Influence of Fiber Orientation on the Natural Frequencies of Laminated Composite Beams

Transcription

1 Intrnatonal Journal of Engnrng Rsarch And Advancd Tchnology (IJERAT) DOI: E-ISS : Vol. (9) Sp -7 Influnc of Fbr Orntaton on th atural Fruncs of Lamnatd Compost Bams Imad-Eldn Mahmoud Mahd, Osama Mohammd Elmard Sulman Assstant Profssors, Dpartmnt of Mchancal Engnrng, Faculty of Engnrng and Tchnology, l Vally Unvrsty, Atbara, Rvr l Stat, Sudan ABSTRACT In ths study, th ffct of th fbr orntaton on th natural fruncs of fr vbraton of lamnatd compost bams was nvstgatd. Th problm s analyzd and solvd usng th nrgy approach whch s formulatd by a fnt lmnt modl. In that modl, a thr-nodd lmnt wth thr dgrs of frdom at ach nod s assumd. umrcal rsults wr obtand for symmtrc and non-symmtrc lamnatd bams, and vrfd by comparsons wth othr rlvant wors. Also, som spcal cass, whch rlatd to th ttl, wr studd and prsntd. Th angl of fbrs orntaton, whch masurd from th longtudnal as of th bam, s vard from o o to9. It s found that both symmtrcally and ant-symmtrcally lamnatd bams of smlar sz and nd condtons hav ual natural fruncs whch, gnrally, dcras as th angl of orntaton ncrass. Kywords: Fnt Elmnt Mthod, Frst Ordr Shar Dformaton Thory, Fr Vbraton, Lamnatd Composts, atural Fruncs.. ITRODUCTIO Compost matrals ar thos formd by combnng two or mor matrals on a macroscopc scal such that thy hav bttr ngnrng proprts than th convntonal matrals, for ampl, mtals. Som of th proprts that can b mprovd by formng a compost matral ar stffnss, strngth, wght rducton, and corroson rsstanc, thrmal proprts, fatgu lf, and war rsstanc. Most man-mad compost matrals ar mad from two matrals: a rnforcmnt matral calld fbr and a bas matral, calld matr matral. Lamnar composts ar thos havng altrnatng layrs of matral bondd togthr n som mannr and nclud thn coatngs, thcr protctv surfacs, claddngs, bmtallc, lamnats, and sandwchs. Lamnatd compost bams ar ncrasngly bng usd n many ngnrng applcatons n th flds of mchancal and cvl ngnrng, transportaton vhcls, marn, avaton and arospac. Pag

2 Imad-Eldn Mahmoud Mahd t al.,influnc of Fbr Orntaton on th atural Fruncs of Lamnatd Th paprs, whch ar prsntd hr as rfrncs, addrss th problm of th fr vbraton of lamnatd compost bams. A thortcal analyss of th vbraton of compost bams wth sold cross sctons was also prsntd by Toh and Huang [], Chandrashhara t al. [], Abramovch []. In thos analyss, th uatons of moton wr basd on a Tmoshno bam modl (shar dformaton consdrd). umrcal rsults showd th ffct of th shar dformaton and fbr orntaton on th natural fruncs. Agan, Abramovch and Lvshts [4] prsntd act solutons for th fr vbraton of non-symmtrcally lamnatd cross-ply compost bams. Marur and Kant [5] and [6], and McCarthy t al [7] appld hghr ordr shar dformaton thors to solv th problm of th fr vbraton of compost bams. Th frst-ordr shar dformaton thory was usd by Tboub and Hajla [8] to analyz th fr vbraton of gnrally layrd compost bams. Hodgs t al. [9] prsntd two dffrnt mthods, whch wr smpl analytcal mthod and fnt lmnt mthod for th prdcton of th natural fruncs and mod shaps of compost bams. In addton to th rfrncs mntond abov, rfrncs [], [], and [] appld dffrnt tchnus of th fnt lmnt mthod for th sam problm. Th thory usd n th prsnt papr coms undr th catgory of dsplacmnt thors as classfd by Phan and Rddy [], Osman and Sulman [4] and [5]. In ths thory, whch s calld frst ordr shar dformaton thory (FSDT), th transvrs plans, whch ar orgnally normal and straght to th mddl plan of th bam, ar assumd to rman straght but not ncssarly normal aftr normal aftr dformaton, and consuntly shar corrcton factors ar mployd n ths thory to adjust th transvrs shar strss, whch s constant through thcnss.. MATHEMATICAL FORMULATIO Som assumptons wr mad n ths analyss, whch ar: () All layrs bhav lastcally; () Dsplacmnts ar small compard wth th bam dpth; () Prfct bondng sts btwn layrs; (4) Th lamnat s uvalnt to a sngl ansotropc layr; (5) Th bam s flat and has a rctangular scton and vbrats n a vacuum; and (6) All nds of dampng ar nglctd. Fgur blow shows th gomtry of a bam drawn n th thr dmnsons X, Y, and Z or,, and. Y, h X, b L Z, Fgur Th gomtry of a lamnatd compost bam. Th tm-dpndnt aal and transvrs dsplacmnts flds ar: U W, z, t u, t z, t, z, t w, t () Publcaton Impact Factor (PIF):.598 Pag

