A FINIE ELEMEN MODEL FOR COMPOSIE BEAMS WIH PIEZOELECRIC LAYERS USING A SINUS MODEL S.B. Bhsht-Aval * M. Lzgy-Nazargah ** Dpartmnt of Cvl Engnrng Khajh Nasr oos Unvrsty of hnology (KNU) hran, Iran ABSRAC In ths stdy, fnt lmnt modlng of ompost bams wth dstrbtd pzoltr snsors and atators whh s basd pon a opld ltromhanal modl has bn onsdrd. For modlng of mhanal dsplamnt throgh th thknss, a sns modl that satsfs ontnty ondtons of transvrs shar strsss and th bondary ondtons on th ppr and lowr srfas of th bam has bn mployd. In th prsntd modl, th nmbr of nknowns s not dpndnt on th nmbr of layrs. h varaton of ltr potntal n ah pzoltr layr has bn modld sng layr-ws thory. By applyng th vrtal work prnpl (VWP), a formlaton has bn dvlopd for a two-noddd Hrmtan-(n + ) layr-ws noddd lmnt for a n-layrd bam. h VWP lads to a drvaton that old nld dynam analyss. Howvr, n ths stdy only stat problms hav bn onsdrd. Comparson of rslts obtand from ths formlaton wth avalabl works n th ltratr, dmonstrats ffny of proposd modl n analyss of lamnatd bams ndr mhanal and ltral loadngs. Kywords : Fnt lmnt, Compost bams, Pzoltr layrs, Sns modl, ransvrs strss.. INRODUCION In th past tn yars, smart strtrs hav attratd sgnfant attnton of many rsarhrs n th fld of atv ontrol and dynam. By dvlopng smart strtrs, pzoltr matrals hav bn sd xtnsvly as dstrbtd snsors and atators. h opld ltromhanal proprts of pzoltr matrals and thr lghtnss mak thm stabl for s n advand smart strtrs. A pzoltr matral gnrats an ltral harg whn sbjtd to a mhanal dformaton (drt pzoltr fft), and onvrsly, th pzoltr matral dforms whn sbjtd to an ltral fld (nvrs pzoltr fft). hrfor, pzoltr matrals ar apabl of atng as dstrbtd snsors and atators. Varos mathmatal modls hav bn prsntd to prdt th bhavor of strtrs ontanng pzoltr snsors and atators. hs modls an b lassfd nto two broad grops: Indd stran modls and opld ltromhanal modls. In th ndd stran modls, approxmat thors hav bn mployd to norporat th pzoltr ffts [-4]. In ths modls, th ltr potntal s ngltd as a stat varabl n th formlaton. Howvr, th ndd stran modls nontr lmtatons that ars from th s of for approxmaton to rprsnt th pzoltr strans. hs approxmat rprsntaton s not abl to aptr th opld mhanal and ltral rspons and lmts ths modls only n prdtng th atator bhavor of pzoltr matrals. h opld ltromhanal modls ar abl to prdt both th snsor and atator bhavor of pzoltr matrals, d to norporaton both th dsplamnts and ltr potntal as stat varabls n th formlaton. W and hs ollags sd an analytal approah to obtan thr-dmnsonal (3D) xat soltons of mltlayrd pzoltr plats/shlls. h formlaton of ths rsarhrs s basd on 3D pzoltrty and th mthod of prtrbaton has bn sd n thr drvatons [5-7]. Som rsarhrs sd 3D fnt lmnt to analyz th strtrs ontanng pzoltr snsors and atators [8-]. Althogh 3D fnt lmnt analyss gvs arat rslts, bt rqrs hg omptatonal ffort. Morovr, th omptatonal nvolvmnt boms hgh whn pzoltr layrs ar too thn ompard to th * Assstant Profssor, orrspondng athor ** Ph.D. stdnt Jornal of Mhans, Vol. 6, No., Jn 49
