instability of laminated cylindrical shells

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Pergamon Inl. J. Non-Ltnear bfechnnicr. Vol. 33 No. 3 pp. 407415 1998Q 1997 Ekvier Science Ltd

All rights reserved. Printed in Great Britain0020-7462198 17.CKl+ 0.W

PII: SOO20-7462 97)00030-9

INSTABIL ITY OF LAMINATED CYLINDRICAL SHELLSWITH MATERIAL NON LINEAR EFFECT

A. Tabiei and Y. JiangDepartment of Aerospace Engineering and Engineering Mechanics, University of Cincinnati,Cincinnati, OH 45221-0070, U.S.A.

Receiv ed for publicati on 7 May 1997)Abstract-An anaiytical methodology is presented to study the post-buckling behavior of laminatedcylindrical shells under axial compression and lateral pressure. Non-linear inplane stress-strainrelations for transverse (&) and shear (G,,) modulus is considered. The user defined materialsubroutine UMAT in finite element analysis package, ABAQUS, with an updated Lagrangianformulation, and large displacement and small strain kinematic relations, is employed in this study.Numerical analyses are conducted to demonstrate the effect of material non-linearity and stackingsequence on the post-buckling behavior of laminated shells. 0 1997 Elsevier Science Ltd.Keywords strain softening, finite element, shell instability

INTRODUCTIONThe compressive behavior of fiber-reinforced composite shells has been the subject ofinvestigation for several decades. On the structural level, most of the compressive behavioranalyses of composite laminates have been limited to linear-elastic material properties Cl],and with a few to material shear non-linearity [2,3]. Examining typical stress-strainrelations of laminated composites [4], apart from severe non-linearity in in-plane shear, ithas been found that a lamina exhibits non-negligible non-linearity in transversestress-strain relations. Therefore, transverse and shear non-linearity must be accounted forin the analysis.

The kind of material non-linearity demonstrated in experiments is generally strainhistory dependent. However, as pointed out by Schapery [S], the inelasticity for manycomposites is neither small enough to neglect nor large enough to use classical plasticitytheories in which unloading follows the initial moduli. This non-linear behavior mayarise from a variety of mechanisms including, microcracking, shear banding, interfacedebonding, etc. A common approach to treat the material non-linearity is to use classicalelastic orthotropic stress-strain relations for laminae with secant moduli. In an effort tocharacterize the non-linear inelastic behavior of composite materials Schapery [S, 61has developed an inelastic in-plane constitutive relation for laminae. These inelasticrelations form the basis of the present study. In Schaperys model the material non-linearityis characterized by a structural parameter and it is assumed that transverse Youngsmodulus and shear modulus are the only secant moduli to account for the materialnon-linearity.

In the present work the considerations of material non-linearity are incorporated, usingthe user defined material subroutine UMAT option, in the commercial finite elementsoftware package ABAQUS [7] which is employed in generating results.

PROBLEM DESCRIPTIONThe structure under consideration is cylindrical laminated composite shell with symmet-ric stacking sequence subjected to axial compression or lateral pressure. The following

Contributed by G. J. Simitses.407


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408 A. Tabiei and Y. Jiangassumptions are made in the present analysis:

Each lamina is transversely isotropic and homogeneous; secant moduli E2 and G12 ac-count for the material non-linearity, and transverse shear non-linearity is neglected.Layers are perfectly bonded together.The initial geometric imperfections were given by superimposing a small imperfectionin the form of the weighted sum of several buckling modes (eigenvectors) on the initialgeometry.

Non-linear stress-strain relationsThe constitutive relations employed in the shear deformable shell finite elements are

izj-1:: i:: S6jfZj (1)

(2)where s11=&

S 122 =-E22

(3)

(4)

(5)

and constants 01~ nd t12 are the shear correction factors.In equation (l), compliances S22 and Se6 are supposed to be stress state (or strain state)dependent. Schaperys model is adopted herein and compliances Sz2 and Sh6 are assumed to

be functions of an internal state variable S as follows:s22 = S22(S) (7)

6 = s6 ) (8)

Fig. 1. Stress-strain urve for uniaxial oadingarbitrary (x) direction.


