A COMPARISON OF MEMBRANE SHELL THEORIES OF HYBRID ANISOTROPIC MATERIALS S. W. Chung* School of Achitectue Univesity of Uth Slt Lke City, Uth, USA S.G. Hong Deptment of Achitectue Seoul Ntionl Univesity Seoul, KOREA ABSTRACT Membne shell theoies e simple nd widely used but lso ce must be tken to pevent secondy bending moments due to the unblnced ngement of lmintes of nisotopic mteils. At times, bending theoy my hve to be dopted nd the cuent design codes, such s ASME, API nd ACI must be eviewed fo the cse of nisotopic mteils. The stesses nd stins cn be significntly diffeent between the pue membne nd bending theoies. This ppe deives membne type shell theoy of hybid nisotopic mteils, govening diffeentil equtions togethe with the pocedues to locte the mechnicl neutl xis. The theoy is deived by fist consideing genelized stess stin eltionship of thee dimensionl nisotopic body which is subjected to 2complince mtix nd then non-dimensionlizing ech vible with symptoticlly expnsion. Afte pplying to the equilibium nd stessdisplcement equtions, we e llowed to poceed symptotic integtion to ech the fist ppoximtion theoy. Also possible secondy moments due to the unblnced built up of lmintion e quntifibly expessed. The theoy is diffeent fom the so clled pue membne o the semi-membne nlysis. Key Wods: Hybid nisotopic mteils; Asymptotic integtion; Length scles; Membne Stesses; Secondy Bending moments. *Coesponding utho, smuelchung00@gmil.com INTRODUCTION Shell theoies nd its design nd mnufctuing technology e becoming moe impotnt ecently s the oute spce explotion being moe ctive. It nges fom deep wte submines, spce vehicles, to the dome type humn esidences in the Moon o Ms. The membne theoy of shell is simple nd been existing fo genetions since Tusdell nd Goldenveise hve theoeticlly fomulted s shown in the Refeences (7) nd (8). The mechnics of composites e complicted comped to the odiny conventionl mteils such s steel nd othe metllic bnds but composites possess such chcteistics s high stength/density nd modulus/density tios, which will llow flight vehicles moe efficient nd incesed distnce. The filments embedded in the mtix mteils of composites give dditionl stiffness nd tensile stength. They cn be nged bitily so s to mke stuctue moe esistnt to lodings. As the mechnicl popeties of composites vy depending on the diection of the fibe ngement, it is necessy to nlyze them by use of n nisotopic theoy. Also the cuent design codes including ASME, API nd ACI, Refeences (5) to (8), e ll bsed on membne theoy fo isotopic mteils. Pogessive Acdemic Publishing, UK Pge 83 www.idpublictions.og
Pessue vessels of composite mteils e, in genel, constucted of thin lyes of diffeent thickness with diffeent mteil popeties. The popeties of nisotopic mteils e epesented by diffeent elstic coefficients nd diffeent coss-ply ngles. The coss-ply ngle,, is the ngle between mjo elstic xis of the mteil nd efeence xis ( Figue nd 2). The vition in popeties in the diection of the thickness implies non-homogeneity of the mteil nd composite stuctues must thus be nlyzed ccoding to theoies which llow fo nonhomogeneous nisotopic mteil behvio. Ou tsk is to fomulte theoy fo shell of composite mteils which e non-homogeneous nd nisotopic mteils. Accoding to the exct thee-dimensionl theoy of elsticity, shell element is consideed s volume element. All possible stesses nd stins e ssumed to exist nd no simplifying ssumptions e llowed in the fomultion of the theoy. We theefoe llow fo six stess components, six stin components nd thee displcements s indicted in the following eqution: i, j,2,3 kl,,2 () C k l kl w h e e nd kl e stess nd stin tensos espectively nd C kl e elstic moduli. Thee e thus totl of fifteen unknowns to solve fo in thee dimensionl