man strtr. hs staton s qt ommon n many pratal problms. hs ths knd of approah s nadqat n handlng a ral problm. In ordr to mprov omptatonal ffny layr-ws thors wr prsntd by Garao t al. [], Gara Lag t al. [,3], Hylgr and Saravanos [4], and Saravanos t al. [5]. In spt of th fat that layr-ws thors mprov th omptatonal ffny, n ths mthods th nmbr of nknowns ar dpndnt on th nmbr of layrs. So, layr-ws thors ar also rathr xpnsv approahs. In ordr to ovrom ths lmtatons, th onpt of mxd thory was prsntd [6,7]. In ths thory, th modlng of mhanal omponnts s basd on sngl layr plat thory whras ltr potntal s modld wth layr-ws thory. Bas of sng sngl layr thory for mhanal omponnts n mxd thory, th nmbr of nknowns s ndpndnt of th nmbr of layrs. Bas of xprssng th varaton of dsplamnt omponnts throgh th thknss by ontnos fntons, all th stran omponnts obtand from sngl layr thory ar ontnos and all th strss omponnts ar dsontnos at th layr ntrfas. In a lamnatd strtr, n-plan strans and ot of plan strsss ar ontnos at th layr ntrfas whras n-plan strsss and ot of plan strans ar dsontnos [8-]. hrfor, sngl layr thory s not abl to modl th bhavor of a ompost strtr proprly. In ordr to ovrom th drawbak of sngl layr thory, th da of rfnd thory was prsntd by rsarhrs [,]. Frst, rfnd frst-ordr shar dformaton thory (RFSD) was prsntd. In RFSD, th varaton of n-plan dsplamnt throgh th thknss of plat s p-ws lnar whl transvrs dsplamnt s onstant aross th thknss. Bhaskar and Varadan [3], D Sva [4], L and L [5], Parmrtr and Cho [6] ombnd th onpts of RFSD and hgh shar dformaton thory (thrd-ordr SD) of Rddy [7] to dvlop a nw lass of rfnd thory dfnd as rfnd hghr ordr shar dformaton thory (RHSD). In RHSD, th transvrs shar strsss ar ontnos at th layr ntrfas and hav a p-ws parabol varaton aross th lamnat thknss. h transvrs strsss ontnty nsrs dsontnty of transvrs strans at th ntrfa of two layrs whh hav dffrnt rgdty. Morovr, th transvrs shar strsss satsfy th bondary ondtons on th ppr and lowr srfas. Dstrbton of potntal ltr n pzoltr layrs (atators and/or snsors) s rprsntd by th sal layr-ws thory. In ths stdy a nw lmnt has bn proposd that has th sam nmbr of mhanal and ltral dgr of frdom wth mxd modl of Ch t al. [8]. In omparson to thr mxd modl, or proposd formlaton nsrs ontnty rqrmnts of th transvrs shar strsss. In or nw modl, th varaton of mhanal dsplamnt aross th thknss s basd on a sns modl [9]. hs modl satsfs fr ondton of th transvrs shar strsss on th top and bottom srfas of bam as wll as ontnty ondton of th transvrs shar strsss at th layr ntrfas. It has only thr ndpndnt gnralzd dsplamnts: wo dsplamnts and on rotaton. It has C ontnty xpt for th transvrs dsplamnt assoatd wth bndng whh has C. Conrnng th ltr potntal, th throgh thknss varaton s modld by th sal layr-ws thory. A omptr od whos algorthm s basd on th prsntd modl has bn dvlopd. Obtand nmral rslts show a good agrmnt wth othr pblshd rslts.. BASIC PIEZOELECRIC EQUAIONS AND VIRUAL WORK