Buckling Analysis of Piezolaminated Plates Using Higher Order Shear Deformation Theory

Transcription

1 Intrnational Journal of Composit Matrials 03, 3(4): 9-99 DOI: 0.593/j.cmatrials Buckling Analsis of Pizolaminatd Plats Using Highr Ordr Shar Dformation hor Rajan L. Wankhad, Kamal M. Bajoria Dpartmnt of Civil Enginring, Indian Institut of chnolog, Bomba, Mumbai, India Abstract Buckling analsis of pizolaminatd plat subjctd to combind action of lctro-mchanical loading is considrd for th prsnt work. Highr ordr shar dformation thor basd finit lmnt mthod is usd to prform th analsis. An isoparamtric ight nodd lmnt is mplod in th finit lmnt formulation. Studis ar conductd for diffrnt pl orintation and plat aspct ratios of pizolaminatd plat with diffrnt lctric condition of th pizolctric lar. Buckling loads ar found out for simpl supportd pizolctric laminatd plats. Pizolaminatd plats consist of cross pl smmtric and antismmtric orintation of laminats with pizolar attachd at th top and bottom of th plat. Rsults prsntd for buckling analsis of plats ar vrifid with othr numrical solution availabl in th litratur and furthr rsults for futur rfrncs ar providd. Kwords Buckling Analsis, Pizolaminatd Plat, Highr Ordr Shar Dformation hor. Introduction Ovr th last two dcads plnt of work has bn carrid out on analsis of pizolctric laminatd structurs including smart bams, columns and plats. h incrasd us of pizolaminatd smart structurs in structural application is stimulating dvlopmnt of diffrnt mthods and solution tchniqus for th accurat analsis of pizolaminatd structurs. Som of ths mthods ar Galrkin mthod, Finit lmnt mthod, and Rdd s highr ordr shar dformation thor. Also th xprimntal invstigation of smart pizolaminatd structurs has attractd som attntion. h nd of such smart structurs with high stiffnss and low wight in nginring applications has ld to th gradual rplacmnt of man isotropic matrials/structurs with light wight pizo-composits. h pizolctric matrial posssss a coupld lctromchanical proprt having a dirct and convrs pizolctric ffct. h availabilit of ths pizolctric cramics in th form of thin sht maks thm wll suitd for us as distributd snsors and actuators to control structural rspons in shap, vibration and buckling. Brlincourt t al.[] and Mason[] discussd th fundamntals of pizolctricit in brif. akagi[3] prsntd dfinition of smart matrials and structurs with notabl argumnts and dtaild commnts. h limitations of classical plat thor and first ordr shar dformation Corrsponding author: rajanw04@gmail.com (Rajan L. Wankhad) Publishd onlin at Copright 03 Scintific & Acadmic Publishing. All Rights Rsrvd thoris forcd th dvlopmnt of highr ordr thoris to avoid th us of shar corrction factors, to includ corrct cross sctional warping and to gt th ralistic variation of th transvrs shar strains and strsss through th thicknss of plat. zou and Zhou[4] studid linar dnamics and distributd control of pizolctric laminatd continua. Donthirdd and Chandrashkhara[5] dvlopd a mathmatical modl basd on a larwis thor for laminatd composit bams with pizolctric actuators to stud its shap control. Soars t al.[6] analsd pizolaminatd composit plats using rfind finit lmnt modls basd on highr ordr displacmnt filds and studid structural optimization. Artl and Bckr[7] invstigatd numricall th ffct of lctromchanical coupling on th intrlaminar strsss and th lctric fild strngths at fr dgs of pizo laminatd plats. Oh and L[8] studid gomtric nonlinar analsis of clindrical pizolaminatd shlls basd on th multifild larwis thor using finit lmnt mthod for th thrmal and pizolastic loading. zou and Gadr[9, 0] carrid out thortical analsis for vibration control of a multilard thin shll coupld with pizolctric actuators. Batra and Liang[] analsd th stad stat vibrations of a simpl supportd rctangular lastic laminatd plat with mbddd PZ lars. h usd thr dimnsional linar thor of lasticit. Oh t al.[] workd on postbuckling and vibration charactristics of pizolaminatd composit plat subjct to thrmo-p izolctric loads. Shn[3, 4] prsntd post buckling of shar dformabl laminatd plats with pizolctric actuators undr complx loading condition including thrmo-lctro-mchanical loading. Wang[5]