elsticity poblem. On the othe h n d, thee equilibium equtions nd six stin displcement equtions cn be obtined fo volume element nd six genelized Hook's lw equtions cn be used. A totl of fifteen equtions cn thus be fomulted nd it is bsiclly possible to set up solution fo theedimensionl elsticity poblem. It is howeve vey complicted to obtin unique solution which stisfies both the bove fifteen equtions nd the ssocited boundy conditions. This led to the development of vious theoies fo stuctues of engineeing inteest. A detiled desciption of clssicl shell theoy cn be found in vious efeences [-2]. In the fist pt of this ticle, the symptotic expnsion nd integtion method is used to educe the exct thee-dimensionl elsticity theoy fo non-homogeneous, nisotopic cylindicl shell to ppoximte theoies. The nlysis is mde such tht it is vlid fo mteils which e nonhomogeneous to the extent tht thei mechnicl popeties e llowed to vy with the thickness coodinte. The deivtion of the theoies is ccomplished by fist intoducing the shell dimensions nd s yet unspecified chcteistic length scles vi chnges in the independent vibles. Next, the dimensionless stesses nd displcements e expnded symptoticlly by using the thinness of the shell s the expnsion pmete. A choice of chcteistic length scles is then mde nd coesponding to diffeent combintion of these length scles, diffeent sequences of systems of diffeentil equtions e obtined. Subsequent integtion ove the thickness nd stisfction of the boundy conditions yields the desied equtions govening the fomultion of the fist ppoximtion stess sttes of non-homogeneous nisotopic cylindicl shell. Fomultion of Cylindicl Shell theoy of Anisotopic Mteils Conside non-homogeneous, nisotopic volume element of cylindicl body with longitudinl, cicumfeentil (ngul) nd dil coodintes being noted s z,,, espectively nd subjected to ll possible stesses nd stins ( Figue ). The cylinde occupies the spce between Pogessive Acdemic Publishing, UK Pge 84 www.idpublictions.og
h nd the edges e locted t z 0 nd z L. Hee, is the inne dius, h the thickness nd L the length. Assuming tht the defomtions e sufficiently smll so tht line elsticity theoy is vlid, the following equtions goven the poblem:,,, 0 z z z z,,, 0 z z z,,, 0 (2) z z u S S S S S S z, z z 2 3 4 5 z 6 z u, u S2 z S22 S26 z u, z S3z S36 z (3) u, u z u S4z S6 z uz, u, z S5z S36 z u, z uz, S6z S66 z In the bove Equtions (2) e equilibium equtions nd (3) stess-displcement eltions. In tht u, u, u z e the displcement components in the dil, cicumfeentil nd longitudinl diections, espectively,,, z the noml stess components in the sme diections nd z, z, e the she stesses on the - z fce, - z fce, - fce espectively (Figue ). A comm indictes ptil diffeentition with espect to the indicted coodintes. The S s i, j,2,,6 in the eqution (3) e the components of complince mtix nd epesent the diectionl popeties of the mteil. Complete nisotopy of the mteil is llowed fo nd thee e thus 2 independent mteil constnts. We e not llowed to illuminte ny of those components since the mteil popeties e depending on the mnufctues set up nd diffeent gvity envionment in cse of eospce vehicles. Also the complince mtix is symmetic, S S, nd the components cn be expessed in tems of engineeing constnts s follows: ji S S S ii E, i,3 i, (, 2 Ei 44 23 i, j 2,3, i j (4) G Pogessive Acdemic Publishing, UK Pge 85 www.idpublictions.og