PRINCIPLE [3] h govrnng qatons for a pzoltr body of volm Ω and rglar bondary srfa S, an b wrttn as: σ + f =ρ () D j, j, q = () whr, f, q, ρ, ar mhanal body for omponnts, ltr body harg and mass dnsty rsptvly. σ j and D ar th symmtr Cahy strss tnsor and ltr dsplamnt omponnts. h followng onvrs and drt lnar pzoltr onstttv qatons rlat strss and ltr dsplamnt omponnts to th lnar symmtr Lagrang stran tnsor omponnts ε j and ltr fld omponnts E : σ j = Cjkl εkl kj Ek (3) D = kl ε kl +χ k Ek (4) C jkl, kj and χ k dnots last, pzoltr and dltr matral onstants. h stran tnsor and ltr fld omponnts ar obtand from mhanal dsplamnts and ltr fld potntal throgh th followng rlatons: ε j = (, j + j, ) (5) E = (6), Essntal or natral mhanal and ltral bondary ondtons or a ombnaton of thm mst b satsfd on bondary srfa S: or or = U (7a) σ n = F (7b) j j = V (8a) Dn = Q (8b) 5 Jornal of Mhans, Vol. 6, No., Jn
whr U, F, V, Q and n ar rsptvly prsrbd mhanal dsplamnt and srfa for omponnts, ltr potntal and srfa harg, and otward nt normal vtor omponnts. Usng admssbl vrtal dsplamnt δ and potntal δ, Eqs. () and () gv th followng xprsson: Ω ( σ + f ρ ) δ dω+ ( D q) δdω = (9) j, j, Ω Intgratng ths qaton by parts and sng dvrgn thorm gvs: σ δ dω+ σ n δ ds+ f δ dω ρ δ dω j, j j j Ω S Ω Ω, Ω S Ω D δ dω+ D n δds q δdω= () Usng th symmtry proprty of th strss tnsor, th natral bondary ondtons (7b), (8b) and rplang th ltr fld-potntal Eq. (6) nto Eq. () lads to: j j Ω S Ω σ δε dω+ F δ ds + f δ dω+ f δ ρ δ dω+ D δe dω QδdS qδdω= Ω Ω S Ω whr f s th omponnts of onntratd load. 3. MAHEMAICAL FORMULAION 3. Gomtry and Coordnat Systm () In ths stdy, th assmd ompost bam s mad of NC layrs of dffrnt lnarly last matrals. It has a nform rtanglar ross ston of hght h, wdth b and ts lngth s L. Eah layr may b pzoltr (atator and/or snsor) or non-pzoltr. In Fg., th lamnatd bam has bn shown n a Cartsan Coordnat Systm (x, y, z). 3. Constttv Rlatons Consdrng only bndng abot th y axs s takn nto aont, and th ross ston s symmtr abot th z axs, th onstttv qaton of pzoltr matral n ts matral-axs systm an b wrttn as: σ s = Cs ε s () Fg. h lamnatd bam and o-ordnat systm 3 3 33 3 C = 33, = 33, 55 5 χ χ= 33 χ 33 + 33 In th abov qatons, σ and D ar rsptvly th omponnts of strss and ltr dsplamnt. ε j and E j ar th omponnts of stran and ltr fld rsptvly and j, k j and j ar th orrspondng last, prmttvty and pzoltr onstants. h onstttv qaton of a lamna n a ommon strtral axs systm an b xprssd as: whr σ = Qε (3) σ= [ σ τ D D ] x xz x z ε = [ ε γ E E ] x xz x z C Q = Cs = χ whr σ s = [ σ σ 5 D D3] ε s = [ ε ε5 E E3] C C χ s =, In th abov qaton, s th transformaton matrx. In ths work th non-pzoltr matrals s assmd to b orthotrop and th gnral typ of pzoltr matrals s orthorhomb-lass mm. 3.3 Dsplamnt and Stran Flds h dsplamnt fld sd n ths modl s gvn by [9] Jornal of Mhans, Vol. 6, No., Jn 5