2 Intrnational Journal of Composit Matrials 03, 3(4): invstigatd lastic buckling of column structurs with a pair of pizolctric lars surfac bondd on both sids of th columns. h govrning quation coupling th pizolctric ffct was drivd basd on th assumd distribution of th lctric potntial in th flxural dirction of th pizolctric lar and an ignvalu problm was solvd using th dirct diffrnc mthod. Liw t al[6] carrid out stabilit of pizolctric FGM rctangular plats subjctd to non-uniforml distributd load, hat and voltag. For carring out analsis lmnt fr Galrkin mthod was mplod. In th prsnt work stabilit analsis of pizolaminatd plats is carrid out considring highr ordr shar dformation thor. Pizo laminatd plat studid is subjctd to combind action of lctromchanical loading. Linar through-th-thicknss lctric potntial distribution is assumd for ach pizolctric sub-lar. mpratur fild is assumd to b uniform for th orthotropic lars of th laminat and for pizolar. Finit lmnt formulation considrs an isoprimtric ight nodd rctangular lmnt. Paramtric studis ar conductd to dmonstrat th influnc of boundar condition, pl orintation and plat aspct ratio on stabilit of pizolaminatd plat with diffrnt lctric condition of th pizolar. umrical rsults ar prsntd considring simpl supportd and clampd pizolctric laminatd plat.. Finit Elmnt Formulation Fin it lmnt formu lation is basd on highr ordr shar dformation thor which do not nd shar corrction factor as that of first ordr shar dformation thor. And hnc for considring th ffct of shar dformation displacmnt fild of HOS is assumd as follows 3 u u0 zθ x z u0 z θ x 3 v v0 zθ z v0 z θ w w 0 () Whr, u, v and w ar th displacmnt of an point in th plat domain in x, and z dirction rspctivl. u 0, v 0 and w 0 ar th displacmnt of midpoint of normal. θ x, θ ar th rotations of normal at th middl plan in x and dirction about and x axis rspctivl. u 0, v0 w0, θ x and θ ar highr ordr trms which accounts cubic variation of normal. Figur. shows a pizolaminatd composit plat providd with pizolctric patchs at top and bottom ssurfac of th plat. Fibr can b orintd with rfrnc to th horizontal axs and is modlld in finit lmnt formulation. Figur. Pizolaminatd composit plat providd with pizolct ric pat chs at t op and bott om surfac

3 94 Rajan L. Wankhad t al.: Buckling Analsis of Pizolaminatd Plats Using Highr Ordr Shar Dformation hor Strain-Displacmnt Rlations h strains associatd with th displacmnt modl as pr highr ordr shar dformation thor ar givn b 3 ε x ε x0 zχ x z ε x 0 z χ x 3 ε ε 0 zχ z ε 0 z χ Whr, z 3 z0 zχ z z ε z 0 z χ ε ε x 3 x0 zχ x z ε x 0 z χ x γ ε 3 γ xz ϕ x zχ xz z ϕ x z χ xz 3 z ϕ zχ z z ϕ z χ z () γ [ ε ε ε ε ] u v 0 0 x0 0 z0 x0 θ x u0 v0 θ θ x θ θ x x z x w0 χ χ χ χ u v uo v [ ε x 0 ε 0 ε x 0 ] [ χ χ χ ] θ θ x θ θ x x x [ φ φ ] x [ ψ ] w w 0 0 3θ x 3θ θ z θ z ψ x u0 v0 (3) Shll Elmnt and Shap functions 8-nod rctangular lmnt is mplod as shown in figur Figur. Eight-noddd rctangular lmnt Shap functions ar givn as, For cornr nods: i ( ξ, η) ( ξξi ) ( ηηi )( ξξi ηηi ), 4 for i,,3,4 For middl dg nod 5( ξ, η) ( ξ )( η) 7 ( ξ, η) ( ξ )( η) 6 ( ξ, η) ( ξ )( η ) 8( ξ, η) ( ξ )( η ) (4) Displacmnt fild h displacmnt fild associatd with th ight noddd lmnt can b writtn as u v 8 i 8 i 8 i i u i i v i w i w i (5) Strain within th Elmnt Strains associatd with th displacmnt fild can b writtn as follow, a. Middl plan mmbran strains L εp εp εp (6) 3 in which L ε p and ε p ar linar and non-linar componnts of middl plan mmbran strains and combining linar and non-linar trms mmbran strains ar givn as, u w x v w ε p (7) 3 u v w w. x Curvatur strains/bnding strains Curvatur strains ar linarl rlatd to bnding displacmnt as, ε L b ε xxb εb ε xb

