EFFECTS OF FIBER ORIENTATION FOR THE BUCKLING OF LAMINATED FIBER REINFORCED POLYMER CYLINDRICAL SHELLS

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1 EFFECTS OF FIBER ORIETATIO FOR THE BCKIG OF AMIATED FIBER REIFORCED POYMER CYIDRICA SHES M. Yanagia 1, Seishi Yaaa an K.K. Bhetwal 1 1 Grauate Stuent, Dept of Architecture an Civil Engrg, Toohashi niversit of Technolog, Toohashi , Japan. Eail: 6364@eu.ic.tut.ac.jp Professor, Hea of Dept, Dept of Architecture an Civil Engrg, Toohashi niversit of Technolog, Toohashi , Japan ABSTRACT For 6-pl lainate fiber reinforce poler (FRP) clinrical shells uner copression, a classical, non-linear an reuce stiffness buckling analses have been carrie out. In aition, ssteatic nonlinear Ritz analsis for initiall iperfect FRP shells has provie the scatter of elastic buckling loas which ehibit well efine lower bouns for increasing aplitues of initial iperfections. The present reuce stiffness criteria of FRP clinrical shells is shown to be consistent with the lower bouns isplae b iperfect, nonlinear buckling analsis. For asetric lainations, the nature of the ebrane-bening coupling strain coponent is ae clear b eonstrating the classical analsis, energ analsis, an reuce stiffness equation is ecie to be base upon previous reuce stiffness concept. After that, the effects of the angle of fiber orientation an lainate constitution on buckling loas an behaviour will be ae clear about asetric lainations involve in the setric one. KEYWORDS Clinrical Shell, Buckling, Coupling Energ, Reuce Stiffness Metho, Angle of Fiber Orientation. ITRODCTIO It is well-known that isotropic etal clinrical shells uner copression have buckling behaviour which is ver sensitive to initial geoetric iperfections. In the case of orthotropic FRP aterial, the angles an ispositions of fiber orientations, as well as the agnitues of an iperfections, have been suggeste to affect the buckling behaviour (e.g. Matsuoto an Yaaa et al. 7; Wang an Croll 7). In this paper the interactions between the angles an ispositions of fiber orientations an the iperfections are ae clear through the reuce stiffness buckling analsis in which iperfection effects are intrinsicall consiere. This paper shows that on the basis of the present reuce stiffness criteria the buckling of FRP lainate clinrical shells uner copression are largel insensitive ue to fiber orientation. For longer aial wave-length oes the potential reuctions in buckling loas are shown to be ost iportant for esign. AAITICA METHOD Classical critical analsis The geoetr an coorinate sste of a lainate shell are shown in Figure 1 with thickness, t, length,, an raius, R. The shell is sipl supporte as escribe in Eq.1 an subject to aial force P. w =, w =, u =, v = at =, (1) P u wv z t R Figure 1. Definition for clinrical shell geoetr an coorinate 41

2 Displaceent functions u, v, w can be given as bi-haronic functions which satisf the bounar conition of Eq.1, u = u cos i/r cos jπ / i,j ( ) ( ) ( ) sin ( ) ( ) sin ( ) v = v sin i/r jπ / i, j w = w cos i/r jπ / i, j where u i, j, v i, j an w i, j are non-iensional aplitues of each isplaceent functions; i an j are respectivel the circuferential full-wave nuber an the longituinal half-wave nuber. In this stu, Donnell-Mushtari-Vlasov tpe of isplaceent-strain relationships have been aopte as u v w 1 u v ε =, ε =, ε = + R w w w κ =, κ =, κ = (3) 1 w 1 w 1 w w ε =, ε =, ε = where ε, ε, ε are ebrane strain coponents, an κ, κ, κ bening strain coponents. Superscripts of an enote respectivel the linear an the quaratic isplaceent coponents of strain (in Eq.3) or stress (in Eq.4). Base upon the classical laination theor the constitutive relations corresponing to the linear increental strain coponents an those corresponing to the nonlinear increental strain coponents are n ε A11 A1 A1 B11 B1 B16 ε n A1 A A6 B1 B B 6 ε = n A A A B B B n A11 A1 A16 ε, B11 B1 B16 D11 D1 D16 n = A1 A A6 ε (4) κ B 1 B B6 D1 D D κ n A A A ε B16 B6 B66 D16 D6 D66 κ where n, n, n are ebrane stress resultants, an,, are bening stress resultants. Constants Aij, Bij an Dij (i,j=1,,6) are