dw( x) dw( x) U( x, z, t) = ( x, t) z + +ψx ( x, t) dx dx ( NC) f( z) + α z+ g( z) + ( z z+ ) H( z z+ ) W( x, z, t) = w( x, t) (4) U(x, z, t) and W(x, z, t) ar th horzontal and vrtal dsplamnts rsptvly. (x, t) and w(x, t) ar th md-plan horzontal and vrtal dsplamnt rsptvly. ψ x (x, t) s th shar bndng rotaton arond th z axs. H s Havsd fnton, f (z) and g(z) ar dfnd by h πz h πz f( z) = sn g( z) os h = π π h πz πz f ( z) = os g ( z) = sn h h α ar th ontnty offnts obtand from a lnar systm. h allaton of ths offnts s dtald n Ston A. h oordnat systm of lamnatd bam has bn shown n Fg.. Usng th sal dfnton of stran (s Eq. (5)), th stran qatons an b drvd from Eq. (4) as follow: Fg. oordnat systm of lamnatd bam 3.4 h Layr-Ws hory for Eltr Potntal Basd on sssfl xprns of Shkh t al. [7], Cho and Oh [3-33], and opdar t al. [34] n assmng pws lnar ltr potntal along th transvrs drton, lnar ltr potntal has bn sd n ths stdy. hrfor, ltr potntal anywhr n th th layr an b xprssd as ( x, z, th layr) = ( x) L ( z) + ( x) L ( z) (7) d + d d w dψ x d w ε =ε x = z + F( z) + dx dx dx dx dw ε 5 = γ zx = ψ x + S ( z ) dx (5) whr z z z z L z = L z = + d ( ), ( ) z z+ z+ z whr ( NC) Fz () = f() z+ α z+ gz () + ( z z+ ) Hz ( z+ ) ( NC) Sz () = f () z+ α + g () z+ Hz ( z+ ) + ( z z ) ( z z ) + δ + and δ(z z + ) s frst drvatv of H(z z + ). h stran and dsplamnt Eqs. (4) and (5) an b xprssd n th followng matrx form: = A, ε = L (6) whr = [ U W], = [ w ψx], ε = [ ε ε 5], and d d z + F( z) F( z) A = dx dx, d d d d z + F( z) F( z) L dx dx dx dx = d Sz ( ) Sz ( ) dx and (x) and + (x) ar potntal fntons at th th and ( + )th ntrfas rsptvly. h ltr fld an b drvd from sal dfnton of ltr fld (s Eq. (6)): Ex x E = E = z layr z d d Ld ( z) L ( z) dx dx ( x) = = L dld ( z) dl ( z) + ( x) dz dz 4. FINIE ELEMEN FORMULAION (8) h mhanal and ltral lmnts bng onsdrd hav bn shown n Fg. 3. h thr mhanal varabls an b xprssd sng for mhanal nodal varabls as follows: = N (9) 5 Jornal of Mhans, Vol. 6, No., Jn
xprssd as: E L L N B = = = (5) Sbstttng Eq. (3) and Eqs. (3) ~ (5) nto Eq. () and assmblng th lmnt qatons ylds gnral dynam qaton of moton: [ M ] [ ] [ K ] [ K ] [ ] [ F] [ ] + [ K ] [ K] = [ ] [ F] (6) Fg. 3 (a) Mhanal two-noddd bam lmnt; (b) Eltral fv-noddd layr-ws lmnt whr = { w ψx ( dw/ dx) w ψ x ( dw/ dx) }, shap fnton matrxn whh N, N ar th Lagrangan shap fntons dfnd as: N = N ( ξ ) = ( ξ )/, N = N ( ξ ) = ( +ξ )/ () and th Hrmtan shap fntons ar N 4 N = N( ξ ) = ( ξ )( + ξ) 4 N 4 N N ( ) ( )( ) 4 = N( ξ ) = ( ξ ) ( + ξ), = N( ξ ) = ( ξ ) ( + ξ), = ξ = ξ + ξ () whr ξ s th loal oordnat dfnd as: x x ξ= x x () Usng Eqs. (9) and (6), th dsplamnts vtor and th stran vtor an b xprssd as follows: = A = A N = N, ε= L = L N = B (3) whr th N and B ar th dsplamnt ntrpolaton matrx and stran ntrpolaton matrx rsptvly. h vtor an b wrttn as: = N (4) N N whr N = and = [ ( + ) N N ( + )] From Eqs. (8) and (4), th ltr fld E an b h matrs and vtors n th abov qaton ar th mass matrx M = ρn NdV, th last matrx K V V = B CB dv, ltromhanal oplng ma- trx K = B B dv, th prmttvty matrx K = B χb dv, V N f dv V harg vtor V + N FdS S