4 Intrnational Journal of Composit Matrials 03, 3(4): Shar strains, hus Hnc, θ x x θ z θ θ x γ γ γ θ x θ xz z ε w w x ε p 3 ε 3 6 b L ε p ε p L ε b 0 ε ε L ε z.k (8) (9) (0) () and shar strains can sparatl b writtn as, γ xz γ () γ z In which { ε } { ε } and { γ } ar mmbran, bnding and p, b shar componnts of strains rspctivl. is combind 6 strain vctor of mmbran and bnding strains. γ is a vctor containing shar strains. Subscript p stands for in-plan, b for bnding, L for linar and subscript stands for non-linar. Displacmnt-strain rlation Strains ar rlatd with displacmnts using strain displacmnt matrix as follows, ε B δ (3) Elctro-Mchanical Coupling For pizolaminatd plats two constitutiv rlationships xist including th ffct of mchanical and lctrical loading as givn b q. 4. Variation of tmpratur ffct is nglctd in formulation. p D ε g E { } [ ]{ } [ ]{ } ε t p { } [ C]{ ε} [ ] { E } σ (4) And hnc, σ x Q Q Q Q 0 0 ε 3 4 x σ Q Q Q Q ε σ Q Q Q Q 0 0 ε z z σ Q Q Q Q 0 0 ε x x σ Q Q (5) xz ε xz σ Q Q z ε z 0 0 z 0 0 p z E x 0 0 z3 p E p E z x6 0 0 Whr, {D} is lctric displacmnt vctor,[] is dilctric g is th σ is th prmittivit matrix, ε is th strain vctor, { } dilctric matrix. { E} is th lctric fild vctor, [ ] strss vctor and[c] is th lastic matrix for constant lctric fild. Elctrical Potntial Function φ a and φ s ar th lctric displacmnt at an point in th actuator and th snsor lars, rspctivl, th lctrical potntial functions in trms of th nodal potntial vctor ar givn b φ a [ pa ] { φa } Whr, [ ] pa φ [ ] { s ps φs } and [ ] (6) ps ar th shap function matrics for th actuator and snsor lars, rspctivl. { φ a } and { φ s } ar th nodal lctric potntial vctor for th actuator and snsor lars rspctivl and can b givn as follow. { } { } φ a φa φa φ... a3... φan { φ } { φ φ φ... φ } s s s s3... sn (7) Stiffnss Matrix Equations El mnt stiffnss matrix can b writtn as, [ K ]{ δ } [ Kσ ] { δ } [ F ] [ Fac ] (8) In which [ K ] [ K d ] [ K aa ] [ K aa ][ K ad ] [ K ds ][ K ss ] [ K sd ] { F } [ K ][ K ] { Q } Whr, (9) ac (0) da aa [ K d ] [ B] [ C][ B] dv, V [ da ] [ K ad ] [ B] [ ][ Ba ] K dv, Va a