respectivel the ebrane, the ebrane-bening coupling an bening stiffness coponents for the lainate plate as aopte in the Halpin-Tsai-equations (e.g. Jones 1999; Matsuoto et al. 7; Wang an Croll 7). Here we assue that the effect of etension-shear coupling stiffness A 16 an A 6 are equal zero. For a circular clinrical shell subject to unifor aial copression the pre-buckling state prior to bifurcation point can be taken as ai-setric ebrane for the linear buckling analsis. Hence the funaental stress resultants an the strain coponents are (,, ) E E E E P E E E n n n n =,,, ( ε, ε, ε ) = ( X1, X, ) (5) πr X where X 1, X an X are as follows, A11 A1 B11 B1 A B1 B A1 B1 B A1 A B1 B X1 = B1 D11 D1, X = B11 D11 D1, X = (6) B 11 B 1 D 11 D 1 B D1 D B1 D1 D B B D D 1 1 The quaratic ter of total potential energ Π are use in classical buckling analsis. The quaratic orer strain energ coponents are convenientl broken own as follows, Π = + b + bb + V + V 1 πr = ( ) n ε + nε + nε 1 πr b = ( ε + ε + ε ) n b nb nb 1 πr (7) bb = ( κ + κ + κ ) b b b 1 πr E E V = ( ε + ε ) n n PV 1 πr E E V = ( ε + ε ) n n PV () 4

3 where, is the ebrane, b the ebrane-bening coupling, bb the bening, V the linearize ebrane associate with aial irection, an V the linearize ebrane associate with circuferential irection strain coponents. The stationar of total potential energ a be represente as ( V ) + V δπ = δ + b + bb + P = (8) c P Fro this, the classical critical loa Pc an the associate with buckling oes u i, j, v i, j, wi, jhave been copute. If the circuferential full-wave nuber woul be aopte to be equal to zero in Eq., we can obtaine the ai-setric buckling value Ps as follows; 8π 1 A 1 Ps = ( AA 11 A1 )( AD B11 ) + B11 B1 (9) 1+ A11X1 A1X A11 A11 Reuce stiffness analsis The reuce stiffness etho is base upon the concept in which the reuction of the buckling loa capacit occurs through the effects of initial geoetric iperfections ue to the losses of the initiall positive ebrane strain energ coponents. The previous stu of the reuce stiffness theor has shown that bb, an V in Eq.7 are positive, while V is negative. That is, V contributes the estabilizing coponent in the buckling phenoena (e.g. Yaaa an Croll 1999), then the reuce stiffness buckling solution a be obtaine through the following equation (e.g. Yaaa an Croll 1993; Batista an Croll 1979; Croll 1995; Yaaa an Croll 1989). V δ b + Pc = (1) P where P c is the reuce stiffness critical loa. In the case of the setric laination, an V are preliinar analze to be positive, while V is negative; these results are the sae as for isotropic clinrical shells in Matsuoto an Yaaa et al. 7. On the other han, in the case of asetric laination, the nature of the ebrane-bening coupling strain coponent is ae clear. ADOPTED GEOMETRIC PARAMETER AD AMIATIOS b For consistenc the aopte geoetric paraeters have been aopte fro Yaaa an Croll 1999, as =. 51, R = 45 R t (11) Fro the previous stu in Wang an Croll 7, the Young s oulli E F an E P, Poisson s ratios µ F an µ P an volue fractions V F an V P, for fiber an poler atri, respectivel, are taken as follows: EF = 7GPa, µ F =., VF =. 5 Ep = 35GPa., µ P = 34., VP = 5. (1) This paper eals with the lainate constitutions as shown in Table 1; the asetric lainate constitution [θ,,, θ,, ] eans that the laina of outsie shells has angle of fiber orientation θ, then akes up,, θ, an fro outsie shells to insie shells for thickness irection; θ represents the orientation of the lainate fibers relative to the ais. In aition the thickness of each laina aopte to be of 1 consiering current FRP prouction in Japan, consequentl the total shell thickness t is 6. Table 1. Aopte lainate constitutions Setric [,, θ, θ,, ] Asetric [ θ, θ,,,, ] [, θ, θ,,, ],,, θ, θ, ] [,,,, θ, θ ] RESTS AD DISCSSIOS Figure an 3 show the results of the asetric lainate constitution [45,45,,,, ], with the horizontal ais inicating the circuferential full-wave nuber i. P c, j is taken to be the iniu loa of classical buckling loas associate with varing longituinal half-wave nuber j. On the classical critical loa spectra in Figure., P c = 1.81 M is the iniu of P c, j, an the corresponing buckling oe occurs at (ic, jc) = (18.4, 3). Figure 3 shows the energ spectra of the longituinal half-wave nuber jc = 3 coponent; its vertical ais represents their quaratic increental energies noralize b (4D 11 π/r). The previous stu in 43

