14. MODELING OF THIN-WALLED SHELLS AND PLATES. INTRODUCTION TO THE THEORY OF SHELL FINITE ELEMENT MODELS

Transcription

1 4. ODELING OF IN-WALLED SELLS AND PLAES. INRODUCION O E EORY OF SELL FINIE ELEEN ODELS Srő: Dr. András Skréns Dr. András Skréns BE

2 odlng of thn-walld shlls and plats. Introducton to th thor of shll fnt lmnt modls 4. ODELING OF IN-WALLED SELLS AND PLAES. INRO- DUCION O E EORY OF SELL FINIE ELEEN ODELS 4. Plat and shll thors Plan structurs ar calld plats f th thcknss of structur s sgnfcantl lss than th othr dmnsons morovr f th structur s loadd prpndcularl to ts plan. h plat can b boundd along ts sds b an optonal gomtrcal objct; th knmatc boundar condtons can b varous (pont-supportd rgdl or lastcall supportd along th sds smpl supportd tc.) []. h plat can b consdrd as th tnson of a bam n two dmnsons bcaus both mpls th domnanc of th bndng load and most commonl th load s ntroducd transvrsl. Nvrthlss thr ar sgnfcant dffrncs too snc.g. th flur of th bam can b thr straght or curvd on th othr hand th mdplan of a plat s alwas flat. If th mdplan of th plat s curvd thn t s no longr plat but a shll []. In th squl w ovrvw th most mportant dtals of th thor of plats and shlls. 4. h basc quatons of Krchhoff plat thor h Krchhoff plat thor s oftn calld th thor of thn plats. W not that f th plat s rlatvl thck thn th transvrs shar dformaton can b consdrd too. h rlvant plat soluton s provdd b th ndln plat thor []. 4.. Dsplacmnt fld Basd on Fg.4. w nvstgat th dsplacmnt of a pont of th mdplan of an lastc flat plat [3]. h dsplacmnt fld can b capturd b thr componnts: th transvrs dsplacmnt along and th rotatons about and..: β u = α (4.) w whr α = α() s th rotaton about as β = β() s th rotaton about as and w = w() s th transvrs dsplacmnt. Fg.4..Dsplacmnt of a pont n th mdplan of a flat plat. Dr. András Skréns BE

3 Alfjtcím Stran componnts Assumng small strans w can calculat th stran componnts b usng th strandsplacmnt quaton dfnd n scton b Eq.(.4) [4]: u v ε = = β ε = = α ε = 0 (4.) u v u w v w γ = + = ( β α ) γ = + = β + w γ = + = α + w whr for th sak of smplct - th drvatvs wth rspct to and ar ndcatd n th subscrpt. In th squl w assum that th cross scton plans rman flat and th outward normal of ach cross scton s prpndcular to th cross scton plan aftr th dformaton. hs assumpton s calld Krchhoff-Lov hpothss []. From th lattr t follows that n th plans prpndcular to th mdplan of th plat th shar strans ar qual to ro: γ = γ = 0 β = w and α = w. (4.3) Utlng th formr w obtan from Eq.(4.) that: w u = w. (4.4) w h stran componnts bcom: ε ε γ. (4.5) = w = w = w Consquntl n th mdplan ponts ε = 0. Accordng to th Krchhoff plat thor undr th assumpton of small strans th componnts of th dsplacmnt and stran fld can b dfnd b w() Strss fld forcs and momnts n th mdplan Assumng plan strss stat w prss th strss componnts b Eqs.(.8) and (4.5): E σ = [ ε + νε )] = E( w + ν w ) = A (4.6) ν E σ = [ ε + νε ) ] = E( w + ν w ) = B ν E τ = γ = E ( ν ) w = C ( + ν ) whr E = E/(-ν ) A B and C ar constants. Smlarl to th thor of bams subjctd to bndng th strss dstrbutons ar gvn b lnar functons along th thcknss drcton as t s shown b Fg.4.. Dr. András Skréns BE

