Nonlinear analysis of doubly curved composite shells of bimodular material

0961-9526/93 56.00 + .oo Pcmamon Pm Ltd

Composites Engineering, Vol. 3. No 5. pp. 419-435, 1993.

Printed in Great Britain.

NONLINEAR ANALYSIS OF DOUBLY CURVED COMPOSITE SHELLS OF BIMODULAR MATERIAL D. BRUNO, S. LATO and R. ZINNO Department of Structural Engineering,

University of Calabria, Cosenza, Italy

(Received 24 April 1992; final version accepted 19 August 1992) Abstract-An analysis of doubly curved shells, based on Sanders’ theory, is developed to study shear-deformable laminated composite shells made of bimodular materials. The theory takes into account geometric and material nonlinearities. Material nonlinearities are modeled by the fibergoverned Bert scheme. Thin and thick doubly curved shells subjected to sinusoidal or uniformly distributed loading are analyzed. Closed-form and numerical FEM solutions are compared for simply supported cross-ply doubly curved shells subjected to sinusoidally distributed loading. The numerical examples presented demonstrate the effectiveness of the FEM procedure adopted and show the influence of boundary conditions, lamination scheme, and geometric and material nonlinearities.

INTRODUCTION

The study of the nonlinear behavior of shell structuresis a topic of considerable interest and practical importance. In fact, shellsof revolution are important structural componentsin all industrial applications,and especiallyin nuclear,aerospaceand petrochemical engineering.In particular, multilayered compositesare appropriatestructural materialsin weight-sensitiveaerospaceapplicationswherehigh strength-to-weightratios are needed. A largebody of technicalliterature hasbeenpublishedon the subject.Much of this wasoriginally developedfor thin shells,basedon Kirchhoff-Love kinematicassumptions that straightlines normal to the undeformedmidsurfaceremainstraightand normal to it after deformation and undergono thicknessstretching.As is well known, the classical laminatetheoryunderpredictsthe deformationenergybecausetransverseshearstrainsare neglected. Dong et al. (1962)useda small-displacementKirchhoff-Love theory for the bending analysisof thin anisotropicplatesand shellsand 5 yearslater, Bert (1967)formulated a linear shell theory similar to the classicalplate theory. Yang (1973)appliedthe classical theory to shell geometrieswith constantcurvatures. The importanceof sheareffectsin shellswasdiscussedby Koiter (1960),who pointed out that refinements to Love’s first approximation theory of thin elastic shells are meaningless,unlessthe effects of transverseshearand normal stressesare taken into accountin the theory. The effectsof transversesheardeformationbecomemore pronouncedwhenthe shell becomesthicker comparedto its radii of curvatureand in-plane dimensions.This also occursin the caseof laminated compositeshells becauseof their low transverseshear moduluscomparedto their Young moduli. Reissner(1945,1949)studiedtheseeffectson elasticplates and on sandwich-typeshells, while Mindlin (1951)included rotary inertia terms in the dynamic analysisof plates(First-OrderShear-DeformationTheory). Reddy(1984)appliedthe shear-deformationtheoryto both singly and doublecurved anisotropicshell structures.Reddyand Chandrashekhara(1985)solvedboth cylindrical and sphericallaminated shell casesaccountingfor transversesheardeformation. In addition, somecompositematerialsexhibit a phenomenonknown asbimodulurity becausetheir elasticpropertiesin tensiondiffer from thosein compression. The analysisof thesematerialsdatesback to Saint-Venant(1864).It wascontinued by Timoshenko(1941)who consideredflexural stresses,while Ambatsumyan(1%5) introducedthe terminology bimodulusand extendedthe conceptto two-dimensionalanalysis. 419

D. BRUNO

420

et

al.

The analysisof structuresfabricatedfrom thesematerialsis usually more difficult than that of conventionalcomposites,i.e. compositesfabricatedwith laminaepossessing equalvaluesof tensileandcompressivemoduli, sinceit is not known a priori which parts of the structureare in tensionand which are in compression.The resulting governing equationsfor bimodular structuralelementsin bendingdependon the unknown position of the neutral surface;thereforethe analysismust be carried out iteratively rather than directly. A macroscopicmaterialmodelappropriatefor bimodulusfiber-reinforcedcomposites, usually calledthefiber-governedmodel, was proposedby Bert (1977)and subsequently found to agreewell with experimentalresults. In this model, the different behaviorin tensionand in compressionis usually modeledusinga bilinear stress-strainrelationship with moduli Et and EC,which satisfactorilyapproximatethe actualnonlinearbehaviorof materials, and the governing elastic constantsdependexclusivelyon the sign of the deformationsin the fiber direction. Bert and V. S. Reddy (1982)publisheda paper on cylindrical shellsof bimodulus compositematerialswhich presenteda closed-formsolution for the caseof simply supported cylindrical thin panelsunder sinusoidallydistributed loading. Ramana,Murthy and Rao (1985)implementeda finite elementanalysisof bimoduluscompositethin shells of revolution. In the presentwork, a shear-deformationversionof the Sanderstheory is developed to analyzethin and thick compositeshellsaccountingfor von Karman strains. Closedform solutions are given for simply supportedcross-ply thick and thin shells under sinusoidallydistributedloads. Following this, finite element solutions are given for thick shells under general loading, variousboundaryconditions and lamination schemes. The particular caseof thin shellsis studiedusing a penalty function method. GOVERNING