3 Intrnatonal Journal of Engnrng Rsarch And Advancd Tchnology Vol. (9) Sp -7 Whr, u and w ar th aal and transvrs dsplacmnts at th md-plan, z s th prpndcular dstanc from th md-plan to th layr plan, s th rotaton of a plan aftr dformaton, and t s th tm. Th strandsplacmnt rlatons ar: U u z W U w 5 z () Whr th subscrpts hav th sam manngs as thos usd n -D lastcty formulaton,.. s th aal or longtudnal stran, and 5 s th through-thcnss shar stran. Th strss-stran rlatonshp of a lamna can b shown as: j C () Whr, T 5 C C j (4) C 55 t 5 Th lastc constants C 55 for orthotropc bams can b prssd as: C and C C 55 C C 44 4.cos C.sn C 55.sn.cos 4 ( C. C 66 ).sn.cos (5) Whr; C (6) 66 G; C44 G; C55 G; C C C Applyng th nrgy approach for th bam lmnt shown n Fgur, th stran nrgy stord s gvn by: U s T dv (7) Whr, dv bddz and th subscrpt,, mans on lmnt. u u u w w w L X Fgur A -nodd fnt lmnts wth nn dgrs of frdom and shap functons. Pag

4 Imad-Eldn Mahmoud Mahd t al.,influnc of Fbr Orntaton on th atural Fruncs of Lamnatd Also, th ntc nrgy s found as follows; W U K. E. b W U d dz (8) t t Th dgrs of frdom at ach nod ar; th aal dsplacmnt u, dflcton w, and rotaton, and can b wrttn n trms of thr nodal valus as follows: u, w, u, w, Whr, (9) s shap functon and assumd as a scond-ordr polynomal as: a b c,, () Th constants a, b c, and can b computd for ach lmnt from th followng data: at,, () Ens. (7), and (8) lads to th fnal form of th non-dmnsonal lmnt stffnss and nrta matrcs [K],and [M] rspctvly, whch ar gvn by: K A A s B A s D A s 7 Symmtrc A B A4 A s A s 4 A s 4 B A s 6 D A s 8 B4 A s 7 D4 A s A B A5 B5 A6 A s A s 5 A s 5 A 8 A s 6 B A s 9 D A s 9 B5 A s D. 5 A s B6 A s D6 A s () M Symmtrc () Whr, s th aspct rato (lngth to thcnss rato), and; Publcaton Impact Factor (PIF):.598 Pag 4

