BUCKLING AND REDUCED STIFFNESS CRITERIA FOR FRP CYLINDRICAL SHELLS UNDER COMPRESSION

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1 Asia-Paifi Conferene on FRP in Sruures (APFIS 7) S.T. Smih (ed) 7 Inernaional Insiue for FRP in Consruion BUCKLIG AD REDUCED STIFFESS CRITERIA FOR FRP CYLIDRICAL SHELLS UDER COMPRESSIO K. Masumoo 1, S. Yamada 1, H.T. Wang and J.G.A. Croll 3 1 Toohashi Universi of Tehnolog, Toohashi, Japan. ken-masu@s.urp.u.a.jp Universi College London, U.K. 3 Universi College London, U.K. ABSTRACT This paper deals wih bukling of hin-walled lindrial shells under ompression. I is well-known ha aiall ompressed lindrial shells have bukling behaviour whih is ver sensiive o iniial geomeri imperfeions. However, urren approahes using mahemaial algorihms o opimise he linearised lassial riial loads wih respe o man design variables, generall ignore he poenial reduions in elasi load arring apaiies ha resul from he iniial imperfeions. The presen major problems for inorporaion ino design proesses and ofen involve eessive ompuaional effor. Adoping 6-pl smmeri, glass-epo lindrial shells, his paper arries ou lassial bukling analsis for perfe shells and nonlinear analsis for imperfe shells. In addiion he presen paper applies an alernaive lower bound design onep alled he redued siffness mehod. B onfirming he orrespondene beween he lower bounds of nonlinear bukling analsis and he redued siffness analsis, i suggesed ha he redued siffness mehod ould provide an imporan basis for design. KEYWORDS Clindrial Shells, Bukling, Imperfeion Sensiivi, Redued Siffness Mehod, Angle of Fibre Orienaion. ITRODUCTIO In man siuaions FRP omposies provide opporuniies for enhaned effiien, primaril beause of heir high srengh-o-weigh raios and orrosion resisane. These advanages are signifian for he bridges, sorage anks, pressure vessels, and espeiall for he reduions in weigh for airraf and oher aerospae sruures. The presen paper underakes bukling of orhoropi lindrial shells under ompression. In his area one of he mos imporan aspes is ha real lindrial shells alwas have iniial geomeri imperfeions. These imperfeions pla a defining role as o he reduions of elasi bukling apaiies from he upper bound provided b he lassial riial loads (Yamaki 1984). Moreover differen design parameers and differen bukling modes will ehibi ver differen levels of imperfeion relaed knok-down o he lower bound, bukling loads. One possible wa o akle his dilemma is o predi he wors effes of iniial imperfeions on he bukling apaiies of omposie shells. The redued siffness mehod is suh a lower bound design philosoph ha is able o predi he wors possible effes of iniial imperfeions. I is based on he phsial argumen ha he reduions in he bukling loads of shells resul from he loss of iniial sabilizing membrane energ. B eliminaing hese membrane energies from he lassial riial analsis allows a lower bound o he imperfeion sensiive bukling loads of shells o be obained (e.g. Yamada and Croll 1993, 1999; Yamada e al. 6). Using nonlinear analsis his paper will demonsrae ha for inreasing ampliudes of iniial imperfeions he elasi bukling loads ehibi well defined lower bounds. The presen sud onenraes he effes of angle of fibre orienaion (Wang and Croll 7), and onfirms he oeisene beween he heoreial lower bound predied b he redued siffness analsis and he lowes bukling loads ehibied b nonlinear numerial eperimens. For his reason i is suggesed ha he redued siffness mehod ould provide a signifian onribuion o more effeive opimizaion design. METHOD OF AALYSIS The geomer and oordinae ssem of a laminaed shell are shown in Figure 1 wih hikness,, lengh, L, and radius, R. The shell is simpl suppored as desribed in Eq. 1 and subje o uniform aial sress. 46