4 4 odlng of thn-walld shlls and plats. Introducton to th thor of shll fnt lmnt modls Fg.4.. Dstrbuton of th strsss along th thcknss drcton of a dffrntal plat lmnt. h strss coupls n th mdplan of th plat ar calculatd b ntgratng th strsss ovr th thcknss [3]: t / t / = σ d = A d = IE( w + ν w ) (4.7) / t / / t / = σ d = B d = IE( w + ν w ) / / t / t / = τ d = C d = IE( ν ) w / / t / t / = τ d = C d = IE( ν ) w / / whr s th bndng momnt along as s th bndng momnt along as and ar th twstng momnts. orovr: 3 t I = (4.8) whch s smlarl to bams th scond ordr momnt of nrta of th cross scton. h strss coupls n th mdplan of th plat ar dmonstratd n Fg.4.3a. h rlatonshps btwn strsss and strss coupls (bndng and twstng momnts) basd on Eqs.(4.6) and (4.7) ar: σ = σ = τ =. (4.9) I I I For th qulbrum of a dffrntal plat lmnt transvrs shar forcs ar rqurd. ransvrs shar forcs ar shown b Fg.4.3b and th can b calculatd usng th followng formula [3]: Dr. András Skréns BE

5 Alfjtcím 5 t / Q = τ d = / t / Q τ d. (4.0) / Fg.4.3. Strss coupls n th mdplan of a thn dffrntal plat lmnt (a) and ts qulbrum n th cas of transvrs shar forcs and dstrbutd load (b) h qulbrum and govrnng quaton of thn plats h homognous qulbrum quaton wth rspct to th strss fld has alrad bn ntroducd n scton. [4]: σ = 0 (4.) of whch frst componnt quaton s: σ τ τ + + = 0. (4.) Intgratng th quaton wth rspct to lds: Ad Cd τ τ = 0 (4.3) and: 0 A + C = τ τ (4.4) fnall: 0 σ + τ = τ τ. (4.5) Nt w ntgrat Eq.(4.5) wthn th rangs of t/ and t/: t / t / t / 0 d d d σ + τ = ( τ τ ( )) / / / (4.6) whr τ 0 s an ntgraton constant. A possbl soluton for τ whch satsfs vn th dnamc boundar condtons s [3]: = 0 4 τ τ. t (4.7) In fact Eq.(4.7) gvs th dffrnc btwn th ara undr a rctangl and a parabola whch s /3 of th total ara. Accordngl f t s multpld b two thn mathmatcall w obtan th ara undr th parabola that s from Eq.(4.6) w hav: Dr. András Skréns BE

6 6 odlng of thn-walld shlls and plats. Introducton to th thor of shll fnt lmnt modls t / / 0 t / ( τ τ ( )) d = τ d = Q (4.8) / whch s not ls than th shar forc along as gvn b Eq.(4.0). akng Eqs.(4.6) (4.9) and (4.8) back nto th qulbrum quaton w obtan: Q = 0. (4.9) h scond componnt quaton and th corrspondng qulbrum quaton n trms of th strss coupls and transvrs shar forc ar: τ σ τ + + = 0 (4.0) + Q = 0 and: =. (4.) From th thrd componnt quaton of Eq.(4.) w obtan th followng: τ τ σ + + = 0. (4.) W ntgrat Eq.(4.) wthn th rangs of t/ and t/ wth rspct to : t / t / t / τ τ σ d + + = 0 d d / / / (4.3) and: t / t / t / + + = 0 τ d τ d dσ. (4.4) / / / Basd on Eq.(4.0) th frst two trms ar th shar forcs Q and Q th thrd on s n accordanc wth th dnamc boundar condton th ntnst of th dstrbutd load p prpndcularl to th mdplan..: Q Q + + p = 0. (4.5) Summarng th qulbrum quatons w hav: Q = 0 (4.6) + Q = 0 Q + Q + p = 0. o drv th plat quaton w rarrang th frst two quatons: Q = (4.7) Q = +. akng thm back nto th thrd of Eq.(4.6) w obtan th followng: + + p = 0. (4.8) B th hlp of Eq.(4.7) w hav: IE( w + ν w + w + ν w + ( ν ) w ) = p (4.9) Dr. András Skréns BE

