Free Vibration Analysis of Stiffened Laminated Plates Using a New Stiffened Element

Transcription

1 TEHNISHE MEHANIK, Band 8, Hft -4, (8), 7-6 Manusriptingang: 7. Otobr 7 Fr Vibration Analysis of Stiffnd Laminatd Plats Using a Nw Stiffnd Elmnt Tran Ich Thinh, Ngo Nhu Khoa A nw 9-nodd rctangular iffnd plat lmnt for th vibration analysis of laminatd iffnd plats basd on Mindlin s dformation plat thory has bn dvlopd. Th iffnd plat lmnt is a combination of basic rctangular lmnt and bam bnding componnt. Th iffnd plat lmnt has bn dvlopd to includ th ffcts of transvrs shar. Th lmnt can accommodat any numbr of arbitrarily orintd iffnrs and obviats th us of msh lins along th iffnrs. Fr vibration analyss of iffnd laminatd plats hav bn carrid out with this lmnt and th rsults ar compard with thos publishd. Th finit lmnt rsults show vry good matching with th xprimntal ons. Introduction Stiffnd composit plats ar usd xtnsivly in many indurial ructurs: Arospac ructurs, ship hulls, bottom shll-machin girdr plat of composit boat tc. So dtrmination of th natural frquncis of th iffnd composit plats is a practical dmand, in ordr to solv th anti-vibration problms whn th ructurs ar subjctd to priodic xciting loads. A numbr of analytical and numrical modls for th analysis of iffnd laminatd plats hav bn proposd in th litratur such as Kir (96), Satsangi (987), Muhopadhyay (989), Kolli (996), Satish Kumar (), Gangadhara Pruy () tc. Among all th numrical mthods, th finit lmnt mthod (FEM) has bn found to b rasonably accurat with lss complxity to modl iffnd plats. A mor accurat modl is achivd by rprsnting th plat and iffnrs sparatly and maintaining compatibility btwn thm. Thomson (988) t al. and Satsangi (987) usd 8-nodd rctangular plat lmnts and assumd that th iffnrs follow th sam displacmnt fild as that of th plating. Th FE modl of Kolli (996) consis of th 9-nodd rctangular plat lmnt and -nodd bam lmnt, th bams ar placd along th plat nodal lins. Edward t al. () usd a iffnd plat lmnt that is composd of a rctangular 9-nodd rctangular plat lmnt and a numbr of -nodd iffnr lmnts placd within th plat lmnt and paralll to th lmnt dgs. Gangadhara Pruy () udid linar atic analysis of composit hat-iffnd laminatd shlls using 8-nodd rctangular plat lmnt and -nodd bam lmnt. Th xiing finit lmnt tchniqus imulat th iffnr to pass along th plat nodal lins. In ths udis, no discussion has bn mad for th iffnrs of various shaps and having arbitrary orintation in th plat. To ovrcom this problm, Satish Kumar and Muhopadhyay () hav dvlopd a iffnd triangular plat lmnt for th analysis of laminatd iffnd plats. This basic plat lmnt is a combination of Allman s plan rss triangular lmnt and discrt Kirchhoff-Mindlin plat bnding lmnt. Howvr, th intrpolation of displacmnts in thir modl is vry complx and th natural frquncis obtaind ar lss accurat than thos publishd. In an attmpt to fficintly solv this problm, w dvlopd a nw 9-nodd iffnd rctangular laminatd lmnt. This lmnt can accommodat any numbr of arbitrarily orintd iffnrs and is compltly fr from th usual conraints on th msh division of th iffnd plats. 7

2 Stiffnss Matrix of th Stiffnd Plat Elmnt Th displacmnt fild basd on th fir-ordr shar dformation plat thory is givn by u( x, z, u v( x, z, v w( x, z, w + zθ + zθ y x (a) Th displacmnt fild of iffnr u z, t ) u t ) + zθx t ) w t ) w hr, x is x-axis of iffnr. In gnral, ( x ) ϕ x, as shown in Figur. (b) rfrnc plan of plat 4 7 M t M y rfrnc plan of iffnr y ϕ M t 8 9 x x 5 6 Figur. Nin-nodd rctangular iffnd plat lmnt Th lmnt iffnss matrix of iffnd plat is dtrmind by whr p K K + K () p K and K ar th lmnt iffnss of flat plat and iffnr rspctivly and givn by Ngo (7) K BAB + BBB + BBB + BDB + BAB ' p T T T T T St T T b lg K B T D T B dx ds A, B, D and A ar popular matrics (Tran, 994); D is th rigidity matrix of th iffnr; b is th width of iffnr, B i is th rain-displacmnt matrix of th plat (Ngo, 7). [ B ] [ L ] N [ L ] N...[ L ] N i (,,) i i i i 9 N i ar quadratic shap functions; B is th rain-displacmnt matrix of th iffnr () 8

