Stress and Strain Analysis of Curved Beams of Fibre Reinforced Plastic

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1 Aa Mehana loaa 4 (): DOI: ress and ran Analss of Cured Beams of Fbre enfored Plas Peer INZ * (D) peerhenze@hs-wsmarde BIOGAPICAL NOT Prof Dr-Ing Peer enze s a professor of engneerng mehans a he ohshule Wsmar Uners of Appled enes Tehnolog Busness and Desgn In 980 he graduaed a he Uners of annoer Deparmen of Cl ngneerng spealsaon: he sruural mehans Afer ears spen n professonal engneerng posons n arous engneerng ompanes he beame he Dean of he Faul of ngneerng enes a he ohshule Wsmar durng ears 004 o 006 and n he ne me perod from 006 o 009 he was he ead of he Deparmen of Mehanal ngneerngproessand nronmenal ngneerng KY WOD ghl Cured Beams Ber enfored Plas Classal Lamnae Theor ABTACT Ths paper eends he heor of Lepholz [] abou srongl ured beams o ansorop ber maerals For hs purpose he lassal lamnae heor as proposed b alpn [] and ohers has been appled o ured beams The lamnae s made up of seeral orhorop laers wh a dened orenaon (angle {) ges he lamnae ansorop properes The mehod presened here an deermne sress n and aross he ber as a resul of nernal fores INTODUCTION Fber-renfored plass found beause of hgh ebl hgh spe srengh and sness of an nreasngl prealen For he modern new maeral jonng ehnques suh as bondng and res or loops hae been pushed o he fore Loops are ofen seen as one-dmensonal bu hghl ured omponens whh are omposed of seeral derenl orened orhorop laers ah laer onsss of orened bers made of glass or arbon embedded n plas (epo or poleser resn) as he mar The ber olume fraon whh nreases he sness and srengh s aordng o he manufaurng proess 0-60 % The problem of sress alulaon of hghl ured beams has been proessed alread b Lepholz (969) [] for sorop maerals Ths paper wll ge an eenson o ansorop maerals Assumpons and Condons The nesgaon s based on he followng assumpons: The beam s loaded onl b bendng momen and aal fore The ransersal bendng momen shear fore and orque are no aken no aoun The lamnae onsss of seeral laers The laer onsss of dreed parallel lamens 08 VOLUM 4 No 00 Unauhenaed Download Dae 088 6:48 PM

2 Aa Mehana loaa Journal publshed b Faul of Mehanal ngneerng - Tehnal Uners of Koše surrounded b a mar The mehanal behaor of he lamnae depends srongl from sequene and angle of lamens Fber and mar are onsdered as a homogeneous maeral wh an deal ompound aemens abou sress and he adheson beween ber and mar are no possble The generalzed ooke s law for orhorop maeral s ald Bernoull s hpohess s appled Wh hese ondons a relaonshp beween beam sress resulans and srans wll be deeloped whh allows o deermne sress of eah laer ooke s Law for Orhorop Maeral Orhorop maeral are desrbed n onras o sorop maerals wh four maeral onsans []: modulus of elas n ber dreon modulus of elas perpendular o ber dreon G shear modulus of elas n plane major Posson ao mnor Posson ao ooke s law desrbes he relaonshp of sress and sran here ooke s law for plane sress sae s gen: Q Q 0 e > > Q Q 0 > e 0 0 Q Q f wh Q - Q - Q Q - Q G 66 - The mar Q s alled he sness mar he ndes and ndae here ha s relaed o he maeral aes The eors and f are he sress respeel he sran eor boh relaed o maeral as The relaonshp of sran and sress s gen n equaon () The mar s alled omplane mar Of ourse he relaonshp s Q - s ald () e 0 > e > 0 > f s Wh ; - ; - ; ; 66 G ness mar as well as omplane mar are smmer her sruure ndaes ha no ouplng beween normal sress and shear sran ours Transformaons To sud he behaor of he enre lamnae boh he sffness mar and he omplane mar of a laer are ransformed n a ommon oordnae ssem Ths ommon oordnae ssem s alled he -ssem The angle beween he ommon -as and he ber dreon of he h laer s { The sress eor s ransformed b equaon (3) The supersrp r denoes he --ssem > > T (3) r ne we use engneerng shear sran nsead of ensor shear sran s essenal for ε a seond ransformaon relaonshp e > > e e T f e (4) f fr wh s s T > s - s - s s s - s - T > s s s - s () (5) (6) Unauhenaed Download Dae 088 6:48 PM 09