F th mhanal load vtor F = =. N QdS S + N f and th appld 5. NUMERICAL RESULS In ordr to valdat th proposd modl, th obtand nmral rslts hav bn ompard wth th xprmntal and nmral rslts of othr rsarhrs. h followng as stds hav bn onsdrd: () hr-layr antlvr bam mad of pzoltr and non-pzoltr matrals. () PVDF bmorph bam. h nmral rslts ar obtand from a MALAB program whos algorthm s basd on th proposd modl dsrbd n th prvos stons. In th followng as stds, th polarzaton of th pzoltr layrs s algnd n th transvrs drton of ths layrs nlss statd othrws. 5. hr-layr Cantlvr Bam Mad of Pzoltr and Non-Pzoltr Matrals h top and mddl layr of ths thr layrd antlvr bam s rsptvly mad of pzoltr matral and adhsv. Its bottom layr (sbstrat) s sotrop almnm or Gr/poxy ompost 3/934. Howvr, w hav sd both matrals for th bottom layr for omparson prposs. h gomtr data and matral proprts of ths bam hav bn shown n abl. hs layrd strtr has ban stdd by Saravanos and Hylgr [35], and Ch t al. [8]. Cas I (hr-layr atv antlvr): In ths ston, th atator apablty has bn nvstgatd. By applyng a.5kv voltag aross th thknss drton, th pzoltr layr wll at as an atator, whh wll bnd th bam. As shown n Fg. 4, ths bam has bn modld by sng fv lmnts of qal lngth and for layrs. D to th xstn of potntal Jornal of Mhans, Vol. 6, No., Jn 53
abl Proprts for thr-layr antlvr (data obtand from [8]) Almnm 3/934 Adhsv PZ-4 6.89.338 6.9 8.3 E 9 Pa 6.89.76 6.9 6.45 E 33 9 Pa Posson rato 3.5.4.4.43.76 5.65.46.56 G 3 9 Pa D 3 El. Prm. χ El. Prm. χ 33 3.989.656. MV.35965 8 FV.5 8 FV Lngth L.54.54.54.54 M hknss h.54.54.54.54 m Wdth b.54.54.54.54 m Dflton (mm) -.5 -. -.5 -. -.5 -.3 -.35 -.4 -.45 Fg. 5 Dflton (mm)..4.6.8. Normalzd Dstan (x/l) Ch t al. 3 Sar./Hy. 3 prsnt 3 Ch t al. Al Sar./Hy. Al prsnt Al Dflton ndd by pzoltr layr along th normalzd lngth of th antlvr.8.6.4..8.6.4...4.6.8. Normalzd Dstan (x/l) Fg. 6 Dflton d to load at antlvr tp Ch t al. Al Sar./Hy. Al Ch t al. 3 Sar./Hy. 3 prsnt Al prsnt 3 5 Fg. 4 hr-layr atator/snsor antlvr bam Eltr Potntal (V) 5 5 Sar./Hy. Al Ch t al. Al prsnt Al Sar./Hy. 3 Ch t al. 3 prsnt 3 dffrns aross th pzoltr thknss, pzoltr layr s dvdd nto two layrs. h obtand rslts from th prsnt mthod hav bn ompard wth rslts from othr rsarhrs n Fg. 5. h proposd modl, whh nsrs ontnty ondton for th transvrs shar strsss, has a hgh agrmnt wth mxd modl of Ch t al. Cas II (hr-layr atv/snsory antlvr): In ordr to nvstgat th snsory apablty, a mhanal load of N pwards was appld to th fr tp of layrd bam. D to th ltro-mhanal proprts, a potntal dffrn wll appar aross th thknss of pzoltr layr. h md-plan dfltons along th bam and th total voltag aross th thknss of th pzoltr hav bn plottd n Fgs. 6 and 7, rsptvly. From ths fgrs, th dflton and th voltag dstrbton prdtd by th prsnt formlaton hav a vry good orrlaton wth rslts of Ch t al. [8]. 