4 Matsuoto an Yaaa et al. 7 has alrea iscusse that bb, an V are positive while V is negative. Meantie the ebrane-bening coupling coponent b varies fro positive to negative as the circuferential full-wave nuber increases, however, the total ebrane coponent ( + b ) is shown to positive in the buckling oe i c. As a result, in the present reuce stiffness analsis, all the linear ebrane coponents are eliinate as where P c is the corresponing reuce stiffness loa P M j = j = 5 P c j c V δ + = P bb Pc Figure 4 an 5 are of setric laination [,,45,45,, ]; Figure 4 shows the various spectra of the reuce stiffness buckling loas copare with the classical ones. For various aial half-wave nubers j the iniu classical critical loa a be eterine, then the corresponing circuferential full-wave nuber is obtaine as c ( ) = 1.81M = 3 j = 1 Figure 3. Contributions of energ coponents in critical oe =3 for [45? 45????? j c i j. After that, its reuce stiffness loa associate with ic ( ) ( D11 R) 3 Energies 1 / 4 π / + b j c i c = 18.4 i i -6 c = 18.4 i Figure. Classical loas spectra for asetric [45,45,,,, ] j is calculate as P c, j using Eq.13. Consequentl, fro all the calculate P c, j, the iniu value can be selecte as efining the reuce stiffness criterion Pc as epicte on Figure 4. Figure 5 is representative of the iperfect equilibriu paths having initial geoetric iperfections with a single aial half-wave nuber f=1 an circuferential wave nuber b=15 obtaine b using non-linear buckling analsis that is showe in Matsuoto an Yaaa et al. 7. Its horizontal ais represents the total eflection with a single aial half-wave nuber j=1 an circuferential wave nuber i= bb V V b = 3 (13) 4. P M j = 3.. P c j c =.13M = j = 1 j = 5 j = 1 P : w / t =. P : w / t =. P : w / t =.5 P M. P : w / t =.1 P : w / t =. P : w / t =.4 P : w / t =.6 P : w / t =.8 P : w / t = P c =.65M j = j c = 1 i i c = 18.6 c = 15 iorb Figure 4. Classical an RS loas spectra for setric [?? 45? 45??? 1. w15,1 + w15,1 ) / t ( Figure 5. on-linear buckling loas versus total eflection in haronic i= b=15, j=1 for setric [?? 45? 45??? The effects of the angle of fiber orientation on the classical an reuce stiffness buckling loas in the case of the representative setric laination [,, θ, θ,, ] are shown in Figure 6. The black line enotes the 44

5 classical buckling loas P c, j, the gra line the reuce stiffness critical buckling loas P c, j an the re line the ai-setric buckling loas P s. In aition, the various ots inclue in Figure 6 are the peak loas on the paths siilar to those of Figure 5 an represent the buckling loas for iperfect shells. Shown in Figure 6 for setric lainations are that the lower liits the nonlinear buckling loas closel correspons with the reuce stiffness buckling loas. Therefore, the lower boun preictions of buckling obtaine using the present reuce stiffness estiation proceure are suggeste to provie a useful basis for practical an safe esign of lainate orthotropic cliners, just as it i for the previousl consiere isotropic cliners (e.g. Yaaa an Croll 1999; Yaaa an Croll 1993; Batista an Croll 1979; Croll 1995; Yaaa an Croll 1989). P M P s 3. P / t =. P / t =.4.5 P / t =. P / t =.6. j = P / t =.5 P / t =.8 1. j = P / t =.1 P / t = 1..5 j = 1 P / t =. θ Figure 6. Buckling loas versus angle of fiber orientation for setric [?? θθ,,?? Figure 7 shows for four cases of angle of fiber orientation for asetric lainations the variations of the classical an reuce stiffness buckling loas. It woul suarize fro the results of the classical buckling analsis for perfect shells in Figures 6 an 7 that the buckling oe changes to the ai-setric one with a high orer nuber for θ 45 eg. laination at the insie skin wall of shell, for instance, j = 5 or 6 in the present proble as shown in Figure 7() obtaine fro Eq.9. P M 3. s.5. j = 1..5 j = j = 1 θ (a) Asetric [ θθ,,???? P M j = 6 3. P s.5 j = 5. j = 1. j =.5 j = 1 θ (c) Asetric [??? θθ,,? P s 3. P M P j = j =.5 j = 1 θ (b) Asetric [?,,??? P M j = 6 j = 7 3. j = 5.5. j = j = 3 1. j = 5.5 j = j = 1 θ () Asetric [????, ] Figure 7. Buckling loas versus angle of fiber orientation for asetric lainations The angle of fiber orientation versus classical buckling loas an reuce ones of all aopte lainate constitution are shown in Figure 8. Fro the reuce stiffness analsis, we unerstan that optial angles of 45