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Instability of laminated cylindrical shells 409where S is shown as the shaded area in Fig. 1. This internal state variable is related to thetotal work density and the total strain energy by the following equation:

s=w,-w (9)where W is the strain energy density and W is the total work input. Introducing anotherform of the internal state variable, namely S,, which is the cubic root of S (for details seeRef. [6]) and through off-axis tests, Sz2 and Ss6 can be determined as polynomial functionsof S, by curve fitting techniques.It has been found that Sz2 and Se6 are related by the following equation:

S.5&) = S66 0) + CC~22cc) - S, , O)1 10)where C is a material constant. This constant is found to be equal to 3.66 for HerculesAS413502 Carbon/Epoxy composites [S].Because of the fact that El and v12 are assumed to be constants the internal state variableS, can be expressed as a function of an effective stress

112 (11)where

cr J (0; + c0:2)12 (12)Thus equations (11) and (12) provide a convenient way to express the compliances S22and S66 as functions of the effective stress co instead of S, in the numerical analysis asfollows:

s22 =S22bJ o) (13)s66 = S66kO) (14)

Based on the constitutive relations prescribed above, the incremental stress-strain relationsfor orthotropic lamina in the material coordinates can be written as

dsl = Slldai + Slzda2 + dSllal + dSr2c2 (15)da2 = Si2dal + S2zdc2 + dSlzol + dSz202 (16)

dy, = S66daiz + dS66c12 (17)d 6 cdS22-= -da0 do,dSz2 =2($dts2 +sdc12)0 2 12

(18)(19)

(20)and

where the matrix with superscript t represents the tangent stiffness matrix, and subscriptIZ represents the nth load incremental step. The corresponding components of tangent


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410 A. Tabiei and Y. Jiangstiffness matrix of equation (21) are

where

c11 = a22a33 43A

c12 - aLra33al2a23Cl6 =balla33

c22 = 7

al la23C26 = - Ac66 =

a22a11 --a:2A

all = S1la12 -S 12

d z ataz2 =Sz2 --da0a0az3 =

cdS22 2al2- g-o,

d 6 a?2a = S66 C da0 a0

(23)

(24)(25)

(27)

(28)

(29)(30)(31)

(32)(33)

andA=ai1az2a33 -ail& -a33a:2 (34)

It must be pointed out that in the derivation of equations (23)-(28), the relationd&/da0 = C dSz2/dao is applied. As a result the tangent stiffness matrix is symmetricwhich is of considerable practical significance since lack of symmetry requires specialtreatment in the solution by finite element method and may cause difficulties in thenumerical solution.

RESULTS AND DISCUSSIONSIn the following the geometry under consideration is a simply supported imperfectcylindrical shell with two different imperfection amplitudes t = 0.1 and 1. The layer

Table 1. Linear moduli and strength parameters of AS4 Carbon/Epoxy lamina (Unit: Msi)El = 18.2 E2 = 1.34 G,r = 0.194 V 2 = 0.334 Gzs = 0.391Lamina thicknesst = 0.0125 in. Radius = 7 in. Length = 21 in.

Table 2. Polynomial coefficients for S,, and &sExponent 0 1 2 3 4 5 6522 0.141 5.46S66 1.26 21.6 180 -8000 32.e4 - 5.28e4 3.16el- 260 14,100 125.e3 - 4.89e6 4.36el


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Instability of laminated cylindrical shells 411thickness is taken to be 0.0125 in. The non-linear response of laminated composite shells ismodeled by an updated Lagrangian formulation with large displacement and small strainkinematic relations. Four-nodded 6 DOF shell elements (S4R) are used. The modified Riksmethod is used to construct the equilibrium path.

The shell geometries, lamina properties, and polynomial coefficients in Tables 1 and 2are used to generate results. The polynomial coefficients were obtained based on curvefitting the experimental data from [S]. Mesh convergence study indicates that using 80- 100elements in circumferential by 20-30 elements in longitudinal direction yields an acceptableconvergence state. Using ABAQUS, the first three eigenvalues and corresponding eigen-vectors are obtained for each cylinder. The lowest eigenvalue represents either theaxial compression buckling load or the lateral pressure buckling load for a perfect cylin-der. The initial geometric imperfection were taken by superimposing a small imperfection inthe form of the weighted sum of several buckling modes (eigenvectors) on the initialgeometry.

Fig. 2. Normalized post-buckling behavior, shells under axial compression.

Fig. 3.