5. PVDF Bmorph Bam h sond as stdy s a antlvr pzoltr bmorph bam wth two PVDF layrs bondd togthr and polarzd n oppost drtons. hs partlar xampl and ts rlatd dmnsons hav bn shown n Fg. 8. hs bmorph bam has bn stdd by th followng rsarhrs: Hwang and Park [36], zo and Y [37], Ch t al. [8], zo and sng [9], Slman and Vnkayya [38], Corra t al. [39], and Fknaga t al. [4]...4.6.8. Normalzd Dstan (x/l) Fg. 7 Voltag dffrn aross th pzoltr layr Fg. 8 Pzoltr bmorph bam In ordr to ompar th prsnt rslts wth rslts obtand by ths rsarhrs, th followng mhanal and pzoltr proprts ar sd for th PVDF bam: E = E = E 33 = GPa, G = G 3 = G 3 = GPa, ν = ν 3 = ν 3 =., 3 = 3 =.46C/m, and χ = χ 33 =.6 9 F/m. h 33 offnt s assmd to b zro. Also th bam has bn dsrtzd nto fv qal lmnts for omparson prposs. Cas I (Atator modl): By applyng a total V voltag aross th thknss drton, PVDF bmorph bam wll at as an atator. h ndd dfltons an b fond n abl. For omparson, th rslts 54 Jornal of Mhans, Vol. 6, No., Jn
abl Comparson of dfltons ndd by atators Dfltons ( 4 mm) Loaton x (mm) 4 6 8 Prsnt.38.55.4.8 3.45 Mxd modl (Ch t al. (999)).38.55.4.8 3.45 Plat FE (nn-nod lmnt) (Fknaga t al. ()).39.553.4. 3.45 Plat FE (for-nod lmnt) (Fknaga t al. ()).39.553.4. 3.45 hory (zo and sng (99)).38.55.4. 3.45 Plat FE ( DOFs) (Corra t al. ()).38.55.4. 3.45 Plat FE (9 DOFs) (Corra t al. ()).38.55.4. 3.45 Shll FE (FDS) (zo and Y (996)).3.58.9. 3.3 Plat FE (FDS) (Slman and Vnkaya (995)).4.55.4. 3.45 Analytal (Corra t al. ()).3.5.4. 3.6 Exprmntal (zo and sng (99)) 3.5 obtand from Ch t al. [8], zo and Y [37], zo and sng [9], Slman and Vnkaya [38], Corra t al. [39], and Fknaga t al. [4] hav bn mployd. From ths tabl, t an b fond that th prsnt rslts ar n vry good agrmnt wth prvos thortal and xprmntal rslts. Cas II (Snsor modl): h snsor apablty has bn nvstgatd n ths ston. h voltag dstrbton for an mposd tp dflton of mm has bn plottd n Fg. 9. In ths fgr, th prsnt rslts hav bn ompard wth th rslts obtand from Hwang and Park [36], Ch t al. [8], and zo and sng [9]. Bas Hwang and Park sd fv ps of sparat ltrod, th snsor voltag n Fg. 9 has a stp dstrbton. Howvr, t s possbl to masr th snsor voltag sng pont ltrods. Masrng th snsor voltags va ths typs of ltrods lads to a snsor dstrbton as shown by th rslts of zo t al., Ch t al. and th prsnt modl. Fgr 9 also ndats a good agrmnt btwn prvos rslts and th prsnt modl. 6. CONCLUSIONS In th prsnt papr, fnt lmnt modlng of pzoltr ompost bams wth dstrbtd snsor and atator layrs s onsdrd. h varaton of mhanal dsplamnt aross th thknss has bn modld by a sns modl whh nsrs ontnty ondton for th transvrs shar strsss as wll as th bondary ondtons on th ppr and lowr srfas of th bam. Any pzoltr layr n th lamnatd strtr an fnton as snsor or atator. hs modl has th dstnt advantag of nvolvng nknowns at th bam md-plan only. For th ltral potntal, th throgh thknss varaton was modld by th layr-ws thory. h proposd lmnt has th thr ndpndnt gnralzd dsplamnts: wo dsplamnts and on rotaton. It has C ontnty, xpt for th transvrs dsplamnt whh has C. A thr-layr antlvr bam mad of pzoltr and non-pzoltr matrals and a PVDF bmorph bam hav bn mployd for valdaton. A