7 Alfjtcím 7 whch aftr a smpl rarrangmnt hav th form of [5]: w w w p + + = (4.30) 4 4 IE or: p w ( ) =. (4.3) IE Consquntl th govrnng quaton s a fourth ordr partal dffrntal quaton wth th propr knmatc and dnamc boundar condtons. hat mans that th problm of plats subjctd to bndng s a boundar valu problm. 4.3 Fnt lmnt quatons of thn plats For th fnt lmnt soluton of th problm of thn plats subjctd to bndng w collct th stran and strss fld componnts nto vctors and w assum plan strss stat [6]: ε = ε ε γ (4.3) [ ] [ σ σ τ ] σ =. Basd on Eq.(4.5) th stran componnts can b wrttn as: ε = κ (4.33) whr κ s th vctor of curvaturs: κ w κ = κ = w. (4.34) κ w Incorporatng th matral law w formulat th vctor of strss componnts as: str σ = C ε. (4.35) h stran componnts can b obtand b a two-varabl functon w() th fnt lmnt ntrpolaton of th w() functon dpnds on th lmnt tp and th chosn dgrs of frdom but t can alwas b formulatd n th form blow: w ( ) = A λ (4.36) whr A s th vctor of unknown coffcnts λ s th vctor of bass polnomals. h vctor of nodal dsplacmnts s: u = A (4.37) whch for ampl n th cas of a trangl lmnt wth thr nods bcoms: u = w α β w α β w α. (4.38) [ ] 3 3 β 3 In Eq.(4.37) matr can b calculatd basd on th appromat w() functon and Eq.(4.). h α and β paramtrs ar th rotatons about th as and n th corrspondng nods whr = 3. From Eq.(4.37) w hav: A =. (4.39) u Gnrall spakng th vctor of stran componnts can b dtrmnd usng th strandsplacmnt matr: ε = Bu (4.40) whr Eq.(4.40) can b rformulatd utlng Eqs.(4.5) (4.37) and (4.39) as follows: Dr. András Skréns BE

8 8 odlng of thn-walld shlls and plats. Introducton to th thor of shll fnt lmnt modls u ε = RA = R (4.4) whr matr R stablshs th rlatonshp btwn th vctor of stran componnts and th vctor of unknown coffcnts ts dmnson s lmnt dpndnt. Consquntl w hav: B = R. (4.4) Followng th dfnton b Eq.(.9) w formulat th lmnt stffnss matr as: str K = B C BdV. (4.43) V h dmnson of th lmnt stffnss matr dpnds on th numbr of nods and th numbr of nodal dgrs of frdom. Smlarl to th plan mmbran lmnt th vctor of forcs s composd as th sum of svral trms. h most common s th dstrbutd (surfac) load and concntratd forc. B formulatng th work of trnal forcs w drv th forc vctor rlatd to th dstrbutd load: W = p w( ) da = u F (4.44) Ap whr p s th ntnst of th dstrbutd load prpndcularl to th mdplan of th plat w() s th appromat functon of th dflcton surfac accordng to Eq.(4.36). h vctor F p can b dtrmnd basd on th vctor of nodal dsplacmnts. In th cas of concntratd loads consdrng.g. a trangular shap plat lmnt wth thr nods at ach nod thr can b a forc prpndcularl to th plat surfac and vn concntratd momnts actng about th and as rspctvl: F c = [ F F F 3 3 3]. (4.45) hus th vctor of forcs bcoms: F = F p + F c. (4.46) Evntuall th fnt lmnt qulbrum quaton for a sngl lmnt and for th whol structur s: K u = F K U = F. (4.47) Smlarl to th plan mmbran lmnts thr s larg numbr of plat bndng lmnts. hs lmnts wll b rvwd n scton Basc quatons of th tchncal thor of thn shlls In that cas whn th mdplan of a thn-walld structur s not flat but curvd thn w talk about shlls. h analtcal nvstgaton of shlls rqurs consdrabl complcatd mathmatcal computatons. hrfor n th squl onl th most mportant quatons wll b prsntd Gomtrcal quatons Du to th fact that th mdsurfac of shlls s curvd w nd to ntroduc curvlnar coordnat sstms as t s shown b Fg.4.4. p Dr. András Skréns BE

