the ecessary algebrac relatos oe has to obta the algebrac forms of the tegral equatos (7), (8) ad (12), as follows: θ b θ b θ χ θ θ =-Hb U + Gb P + Sb TN + B b (15) χ b χ b χ χ χ χ =-Hb U + Gb P + Sb TN + B b (16) N s N s N θ N N =-Hs U + Gs P + Ss Wθ + B s (17) where θ, χ ad N are vectors cotag rotato, curvature ad membrae teral force cremets at the doma odes defed by the adopted dscretzato. After applyg the boudary codtos equatos (13), (14) ad (15) (17) become: w b w w χ Ab X = Fb + Sb T N (18) u s u u θ As X = Fs + Ss W θ (19) θ b θ θ χ θ =-Ab X + Fb + Sb T N (20) χ b χ χ χ χ =-Ab X + Fb + Sb T N (21) N b N N θ N =-As U + Fs + Ss W θ (22) y y y where the matrces A x are coveetly bult up usg colums of H x ad G x y correspodg to ukow boudary values, ad F x s a vector collectg all cotrbutos of prescrbed doma ad boudary values. WIT Trasactos o Modellg ad Smulato, Vol 44, ISSN X (o-le)

7 Boudary Elemets ad Other Mesh Reducto Methods XXIX 219 Solvg eqs (18) ad (19) ad replacg to eqs (20) (22) gves: where θ θ χ θ = Nb + Qb T N (23) χ χ χ χ = Nb + Qb T N (24) N N θ N = Ns + Qs W θ (25) y y z y Nx =-Ax Mx + F x ad z z -1 z x x x M = A F ad z z -1 z x = x x correspodg boudary algebrac equato used to compute y y z y x =- x x + x Q A Q S, wth Q S ad z gve by the superscrpt of the z w Q x,.e. Q x from u eq (18) or Q x from eq (19). Equatos (23) (25) are the ecessary relatos to solve a geometrcal olear plate problem. However, oe has to treat correctly the cremets T p. We ca frst fd the rates of the destes eqs (6) W ( p ) ad ( ) jk ad (11) ad obta the cremetal forms of T ad W for a gve tme terval t, as follows: T = N χ + N χ (26) 5 System soluto W = θ θ + θ θ (27) Equatos (23) - (25) represet the o-lear system to be solved terms of the cremets θ, χ ad N. Replacg the cremets T ad W accordg to eqs (26) ad (27) oe has: θ θ θ F( θ θ, χ, N )=- θ + Nb +Qb N χ +Q b N χ =0 (28) χ χ χ F( χ θ, χ, N )=- χ + Nb +Qb N χ +Q b N χ (29) N N N F N( θ, χ, N )=- N+ Ns +Qs θ θ+qs θ θ (30) The above o-lear system of equatos s solved by applyg the Newto- Raphso s scheme. Wth a tme cremet t = t t a teratve process +1 may be requred to acheve the equlbrum. The, from the soluto at terato the ext try at terato (+1) s gve by: { +1 } { } { = + δ } { +1 } { } { = + δ } N N N (31) θ θ θ (32) WIT Trasactos o Modellg ad Smulato, Vol 44, ISSN X (o-le)