5 Intrnatonal Journal of Engnrng Rsarch And Advancd Tchnology Vol. (9) Sp -7 Pag 5 d d d d d d d d d d d d d d d d d d d d d (4) For lamnatd bam wth (n) layrs, th constants n En. () ar:,,,, 55 n Z Z s s n Z Z bdz C K A bdz C z z D B A (5) Th constants A, B, and Dar th tnsonal, couplng, bndng stffnsss rspctvly, whl th constant As s th shar coffcnt, and Ks s th shar corrcton factor (tan or 6 5 ). Th ndvdual lmnt stffnss and nrta matrcs [K] and [M] must b lnd togthr or assmbld to charactrz th unfd bhavor of th ntr bam. Thrfor, th global stffnss and nrta matrcs ar gvn rspctvly by, n n M M K K (6) Whr, s th total numbr of bam lmnts. Th bam nd condtons whch consdrd ar; () Clampd-fr bam (CF), () Hngd-hngd bam (HH), () Clampd-clampd bam (CC), (4) Hngd-clampd bam (HC), (5) Hngd-fr bam (HF), and (6) Fr-fr bam (FF). Each bam has thr movabl nds or mmovabl nds. In th formr group, th aal dsplacmnt at both bam-nds s consdrd, whl nglctd for th lattr group. Th soluton can b obtand aftr th ncorporaton of nds condtons whch wll modfy both stffnss and nrta matrcs. Thus, th non-dmnsonal natural fruncs can b dtrmnd from th rlaton: I K M (7)

6 Imad-Eldn Mahmoud Mahd t al.,influnc of Fbr Orntaton on th atural Fruncs of Lamnatd Whr, I s an dntty matr, and s th non-dmnsonal natural fruncs, whch can b computd by computng th suar root of th gnvalus of th matr M K MATLAB was usd). usng a sutabl computr program (Hr. UMERICAL RESULTS AS/5-6 grapht-poy matral was usd for all numrcal rsults bcaus of ts wd applcatons n modrn ndustrs. Th mchancal proprts of ths matral ar tabulatd n Tabl (). Tabl Mchancal Proprts of AS/5-6 grapht-poy matral Proprty E E G G G Posson s rato Dnsty Magntud 45 G/m 9.6 G/m 4. G/m 4. G/m.4 G/m. 5 g/m In ordr to chc th valdty of th prsnt mthod, som comparsons wr prformd. Ths comparsons wr slctd to covr th cass of symmtrcally and non-symmtrcally lamnatd bams. Tabl () shows comparsons wth Chandrashhara t al. []. Th comparson shows th dffrncs btwn th non- dmnsonal fundamntal fruncs [. L E h ] of symmtrc [ / / / ] angl-ply bams wth dffrnt boundary condtons for aspct rato of (L/h=5). Th angl vars from o to o 9 wth a stp ( o 5 ). Th comparson showd a dffrnc of lss than.6% for th angl zro, ncrasng to about.% for hghr valus of th angl. Tabl on-dmnsonal fundamntal fruncs [. L E h ] of symmtrc [ / / / ] anglply bams (L/h = 5). (dg.) Bam Typ Hngdhngd Clampd- Clampd Hngd- Fr Fr-fr prsnt Rf. [] Prsnt Rf. [] prsnt Rf. [] Prsnt Rf. [] Publcaton Impact Factor (PIF):.598 Pag 6