9 Alfjtcím 9 Fg.4.4. Coordnat lns and unt bass vctors of th mdsurfac of a shll. h two-paramtr rprsntaton of th mdsurfac of shlls can b formulatd n th form of a vctor quaton [4]: R = R( q q ) (4.48) whr: X = X ( q q ) Y = Y ( q q ) Z = Z( q q ) (4.49) ar th global coordnats R s th poston vctor of a pont n th q and q ar th gnral or curvlnar coordnats of th surfac (s Fg.4.4). If th paramtrs tak on th valus q = constant and q = constant w obtan th q and q coordnat lns. h tangnt unt vctors and th arc lngths ds of th coordnat lns ar: R = = R ds = dq (4.50) q whr: = R = (4.5) ar th so-calld Lamé paramtrs [] or mtrc coffcnts [4]. In th followngs w assum that th local coordnat as ar mutuall prpndcular at ach pont and th curvlnar sstm s orthogonal.. = 0. h outward unt normal vctor of th mdsurfac bcoms: n =. (4.5) h trad of unt orthogonal vctors [ n] dtrmns an orthogonal curvlnar coordnat sstm at an actual pont P. h curvatur and th torson of coordnat lns ar gvn b th Frnt formula [7]: R R = n = n = n R = (4.53) R S q R = n = n R R S S whr R and R ar th rad of curvatur. If R = 0 thn th q and q lns ar th lns of prncpal curvaturs on th mdsurfac morovr th drctons of th unt bass vctors and ar th prncpal drctons. h curvatur of th mdsurfac s a tnsor quantt. If th drctons of vctors and ar not th prncpal drctons thn th angl whch dtrmns th prncpal drctons can b obtand b: Dr. András Skréns BE

10 0 odlng of thn-walld shlls and plats. Introducton to th thor of shll fnt lmnt modls ' / R tgα =. (4.54) ' ' / R + / R h valus of th prncpal curvaturs ar [7]: cos α sn α sn α = + + (4.55) ' ' ' R R R R sn α cos α sn α = + = 0. ' ' ' R R R R R In th followngs w nvstgat th spcal cas whn th drctons of unt bass vctors concd wth th prncpal drctons. h drvatvs of th unt bass vctors of th coordnat sstm on th mdsurfac ar [4]: j j j = j n j = j n j = j j j =. R R j (4.56) Pont P * s locatd on a surfac paralll to th mdsurfac and th dstanc of pont P * from pont P s gvn b coordnat masurd along th normal vctor n. Basd on Fg.4.4 th poston vctor of pont P * s: * R = R + n. (4.57) h unt vctors ar ndpndnt of coordnat v.: * =. (4.58) h drvatv of th poston vctor of pont P* can b wrttn as: * R = R + n = ( + ). (4.59) R Consdr th followngs: * * = ( + ) and ds = ds ( + ) =. (4.60) R R whch ar th Lamé paramtrs and arc lngths wth rspct to pont P* Strss rsultants and coupls qulbrum quatons Fg.4.5 shows th strsss on th boundar plans of a dffrntal shll lmnt whl Fg.4.6 prsnts th strss rsultants and coupls (ntrnal forcs and momnts) on th mdsurfac of th dffrntal shll lmnt wth dmnsons of ds ds. j Dr. András Skréns BE

11 Alfjtcím Fg.4.5. Strss componnts on th boundar plans of a dffrntal shll lmnt. Fg.4.6. Intrnal forcs and momnts n th mdsurfac of a dffrntal shll lmnt. W must consdr th rlatonshp btwn th arc lngths ds and ds * gvn b Eq.(4.60) whn w stablsh th rlatonshp btwn th strsss actng on th dffrntal shll lmnt wth thcknss t and th ntrnal forcs momnts on th mdsurfac of th shll lmnt. h strss rsultants and strss coupls actng on th curv wth outward normal ar: t / = ( + ) R / t / N σ d N = τ ( + ) d N Q = R τ 3( + ) d R t / = ( + ) R / / t / t / / σ d = τ ( + ) d (4.6) R / whr N s th n-plan normal forc N and N ar th n-plan shar forcs Q s th transvrs shar forc s th bndng momnt and ar th twstng momnts rspctvl. It must b takn nto consdraton that although th rcproct law of shar strsss mpls τ = τ n th quatons abov N N and whch can b pland b th fact that th rad of curvaturs ar n gnral not qual to ach othr..: R R. h dvlopmnt of qulbrum quatons stablshng th qulbrum btwn trnal loads and ntrnal forcs and momnts n th shll structur s also vr compl- Dr. András Skréns BE