5 96 Rajan L. Wankhad t al.: Buckling Analsis of Pizolaminatd Plats Using Highr Ordr Shar Dformation hor [ aa ] [ Ba ] [ g][ Ba ] dv Va [ K ds ] [ K sd ] [ B] [ ][ Bs ] K, Vs and [ ss ] [ Bs ] [ g][ Bs ] Vs dv K dv () hus th lmnt quation in th global stiffnss matrix can b writtn as [ K ]{ } [ K ] { δ } [ F ] [ ] δ σ () F ac Equilibrium and incrmntal Equations Virtual displacmnt princip l is mplod to obtain quilibrium quation. Equilibrium btwn intrnal and xtrnal forcs has to b satisfid. If Ψ rprsnts th vctor of th sum of th intrnal and xtrnal forcs. { Ψ } { R} { P} (3) Whr, {R} rprsnts th xtrnal forcs du to imposd load and {P} is a vctor of intrnal rsisting forcs. h quilibrium stat is achivd whn { Ψ } 0. { Ψ } { R} { ε} { σ} dv Va { } p E s { Ds} dv { ε } { σ 0 } dv Vs V p a E Da dv V (4) Whr V, V a and V s ar th ara of th ntir structur, snsor lar and actuator lar rspctivl. Considring th work don b xtrnal forcs du to th applid surfac traction and applid lctric charg on actuator th quation for xtrnal work don can b writtn as { R} { u} σ ( x, ) da ϕa qa ( x, ) da (5) A A Stabilit quation h gomtric stiffnss matrix is associatd with th x and coordinats. h charactristic quation of stabilit is writtn as, ([ K ] λ [ Kσ ]) { δ} F (6) ([ K ] λ [ Kσ ]) ({ δ} { dδ} ) { F} (7) Whr, th lowst magnitud of ign valu givs critical buckling load and th vctor { dδ} rprsnts th buckld mod shap. And th ign valu problm is solvd to gt buckling loads. K λ Kσ dδ (8) ([ ] [ ]) { } 0 3. Rsults and Discussions Buckling analsis of pizolaminatd plats hav bn carrid out for diffrnt pl orintation considring smmtric and antismmtric cross-pl lamination schms. Buckling of smart pizolaminatd plat is xamind subjctd to uniform axial comprssiv forc at th dgs. Furthrmor for pizolaminatd plat, critical mchanical buckling loads ar obtaind for diffrnt thicknss to span ratios of pizolaminatd plats and for closd and opn loop lctric condition. 3.. Buckling of Simpl Supportd S quar Multilard Plat (0 0 /90 0 ) n _ abl. on-dimnsional critical buckling load ( σ cr / ) for simpl supportd multilard cross pl laminat d plat b E h a/h thor Lar Dgr of Orthotrop ( E /E ) prsnt Own and Li [7] prsnt Own and Li [7] prsnt Own and Li [7] prsnt Own and Li [7] prsnt Own and Li [7] prsnt Own and Li [7] prsnt Own and Li [7] prsnt Own and Li [7] prsnt Own and Li [7] prsnt Own and Li [7]

6 Intrnational Journal of Composit Matrials 03, 3(4): Effct of orthotrop is studid for simpl supportd squar multilard plat subjctd to a uniform uniaxial comprssiv forc at th dgs. Laminatd plat is having cross pl orintation of lars. Critical buckling loads ar tabulatd in abl for ffct of orthotrop of individual lars, th ratio of span to thicknss (a/h) and th numbr of lars of simpl supportd squar multilard composit plats. Matrial proprt adoptd is as follow. E E3 G G3 3to 40,, 0.6, E E E E G E and µ µ 3 µ Rsults ar found in good agrmnt with that of rfind fin it lmnt solution givn b Own and Li [7]. 3.. Buckling of Simpl Supportd Squar Pizolctric Laminatd Plat (p/0 0 /90 0 /90 0 /0 0 /p) h pizolctric laminat with pizolar attachd at th top and bottom of th plat is subjctd to a uniaxial uniform dg comprssiv forc x. Hnc a squar pizolctric laminatd plat of thicknss of 0.0 m and sid lngth a is having simpl supportd boundar on all four dgs. h laminat consists of a (p/0 0 /90 0 /90 0 /0 0 /p) Graphit-Epox sublaminat providd with two PZ-5A attachd on outr surfac of th plat. Each pizolctric lar has thicknss of 0.h, whras ach lastic lar has a thicknss of 0.h. Elastic and pizolctric matrial proprtis ar as shown in tabl. abl. Elast ic and pizolct ric proprtis for Graphit-Epox and PZ-5 Elastic Proprtis Pizolctric Proprtis Proprt is Grphit-Epox PZ -5A Proprt is Grphit-Epox PZ -5A E ( Gpa) d 3 (0 - m/v) 0-7 E ( ) d 3 (0 - m/v) 0-7 E 33 ( Gpa) d 33 (0 - m/v) G ( ) d 5 (0 - m/v) G 3 ( GPa).87. d 4 (0 - m/v) G 3 ( GPa) 7.7. g (0-8 F/m) g (0-8 F/m) g 33 (0-8 F/m) h ffct of span to thicknss ratio on critical uniaxial buckling load ( x a /E.h 3 ) of simpl supportd squar laminatd plat with diffrnt lctric condition is shown in figur 3. cr a/h Opn Loop Closd Loop Without pizo ffct Fi gur 3. Effct of span to thicknss ratio on normalizd critical buckling load of simpl support d smmt ric cross pl pizolaminatd plat (p/0 0 /90 0 /90 0 /0 0 /p) Fro m figur 3 it can b obsrvd that critical buckling load incrass for pizolctric ffct. Also pizolctricit has littl ffct on th buckling load of th laminats with closd-circuits than that of opn loop. hr is an incras of 7.9 % in critical buckling load for pizolaminatd plat than that of without pizoffct. Hnc critical buckling load can b considrabl incrass if plat is subjctd to pizoffct. For lssr vaus of a/h ratio % variation in critical buckling load is found to b 3.3 % for opn loop lctric condition than that of closd loop Buckling of Si mpl Supportd Squar Pizolctric Laminatd Plat (p/0 0 /90 0 /0 0 /90 0 /p) Simpl supportd squar pizolaminats with pizolar attachd at th top and bottom of th plat is subjctd to a uniaxial uniform dg comprssiv forc x. Pizolctric laminatd plat is having a thicknss of 0.0 m and sid lngth a. h laminat consists of antismmtric pl orintation of Graphit-Epox sublaminat providd with two PZ-5A attachd on outr surfacs of th plat. Matrial proprtis of Graphit-Epox and PZ-5A ar sam as