7 Intrnatonal Journal of Engnrng Rsarch And Advancd Tchnology Vol. (9) Sp -7 Clampd- Fr Clampd- Hngd prsnt Rf. [] Prsnt Rf. [] Tabl () prsnts a comparson wth Abramovch and Lvshts [4] of th non-dmnsonal fruncs of symmtrc [/9/9/] cross-ply grapht-poy bams for aspct rato of (L/h = ). Th bams consdrd ar hngd-hngd, fd-fr, and fd-fd wth mmovabl nds. Hr, th authors ntroducd th scondary ffct of couplng btwn bndng and torson n thr analyss, whch s small, compard wth th othr scondary ffcts. For th hngd-hngd bam, a prcntag dffrnc of lss than.6% was rcordd for th fundamntal fruncy, and lss than.54% for both fd-fr and fd-fd bams. Ths dffrnc was obsrvd to ncras as th mod ordr ncrass (lss than.4%) for th svnth mod for all bams consdrd. In addton, Tabl shows th mods wth prdomnanc of longtudnal vbraton. Tabl on-dmnsonal fruncs [. L E h ] of symmtrc [/ 9/ 9/ ] cross-ply bams (L/h = ) Mod Hngd-hngd (Immovabl) Fd-fr (Immovabl) Fd-fd (Immovabl) o. Prsnt Rf. [4] Prsnt Rf. [4] Prsnt Rf. [4] *.9* *.7* *.7* (*) Mods wth prdomnanc of longtudnal vbraton. Th last comparson was carrd out wth th rsults of Marur and Kant [6]. Tabl (4) compars th frst s mods of th non-dmnsonal fruncs [. L E h ] of symmtrc [ /9/9/ ] cross-ply, grapht-poy, clampd-fr bam wth aspct rato of (L/h = 5). Th authors appld th hghr-ordr shar dformaton thory (HOSDT) n th analyss, whras, th prsnt mthod uss frst-ordr shar dformaton thory. Th comparson shows a prcntag dffrnc of lss than.% for th fundamntal mod of vbraton. Ths dffrnc ncrass wth th mod ordr to lss than.6% for th sth mod. Tabl 4 on-dmnsonal natural fruncs [. L E h ] of symmtrc [ /9/9/ ] cross-ply clampd- fr bam. (L/h = 5). Mod o. Prsnt Rf. [6] Pag 7

8 Imad-Eldn Mahmoud Mahd t al.,influnc of Fbr Orntaton on th atural Fruncs of Lamnatd * (*) Mods wth prdomnanc of longtudnal vbraton. It s obvous from th numrcal rsults gvn n Tabl () abov, and Tabl (5) blow, that th valus of nondmnsonal natural fruncs of varous bams gnrally dcras as th angl of orntaton of fbrs wth rspct to th longtudnal as of th bam s ncrasd. Tabl 5 Th frst thr non-dmnsonal mods of fr vbraton of symmtrc / / / lamnatd bams Angl Mod o. Bam typ wth mmovabl nds. L/h=. CF HH CC HC HF FF o o o Smlar valus of fruncs for symmtrc / / / lamnatd bams wth mmovabl nds and aspct rato of ar plottd aganst th angl of orntaton for th rang from up to 9 dgrs n Fg. () to Fg. (8). Th nflunc of fbr orntaton bcoms mor notcabl as th mod ordr ncrass, and sgnfcant varatons of fruncs wr obsrvd up to an angl of appromatly 7 dgrs. Byond ths angl, th varatons n th fruncs ar vry small. Incrasng angl of orntaton to mor than 7 dgrs lads to ncras th couplng btwn bndng and strtchng ffct, whch causs th lamnatd bam to b stffr, and thus th varaton n natural fruncs dcrass. In addton, th valus of non-dmnsonal natural fruncs of th longtudnal mods of fr vbraton ar obsrvd to dcras as th angl of fbrs orntaton s ncrasd. 4. CASE STUDIES Som mportant cass, whch rlatd to th fbr orntaton, wr studd and prsntd hr. Th frst cas s rpatd or altrnatng st of layrs n smlar bams. Tabl (6) shows th non-dmnsonal natural fruncs of Publcaton Impact Factor (PIF):.598 Pag 8

9 Intrnatonal Journal of Engnrng Rsarch And Advancd Tchnology Vol. (9) Sp -7 symmtrc 6 / 6 / 6 / 6 hav th sam aspct rato of 5 and 6 / 6 / 6 / 6 lamnatd bams rspctvly. Both bam lamnatons Th frst st of bams contans four layrs orntd as shown n Fg. (9)a, whlst th othr st contans ght layrs as shown n Fg. (9) b. It s obsrvd that ths two sts hav th sam natural fruncs for sam aspct rato. In ths cas, ual valus of th coffcnts A, B, D, and As wll b obtand for both bams and consuntly th rsultng nrta and stffnss matrcs ar th sam for both. (b) 6 / 6 / 6 / 6 (a) 6 / 6 / 6 / 6 Pag 9