2 z u w v R Figure 1 Convenion for shell geomer and oordinae L w u w =, =, =, v = a =, L (1) Classial Criial Analsis Displaemen funion u, v, w an be aken as harmoni funions ha saisf he boundar ondiions of Eq. 1. u = ui, jos( i R) os( jπ L) v = vi, jsin ( i R) sin ( jπ L) () w = wi, jos( i R) sin ( jπ L) where i and j are he irumferenial full-wave and longiudinal half wave number. The Donnell-Mushari-Vlasov approimaions of he linear inremenal srain-displaemen relaions a he riial sae are given as d u d w =, κ d v w d w, κ R (3) d 1 u v d w = +, κ = and he non-linear inremenal omponens dd 1 w dd 1 w dd 1 w w =, =, = (4) Here (,, ) are he inremenal membrane srains; (κ, κ, κ ) he inremenal bending srains. Supersrip of d denoes he linear omponens, and dd he quadrai omponens. Based on he lassial laminaion heor he onsiuive relaions orresponding o he linear inremenal srain omponens are d d d n A11 A1 A B11 B1 B κ d d d n A1 A A 6 B1 B B = + 6 κ d A d d n A6 A 66 B B6 B 66 κ () d d d m B11 B1 B D11 D1 D κ d m = B1 B B d d 6 D1 D D + 6 κ d m B B6 B d d D D D κ and hose orresponding o he nonlinear inremenal srain omponens are dd dd n A11 A1 A dd dd n A1 A A (6) = 6 dd A dd n A6 A 66 APFIS 7 466

3 Following he usual onvenion (n, n, n ) are he sress resulans; (m, m, m ) he inremenal momen resulans. A ij, B ij and D ij (i, j =1,, 6) are respeivel he membrane, bending-membrane oupling and bending siffness of a laminae for whih he omponens adop he Halpin-Tsai equaion (Jones 197 and Wang 7). For a irular lindrial shell subje o uniform aial ompression he fundamenal sae prior o bifuraion poin an be aken as ai-smmeri membrane in he lassial riial analsis. Hene he fundamenal sress resulans are ( n E, E, E n n) = (,, ) (7) Here we assume ha he effe of membrane-bending oupling siffness B ij (i,,,6) an be negleed in he fundamenal sae, and ha he eension-shear oupling siffness A and A 6 equal zero. Using onsiuive relaion Eq. he orresponding fundamenal srain omponens are E E E A A1 (,, ) =,, A11A A1 A11A A1 In he lassial riial analsis he priniple of saionar oal poenial energ gives a ompa and ssemai framework for inerpreing bukling behaviour. Of presen ineres is he quadrai erm of he oal poenial energ from whih he ondiion of he saionari resuls in he eigenvalue problem ha ields he lassial riial load spera. The quadrai erm of he oal poenial energ an be represened as U = UM + UB + VM + VM (8) where U B is he linear bending energ, U M he linear membrane energ, and V M he linearised membrane omponen assoiaed wih aial direion, while V M is assoiaed wih irumferenial direion, ha is 1 π d d d d d d U B = ( ) mκ + mκ + mκ dd 1 π d d d d d d U M = ( ) n + n + n dd 1 π (9) dd E E dd V M = ( n n ) dd + 1 π dd E V M = ( n ) dd Hene he ondiion of he saionari is epressed as follows, and soluion of he eigenvalue problem gives riial load as eigenvalue and bukling mode (u, v, w ) as eigenveor. VM V M δ UM + UB + + = (1) Subsiuing and (u, v, w ) ino Eq. 1, we an onfirm ha he oal poenial energ equal zero VM V M UM + UB + + = (11) onlinear Bukling Analsis In he presen nonlinear bukling analsis displaemen funions are aken o be linear ombinaion of harmoni epressions u = ui, jos( i R) os( jπ L) i j v = vi, jsin ( i R) sin ( jπ L) (1) i j w = w os i R sin jπ L i j i, j ( ) ( ) For onvergene of he nonlinear pos-bukling response, a oal of 64 modes in Eq. 1 in whih eah degreesof-freedom was 8, 17 and 19 for u, v and w, respeivel, was adoped as ompleel he same in Eq. 3 of Yamada and Croll (1999). The iniial geomeri imperfeion is aken o onsis of a single harmoni w = w b, fos( b R) sin ( f π L) (13) In whih b and f represen he irumferenial full-wave number and longiudinal half-wave number for he imperfeion. APFIS 7 467