8 220 Boudary Elemets ad Other Mesh Reducto Methods XXIX { +1 } { } { = + δ } χ χ χ (33) By learzg eqs (28) - (30), usg the frst term of the Taylor s expaso, gves: Fθ Fθ Fθ ( ) F θ θ, χ, N θ χ N δ θ F χ Fχ F χ ( ) Fχ θ, χ, N + δ χ +...= 0 θ χ N F ( ) N θ, χ, N δ N FN FN FN θ χ N where each term of the taget matrx s gve by: (34) [ C ] = θ θ -I Qb N II QbII χ χ χ 0 - II + Q b N II Qb II χ N Q s (I θ + θ I) 0 -II (35) Thus, the correctos to be cumulated durg the teratos are obtaed by solvg eq (34), as follows: 6 Numercal applcato ( θ χ N ) ( ) ( θ χ N ) F θ,, δ θ 1 δ χ =-[ C] F χ θ, χ, N δ N F N,, To check the performace of the proposed formulato we have chose a square plate subjected to a uform load (fgure 2). The plate sde legth s a, the thckess s t, wth the rato t/a = The Posso s rato was assumed ν = 0.3 whle q s the uform appled load. Several boudary codtos have bee aalyzed. We started by assumg smple supported ad clamped plate codtos. For each of ths case the -plae boudary dsplacemets could also be prescrbed equal to zero (IE) or kept free (ME). The results obtaed are show fgures 3 ad 4, for the smply supported plate ad clamped plate respectvely. As ca be see the results are total agreemet wth the oe gve by Ye ad Lu [3]. The obtaed results, compared wth other umercal ad aalytcal solutos demostrated that the proposed formulato s accurate. Rug several other teral ad boudary meshes has also demostrated that the covergece s obtaed quckly. Wth a rather coarse mesh, partcularly to tegrate the doma tegrals, the results are already accurate. The preseted results were obtaed by usg a 160 boudary elemets ad oly 8 doma cells (36) WIT Trasactos o Modellg ad Smulato, Vol 44, ISSN X (o-le)

9 Boudary Elemets ad Other Mesh Reducto Methods XXIX 221 q q a a Fgure 2: Square plate wth uform load (w max /t) (q a 4 /16 D t) Lear soluto ME IE Ye & Lu [3] - IE Ye & Lu [3] - ME Fgure 3: Smply supported plate load dsplacemet curve (w max /t) (q a 4 /E t 4 ) Lear soluto Expermetal [3] ME IE Ye & Lu [3] - IE Ye & Lu [3] - ME Fgure 4: Clamped plate load dsplacemet curve. WIT Trasactos o Modellg ad Smulato, Vol 44, ISSN X (o-le)

10 222 Boudary Elemets ad Other Mesh Reducto Methods XXIX 7 Coclusos A boudary elemet formulato to aalyse vo Kármá plates was proposed. The doma approxmatos were smplfed by reducg the desty of doma tegrals to smple equvalet values reducg the computatoal effort to compute the matrces related wth the doma values. A cosstet taget operator has bee derved ad the Newto process has bee mplemet leadg accurate ad relable solutos usg a very small umber of teratos. Ackowledgmets To FAPESP São Paulo State Research Foudato for the support gve to ths work. Refereces [1] Morjara, M., Ielastc aalyss of trasverse deflecto of plates by the boudary elemet method. Joural of Appled Mechacs-Trasactos of the ASME, 47: 291-, 1980 [2] Kamya, N. & Sawak, Y., A tegral equato approach to fte deflecto of elastc plates. Iteratoal Joural of No-Lear Mechacs, 17(3): , [3] Ye, T.-Q. & Lu, Y.-J., Fte deflecto aalyss of elastc plate by the boudary elemet method. Appled Mathematcal Modellg, 9: , [4] Taaka, M., Matsumoto, T. & Zheg, Z.-D. Icremetal aalyss of fte deflecto of elastc plates va boudary-doma-elemet method. Egeerg Aalyss wth Boudary Elemets, 17: , [5] Wag, W., J, X. & Taaka, M., A dual boudary elemet approach for the problems of large deflecto of th elastc plates. Computatoal Mechacs, 26: 58-65, [6] We, P.H., Alabad, M.H. & Youg, A., Large deflecto aalyss of Resser plate by boudary elemet method. Computers & Structures, 83 (10-11): , [7] Purbolaksoo, J. & Alabad, M.H., Large deformato of shear-deformable plates by the boudary elemet method. Joural of Egeerg Mathematcs 51 (3): , [8] Brebba, C.A., Telles, J.C.F & Wrobel, L.C. Boudary Elemet Techques. Theory ad Applcatos Egeerg, Sprger-Verlag: Berl ad New York, [9] Lete, L.G.S. & Vetur, W.S., Stff ad soft th clusos twodmesoal solds by the boudary elemet method. Egeerg Aalyss wth Boudary Elemets, 29 (3): , WIT Trasactos o Modellg ad Smulato, Vol 44, ISSN X (o-le)