10 Imad-Eldn Mahmoud Mahd t al.,influnc of Fbr Orntaton on th atural Fruncs of Lamnatd Fgur 9 Two sts of bam lamnatons Tabl 6 on-dmnsonal fruncs of a symmtrc [6/-6/-6/6] and [6/-6/-6/6] lamnatd bams wth mmovabl nds, (L/h = 5) Bam typ Mod o. CF HH CC HC HF FF Th scond cas s rvrsd lamnaton ordr of smlar bams. Two lamnaton sts wr consdrd whch ar / 6 / 9 and9 / 6 /. Th rsults of thr non-dmnsonal natural fruncs ar gvn n Tabl (7) for an aspct rato of 5. It s obvous that ths fruncs ar th sam for both sts. 4 Tabl 7 on-dmnsonal fruncs L Eh of a non-symmtrc [/6/9] and [9/6/] lamnatd bams wth mmovabl nds, (L/h = 5). Bam typ Mod o. CF HH CC HC HF FF Publcaton Impact Factor (PIF):.598 Pag 4

11 Intrnatonal Journal of Engnrng Rsarch And Advancd Tchnology Vol. (9) Sp Th last cas s th ncorrctly orntd layr. Th cass studd hr ar symmtrc / / / / / / / lamnatd bams wth mmovabl nds. If th orntaton of frst layr s ncorrctly mad 9o rathr than o, th valus of nondmnsonal fruncs gnrally dcras as could b sn n Tabls (8) and (9) rspctvly. 4 Tabl 8 on-dmnsonal fruncs L Eh of a symmtrc [///////] lamnatd bam wth Mod o. mmovabl nds, (L/h = 5). Bam typ CF HH CC HC HF FF Ths occurs bcaus ncrasng of angl of orntaton caus a dcras n th valus of th natural fruncs as dscussd abov. Th mamum and mnmum prcntag dcrass n th valus of th fundamntal mod ar about 6% and % rspctvly. Ths prcntag ncrass f th mod ordr s ncrasd. It may b pctd that ths dffrncs dcras as th numbr of layrs s ncrasd. 4 Tabl 9 on-dmnsonal fruncs L Eh of a non-symmtrc [9///////] lamnatd bam wth mmovabl nds, (L/h = 5) Bam typ Mod o. CF HH CC HC HF FF Pag 4

12 Imad-Eldn Mahmoud Mahd t al.,influnc of Fbr Orntaton on th atural Fruncs of Lamnatd COCLUSIOS In ths papr, a frst-ordr shar dformaton thory was appld n th analyss, and a fnt lmnt modl has bn formulatd to prdct th non-dmnsonal natural fruncs and to study th nflunc of fbrs orntaton on th natural fruncs. Twlv nd condtons wr studd whch ar clampd-fr, hngd-hngd, clampd-clampd, hngd-clampd, hngd-fr, and fr-fr bams wth mmovabl and movabl nds. Th man conclusons ar: () Smlar bams, whch ar thr symmtrcally lamnatd / / / or ant-symmtrcally lamnatd / / /, hav ual natural fruncs, snc th coffcnts Cand C 55 ar ual for both cass (s En. (5)). () Rptton of a st of layrs n symmtrc or ant-symmtrc smlar bams dos not altr th natural fruncs of th bam..g. / / / ( / / /, whr n,,,.... ) n or S () If th lamnaton ordr of a lamnatd bam s rvrsd, th natural fruncs wll rman th sam..g. (/6/9) or (9/6/). (4) Th natural fruncs of a lamnatd bam gnrally dcras as th fbr orntaton angl ncrass. (5) Th rror n natural fruncs of a lamnatd bam rsultng from an ncorrctly orntd layr wll dcras f th numbr of layrs s ncrasd and/or th angl of orntaton of th ncorrctly orntd layr s dcrasd. ACKOWLEDGEMET Th authors would l to acnowldg wth dp thans and profound grattud Mr. Osama Mahmoud of Danya Cntr for Publshng and Prntng Srvcs, Atbara, who spnt many hours n dtng, r dtng of th manuscrpt n complanc wth th standard format of Intrnatonal Journal of Engnrng Rsarch and Advancd Tchnology (IJERAT). REFERECES [] Toh L.S. and Huang C.C. 'Th vbraton of bams of fbr rnforcd matral', (976), Journal of sound and vbraton 5(4), pp. (467-47). [] Chandrashhara K., Krshnamurthy K. and Roy S. ' Fr vbraton of compost bams ncludng rotary nrta and shar dformaton', (99), Compost Structur 4, pp. (69-79). Publcaton Impact Factor (PIF):.598 Pag 4