omptr Voltag Dffrn aross bam (Volts) 35 3 5 5 5 Fg. 9..4.6.8. Dstan (m) Hwang/Park zo t al Ch t al. prsnt Voltag dffrn aross th thknss of PVDF along th lngth of th antlvr od whos algorthm s basd on th prsnt modl has bn dvlopd. Comparson of nmral rslts from ths formlaton wth othr pblshd rslts shows that th prsnt modlng mthod s stabl n prdtng th bhavor of lamnatd bams ndr mhanal and ltral loadngs. APPENDIX A SINUS MODEL WIH CONINUIY CONDIIONS [9] A. Dsplamnt Fld h bam dsplamnt fld s dfnd as follows: x (, z) = x ( ) + zvx ( ) + f( z) ( x) + gz ( ) v( x) ( NC) wx (, z) = wx ( ) + ( x)( z z ) H( z z ) + + (A.) whr (x), v(x), (x), v (x), ( x ), w(x) ar th nknown fntons. H s th Havsd fnton, f (z) and g(z) ar dfnd by h πz h πz f( z) = sn g( z) os h = π π h πz πz f ( z) = os g ( z) = sn h h Jornal of Mhans, Vol. 6, No., Jn 55
h transvrs shar stran omponnt an b wrttn as: γ ( x, z) = v( x) + w ( x) + f ( z) ( x) + g ( x) v ( x) (A.) 3 ( NC) + ( x) H( z z+ ) A. Bondary Condtons At th bottom srfa z = h/; th followng rlaton mst b satsfd: ( ) () () σ 3 ( z ) = G v( x) + w( x) + v ( x) = vx ( ) + w ( x) = v( x) (A.3) In th sam way, at th top srfa znc + = h/; w hav: layr (k) : σ ( ) = ( ) ( ) + ( ) ( ) ( NC) ( k) ( k) 3 zk+ G f zk+ x x g zk+ k + ( x) layr (k + ): ( NC) ( k+ ) ( k+ ) 3 k+ k+ k ( g zk+ ) x k x ( ) σ ( z ) = G f ( z ) ( x) + ( x) ( ) + ( ) + ( ) For k {, NC }, Eq. (A.8) gvs: σ ( z = z ) = ( NC) 3 NC+ ( NC) ( NC) G v( x) + w ( x) v ( x) + ( x) = ( NC) = v ( x) hn, th followng rlatons an b ddd: (A.4) ( NC) ( k) G f ( zk+ ) ( x) + ( x) ( g ( zk+ ) ) k ( k+ ) + ( x) = G ( f ( zk+ ) ( x) ( NC) k ( k+ ) k + ( x) g ( z ) + ( x) + ( x) (A.9) ( NC) vx ( ) = w ( x) ( x) ( NC) v ( x) = ( x) (A.5) Sbstttng Eq. (A.5) n Eq. (A.), th bam dsplamnt fld an b rwrttn as: x (, z) = x ( ) zw ( x) + ( xf ) ( z) ( NC) z+ g ( z ) + ( z z+ ) H ( z z+ ) wx (, z) = wx ( ) (A.6) Unknown fntons ar now sd to nsr th ontnty ondtons at ah layr ntrfa. A.3 Contnty Condtons for th ransvrs Shar Strss From Eq. (A.6), th transvrs shar stran an b xprssd as: ( NC) ( NC) γ 3( x, z) = f ( z) ( x) + ( x) + g ( z) + ( x) H( z z ) + (A.7) In ordr to nsr ontnty ondton for th transvrs shar strss, w mst hav: σ ( z = z ) =σ ( z = z ) (A.8) ( k) ( k ) 3 k+ 3 k+ Eqaton (A.9) an b wrttn ndr th followng form: ( k) ( k+ ) ( G G ) f ( zk + ) = G ( x) + ( G G ) ( k+ ) ( k+ ) ( k) k + ( NC) k ( x) ( g ( zk+ ) ) ( x) (A.) Fnally, a systm of NC qatons s obtand for th NC nknown fntons ( x ). hs systm an b xprssd as: [ A][ U] = [ B] t whr [ U] = [... NC ] and th offnts A kl of [A] for k {, NC } ar: ( k+ ) ( k) ( )( k + ) l < k: A = G G g ( z ) + kl ( ) l = k A = G + G + G G g z ( k) ( k+ ) ( k+ ) ( k) : kl ( k + ) ( k+ ) ( k) ( )( k + ) l > k: A = G G g ( z ) kl whl [B] s dfnd by offnts: ( k) ( k+ ) ( ) B = G G f ( z ) k k+ h solton of ths systm ylds ( x) =α ( x) whr offnts α ar xprssd from 56 Jornal of Mhans, Vol. 6, No., Jn
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