3 and s s Tf > s - s - s s s - s - Tf > s s s - s wh: os{ and s sn { (9) Wh equaon (3) o (8) he ransformaon of sffness and omplane mar wll be performed: Qr T Q T f (0) and r T T () - f The ransformed sness and omplane mares are sll smmer bu full populaed Tha s wh a ouplng of normal sress and shear sran an be obsered (7) (8) LAMINATD CUVD BAM I s known ha n hghl ured beam loaded b pure bendng he neural as does no ondes wh he enrod Therefore he frs queson s o fnd ou he zero sran poson For hs purpose n Fg a dfferenal pee of he beam s shown The desgnaons are: s - dsane of ener of uraure o enrod - dsane of ener of uraure o neural as - oordnae sarng a ener of uraure zr - oordnae n dreon sarng a neural as 9d{ - nrease of uraure due o bendng momen b - wdh of beam Coordnae sem The orgn of he oordnae s he ener of uraure s he dsane from he ener of uraure o he neural as The orgn of he loal beam oordnae ssem s he neural as of a pure bendng loaded beam zr s dreed ouwards s followng he beam as The fber angle { s measured from he -as ress - ran elaon of a Plane Beam The beam heor assumes ha sresses perpendular o he as of he beam n omparson o hose n beam s as dreon are neglgble herefor he sress eor s gen b: T r 6 0 () The sran ndued b a un sress σ s he frs olumn of omplane mar: T f r u r6 0 (3) The sran f ndued b a un sress s he elemen n he frs olumn and he frs row of he omplane mar r n global oordnae ssem Therefore he nerse of he mar elemen r an be regarded as a generalzed modulus of elas n -dreon of he h laer u () (4) fr r () r 0 VOLUM 4 No 00 u The denoes ha s a sress for un sran for he h laer Fg Bendng of hghl ured beam Deermnaon of Poson of Neural As arng pon s he assumpon ha a pure bendng momen and no aal fore s ang The aal fore s: N # da 0 (5) Due o Bernoull s hpohess an assumpon abou he dsplaemen u an be made: u Od{ z r (6) wh zr - (7) The sran f s: Unauhenaed Download Dae 088 6:48 PM

4 f Od{ z r d{ Aa Mehana loaa Journal publshed b Faul of Mehanal ngneerng - Tehnal Uners of Koše N # da () so ha he sress beomes: f N # A Od{ zr d{ The aal fore of one laer s he negral oer he laer area he aal fore of he lamnae he summaon oer all laers: Od{ ^ - h da (8) d{ Od{ The erm and remans onsan and s shfed n d{ fron of he summaon he erm remans onsan for one laer and s shfed n fron of he negral: Od{ - N # da 0 (9) d{ A The summaon s se o zero; - # da 0 A da ; # - # da A A # A The frs negral ges he ross seon area A of he h laer he seond leads o da b 6 + wh he borders of laer Ths leads o he equaon o deermne : A - b (0) A b 6 + The radus of neural as ges: T ULTANT A INTGAL OF T Aal Fore The aal fore of a beam s () usng ooke s law n onneon wh equaon (4) leads o # N f da (3) ran s smpl spoken new lengh - old lengh f old lengh The sran f s omposed of a onsan share f 0 and a arable poron due o he nrease n uraure l For hs seond par s aken no aoun ha also he old lengh s arable wh zr Takng hs no aoun we oban Od{ z O d { f z fo + r fo + r (4) Inrodung equaon (4) n equaon (3) leads o # 0 m N O d { z r f + da (5) The sum s dded no wo pars: N O d{ zr ; f0 # da + # mda (6) The frs par s: b 6 ln@ + d ^ - h O { # m da f 0 n he seond par zr wll be replaed b - and 9d{ and aken before he summaon: 0 Now ou an see ha he epresson s zero beause was he ondon whh was used o deermne Ths allows o alulae he aal fore: + N b 6 f0 (7) The erm Unauhenaed Download Dae 088 6:48 PM