S S 55 66 G3 G 2 In eqution (4) the E i s e the Young s moduli in tension long the i diection nd ע nd G e the Poisson s tio nd she moduli in the i-j fce, espectively. Eqution (4) implies nisotopic popety of the mteil only, mteil to be non-homogeneous, diffeent popeties of ech lye of the shell, we will llow the mteil popety vition in the dil diection s follows: S S (5) The bove eqution is unique nd diffeent fom most of conventionl theoies, including Reddy s, Refeence (), which input the engineeing constnts tificilly fom the beginning, while we tke the existence nd mgnitude of components only by ppoximtion theoy of the symptotic expnsion. The pincipl mteil xes ( ', ', z ' ) in genel do not coincide with the body xes of the cylindicl shell (,, z ). If the mteil popeties S with espect to mteil xes specified, then the popeties with espect to the body xes e given by the following tnsfomtion equtions: ' 2 ' 2 6 26 ' 2 ' 2 S6 sin S26 cos sin 2, ' ' S 2 26 S6 sin 2 cos 2, S S cos 2S S sin cos S sin ' 4 ' ' 2 2 ' 4 2 66 22 S cos S sin sin 2, S S cos 2S S sin cos S cos ' 4 ' ' 2 2 ' 4 22 2 66 22 S S S S 2S S sin cos ' ' ' ' ' 2 2 2 2 22 2 66 S S 4 S S 2S S sin cos 2 S S sin 2 cos 2, ' ' ' ' ' 2 2 ' ' 66 66 22 2 66 26 6 ' ' 2 ' 2 ' ' ' 2 2 2 S6 S22 sin S cos 2S 2 2 S66 cos 2 sin 2 S6 cos cos 3sin S sin 3cos sin, ' 2 2 2 26 ' 2 ' 2 ' ' S26 S22 cos S sin 2S 2 2 S66 cos 2 sin 2 S ' 2 2 2 ' 2 2 2 6 26 S sin 3cos sin cos cos 3sin. (6) Pogessive Acdemic Publishing, UK Pge 86 www.idpublictions.og
whee is the ngle of nisotopic oienttion between the z ' nd the oiginl coodinte z xes. Fo the cse of n othotopic mteil, whee the mjo nd mino elstic xes e 90 Degee, the tnsfomtion equtions (6) e educed to eqution (7): 4 4 cos 2 2 2 sin sin cos ', E2 E G E E2 4 4 sin 2 2 2 cos sin cos ', E2 E G E E2 2 ' sin 2 G G E2 E2 G ' ' E E 4 E E 2 G ' ' ' E2 2 ' E (7) The invints e expessed by E ' ' ' ' ' E2 2 E E E2 2 E G ' 4 ' ' E G 4 E The shell is subjected to unifomly distibuted tensile foce then the boundy conditions e s follows: 0, z, 0 h z z (8) We will find it convenient to wok with stess esultnts the thn the stesses themselves. These stess esultnts which e foces nd moments pe unit length e obtined by integting with espect to the thickness coodinte. They e: N N N N z z z d z d d h h h d zd h d z d d (9) h M d d Pogessive Acdemic Publishing, UK Pge 87 www.idpublictions.og
h d M z z d d z h M d d M z h z d z d d In the equtions (9) vible denotes the inne dius of the cylindicl shell nd d the distnce fom the inne sufce to the efeence sufce whee the stess esultnts e defined. Note tht N z nd N z nd M z nd M z espectively e diffeent. This is due to the fct tht the tems of the ode of thickness ove dius e not neglected compe to one in the integl expessions. Fomultion of Boundy Lye Theoy The pocedue used to fomulte the shell theoy hee is bsiclly to educe the thee dimensionl equtions to two dimensionl thin shell equtions nd we will use the symptotic integtion of the equtions (2) nd (3) descibing the cylindicl shell. As fist step to integting equtions (2) nd (3), we mke them non-dimensionlized coodintes s follows: X z L, Y h, whee L nd e quntities which e to be detemined lte. Next the complince mtix, the stesses nd defomtions e non-dimensionlized by the use of epesenttive stess level σ, epesenttive mteil popety S nd the shell dius, s follows: S S S z,,, z z, z z u S, u S, uz Sz () whee the dimensionless displcements nd stesses e functions of x, y nd. These vibles togethe with thei deivtives with espect to x, y nd e ssumed to be of ode unity. The pmetes L nd intoduced in eqution (0) e thus seen to be chcteistic length scles fo chnges of the stesses nd displcements in the xil nd cicumfeentil diections, espectively. It is convenient t this point to define wht is hee ment by the concept of eltive ode of mgnitude. Conside smll pmete, is less thn. With espect to n bity domin n D of the cylinde, M is sid to be of ode eltive to second quntity M 2 (0) M M (2) n 2 Pogessive Acdemic Publishing, UK Pge 88 www.idpublictions.og