3 x y y x B y x N (4) x y y x T is th rain transformation matrix of th iffnr (Ngo, 7) cos ϕ sin ϕ sinϕ cos ϕ sin ϕ sinϕ sinϕ T sinϕ cosϕ (5) Th rigidity matrix of th iffnr du to paralll laminations: Th rigidity matrix of th iffnr is xprssd as A B D B D (6) A' whr, th cofficints of rigidity matrix ar calculatd by ignoring th rsss in th width or y dirction of iffnr ( σ τ τ ) but not th rains ( γ γ ) ' i j y xy xz A B D A' ε : y xy n [ ' ' ( ' ' + ' 6 ' 6 ) ' ( ' 6 ' + ' 66 ' 6 )] ( + ) z z n [ ' ' ( ' ' + ' 6 ' 6 ) ' ( ' 6 ' + ' 66 ' 6 )] ( + ) z z n [ ' ' ( ' ' + ' 6 ' 6 ) ' ( ' 6 ' + ' 66 ' 6 )] ( + ) z z n ' 55 ( ' ) 45 ( z z ) ' 44 + (i, j,,, 4, 5 and 6) ar th iffnss cofficints of th iffnr matrial and yz (7) ' ' 6 ' 6 ' 66 ' ' 6 ' 6 ' 66 (8) Th rigidity matrix of th iffnr du to prpndicular laminations: Th rigidity matrix of th iffnr is xprssd as 9

4 ( ) whr d s is th hight (dpth) of th iffnr. d s n d s D ( ) (9) ( 6 ) ds ( 66 ) d s Mass Matrics Th intic nrgy of plat lmnt with diributd mass is givn by whr T T ρ dv u u () V T u [ uvw,, ] u v w θ x θ y () z z W can xprss u in trms of th nodal displacmnts, a, by using shap functions N. T * T T * T T {} {} {}[ ] [ ]{} {}[ ]{} d ρ d ds ρ ds a N N a a M a () whr [ M ] is th mass matrix of plat lmnt S S and with and [ *] T * [ ] [ ] ρ [ ] M N N ds () S I I Sym ρ I (4) I I I I n + ( I I, I ) (, z, z ) h, ρ dz (5) h N N N N9 N N N 9 N N N9 N N N9 N N N 9 (6) Th intic nrgy of iffnr mmbr with diributd mass is givn by

5 u can b xprssd by T T ρ dv u u (7) V T u u, w u w θ x (8) z Ths trms in plat lmnt coordinats ar xprssd as u w cos u v z sinϕ w θ θ ϕ sinϕ z cosϕ Th displacmnt compatibility btwn th iffnr and th plat is nsurd by th bam lmnt s displacmnt fild which is intrpolatd from plat lmnt s nods x y (9) u δ u v δ v w w θ x θ x θ y θy d whr s + h δ is ccntricit d s is dpth of iffnr, and h is thicnss of plat. Finally () and [ ] T T T T T b { } [ ]{} {} [ ] {} {}[ ] {} d ρ d dx b ρ dx a N N a a M a () l M is th mass matrix of iffnr l whr and [ ] ( ρ ) T [ ] bl [ ρ] M N N () cos ϕ cos ϕ sin ϕ z cos ϕ z cos ϕ sin ϕ n h + sin ϕ z cos ϕ sin ϕ z sin ϕ ρ dz () h Sym z cos ϕ z cos ϕ sin ϕ z sin ϕ δ δ N [ N] ()

6 4 Rsults and Discussions 4. Validation of th Modl Exampl In ordr to chc th rliability and accuracy of th prsnt lmnt, w considr th fr vibration of a simply supportd blad iffnd plats. Th variation of natural frquncis with iffnr ccntricity ar prsntd for spcially orthotropic cross-ply laminats with thr qually spacd iffnrs (Figur ). Th gomtry of th iffnd laminats ar a b h 4.4 (mm ), th lamination of plat and iffnrs is (9 / /9 ) T. Th ply proprtis ar E 9.7 GPa; E.5 GPa; G G.95 GPa; G.56 GPa; ν.9; ρ47g/m for both th plat and th iffnr. Th width of th iffnr is mm. Th numrical rsults ar compard with thos of Biswal and Ghosh, (994) (4-nodd rctangular lmnt with 7 d.o.f at ach nod) and hao and L (98). Th variation of natural frquncis with iffnr half-dpth (Figur. ) shows bttr agrmnt with thos of Biswal and Ghosh than thos of hao and L. This may b du to nglcting of th shar ffcts by hao and L. h.4 δ d s Figur. Laminatd iffnd plats, ccntricity is variabl 5 5 f FREQUNEY (Hz) 5 f Prsnt Satish Kumar [] hao and L [5] f 5 f Half dpth of iffnr (mm) Figur. Frquncy vs. half-dpth of iffnrs