5 + AA b 6 f0 (8) s alled aal sffness BNDING MOMNT M Y bf M O d{ b VOLUM 4 No 00 0 # - # ^ - h d M d b O { lnb (3) d DD DD b ln B (33) + M Od{ DD (34) N AA f 0 (9) M zda r (30) M # Inrodung equaon (4) and ooke s law no quaon (30) leads o # ` j z d z f0 r + O { r z r da (3) earrangng and subsung zr - leads o # 8 B M b d f0 ^ - h+ O { ^ - h The negral has wo pars he frs s d Ths par s zero beause of equaon (9) Therefore bendng momen s: and leads o The bendng sness s: Wh equaon (3) and equaon (33) he bendng momen ges: INTNAL FOC TAIN LATIONIP quaons (30) desrbes he relaons beween he fores and sran equaon (34) of bendng momen and uraure There s no ouplng beween aal fore and bendng or beween bendng momen and sran Ths s due he fa ha he orgn of oordnae ssem was no pu n he geomer enrod of he ross seon bu n he neural as of a pure bendng loaded beam ran n neural as and 9d{ s ompued b: f n 0 (35) AA and M Od{ (36) DD The normal sress s alulaed b equaon (4) n ombnaon wh ooke s law: - f0 + Od{ m (37) resses n and perpendular o ber and shear sress are alulaed b equaon (3) The sresses n maeral ssem are he bas o ompue Tsa-Wu s falure rera reron [] or Puk s falure rera [3] for assessng he srengh CONCLUION A loop onss of 8 laers of glas ber and epo resn wh he smmeral sakng: ah laer of he beam has he hkness of 0:5 mm he nner radus s r 3 mm The loadng s aal fore of N 000 N and bendng momen M Nmm The maeral s desrbed wh: ber olume rao { 06 modulus of elas n ber dreon: Nmm modulus of elas perpendular o ber dreon: 500 Nmm shear modulus: G 6000 Nmm major Posson s rao: 08 wdh b 0 mm aluaon of equaon () ges 46 mm The aal sness aordng equaon (8) ges AA N he bendng sness DD Nmm In Fg he sresses and n he maeral ssem oer he radus are shown In he 0-degree laers (rs and las laer) ou an see he non-lnear ourse of he sress ( ) s eden In he 90 laers (seond and seenh laer) here are onl sresses perpendular o he ber dreon ( ) whle he Unauhenaed Download Dae 088 6:48 PM

6 Aa Mehana loaa Journal publshed b Faul of Mehanal ngneerng - Tehnal Uners of Koše sress n ber dreon and shear sress ansh CONCLUION In he presen paper he generalzaon of he heor of sorop ured beams has been eended o ansorop maerals Of speal sgn ane s he hoe of he orgn of he oordnae ssem If one pu n he neural as no ouplng of normal fore and bendng momen ours FNC [] alpn Prmer on Compos Maerals Analss Lanaser PA 984 [] Lepholz Fesgkeslehre für den Konsrukeur ugar 968 [3] Puk Fesgkesanalse on Faser-Mar-amnaen Münhen Wen 996 [4] MhaelubrehsWegener Dmensoneren m Fasererbundwerksoffen Münhen Wen994 [5] Alenbah Alenbah Kssng Mehans of Compose ruural lemens prnger Berln 004 [6] MhaelWegener: nführung n de Tehnologe der Fasererbundwerksoe Münhen Wen 989 [7] Flemng Fasererbundbauese Berln prnger 995 [8] hürmann Konsrueren m Faser-Verbundwerksoen Berln prnger 005 Unauhenaed Download Dae 088 6:48 PM 3