12 odlng of thn-walld shlls and plats. Introducton to th thor of shll fnt lmnt modls catd. hrfor w prsnt onl th rsultng quatons. h qulbrum quatons n th cas of strss rsultants ar [7]: Q ( N) + ( N ) + N N ( + p) = 0 R (4.6) Q ( N) + ( N) + N N ( + p) = 0 R N N ( Q ) + ( Q ) + ( + p3) = 0 R R whr p and p ar th tangntall dstrbutd loads along drctons and p 3 s th dstrbutd load prpndcularl to th shll mdsurfac. h qulbrum quatons n th cas of strss coupls and momnt of strss rsultants ar: ( ) + ( ) + Q = 0 (4.63) ( ) + ( ) + + Q = 0 + N N = 0 (4.64) R R whr n th subscrpt th comma and th numbr rfr to th dffrntaton wth rspct to th corrspondng coordnat Dsplacmnt fld stran componnts Basd on Fg.4.7 th vctor of dsplacmnts and rotatons n a pont P on th shll mdsurfac can b wrttn as: u = u + v + wn β = β β + β 3 n. (4.65) Fg.4.7. Dsplacmnt of a pont on th mdsurfac of a thn shll. In accordanc wth th knmatc hpothss of th shll thor th componnts of vctor u n a pont P * out of th mdsurfac ar [7]: * * * u = u + β v = v + β w = w (4.66).. th ln of matral ponts whch s prpndcular to th shll mdsurfac rmans prpndcular durng th dformaton. h quatons dscrbng th n-plan strans and changs n curvatur ar [7]: Dr. András Skréns BE

13 Alfjtcím 3 ε = u + v + w (4.67) R ε = v + u + w R u v γ = + κ κ β = β + β = β + β κ = + + β 3 R R β whr ε and ε ar th n-plan strans n th drctons of q and q coordnat lns γ s th shar stran rlatd to th chang of angl btwn unt vctors and durng th dformaton κ and κ ar th changs n curvaturs n th drctons of q and q paramtrs κ s th twstng curvatur. h shar strans rlatd to th unt normal and unt vctors ar [7]: u v γ 3 = + w + β γ 3 = + w + β. (4.68) R R W assum that durng th dformaton of shll an actual ln of matral ponts rman prpndcular to th curvd shap of shll mdsurfac accordngl th shar strans gvn b (4.68) ar qual to ro. h knmatc hpothss of shll thor togthr wth th on mntond bfor s calld th Krchhoff Lov hpothss. Undr thss assumptons w hav: u v β = w β = w. (4.69) R R In othr words th addtonal transvrs shar dformaton s nglctd (smlarl to Krchhoff s thor of thn plats). h rotaton about as can b formulatd b th followng prsson [7]: β 3 = [( v) ( u) ]. (4.70) Nvrthlss n most of th cass th rotaton about s nglgbl; thrfor t s not consdrd n th quatons Appromatons wthn th tchncal thor of thn shlls h shll s consdrd to b thn f th thcknss s rlatvl small compard to th smallr radus of curvatur v. []: <<. (4.7) R Consquntl th Lamé paramtrs and th arc lngths on th mdsurfac and out of th mdsurfac ar appromatl qual whch lads to: * * and: ds ds =. (4.7) Dr. András Skréns BE