if eveywhee in D (with the possible exception of some isolted smll egions) the eltionship n m M M 2 n m (3) holds fo suitbly chosen vlue of m, 0m. Accoding to this definition, two quntities e of the sme ode if n 0 in the bove, while quntity is of ode unity when n 0 nd M. Substitution of the dimensionless vibles defined by (0) nd () into the elsticity equtions (2) nd (3) yields the following dimensionless equtions: Stess-displcement eltionships e, y S t 3 z S t 32 S t 33 S t 34 S t 35 z S t 36 z z, y, x S t 5 z S t S t 52 53 S t 54 S t 55 z S t 56 z L l l, y, y (4) l y S t 4 z S t S t 42 43 S t 44 S t 45 z S t 46 z L z, S t z S t 2 S t 3 S t 4 S t 5 z S t 6 z, l y S t 2 z S t S t 22 23 S t 24 S t 25 z S t 26 z l l y, z z, l y S t 6 z S t S t 62 63 S t 64 S t 65 z S t 66 z L l Equilibium equtions e expessed s, t 0 z y t, y z y tz, x l L t, 0 y t t, y y t z, x l L t, 0 y t, y y tz, x t l L (5) whee is the thin shell pmete defines s h (6) The pmete is epesenttive of the thinness of the cylindicl shell. Pogessive Acdemic Publishing, UK Pge 89 www.idpublictions.og
is much less thn (7) The dimensionless coefficients S of the complince mtix in genel e not ll of sme ode. We theefoe ssume tht they cn be expnded in tems of finite sum s: whee the S N i n0 n n 2 S y S y (8) n y e of ode unity o vnish identiclly. Next, we ssume tht ech m displcement components, epesented by the geneic symbol, nd ech stess components m 2 epesented by the geneic symbol, cn be expnded in tems of powe seies in m0 2 M m m y, x, ; y, x, (9) m0 2 M m m t y, x, ; t y, x, (20) m m The nd t e of ode unity. No convegence popeties e ssumed fo seies (9) only 2 symptotic vlidity fo λ. Tht is, if expnsions (9) e teminted t some powe of, the eo in using the expnsions the thn the exct solutions tends to zeo s ppoches zeo. Length scle L nd e s yet bity. Thei choice, s will be seen in the subjects to follow, detemines the type of shell theoy to be identified. Lst step in the pocedue consists of substituting expnsions of the seies nd one of ssumed length scles into the dimensionless elsticity equtions of stess-displcement nd equilibium 2 given by equtions (4) nd (5). Upon selecting tems of like powes in on both sides of ech equtions nd equiing tht the esulting equtions be integble with espect to the thickness coodinte nd be cpble be cpble of identifying the eltions fo ll stesses nd displcement components, we will obtin systems of diffeentil equtions. The fist system of equtions of thin shell theoy nd we will cll it the fist ppoximtion system. We cn howeve obtin stesses nd displcements of ech lye of thickness coodinte, tht cn be n dvntge of the pocedue mong othes. In the following section, the thin shell theoies fo diffeent combintions of length scles cn be deived. Fomultion of Membne Type Theoy (Associted with chcteistic length scles, ) As we obseved the shell geomety is n impotnt fcto fo the fomultion of theoies. The bsic geomety of cylindicl shell e the longitudinl length L, inside dius, totl wll thickness h nd the distnce fom Inne sufce to desied sufce, d. We e inteested hee in deiving the shell theoy ssocited with the cse whee the xil nd cicumfeentil length Pogessive Acdemic Publishing, UK Pge 90 www.idpublictions.og
scles e both equl to the inne dius of the cylinde,, s follows: L, l (2) The eson fo tking the length scles is the longest pcticl dimension of the shell nd we e inteested in developing membne type theoy which equies longe thn the bending chcteistic influentil length ccoding to the clssicl theoy of isotopic mteils, Refeences (6) though (8) nd (7) though (9). On substituting these length scles into the thee-dimensionl elsticity equtions (2) nd (3) nd stess-displcement eltions nd equilibium equtions of (4) nd (5), we obtin: If the symptotic expnsions (9) nd (20) fo