EQUATIONS

Considera differential elementof the doubly curved shell describedin Fig. 1. It is madeup of a finite number of uniform-thicknessorthotropic layers,orientedarbitrarily with respectto the orthogonal shell coordinates((w,,CZ,, , C)chosenso that the cu,and LY,, curvesare lineson the midsurfacec = 0, andthe c curvesarestraightlinesperpendicular to the midsurface. The symbolspXandp,,denotetheprincipal radii of curvatureof the midsurface,while the surfacemetricswill be denotedby yXand y,,. The strain-displacementrelationsof the shear-deformabletheory of doubly curvedshellsassumethe following form: &,=E:+<&;

&y=&;+&*;

2&,,=E40;

2EYL=E50;

2&,=&,0+(x6.

(1)

where: 8 =tE$+wo+-

1 -aw,* ;

( ax> au0 w. i awo2 --ay , &=ay+p,+z ( > atdo au, aw, --. aw, g=-+ aY aX + ay ax ’ PX

2

awe u. 640 = a/Yx + ---; ax

E; =

px

awe 3 wy + YjT - py

and lo is given by q. = *(l/p, - l/p,). Here u. , u. and w, are the displacementsof the midsurfacealongthe ar,, ay and Caxes,and wXand vy arethe rotationsof the transverse

Nonlinear analysis of shells

421

Fig. 1. Geometry of a doubly curved shell.

normals about the CY~ and CY,axes,respectively.Now, it is possibleto calculatethe total tensileforce (perunit length)Ni on the differential elementin the cu,direction,definedby:

whereh is the thicknessof the shell and dA,, = L,, dol,dc is the elementalcrossareaas shownin Fig. 1, Li (i = X,JJ)beingLame’s coefficients.Then, from eqn (3), we obtain: h/2

N,

u15 d[. (4) yx s -h/2 Similarly, we can write the remaining stressresultants.Note that the shearstress resultantsN12andN,, , and the twisting momentsMl2 andM,, , are generallynot equal. But, assumingthat the thicknessof the shellis small whencomparedto the principal radii of curvature[Li/yi = 1 (i = x, u)], previousconsiderationsarenot valid. In fact, we have: Ni2 = N21 = Ns and M,, = M21 = M, and we can write: =

fk+I

(&sMi) = i Qi =

k=l

sj-k

kil

K2

c+‘(l I 9c) dc I*+’

CT~(~) d[

(i = 1,2,6) (i = 4,5)

(5) (6)

s tk

where n is the number of layers(numberedfrom top to bottom), and ck, &+t are the distancesfrom the midsurfacerespectivelyto the lower and uppersurfaceof thekth layer. The constitutiveequationsof the kth lamina are givenby: ,jj(k” (7) 1 = ~,fO$kO, where dik’) and Ejk’) are the stressand strain components,respectively,referredto the lamina material coordinates,and CF) are the elasticconstantsof the kth lamina (I = 1 denotecompressiveproperties,1 = 2 denotetensileproperties). To derivetheshellconstitutiveequations,wewrite the following relationin curvilinear coordinates(ar,, aY,c): @O = c’~Oe(k” (8) IJ J ’ are the elasticconstantsof the kth lamina referredto the curvilinear coordinates. By substitutingthe eqn (8) into eqns(5) and (6), we obtain: where

Chk’)

=

Cp)(cr”,

(jtk))

IQil = [A~jIlEjO)

(i,j = 4, 5).

(10)

For the calculationof the coefficientsA,, B,, Dii the Bert model is used.In particular, with referenceto the kth layer orientedat an arbitrary angle8 (positivein the clockwise direction)with respectto the global coordinatesystem,the valuesof strainsbeingknown,

D. BRUNO et al.

422

it is necessaryto calculate,by a suitablerotation matrix, the strainef in the fiber direction: cf = 8, cos% - 2.~~sin 0 cos13+ sYsin28. (11) The neutral-surfaceposition&,ecanbedeterminedby replacingthe straincomponents of eqn (11) with thosefrom eqn (l), set equalto zero: E: c0s2e - ,560 sin 0 cos e + 8; sin20 Cd = - x1 cos'e - X6 sin 8 cos8 + x2 sin28* (12) Taking into account eqn (2), relation (12) can be particularized for a cross-ply laminatein the following two expressionsrelativeto fibers orientedat the ar,and oYaxes, respectively:

+ Wo/Px + t4 x1 & = _ hJ + wohy+ twg . L = - [uoJ wx,x ’

(13)

VYS

With referenceto Fig. 2, the calculationof A, leadsto:

and similarly for other matricesBij and DQ. ~Compressive region aTensile

property property

regton

Fig. 2. Scheme of a multilayer angle-ply laminate and neutral-surfaces position.