7 Exampl In this xampl, w considr th fr vibration of a cross-iffnd plat. Th gomtry of cross-iffnd plat is a b 54 mm, h.7 mm, d sx d sy 5.4 mm, b sx b sy 6.5 mm. Whr, d sx, b sx is dpth and width of th x-dirction iffnr; d sy, b sy is dpth and width of th y-dirction iffnr. Th following ply proprtis of AS4/5 graphit/poxy composits ar usd: E 44.8 GPa; E 9.67 GPa; G G 4.4 GPa; G.45 GPa; ν.; ρ 89. g/m. Th iffnr laminations ar paralll to th plat midplan. Th frquncis of th fir four mods of various boundary conditions of a (/9) and (45 /-45 ) cross-iffnd plat ar prsntd in Tabl and thy ar compard with thos rportd in (Satish Kumar t al., ), (handrashra t al., 997) and (Ra 998), using msh for full plat. Th fundamntal frquncis shown good agrmnt with thos of Satish Kumar t al.(), whras th frquncis of highr mods compard xcllntly with thos of Ray (998). According to th analysis of Satish Kumar t al.(), handrashra t al. (997) usd th rducd iffnss cofficints in th rigidity matrix of th iffnr which rducd its iffnss. Morovr, thy hav ignord th coupling cofficints in th rain-nrgy of th iffnr lmnt. Thrfor, th fundamntal frquncis with th prsnt lmnt ar highr than that of handrashra t al. Boundary onditions SSSS SS o /9 o 45 o /-45 o Mod Prsnt Rf. [] Rf. [] Rf. [4] Prsnt Rf. [] Rf. [] Tabl. Effct of boundary conditions on natural frquncis (Hz) of cross-iffnd plats 4. Exprimntal Study Lt us considr a iffnd rctangular composit plat mad of glass fibr/polyr with lamination [M/WR8/M/WR8/M45]. Whr, M dnots glass fibr in Mat form with its wight pr unit ara is g/m ; M45 dnots glass fibr in Mat form with its wight pr unit ara is 45g/m and WR8 dnots glass fibr in WR form with its wight pr unit ara is 8g/m.Th sids of plat ar a b 8 5mm. Th plat is rinforcd by 6 longitudinal (n x 6) and 9 transvrs (n y 9) closd sction (ha iffnrs (Figur 4). Th iffnrs wr mad by glass fibr in Mat and hav th sam sizs as follows: b h t.8 mm (Nguyn, 5).

8 Figur 4. Exprimntal st-up and ting procdur for iffnd composit plat Thr fir natural frquncis of uniffnd and iffnd plats subjctd to various boundary conditions wr masurd by using a Multi-vibration masuring machin (DEWE BOOK-DASYLab 5.6.) and ar givn in Tabl. Plats Mod lampd at 4 dgs Uniffnd Stiffnd at x-dgs SS at y-dgs at y-dgs SS at x-dgs Simply Supportd at 4 dgs Tabl. Exprimntal rsults on natural frquncis (Hz) of iffnd composit plat 4. Finit Elmnt Rsults and omparison In this sction, w calculat th natural frquncis for th abov iffnd composit plat by our computr program. Th fir thr natural frquncis will b compard with thos of xprimntal ons. Th laic conants of matrial usd in th calculation wr dtrmind by our mchanical ts (Tran t al., 5). - For Mat layr: E E 4.87GPa; G.5GPa; ν.7 - For WR layr ( and 9 ): E.58GPa; E.64GPa; G.GPa; ν.7 - Thicnss of a M layr, t M.6mm; thicnss of a M45 and WR8 layr, t M45 t WR8 mm Th hat-iffnrs ar modld, as shown in Figur 5. Th laminations of iffnrs ar prpndicular and paralll to th plat midplan. Th frquncis of th fir thr mods of various boundary conditions of a crossiffnd plat using msh for full plat ar prsntd in Tabl and thy ar compard with thos rportd in (Nguyn, 5). Th fundamntal frquncy of clampd plat at 4 dgs shown vry good agrmnt with thos of Nguyn (5), whras th frquncis of simply supportd plat at 4 dgs ar highr than that of Nguyn (5). This may b du to th boundary condition conructd in xprimntal udy (not simply supportd absolutly). 4

