STRESS ANALYSIS OF LAMINATED COMPOSITE CYLINDERS UNDER NON-AXISYMMETRIC LOADING ABSTRACT

Transcription

1 STRESS ANALYSIS OF LAMINATED OMPOSITE YLINDERS NDER NON-AXISYMMETRI LOADING J. Michael Stauc Oa Ridge Natioal Laoatoy* Oa Ridge Teessee 78 ABSTRAT The use of thic-walled composite cylides i stuctual applicatios has see temedous gowth ove the last decade. Applicatios iclude pessue vessels flywheels dive shafts spoolale tuig ad poductio ises. I these applicatios the geomety of a composite cylide is aisymmetic ut i may cases the applied loads ae o-aisymmetic ad moe igoous aalytical tools ae equied fo a accuate stess aalysis. A closed-fom solutio is peseted fo detemiig the laye-y-laye stesses stais ad displacemets ad fist-ply failue i lamiated composite cylides sujected to o-aisymmetic loads. The applied loads iclude iteal ad eteal pessue aial foce toque aial edig momet uifom tempeatue chage otatioal velocity ad itefeece fits. The fomulatio is ased o the theoy of aisotopic elasticity ad a state of geealized plae defomatio alog the ais of the composite cylide. Paametic desig tade studies ca e easily ad quicly computed usig this closed-fom solutio. A compute pogam that was developed fo pefomig the umeical calculatios is descied ad esults fom specific case studies ae peseted. EY ORDS: aisotopic elasticity composite mateials stess aalysis. INTRODTION I the desig ad aalysis of lamiated composite cylides aisymmetic loads ad aisymmetic geometies ae ofte assumed fo developig closed-fom aalytic solutios. I additio the cylide is assumed to have a ifiite legth such that the stesses ae ot oly idepedet of the cicumfeetial coodiate ut also idepedet of the aial coodiate. Solutios have ee fomulated ased o oth the theoy of aisotopic elasticity ad the lamiated shell theoy 4. The lamiated shell theoy povides a accuate solutio fo thi-walled cylides wheeas elasticity solutios ae equied fo a accuate detemiatio of the thee-dimesioal stess states that eist i thic-walled cylides. I oth of these aalytical appoaches futhe simplificatios ae otaied y estictig the composite cylide to e othotopic. *Maaged y Locheed Mati Eegy Reseach opoatio fo the.s. Depatmet of Eegy ude cotact DE-A-9OR44.

2 Thee ae a limited ume of closed-fom solutios fo the case of aisymmetic cylide geometies with o-aisymmetic loads. olla ad Spige ideed a lamiated cylide o cylidical segmet sujected to hygothemal ad mechaical loads that vaied i the adial ad cicumfeetial ut ot i the aial diectio. The theoy of elasticity was used to deive the solutio fo stesses stais ad displacemets i the cylide without ay estictios o ply agle ad lamiatio sequece. The legth of the cylide was assumed to e lage compaed to the wall thicess ad ie ad oute adii such that ed effects could e eglected. The oly estictio o the applied mechaical loads was that they had to e i equiliium. Pagao peseted a geeal solutio fo a cylidically aisotopic cylide sujected to suface tactios that could e epessed y a Fouie seies. The suface tactios had to e idepedet of the aial coodiate ad istet with oveall equiliium of the cylide. O the ed faces of the cylide the suface tactios wee pescied as statically equivalet foce ad momet esultats.. ANALYTIAL FORMLATION. Sigle Laye Solutio The wo of Pagao as suggested y the autho was developed i a fom that could e eteded to aalyze a lamiated composite cylide havig aisotopic layes. This sigle laye solutio is the foudatio fo the cuet wo ad the udelyig assumptios ad asic equatios ae iefly descied i this sectio. A cicula cylide havig a ie adius ad a oute adius is ideed whee the stess field is idepedet of the aial coodiate. Tactio ouday coditios ae applied o the sufaces ad o the ed plaes idepedet of the aial coodiate ad epessed i the fom of a Fouie seies. The titutive equatios fo a mateial havig a sigle plae of symmety with espect to a cylidical coodiate system ae witte as: ε ε ε γ α T α T α T γ γ α T The equiliium equatios i cylidical coodiates ae witte as: F whee the compoets of stess ae fuctios of the ad coodiates ad the comma deotes diffeetiatio. The F ody foce tem is icluded ad fo otatioal velocity is witte as ρω. The compoets of displacemet ae u v ad w i the adial cicumfeetial ad aial diectios espectively ad ae fuctios of ad. The staidisplacemet elatioships ae witte as:

3 Now y elimiatig the stesses ad stais fom Eq. - a geeal solutio fo the displacemets that satisfies the compatiility equatios ca e witte as: 4 whee i ae aitay tats that deped o the ouday coditios. The goveig equatios fo ad ae foud y sustitutig Eqs. ad 4 ito Eq.. whee γ γ γ ε ε ε w v u w v u v w v u u w v u 4 si si si si si. P P T P ρω si si si 4 P a P P

4 ad A solutio to Eq. is sought suject to a set of ouday coditios that ae epessed i tems of a Fouie seies. Due to the otatioal symmety the ouday coditios at the ie ad oute adii ae epessed i the followig fom: 7 I Eq. 7 the tats p i q i ad t i fo ae ot all idepedet as a esult of gloal equiliium fo the cylide. Diect itegatio of the equiliium equatios esults i the followig elatioships etwee the tats i Eq whee * ae the applied stess compoets coespodig to. The emaiig set of ouday coditios ists of the esultat aial foce F toque T ad momet M actig o ay s sectio of the cylide. 9 A geeal solutio fo ad is give y: si si i i i i i i i i t t i q q p π π π M T F si si φ φ φ [ ] [ ] * * * * q q t t [ ] α α α α T T

5 hee φ i i coespod to the paticula solutio ad the emaiig tems ae fo the homogeeous solutio whe P i T. By sustitutig Eq. ito Eq. the homogeeous solutio is give y: ad s A B D s s s A B D s s s s [ ] s [ s ] [ s ] [ s ] [ s s ] [ s ] [ s ] [ s ] [ ] s s The detemiat of the [] mati is set to zeo ad the esult is a chaacteistic equatio that is cuic i s. The oots to this cuic equatio povide the solutio fo the si tats s fo each value of i the Fouie seies. Special cases to the solutio of Eq. occu fo values of equal to ad whee epeated oots ae foud fo s. Fo : A λ A 4 λ ad D D B l D B B a λ Fo the case of : A A l A A A l 4 The paticula solutio i Eq. is foud y diect sustitutio ito Eq. ad is:

6 φ φ Tβ a a a si a si 4 4 ρω 9 φ a si si whee a a 4 β α α α α ad a a 4 ad a ae foud y solvig the followig set of thee simultaeous liea equatios. 4 a a4 a 7 Fially due to the fom of the pescied ouday coditios i Eq. 7 ad eglectig igid ody motios esults i: A B D 8 The solutio fo a sigle laye as descie y the aove equatios is applicale fo all fie oietatios with some mio chages to the equatios fo othotopic ad tasvesely-isotopic ½ layes. Taig the highest ide i the Fouie seies to e M the actual solutio to the polem cotais M uows ad thee ae M idepedet equatios. The uow tats ae 4 A A 4 A ij j B D ad A s s M. Some of the details have ee omitted fo evity hee ut ca e foud i the oigial wo of Pagao.. Lamiate Solutio Fo a lamiated cylide the solutio descied i Sectio.. is applied to each laye ad itefacial cotiuity is ivoed etwee eighoig layes. The ouday coditios at the ie ad oute adii of the cylide ae applied to the ie adius of the fist laye ad the oute adius of the last laye espectively. Let R ad R e the cylide ie ad oute adii espectively ad e the ie ad oute adii of the th laye ad t e the thicess of the th laye. Fo N layes thee ae N- itefaces ad y usig the followig otatio: N R R t 9 the cotiuity equatios fo N- ae witte as:

7 u v u δ v w w whee δ is a pescied itefeece etwee layes. The esultat foce ad momet give y Eq. 9 ae modified to e a summatio ove all the layes ad ae ewitte as: F T M N N N π π π Recall that each laye has M uow tats ad theefoe the solutio to the polem of a cylide havig N layes has NM uow tats. Thee ae M idepedet equatios fom the ouday coditios ad N M fom the cotiuity equatios fo a total of NM equatios. The solutio pocedue is divided ito thee sepaate pats that deped o the ume of tems i the Fouie seies. Fo i the Fouie seies thee ae 4N simultaeous equatios that ae used to solve fo the 4N uows of 4 A ad A 4. Thee ae 4N equatios fom cotiuity of adial stess ad cotiuity of the thee displacemet compoets. The emaiig 4 equatios ae fom the adial stess ouday coditios at the ie ad oute adii of the cylide p ad p ad fom the esultat aial foce ad toque coditios. The N uows of D ad B ae solved fom cotiuity of the two shea stess compoets N equatios ad the two ouday coditios fo shea stess at the ie adius of the cylide q ad t. Fo thee ae N uows with N equatios fom cotiuity of the thee stess ad thee displacemet compoets ad equatios fom the ouday coditios coespodig to the tems of p p q t ad t ad the esultat momet. The uows ae A ad A s s. Fially fo each M a N X N system of equatios is solved fo the NM uows of A s s. Thee ae N M equatios fom the cotiuity equatios ad thee ae M equatios fom the ouday coditios tems of p p q q t ad t. A FORTRAN pogam has ee developed fo pefomig the calculatios descied i the aove solutio pocedue. The pogam is geeal i the sese that moocliic othotopic ad tasvesely-isotopic layes ca e aalyzed ad multiple mateial systems ae acceptale. The laye-y-laye stesses stais ad displacemets ae calculated i oth the gloal cylidical coodiate system ad i the laye picipal mateial diectios. Failue citeia fo detemiig fist-ply failue ae implemeted i the pogam ased o usig Hashi s citeia 7 ad the Tsai-u citeio 8.

8 . DESIGN ASES. Atea Mast omposite mateials may e used to educe the weight ad life-cycle ts as well as miimize deflectios of atea masts. The masts ae ofte sujected to wid coditios that suject the mast to o-aisymmetic loads. The wid velocity causes a eteal pessue that is well estimated y: p. f whee f is the velocity measued at feet aove goud i ope couty fo the fastest mile of wid i.e. the mea wid velocity ove time that coespods to a movemet of ai mile 9. At the top of the cylide the pessue distiutio is give y: p.4 H z g α p 7 A whee H is the height of the cylide z g the gadiet height aove which the wid velocity is pactically tat ad α deped o the teai. Fo ope couty z g equals 74.4 metes ad α equals /7. The tats A ae give i Tale. Tale. ostats fo cicumfeetial distiutio of wid velocity Agle fom A widwad meidia oside the case of a quasi-isotopic gaphite/epoy cylide metes i height sujected to a f wid velocity of 4 m/h. The pescied ouday coditios ae witte as: 7 i i R p R R R i 4 whee α H p. 4 p z g A apet plots i the -plae of the mast ae show i Figs. -4 fo the adial displacemet adial stess hoop stess ad -shea stess espectively. The esults i Fig. idicate that the maimum adial displacemet occus at the cicumfeetial coodiates of - 9- ad 8-degees. Also these esults show that the tue udegoes adial epasio at ad 8- degees wheeas at 9-degess the tue epeieces adial cotactio. Figs. -4 show the adial stess to e a cotiuous fuctio ut the hoop ad shea stesses have discotiuities at

9 the laye itefaces. This is a diect esult of the cotiuity coditios give y Eq.. I additio the shea stess esults as epected ae zeo i the -degee ad 9-degee plies. Figue. Radial displacemet cotou fo composite mast sujected to wid loads. Figue. Radial stess cotou fo composite mast sujected to wid loads.