4 Srain-displaemen relaions are u w w 1 w w = + +, κ v w w w 1 w w + +, κ R 1 u v w w w w w w w = , κ (14) The ses of nonlinear algebrai equaions are obained hrough he saionari of he oal poenial energ wih respe o eah of he displaemen degrees of freedom inluded in Eq. 14. Soluion of hese ses of nonlinear equaions is ahieved using a sep-b-sep proess in whih eiher load or suiable displaemens is used as he onrol parameer. A eah sep a ewon-raphson ieraion is used o provide onvergene o an aepable level of preision. A more omplee desripion of he heoreial model is for he isoropi linders (Yamada and Croll 1993), whih liss he inegraion oeffiiens for all erms up o and inluding he quari (fourhpower) energ erms. Redued Siffness Analsis As evidened in Yamada and Croll (1999), he linearised aial energ omponen V M is negaive in ompressed lindrial shell bukling problems; all he oher energ omponens in Eq. 8 are posiive definie, impling ha all bu V M onribue o he shell s iniial resisane o bukling. The redued siffness mehod is based on he phsial argumen ha mode oupling, aalzed b geomeri imperfeions, resuls in he loss of iniial sabilising membrane energ. B eliminaing U M and V M from Eq. 8 and appling he ondiion of saionari o he redued quadrai form of he oal poenial energ we obain he redued siffness riial load. V M δ U B + = () As will be shown in he following eample, eah hoie of aial half-wave j will resul in a lassial riial load sperum from Eq. 1 ha ehibis a minimum riial load a some value of i. This load is defined as m,j. The lowes of m,j, ourring in a mode ( i, j) = ( im, jm ), is wha is usuall referred o as he lassial riial load, here denoed b m. For eah hoie of j he redued siffness riial spera of Eq. will predi a value of assoiaed wih he irumferenial wave number i orresponding wih he lowes lassial riial load. The lowes of hese redued siffness riial loads m will our in a mode ( i, j) = ( im, jm ) ha ould be differen o he lassial riial mode ( im, j m ). I is his leas value m ha has been shown o represen a lower bound o imperfeion sensiivi (Yamada e al. 6). TYPICAL CASE STUDY Referring o Wang and Croll (7), a 6-pl smmeri glass-epo irular lindrial shell and he maerial properies are adoped E f = 7GPa, μ f =. () Em = 3.GPa, μm =.34 where E and μ are he Young s modulus and he Poisson raio of he omposie maerials, and subsrip f and m denoe fibre and mari respeivel. The fibre volume fraion is. for eah pl and he geomeries as follows (Yamada and Croll 1999) L/ R =.1 (17) R/ = 4 Figure shows he lassial riial load, he nonlinear bukling load and he redued siffness riial load analsis for several hoies of θ. For a shell [θ ] 6, whih epresses he laminae onfiguraions, θ denoes he angle of fibre orienaion and 6 denoes he number of pl, and hikness of eah pl is equal. For he shell [ o ] 6, he minimum riial load m =18.9MPa and he lowes redued siffness riial load m =7.MPa. Aordingl he reduion is almos 6%. While in ase of [ 4 o ] 6 he reduion indiaes almos 8% beause of m =.6MPa and m =4.13MPa. This shows ha he reduion from minimum riial load is ver dependen upon he angle of fibre orienaion. APFIS 7 468

5 Figure 3 shows he effes of hanging he angle of fibre orienaion. Poins show he bukling loads m obained from he nonlinear analsis for he relevan iniial imperfeion. From Figures and 3 he lower bounds of he nonlinear analsis and he redued siffness mehod are equivalen beause he lowes nonlinear analsis load almos orresponds o m. From ha poin of view he redued siffness analsis ehibis well defined lower bound. 4 [MPa] 3 j= 4 [MPa] 3 m =6. m =18.9 w b =. b =.) 1 j= j= 1 m =7. m =7.71 b =.) b =.1) i m =14. i m =17 i m = i m =18 1 b =.) 