the displcements nd stesses e now substituted into equtions (22) nd (23), the following equtions epesenting the fist ppoximtion theoy of the poblem esult upon use of the pocedue outlined in the lst chpte. Note tht both sides /2 of ech eqution e equted in like powes of nd the leding tems my not coespond to m 0 tem. v v (0) y, (0) zy, (0) v, y 0 0 0 (22) v s t s t s t (0) (0) (0) (0) (0) (0) (0) z, x z 2 6 z v v s t s t s t (0) (0) (0) (0) (0) (0) (0) (0), 2 z 22 26 z v v s t s t s t (0) (0) (0) (0) (0) (0) (0) (0), x z, 6 z 62 66 z t t t 0 (2) (0) (0) z, y z, z, x t t t 0 t (2) (0) (0), y, z, x (2) (0) y, t 0 (23) The supescipts indicte the leding tem in ech of the expnsions (8) nd epesent the eltive of mgnitude of the displcements nd stesses. These odes of mgnitude esult fom the intention to obtin system of equtions which is integble with espect to the thickness coodinte y in step-by-step mnne nd the following dditionl esoning: ) The dominnt stess stte in thin shell theoy is the in-plne stess stte. These stesses should be of the sme ode of mgnitude. b) The ode of the displcements is chosen so tht the poduct of the in-plne stins nd the elstic moduli is of the sme ode of mgnitude s the in-plne stesses. Pogessive Acdemic Publishing, UK Pge 9 www.idpublictions.og
c) The choice fo the tnsvese stesses ises fom the fct tht they should contibute tems of the sme mgnitude in the equilibium. Integtion of the fist thee equtions of (23) with espect to y yields v v ( x, ) (0) (0) v v ( x, ) (0) (0) z z (0) (0) v v x (, ) (24) whee v, v z, v e the displcements of the y 0 ( ) sufce. The middle thee equtions of (23) cn be solved fo the in-plne stesses s follows: (0) t z (0) t [ C] 2 (0) t z 2 (25) Hee, C. (i, j =, 2, 3) e the components of symmetic mtix given by s s s C s s s s s s (0) (0) (0) 2 6 (0) (0) (0) 2 22 26 (0) (0) (0) 6 26 66 (26) nd, 2, 2 e the in-plne stin components of the y 0 sufce: (0) v zx, (0) (0) 2 v, v (0) (0) 2 v, x v, (27) On substituting the fist ppoximtion in-plne stess-stin eltions (25) into the lst thee equtions of (23) nd integting with espect to y, we obtin: t T ( x, ) [ A V A ( V V ) A ( V V )] z z 3 z, x 23,, 33, x z, [ A V A ( V V ) A ( V V )] z, xx 2, x, x 3, xx z, x t T ( x, ) [ A V A ( V V ) A ( V V )] 2 z, x 22,, 23, x z, [ A3 Vz, xx A23 ( V, x V, x ) A 33 V, xx Vz, x ( )] t T A V A ( V V ) A ( V V ) 2 z, x 22, 23, x z, (28) whee t z, T, T e the tnsvese stess components of the y 0 sufce nd Pogessive Acdemic Publishing, UK Pge 92 www.idpublictions.og
A y C d (29) 0 In eltions (28) nd in wht is to follow, the supescipts on the displcements hve been dopped. Boundy conditions (8) e to be stisfied by ech tem of symptotic expnsions (8). This yields t t t 0 ( y 0) (2) (2) (2) z t t 0, t p ( y 0) (2) (2) (2) * z (30) Hee, * p is dimensionless pessue defined by * p p/ ( ) (3) Stisfction of conditions (30) by (29) yields T T T 0 (32) z nd the following thee diffeentil equtions fo displcements V, V z nd V A V A ( V V ) A ( V V ) 3 z, x 23,, 33, x z, A V A ( V V ) A ( V V ) 0 z, xx 2, x, x 3, xx z, x A V A ( V V ) A ( V V ) 2 z, x 22,, 23, x z, A V A ( V V ) A ( V V ) 0 3 z, xx 23, x, x 33, xx z, x A V A ( V V ) A ( V V 2 z, x 22, 23, x z, ) p * (33) In the bove equtions A A (34) () To obtin the ppopite expessions fo the stess esultnts we fist non-dimensionlize those defined by (9) s follows: whee N nd M e the geneic symbol fo the foce nd moment stess esultnts, espectively. Assuming it to be possible, we now symptoticlly expnd ech of the dimensionless stess esultnts in powe seies in /2, (35) Pogessive Acdemic Publishing, UK Pge 93 www.idpublictions.og