13 Intrnatonal Journal of Engnrng Rsarch And Advancd Tchnology Vol. (9) Sp -7 [] Abramovch H. 'Shar dformaton and rotary nrta ffcts of vbratng compost bams', (99), Compost Structur, pp. (65-7). [4] Abramovch H. and Lvshts A. ' Fr vbratons of non-symmtrc cross-ply lamnatd compost bams', (994), Journal of sound and vbraton 76(5), pp. (597-6). [5] Marur S.R. and Kant T. ' Transnt dynamcs of lamnatd bams: an valuaton wth a hghr-ordr rfnd thory', (998), Compost Structur 4, pp.(-). [6] Marur S.R. and Kant T. 'Fr vbraton analyss of fbr-rnforcd compost bams usng hghr-ordr thors and fnt lmnt modlng', (996), Journal of sound and vbraton 94, pp. (7-5). [7] McCarthy T.R. and Chattopadhyay A. ' Invstgaton of compost bo bam dynamcs usng a hghr-ordr thory', (998), Compost Structur 4, pp. (7-84). [8] Tboub Y. and Hajla P. ' Fr vbraton of gnrally layrd compost bams usng symbolc computatons', (995), Compost Structur, pp. (-4). [9] Hodgs D.H., Atlgan A.R., Fulton M.V., and Rhfld L.W. ' Fr vbraton analyss of compost bams', (99), Journal of Amrcan Hlcoptr Socty 6, pp. (6-47). [] Banrj J.R. 'Eplct analytcal prsson for fruncy uaton and mod shaps of compost bams', (), Intrnatonal Journal of Sold and Structurs 8, pp.(45-46). [] Kadvar M.H. and Mohbpour S.R. ' Fnt lmnt dynamc analyss of unsymmtrc compost lamnatd bams wth shar ffct and rotary nrta undr th acton of movng loads', (998), Fnt Elmnts n Analyss and Dsgn 9, pp.(59-7). [] Shmp R.P. andanapur A.V. ' A bam fnt lmnt basd on layrws trgonomtrc shar dformaton thory', (), Compost Structurs 5, pp.(5-6). []. D. Phan and J.. Rddy, 'Analyss of lamnatd compost plat usng hghr ordr shar dformaton thory', Intrnatonal Journal of umrcal Mthod n Engnrng, Vol., (985), PP. ( 9). [4] Mahmoud Yassn Osman and Osama Mohammd Elmard Sulman, 'Fr vbraton analyss of lamnatd compost bams usng fnt lmnt mthod', Intrnatonal Journal of Engnrng Rsarch and Advancd Tchnology (IJERAT), Vol., Issu, (7), PP. (5 ). [5] Mahmoud Yassn Osman and Osama Mohammd Elmard Sulman, 'Fr vbraton of lamnatd plats', Intrnatonal Journal of Engnrng Rsarch and Advancd Tchnology (IJERAT), Vol., Issu 4, (7), PP. ( 47). Pag 4