9 d t b Figur 5. Modlling of hat-iffnr. Plats Mod lampd at 4 dgs at x-dgs SS at y-dgs 8.4 (4.5%) 5.74 (.%) Uniffnd 56.6 (5.4%) (.8%) (9.%) (5.9%) 5.84 (.4%) (.%) Stiffnd.6 (.%) (5.6%) (.6%) (.%) Rmar: (...%) dnots th rror prcnt btwn FE rsults and xprimntal ons. at y-dgs SS at x-dgs.9 (.6%) 46.7 (6.6%) 64.5 (5.%) 9.96 (9,%) (.4%) 59.8 (6.6%) Simply Supportd at 4 dgs 9. (4.6%) 5.5 (5.5%) 6.4 (4.8%) 7.44 (9%) (5.4%) 5.74 (4.5%) Tabl. Natural frquncis (Hz) of iffnd plats calculatd by finit lmnt program 5 onclusions In this papr, w hav prsntd a nw 9-nodd iffnd rctangular plat lmnt for th vibration analysis of a laminatd composit plat with rctangular and hat laminatd iffnrs. Th iffnr is lgantly modld and dos not introduc any additional nods. Th plat lmnt accommodats any numbr of arbitrary orintd iffnr lmnts and liminats th usual conraints imposd on th msh division of iffnd plats. Th modl is validatd by comparing with xiing rsults documntd in th litratur. Som problms on fr vibration analyss of laminatd iffnd plats mad of graphit/poxy and glass/polyr ar analyzd with th prsnt lmnt. Morovr, th lmnt has bn vry ffctiv in analysis of both thin and modratly thic plats. Th finit lmnt rsults compar wll with xprimntal ons. It is rcommndd that th prsnt formulation can b usd to dtrmin th fundamntal frquncis rquird in th dsign and analysis of ccntric composit iffnd plats. Acnowldgmnts Th authors gratfully acnowldg th financial support of Miniry of Scinc and Tchnolog Vitnam. 5

10 Rfrncs Biswal K..; Ghosh A.K.: Finit lmnt analysis for iffnd laminatd plats using highr ordr shar dformation thory. omputrs and Structurs (994), 5():6-7. hao..; L J..: Vibration of ccntrically iffnd laminats. Journal of omposit Matrials., (98), 4():-44. handrashara K.; Kolli M.: Fr vibration of ccntrically iffnd laminatd plats. Journal of Rinforcd Plaics and omposits., (997), 6(): Edward A. Sadc.; Samr A. Tawfi.: A finit lmnt modl for th analysis of iffnd laminatd plats. omputrs and Structurs., (), Gangadhara Pruy B.: Linar atic analysis of hat-iffnd laminatd shlls using finit lmnts. Finit lmnts in analysis and dsign., (), 9: 5-8. Kir.L.: Vibration of cntrally placd iffnd rctangular plats. J. R Aro Soc., (96), 65: Kolli M.; handrashhara K.: Finit lmnt analysis of iffnd laminatd plats undr transvrs loading. omposits Scinc and Tchnology., (996), 56:55-6, USA. Muhopadhyay M.: Vibration and ability of analysis of iffnd plats by smi-analytic finit diffrnc mthod. J. Sound Vibr., (989), : 4-5. Ngo Nhu Khoa: alculation of laminatd composit iffnd-plat using finit lmnt mthod. Thainguyn Univrsity - Journal of Scinc and Tchnology., (7), Volum, numbr, (in Vitnams). Nguyn Van Dat: Study th rasonabl conructur of and sym of fishing ship hull to support to vibration. PhD thsis., (5) (in Vitnams). Ray.: Analysis of hat-iffnd composit plats by Finit lmnt mthod. PhD thsis, Indian Initut of Tchnolog Kharagpur, (998). Satish Kumar Y.V.; Muhopadhyay M.: A nw triangular iffnd plat lmnt for laminat analysis. omposit Scinc and Tchnology., (), 6: Satsangi SK.; Muhopadhyay M.: Finit lmnt at analysis of girdr bridgs having arbitrary platform. Int. Ass. for Bridg Struct. Engng., (987), 7: Thomson P.A.; Bttss P.; aldwll J.B.: An isoparamtric ccntrically iffnd plat bnding lmnt. Engng. omput., (988), 5: -6. Tran Ich Thinh: omposit Matrials - Mchanics of Matrials and Structurs. Ed. Education, (994), (In Vitnams). Tran Ich Thinh; Tran Minh Tu: Exprimntal dtrmination of mchanical proprtis of glass/polyr composit. Vitnam Journal of Scinc and Tchnology., (5), Volum 4, N 6. (in Vitnams). Addrsss: Prof. Dr.-Ing. Tran Ich Thinh, Hanoi Univrsity of Tchnolog. Dai o Vit, Hanoi, Vitnam mail: tranichthinh-vl@mail.hut.du.vn Dr.-Ing. Ngo Nhu Khoa, Thainguyn Univrsit Vitnam mail: hoa.ngonhu@gmail.com 6