6 fiber orientation less clear b initial geoetric iperfection becoes large, aitionall if the angle of finer orientation put outsie shells, it less clear in the classical buckling analsis just as there s in the reuce stiffness one. Results shown an others not shown, ake it clear that the initial iperfection sensitivit is ver epenent upon the lainate constitution an the angles of fiber orientation. Optial angles of fiber orientation base upon the classical buckling theor a not be eactl the sae as those base upon the reuce stiffness buckling theor in which the iperfection effects are intrinsicall taken into account. That is, if the optial angle of fiber orientation is eterine fro the classical buckling criteria, as is conventional the case within aerospace inustr, isleaing an even sub-optial esign solutions coul result. It is also worth noting that the present reuce stiffness criteria P c preicts a single half-wave nuber in aial irection j c =1 inepenent of the lainate constitution an the angle of fiber orientation. It is also worth of note that the value of P is roughl constant with respect to the angle of fiber orientation. COCSIO c.5 P M (e) () (c). 1. (a) inear buckling analsis (b) (a) Asetric[ θθ,,???? (b) Asetric[? θθ,,??? (c) Setric[?? θθ,,?? RS buckling analsis () Asetric[??? θθ,,?.5 (e) Asetric[???? θθ, ] θ Figure 8. Buckling loas versus angle of fiber orientation for all aopte lainate constitution In this paper, 6-pl lainate fiber reinforce poler (FRP) clinrical shells uner copression have been consiere an the classical an reuce stiffness buckling analses have been eterine. B coparing these with the preictions fro nonlinear analses of iperfection sensitive buckling loas, it has been eonstrate that the present reuce stiffness estiations provies convincing lower bouns to these buckling loas. It is suggeste that the use of the reuce stiffness etho provies a useful an practical eans for safe esign for lainate orthotropic cliners, just as previous stuies have shown it to be siilarl suitable for isotropic cliner buckling esign. Also, it has been shown that if the optial angle of fiber orientation is eterine fro the classical buckling criteria, as is coonl the case, the results a not provie the best choice an if appropriate iperfection sensitivit stuies are not perfore coul give rise to angerous over estiates of buckling capacit. REFERECES Jones, R. M. (1999). Mechanics of Coposite Materials,n E., Talor & Francis. Matsuoto, K., Yaaa, S., Wang, H.T. an Croll, J.G.A. (7). Buckling an reuce stiffness criteria for FRP clinrical shells uner copression, Proceeings of Asia-Pacific Conference on FRP in Structures, APFIS7, IFC, Vol. 1, Wang. H. an Croll, J. G. A. (7). Buckling esign optiization of fibre reinforce poler shells using lower boun post-buckling capacities,13th Int. Conf. on Ep. Mech., Aleanropoulos, Greece. Yaaa, S. an Croll J. G. A. (1999). Contributions to unerstaning the behavior of aiall copresse cliners,journal of Applie Mechanics,ASME,Vol. 66, Yaaa, S. an Croll, J. G. A. (1993). Buckling an post-buckling characteristics of pressure-loae cliners,journal of Applie Mechanics,ASME,Vol.6,9-99. Batista, R. C. an Croll, J. G. A. (1979). A esign approach for aiall copresse unstiffene cliners, Stabilit Probles in Engineering Structures an Coponents,Applie Science,onon. Croll, J. G. A. (1995). Towars a rationall base elastic-plastic shell buckling esign ethoolog,thinwalle Structures,Vol.3, Yaaa, S. an Croll J. G. A. (1989). Buckling behavior of pressure loae clinrical panels,journal of Engineering Mechanics,ASCE,Vol. 115,