M N N ( x, ) m0 M m0 ( m) m/2 M M ( x, ) ( m) m/2 (36) whee ( m) N nd ( m) M e of the ode unity. Pseudo-Membne Phenomen We e now inteested in fomultion of equtions to be ble to obtin ll the stess esultnts due to membne nd bending ctions. On substitution (35), (36) nd the esults fo in-plne stesses (24) into eltions (9) nd equting tems of like powes in /2 on ech side of the equtions, we obtin the following expessions fo the fist ppoximtion stess esultnts: N N N z z N z A 2 M z B 2 M M M z z (37) whee the supescipt zeo hve been omitted nd B. is defined s follows: y B C ( ) d, B B () (38) 0 nd submtices A nd B e given by Pogessive Acdemic Publishing, UK Pge 94 www.idpublictions.og
(39) A, A2, A3 d / A2, A22, A23 A A3, A32, A33 d / A3, A32, A 33 d d d A B, A2 B2, A3 B3 d / h h h d d d A2 B2, A22 B22, A23 B23 h h h B d d d A3 B3, A32 B32, A33 B33 d / h h h d d d A3 B3, A32 B32, A33 B 33 h h h Note tht d/ cn be witten s d / ( d / h) (40) Fom the esults obtined bove, we chcteize the theoy s follows: ) The ppoch tht this esech took, the symptotic integtion, fo deiving shell equtions is cpble of obtining ll stess components, including the tnsvese components. b) The fist thee equtions of (22) esult fom the eltions fo the tnsvese stins. The vition with espect to y is zeo s shown in the displcements (23) which e independent of y. The stin components of ny point y off the y 0 sufce e thus equl to those of the y 0 sufce, simil to clssicl membne theoy. c) The stess components vy with y becuse s the C, nd the A. e functions of y. d) Equtions (37) show tht moment stess esultnts e poduced due to the nonhomogeneity of the mteil. Fo n isotopic nd homogeneous mteil, the B C A (4) 2 2 C e constnts nd d/h = /2. This yields On substituting this esult into eltions (38) it is seen tht submtix B is equl to zeo nd tht eltions (37) become those of the clssicl membne theoy of shell (zeo moment esultnts). In cse of hybid nisotopic mteils, it is vey e to stisfy ll the components of the submtix [ B ] to be equl to zeo t the sme time. Anothe wy of obsevtion, it is Pogessive Acdemic Publishing, UK Pge 95 www.idpublictions.og
unvoidble to ssocite with some bending moments in ddition to pue membne foces fo lminted nisotopic shell wlls. Theefoe, the nlysis is nmed pseudo-membne theoy. It is diffeent fom the long effective length of Vlsov's semi-membne theoy no the shot effective length of Donnell s theoy. Appliction To demonstte the vlidity of the theoy developed hee, we will choose poblem of lminted cicul cylindicl shell unde intenl pessue nd edge lodings. The shell is ssumed to build with boon/epoxy composite lyes. Ech lye is tken to be tken to be homogeneous but nisotopic with n bity oienttion of the elstic xes. We need not conside the estiction of the symmety of the lyeing due to the non-homogeneity consideed in the oiginl development of the theoy expessed by eqution (5). Thus ech lye cn possess diffeent thickness. We ssume hee tht the contct between lyes is such tht the stins e continuous function in thickness coodinte. As the C e piecewise continuous functions, the in-plne stesses e lso continuous. We would expect them to be discontinuous t the junctue of lyes of dissimil mteils. The tnsvese stesses e continuous functions of the thickness coodinte. Although s mentioned bove the theoy developed cn tke unlimited hybid ndom lyes but fo n exmple, fou lye symmetic ngle ply configution. Fo this configution the ngle of elstic xes is oiented t,,, w I t h the shell xis nd the lyes e of equl thickness. Let the shell be subjected to n intenl pessue p, n xil foce pe unit cicumfeentil length N. The xil foce is tken to be pplied t H such tht moment N( H d) is poduced bout the efeence sufce d. We intoduce dimensionless extenl foce nd moments s follows: N N N( H d) M 2 2 T T 2 3 2 ( d / ) (42) To demonstte the vlidity of the deived theoy, we hve simplified loding nd boundy conditions s follows: Pogessive Acdemic Publishing, UK Pge 96 www.idpublictions.og