$ 0.6x?::Z 0.4

0.2

0.0 L I I I I0.0 0.4 0.6 1.2 1.6 2.0END-SHORTENING /t

d 0.6x$ZZ 0.4

2Linear Material. .f=O 1.----.Nonlinear Material. (=O.l

..... Linear Material, (=I .O-..-Nonlinear Material. t=l.O

0.0 I I I I0.0 0.4 0.6 1.2 1.6 2.0END-SHORTENING /t

Normalized post-buckling behavior, shells under axial compression.


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Instability of laminated cylindrical shells 413

Fig. 6. Deformation of axially compressed ( + 45/ - 45)4, shell with non-linear material.

1.0

0.8

0.6:

?a0.4

0.2

0.0

.... Linear Material, =d.l-Nonlinear Material, (=0-l. -Lmear MaLerial, [= 1.0----.Nonlinear Material, [ = 1 OI I I I I-0.4 -0.2 0.0 0.2 0.4

END-SHORTENING U/tFig. 7. Normalized post-buckling behavior, shells under lateral pressure

Figures 7-9 show the normalized post-buckling behavior of the three shells sub-jected to lateral pressure versus the normalized end shortening. As in the case of axialcompression, both constant material properties and material non-linear behavior arepresented.

CONCLUSIONSIn this investigation a study is undertaken to determine the effect of material non-linearity on limit points and post-buckling behavior of laminated cylindrical shells under


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414 A. Tabiei and Y. Jiang

0.8 -

0.6 -:

0.4i

-0.4 -0.2 0.0 0.2 0.4END-SHOH'l'liJNING/tFig. 8. Normalized post-buckling behavior, shells under lateral pressure.

1.o

0.8

0.6:

0.4

0.2

0.0

I I I I , 1.

. .-Lmear Material,-Nonlinear Material,

-.-.-Linear Material,.-.-...-Nonlmear Material,

{=O.(=O.(=l.(=l

I I 1 I I-0.4 -0.2 0.0 0.2 0.4END-SHORTNING U/t

110.Q

Fig. 9. Normalized post-buckling behavior of ( 45/-45)4,hell under lateral pressure.

axial compression and lateral pressure. The transverse (E,,) and shear (G12) modulusaccount for material non-linearity. Based on the generated results the following observationcan be made:

Material non-linear effect on the limit point could be significant, more pronounced whenthe imperfection amplitude is not large, and is stacking sequence dependent.Material non-linear effect is more significant for cylinders under axial compression thanunder lateral pressure.Shells with angle-ply stacking sequence exhibit the greatest difference in limit point whenconsidering linear and non-linear material behavior.Material non-linearity could drastically change the collapse mode of laminated shells.Other properties that should also be investigated for shells with material non-linearityare the effect of boundary conditions, length to radius ratios, and radius to thicknessratios.


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Instability of laminated cylindrical shells 415Acknowledgements-The research was partially sponsored by the Office of Naval Research, Ship StructureDivision, with Dr Y. D. S. Rajapakse as scientific officer. The valuable discussions and suggestions of professor G.Simitses, principle investigator of the ONR Grant are gratefully acknowledged. Computing support was providedby the National Center for Supercomputer Applications (NCSA).

1.2.

3.4.5.

6.7.

A. W. Leissa, Buckling of laminated plates and shell panels. Technical Report AFWAL-TR-85-3069. Air ForceWright Aeronautical Laboratories, Wright Patterson Air Force Base, OH (1985).S. ST Wang, S. Srinivasan, H. T. Hu and Rami HajAli, Effect of material nonlinearity on buckling andpostbuckling of fiber composite laminates. In Mechanics of Composit e M aterials: N onli neur .Eflects (Edited byM. W. Myer). ASME, AMD-Vol. 159, New York (1993).H. T. Hu, Buckling analyses of fiber composite laminate plates with material non-linearity. ICCM/9 (1993).H. C. Halpin, Primer on Composit e M aterial Anal ysis, pp. 208-212. Technomic. (1992).R. A. Schapery, Mechanical characterization and analysis of inelastic composite laminates with growingdamage. In M echanics of omposite Materials and Structures (Edited by J. N. Reddy and T. L. Teply). ASME,AMD-Vol. 100, New York (1989).R. A. Schapery, Prediction of compressive strength and kink bands in composites using a work potential. Int. J.ofSolidand St ruct ures 32 617), 739-165 1995).ABAQUS U ser M anual. Hibbitt, Karlsson & Sorensen, Version 5.5 (1996).

REFERENCES

instability of laminated cylindrical shells

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