10 Figue. Hoop stess cotou fo composite mast sujected to wid loads. Figue 4. Shea stess cotou fo composite mast sujected to wid loads.. Spoolale Tuig I oilfield applicatios spoolale composite tuig offes advatages ove taditioal tuig mateials i tems of educed weight ad impoved fatigue life. The may diffeet loadig sceaios that ca aise iclude edig stais aial foces iteal ad eteal pessue ad elevated tempeatues. appig the tuig aoud spool diametes that ae typically aoud metes i diamete iduces the edig stai ad

11 the edig of the tuig esults i a o-aisymmetic stess distiutio. To aalyze the spoolig load sceaio a applied esultat ed momet is used. Fo typical tue oute diametes of 8 mm o a -mete spool diamete the edig stai ε is equal to.8%. This is calculated y dividig the distace fom the eutal ais y the adius of cuvatue. The applied momet ecessay to poduce this edig stai is a fuctio of the tue geomety ad lamiatio sequece ad is give y: M ε πε N N E I R 4R 4 4 E whee E ad I ae the th laye aial stiffess ad laye momet of ietia. oside a gaphite/epoy ± agle-ply lamiated tue havig a 8-mm oute diamete ad a -mm wall thicess. The applied ed momet is calculated to e 9 N-m. The picipal stess i the fie diectio is plotted i Fig. at the ie ad oute adii of the tue as a fuctio of the cicumfeetial coodiate. The esults illustate the edig of the tue associated with spoolig whee the stesses ae of equal magitude ut of opposite sig symmetically aout the 9-degee cicumfeetial coodiate. Ie adius Oute adius Picipal stess MPa Theta degess Figue. Picipal stess distiutio i the fie diectio fo spooled load case. 4. ONLSIONS A closed-fom solutio has ee peseted fo desig ad aalysis of a aisymmetic lamiated cylide sujected to o-aisymmetic loads. The solutio is ased o the theoy of aisotopic elasticity ad a assumed geealized plae defomatio state of stess. The pescied ouday coditios ae epessed i tems of a Fouie seies ad ae idepedet of the aial coodiate. sig zeo tems i the Fouie seies epasio teats the special case of aisymmetic loads. The solutio is geeal i that moocliic othotopic ad tasvesely-isotopic layes ae ideed i the fomulatio ad multiple mateial systems

12 may e used. The pescied loadigs iclude aial foce momet toque iteal ad eteal pessue uifom tempeatue chage ad otatioal velocity with itefeece fits. A FORTRAN code was developed fo pefomig the ecessay calculatios ad the code ca e eecuted fom a destop compute. This pemits a computatioally efficiet method fo coductig umeous desig tade studies. The utilizatio of the code was demostated y two desig cases fo composite lamiated cylides sujected to o-aisymmetic loads.. REFERENES. M.. Hye Hydostatic Respose of Thic Lamiated omposite ylides Jl. Reifoced Plastics ad omposites A.. Roy ad S.. Tsai Desig of Thic omposite ylides Jl. of Pessue essel Tech. Tas. of ASME S. A. Amatsumya Theoy of Aisotopic Shells NASA Techical Taslatio NASA TTF J. M. hitey O the se of Shell Theoy fo Detemiig Stesses i omposite ylides Jl. omp. Matl L. P. olla ad G. S. Spige Stess Aalysis of Aisotopic Lamiated ylides ad ylidical Segmets It. J. Solids Stuctues N. J. Pagao The Stess Field i a ylidically Aisotopic Body de Two- Dimesioal Suface Tactios Jl. Appl. Mech Z. Hashi Failue iteia fo idiectioal Fie omposites Jl. App. Mech S.. Tsai ad E. M. u A Geeal Theoy of Stegth fo Aisotopic Mateials Jl. omp. Matl H.. S. Thom New Distiutios of Eteme ids i the ited States Jl. Stuct. Div. ASE 94 ST R. M. Soem ad S. Boa ompaiso of FEA ad lassical Plate Theoy fo Thic- alled omposite Tuig Poc. Eegy Souces Techology ofeece ad Ehiitio ETE98-44 ASME J. M. hitey Stuctual Aalysis of Lamiated Aisotopic Plates Techomic Pulishig ompay Lacaste PA 987 p. 9.