V V V V ( x 0, y d / h), x z v 0, N N, M M ( x l, y d / h) 2 2 ( d / ) N M T z z (43) Hee, l is the dimensionless length of the cylindicl shell. In the theoies developed in the pevious chptes, the distnce d t which the stess esultnts wee defined ws left bity. We now choose it to be such tht thee exists no coupling between N nd K nd M z nd C. As the loding pplied t the end of the shell is xi-symmetic, ll the stesses nd stins e lso tken to be xi-symmetic. We thus cn set ll the deivtives in the expessions fo the stesses nd stins nd in the equtions fo the displcements equl to zeo. Numeicl clcultions e now cied out fo shell of wll of vious hybid lmine. Ech of the lyes is tken to be equl thickness nd thus the dimensionless distnces fom the bottom of the fist lye e given by z S 0, S 0.25, S 0.5, S 0.75, S.0 2 3 4 5 ech lye of the symmetic ngle ply configution (elstic symmety xes y e oiented t (,,, ) is tken to be othotopic with engineeing elstic coefficients epesenting those fo boon/epoxy mteil system, E E G 2 2 5 2.430 MP 5.00 MP 5 5.70 MP Hee diection signifies the diection pllel to the fibes while 2 is the tnsvese diection. Angles chosen wee =0, 5, 30, 45 nd 60. Use of the tnsfomtion equtions (2.6) then yields the mechnicl popeties fo the diffeent symmetic ngle ply configutions. We next pply the following edge lods: N p nd tke p /, H (3 / 4) h nd the efeence sufce we tke d/ h/ 2 Shown in Figs. 4 to Fig. 6 is the vition of the dimensionless dil displcement with the ctul distnce long the xis fo the diffeent theoies. The efeence sufce fo the chosen configution is given by d/ h / 2. The integtion constnts detemined fom the edge conditions. It is lso seen tht wide vitions in the mgnitude of dil displcement tke plce with chnge in the coss-ply ngle. The mximum displcement occus t = 30 degee while the minimum displcement is t = 60 degee. Becuse we hve simplified ll the conditions to be puely membne sttus, membne stess s well s displcements cnnot ccommodte with the edge Pogessive Acdemic Publishing, UK Pge 97 www.idpublictions.og
conditions s shown in the Figue 5. Also shown in the Figue 6 is the pttens of ne edge zone to compe the pue membne theoy ginst bending theoy, which e close to Donnel s theoy fo the cse of isotopic mteil. The esults of bending theoy wee dopted fom the Refeence (2). In ech cse, the displcements incese with incese in up to =30 degee nd theefte decese. CONCLUSION In the pesent nlysis, fist ppoximtion shell theoies e deived by use of the method of symptotic integtion of the exct thee-dimensionl elsticity equtions fo non-homogeneous nisotopic cicul cylindicl shell. The nlysis is vlid fo mteils which e nonhomogeneous to the extent tht thei popeties e llowed to vy with the thickness coodinte (). The fist ppoximtion theoy deived in this nlysis epesent the simplest possible shell theoies fo the coesponding length scles consideed. Although twenty one elstic coefficients e pesent in the oiginl fomultion of the poblem, only six e ppe in the fist ppoximtion theoies. It ws seen tht use of the symptotic method employed in the esech lso yields expessions f o ll stess components, including t h e tnsvese ones. Unlike the pue membne theoy of isotopic mteils, secondy bending moments cn be computed in ssocition of mteil chcteistics of lmintion. The fct tht these expessions cn be detemined is vey useful when discussing the possible filue of composite shells nd lso fo the discepncy between theoeticl membne theoy nd expeimentl esults. Fo design of spce shuttles nd othe vehicles, shell stuctue must be cefully designed fo ll possible loding conditions, extemely high negtive