14 4 odlng of thn-walld shlls and plats. Introducton to th thor of shll fnt lmnt modls Accordngl Eq.(4.6) can b smplfd sgnfcantl: t / N = σ d N = τ d = N Q = τ 3d (4.73) / t / / t / / t / t / / = σ d = τ d =. / It s sn that n ths cas th transvrs shar forcs and torsonal momnts ar qual to ach othr whch volats th qulbrum quatons gvn b Eq. (4.64). hs appromaton s prmttd wthn th tchncal thor of thn shlls. 4.5 ajor stps n th fnt lmnt modlng of shlls In th cours of th fnt lmnt dscrtaton of shlls smlarl to th plan and plat problms w procd th ntrpolaton of th gomtr and th dsplacmnt fld [7]. h vctor of dsplacmnt and rotaton componnts n a pont locatd on th shll mdsurfac s: u = [ u v w] (4.74) β = [ β β β 3 ]. h componnts of ths vctors ar not ndpndnt of ach othr. From Eq.(4.67) w calculat th n-plan strans and th changs n curvatur: ε = [ ε ε γ ] (4.75) κ = [ κ κ κ ]. W collct th n-plan forcs and momnts nto a vctor: N = [ N N N ] (4.76) = [ ]. ransvrs shar forcs Q Q ar not consdrd n th calculaton of th dformaton. Fnall th vctors of th surfac loads and concntratd forcs and momnts ar: p = p p (4.77) [ p3 ] [ N N Q] [ 0] = N = ρ ρ whr p contans th dstrbutd loads n th drctons of coordnat lns q and q and also th dstrbutd load prpndcularl to th shll mdsurfac ρ N and ρ contan th concntratd forcs and momnts actng n th nods. Usng th vctors gvn b Eqs.(4.75)-(4.77) th total potntal nrg s formulatd as: Π = ( ε N + κ ) dqdq u p dqdq ( u ρ + β N A ρ A S ) ds. (4.78) W assum that th matral of th thn shll s lnar lastc homognous and sotropc. hn th vctor of n-plan forcs and vctor of momnts can b calculatd as follows: 3 str t str N = tc ε = C κ (4.79) whr th consttutv matr assumng plan strss stat s: Dr. András Skréns BE

15 Alfjtcím 5 ν 0 E C str = ν 0. (4.80) ν ν 0 0 Accordngl Eq.(4.78) bcoms: str t str Π = ( ) tε C ε + κ C κ dq dq + A (4.8) u p dq dq ( u ρ + β ρ ) ds A S Utlng th dfnton of th lmnt stffnss matr and th vctor of nodal forcs w can drv th prsson blow: Π = u K u u F (4.8) from whch th fnt lmnt qulbrum quaton for a sngl lmnt (th frst of Eq.(4.47)) can b drvd. As a nt stp w summar th potntal nrg of ach lmnt: Π = Π = U KU U F (4.83) and fnall applng th mnmum prncpl of th total potntal nrg w obtan th structural qulbrum quaton: K U = F. (4.84) For th fnt lmnt modlng of shlls thr s vr larg numbr of lmnt tps. Not onl th flat shll lmnts whch gv mor accurat rsult undr hgh msh rsoluton but also th curvd (.g. clndrcal shll lmnt) and doubl-curvd shll lmnt tps ar avalabl whch appromat bttr both th gomtr and th dsplacmnt fld usng th sam lmnt numbr. h dffrnt plat and shll lmnts ar dscussd n sctons Bblograph [] Imr Bojtár Gábor Vörös Applcaton of th fnt lmnt mthod to plat and shll structurs. chncal Book Publshr 986 Budapst (n ungaran). [] Robrt D. Cook Fnt lmnt modlng for strss analss. John Wl and sons Inc. 995 Nw York Chchstr Brsban oronto Sngapor. [3] Gábor Vörös Lcturs and practcs of th subjct Elastct and fnt lmnt mthod manuscrpt Appld chancs odul Budapst Unvrst of chnolog and Economcs Facult of chancal Engnrng Dpartmnt of Appld chancs 00 Budapst (n ungaran). [4] P Ch Chou Ncholas J. Pagano. Elastct nsor dadc and ngnrng approachs. D. Van Nostrand Compan Inc. 967 Prncton Nw Jrs oronto London. [5] S. moshnko S. Wonowsk-Krgr. hor of plats and shlls. chncal Book Publshr 966 Budapst (n hungaran). [6] Sngrsu S. Rao. h fnt lmnt mthod n ngnrng fourth dton. Elsvr Scnc & chnolog Books 004. N Dr. András Skréns BE

16 6 odlng of thn-walld shlls and plats. Introducton to th thor of shll fnt lmnt modls [7] Eduard Vntsl hodor Krauthammr. hn plats and shlls hor analss and applcatons. arcl Dkkr Inc. 00 Nw York Basl. Dr. András Skréns BE