7 98 Rajan L. Wankhad t al.: Buckling Analsis of Pizolaminatd Plats Using Highr Ordr Shar Dformation hor prvious xampl. Each pizolctric lar has thicknss of 0.h, whras ach lastic lar has a thicknss of 0.h. Figur 4 shows ffct of span to thicknss ratio on critical mchanical buckling load ( x a /E.h 3 ) of pizo laminatd plat with diffrnt lctric conditions. c r Fi gur 4. Effct of span to thicknss ratio on normalizd critical buckling load of simpl support d ant ismmt ric cross pl pizolaminatd plat (p/0 0 /90 0 /0 0 /90 0 /p) Significant incras in critical buckling load is obsrvd for opn loop lctric condition than that of closd loop. hus pizolctricit has littl ffct on th buckling load of th laminats with closd-circuits in this cas. h incras in critical buckling load is found to b 8.09 %,.05 % and 0.9 % for opn loop circuit than that of closd loop for a/h 0, 0 and 00 rspctivl. 4. Conclusions A finit isoparamtric lmnt for th mchanical displacmnt fild is combind with an lctric potntial fild to considr pizolctric ffct. Paramtric stud is conductd to dmonstrat th influnc of boundar condition, pl orintation and plat aspct ratio on stabilit of pizolaminatd plat with diffrnt lctric condition of th pizolar. For smmtric cross pl pizolaminatd plat % incras in critical buckling load is found to b 7.9 % than that of without pizoffct. hus considring pizoffct bucklibg load can b considrabl incrasd. Appndix: Stiffnss Cofficints of th Laminatd Plat According to h Highr Ordr Shar Dformation hor h stiffnss cofficints according to th HOS ar as follows 4 4 ( C C ) sin θ cos θ C θ Q C cos θ 33 sin Q a/h Opn Loop Closd Loop 4 4 ( cos θ sin θ ) ( C C 4C ) sin θ θ C 33 cos ( ) ( ) ( ) 3 ( C C C33 ) sin θ cosθ 3 ( C C C33 ) cos θ sinθ ( C C C C ) sin θ cos Q C C C sinθ cos θ C C C cosθsin θ Q C sin θ C C sin θ cos θ C cos θ Q Q 3 33 C ( cos θ sin θ ) Q 44 C44 cos θ C55 sin Q C sinθ cos ( ) θ C55 C44 sin Q 55 θ C55 cos θ Q ij Q ji for i, j,, 5 Whr, E ( ) x z z C ; x x E ( ) x z z x C ; E ( ) z x x C ; z z E ( ) x x z x z C ; x E ( ) x z x x z C ; x z E ( ) z x z x C x C z G ; x k C C z x z G ; z x G ; k z k And θ 33 ( x x z z zx xz x z xz ) REFERECES [] D. A. Brlincourt, D. R. Curran and H. Jaff, "Pizolctric and pizomagntic matrials and thir functions in transducrs", In: Mason W. P. (Ed.), Phsical Acoustics,, Acadmic w York, pp , 964. [] W. P. Mason, "Pizolctricit, its histor and applications", h Journal of Acoustical Socit of Amrica, vol.70, pp , 98. [3]. akagi, "A concpt of intllignt matrials", Journal of Intllignt Matrials, Sstms and Structurs, vol., pp , 990. [4] zou, H. S., Zhou, Y. H. (995). "Dnamics and control of non-linar circular plats with pizolctric actuators", Journal of Sound and Vibration, 88 (), pp θ

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