nd positive pessue nd tempetue, which demnds futhe ccute shell theoies. In cse the membne theoy seems to be justified, the effect of ll possible secondy bending moments must cefully be exmined s shown in the eqution (37) though (4) of this nlysis. It is moe elistic fo shells of hybid nisotopic mteils of high stength. ACKNOWLEDGEMENTS The symptotic integtion method tht pplied in this fomultion ws fist developed by D. O.E. Wide fo his dynmic theoy of shell stuctues, Refeences (4) nd (5), the uthos concentted on the development of the theoies pplied to pessue vessels nd nlyzed the behvio of shells of hybid lminted composites. Also the esech ws sponsoed by Summit Ptnes in Menlo Pk, Clifoni, USA is gciously cknowledged. Pogessive Acdemic Publishing, UK Pge 98 www.idpublictions.og
REFERENCES () Love,A.E.H, A tetise on the mthemticl theoy of elsticity, Dove Edition, New Yok (2) Donnell, L.H., Bems, Pltes nd Shells, McGw-Hill Book Compny, ISBN0-07- 07593 (3) Reissne, E., The effect of tnsvese she defomtion on the bending of elstic pltes, J. Appl. Mech.2, No A69-77, 945 (4) Johnson, M.W. nd Wide, O.E., An symptotic dynmic theoy fo cylindicl shells,studies Appl. Mth. 48,205, 969 (5) WIDERA, O. E., An symptotic theoy fo the motion of elstic pltes, Act mechnic 9, 54, 970 (6) Vlsov, V.Z., Genel theoy of shells nd it s ppliction in engineeing, NASA TT F- 99, Ntionl Tech. Infomtion Sevice. (7) Tusdell, C., On the Relibility of the Membne Theoy of Shells of Revolution, Bulletin of Ameicn Mthemticl Society Volume 54, No.0 (8) Goldenveise, A.L., Theoy of elstic thin shells, Pegmon Pess (9) Ting, T.C.T., Anisotopic Elsticity, Oxfod Engineeing Science Seies 45, Oxfod Univesity Pess (0) Bimn, V., Extension of Vlsov s Semi-Membne Theoy to Reinfoced Composite Shells, J. Appl Mech 59 (2), 464-464. () Reddy, J.N., Mechnics of Lminted Composite Pltes nd Shells, 2 nd Edition, CRC Pess (2) Chung, S.W. nd Pk, S.M., A Shell Theoy of Hybid Anisotopic Mteils, Intentionl Jounl of Composite Mteils, Volume 6, Numbe, Febuy 206 (3) ASME Boile Code - Boile & Pessue Vessel Code, 205 (4) API Stndd 650, nd 620, Welded Tnks nd Pessue Vessels, 203 (5) ACI 38-: Building Code Requiements fo Stuctul Concete nd Commenty (6) Axeled, E.L., Theoy of Flexible Shells, Noth-Hollnd Seies in Applied Mthemtics nd Mechnics, ISBN 0-444-87954-4(US) (7) Bluwendd, J nd Hoefkke, J.H., Stuctul Shell Anlysis, Spinge, ISBN 978-94-007-6700-3 (8) Clldine, C.R., Theoy of Shell Stuctues, ISBN 0 52 23835 8, Cmbidge Univesity Pess (9) Btdof et l, S.B., A simplified Method of Elstic-stbility Anlysis fo Thin Cylindicl Shells, NACA Technicl Notes No. 34 though 345 (20) Donnell, L.H., Bems, Pltes, nd Shells, McGw-Hill Intentionl Book Compny, 0-07-07593=4 Pogessive Acdemic Publishing, UK Pge 99 www.idpublictions.og
Figue, Exmples of Cylindicl Shells Pogessive Acdemic Publishing, UK Pge 00 www.idpublictions.og
Figue 2. Dimensions, Defomtions nd Stesses of the Cylindicl Shell Pogessive Acdemic Publishing, UK Pge 0 www.idpublictions.og
z ; Longitudinl, ; Cicumfeentil, ; Rdil, L, ; Longitudinl nd cicumfeentil length scles h ; I.D. of cylinde x= z L o= = y= - h ; Totl thickness of shell wll z coodinte Figue 3, Detils of the Coodinte System Pogessive Acdemic Publishing, UK Pge 02 www.idpublictions.og
Figue 4, A Lminted Cylindicl Shell, Mteil Oienttion γ Pogessive Acdemic Publishing, UK Pge 03 www.idpublictions.og
Figue 5, Fibe Oienttions Pogessive Acdemic Publishing, UK Pge 04 www.idpublictions.og
Figue 6, Non-dimensionlized Rdil Displcement of Pue Membne Theoy Pogessive Acdemic Publishing, UK Pge 05 www.idpublictions.og
Figue 7, Compison of Rdil Displcements of the Bending nd Pue Membne Theoies. Pogessive Acdemic Publishing, UK Pge 06 www.idpublictions.og