1 (a) [ o ] 6 (b) [ o ] b =.4) 6 j= 4 [MPa] 4 [MPa] b =.6) j= b =.8) 3 3 b = 1.) b = 1.) j= j= m =.6 m =18.9 j= b = 1.4) j= j= b = 1.6) 1 1 m =7.9 b = 1.8) m =4.13 j= b =.) i m =13.7 i i m =17.3 m = i m = () [ 4 o ] 6 (d) [ 9 o ] [MPa] j= j= j=6 Figure. The lassial riial load, redued siffness riial load and nonlinear load analses for eah angle of fibre orienaion j=7 j= j=6 j=7 j= ( j ) ( jm) ( ) Angle of fibre orienaion( o ) Figure 3. The lower bounds versus he varing angle of fibre orienaion m.).).).1).).4).6) = = = = = = = j=,,, and m, j,,, and.8) 1.) 1.) 1.4) 1.6) 1.8).) = = = = = = = m, j APFIS 7 469

6 In Figure 3 i is worh noiing ha fibre orienaion wih high lassial riial load suh as [ o ] 6 or [ 7 o ] 6 are relaivel low redued siffness load and nonlinear analsis load. This also indiaes ha he effes of iniial imperfeion are large. I is apparen ha opimum design from lassial riial analsis ma be unfavourable for he aual shell. In Figures and 3, he fibres are in all plies in he same direion, while in Figure 4 four pes of la-ups are adoped; Figure 4 shows he omparison of L/R=.1 wih L/R=.48 adoped in Wang and Croll (7) in whih 6-pl smmer was applied o all ases. The square brake denoes he angle of fibre orienaion and he order of laminaion from he ouside o he middle plane of he shell, and s represens smmeri disribuion abou he middle plane. The resuls for he lassial riial loads are onsisen beween hese wo figures, despie he differene of L/R; his is a onsequene of hanges in he riial aial half-wave number ensuring ha he wave lengh in aial direion remains effeivel onsan; The normalised aial o o bukling wave lengh for [,,9 o ] s was L/(Rj)=.6 wih j= for L/R=.1, whih is he same as wih j=8 for L/R=.48. Meanwhile i an be observed ha he redued siffness loads for L/R=.1 are somewha higher han hose for L/R=.48. For shorer lindrial shells he effes of imperfeion on bukling load arring apaiies are redued. 3 1 m, m [MPa] m, m [MPa] 3 (a) (b) () (d) (a) (b) () (d) (a) (b) () (d) 1 (a) (b) () (d) m m o (a) [ θ,, o (b) [, θ, o o () [,,θ (d) [ θ ] 6 o ] s o ] s ] s Angle of fibre orienaion( o ) Angle of fibre orienaion( o ) (a) L/R=.1 (b) L/R=.48 Figure 4. The omparison of lassial riial loads and redued siffness loads on lengh-radius raio COCLUSIOS The presen sud has arried ou hree analses ha are lassial riial analsis, nonlinear analsis and redued siffness analsis, for 6-pl smmeri, glass-epo lindrial shells wih parameer of angle of fibre orienaion. As a resul, he lower bound of he nonlinear analsis and he analial prediion of he redued siffness mehod show lose orrespondene wih he redued siffness rieria providing lose lower bound o he imperfeion sensiive elasi bukling loads for imperfe FRP lindrial shells. The lower bound riial loads due o imperfeion var remarkabl wih he angle of fibre orienaion; he opimised angle based on he redued siffness analsis is shown o be ver differen o he opimisaion b lassial upper bound analsis. REFERECES Jones, R. M. (1999). Mehanis of Composie Maerials, nd Ed., Talor & Franis. Wang, H. and Croll, J.G.A. (7). Bukling design opimizaion of fibre reinfored polmer shells using lower bound pos-bukling apaiies, 13h In. Conf. on Ep. Meh., Aleandroupolis, Greee. Yamaki,. (1984). Elasi Sabili of Cirular Clindrial Shells, orh-holland. Yamada, S., and Croll, J.G. A. (1993). Bukling and posbukling haraerisis of pressrure loaded linders, Journal of Applied Mehanis, ASME, 6, Yamada, S., and Croll, J.G.A. (1999). Conribuion o undersanding he behavior of aiall ompressed linders, Journal of Applied Mehanis, ASME, 66, Yamada, S., Yamamoo,., Croll, J.G.A. and Bounkhong, P. (6). Loal bukling rieria of hin-walled FRP irular linders under ompression, In. Colloquium on Appliaion of FRP o Bridges, JSCE, Toko, Japan, APFIS 7 47