It should be noted that thesevaluesare unknown a priori. Therefore,an iterative proceduremust be adoptedto determinethe neutral surfaceposition. The coefficientsA, for (j = 4,5) are definedby eqn (14)exceptthat CrO are multiplied by the shear-correctionfactor K2 for which we assumethe classicalvalue of 5/6. The total potential energyof the plate, in the absenceof body forces and surface shearforces,can be expressedby: rI=u,+us+v Wa) where: u, = *

sn

(N,&: + N24 + N,&,o + M,X, + &X2 + &Xcj)d~,

WW

Nonlinear analysisof shells

423

In eqns(15),63is the middle surfaceof the shell and80 is its boundary.Further, &, NS, l\;i,, iii, and 0, are specifiedloads acting on X2. The problem of finding the static solution (u,, v,, w,, vX, vu) of the thick-shell equationscanbeviewedasoneof finding the stationarypoint of the total potentialenergy (15a).In this casethe quantitiesA:, Ai and A, . & assumethe values&, ASSand &, respectively. If the classicalthin-shell theory is adopted,the problem can be viewed as one of finding the stationarypoint of functional (15a)subjectedto the constraintconditions:

To incorporate the constraints,one can usethe Lagrangemultiplier method or a penaltyfunction method. In this work, a penaltyfunction methodis usedandthe penalty parametersI, and A2 of eqn (15~)assumesuitably large values. Clearly, when A, and A2-+ 00the constraintsare satisfiedexactly. From this framework the following variational equation of the shell equilibrium problem, correspondingto the stationarycondition of functional (Isa), can be stated:

cm =s

(Nl c3.s:+ N2 de,” + N6 Se: + Ml 6x1 + M2dxz + M6 6~~) dQ

l-l

-s n

q,&vdR -

(zvn624,+ ivs 624,+ A& &In + i&q/, san

+ on dw) ds = 0. (17)

CLOSED-FORMSOLUTION

In this sectiona closed-formsolution for simply supportedcross-plybimodular thick shellsundersinusoidaltransverseload is givenfor the caseof linear strain-displacement relations. Consideringonly linear termsin eqns(2), from the stationarycondition of functional (15a)the following five equilibrium equationsare obtained: $+a(N6

aY

Q4 - tloM6) + 7 x = O

$(N.+&,&)+~+~=o aQ4

aQ5

0

ax+av-

(18)

aMl + --Q4=0 aM6 aX ay ai% + -aM2 - Q5 = 0. aX ay It should be observedthat for symmetric and antisymmetric bimodular cross-ply laminates, We have: ,416= & = Bi6 = & = 016 = DZ6= Aq5 = 0. Note that the coefficientB12 is generallynot equalto zero for bimodular cross-plylaminates.In fact, for unimodular material, the symmetryof the plane-stress-reduced matrix C, guarantees the annuhnentof this coefficient. Instead,for bimodular materials,the arbitrary position of the neutralsurfaceproducesdifferent valuesof C, in eachlamina andin generalthese coefficientsare not zero.

424

D. BRUNOel al.

The simply supportedboundaryconditionsare:

WJ Y) = W&Y) = Ml(O,Y) %(O,Y) = %hY)

= W@,Y)

= Vo(O,Y) = U&Y)

= 0 = 0

N,(x, 0) = N,(x, b) = M*(x, 0) = M*(x, b) = 0

(19)

wo(x, 0) = wo(x, b) = uo(x, 0) = 24()(x,b) = 0 wy(O, Y) = W&Y)

= VAX, 0) = u/,(x, @ = 0.

The sinusoidallydistributedtransverseload is assumedto be of the form: q

= q,sinj9,xsinj?,y;

PI = 5,

jY&= %.

(20)

The symbolsa and b arethe dimensionof the shellmiddle surfacealongits x andy axis, respectively. Boundaryconditions(19)are satisfiedby the following spatialvariation of displacements: uo(x,y) = 0 cosj&x sin/3,y uo(x,y) = B sin&x cos&y wo(x,y) = IV sin/3rx sinjYzy

(21)

w,(x, y) = B cosPI x sin&y vy(x, y) = P sin&x cosj&y. By substitutingeqns(20) and (21)in the equilibrium equations(18) we obtain five equationsin termsof the five unknownamplitudes0, r, m, x, p which canbeexpressed in the following matrix form:

[WV = t4

(22)

which (FJ and (A) denotethe following column vectors: - -

Su

= iWz(&

S,,

= 8:(&

+ tloh +

tto%)

+ B12); + 83322

-

$;

(23)

Nonlinear

analysis

of shells

Sss= S:D,, + 8;D,, + A;-

425

(23)

Notethat the penaltyparametersA, andAZappearin eqns(23).Therefore,the closedform solution should be able to give both thin and thick solutions. In the first casethe penalty parametersAi and A2will assumesuitably largevalues,while in the secondthe penaltyquantitiesA: and A; will be identified with the actual valuesof the shearstiffness terms AU and ASS. Moreover, it shouldbe observedthat coefficientsS, are unknown a priori because they dependon the neutral-surfacepositions. Therefore,an iterativeproceduremust be usedto solveeqn (22). On the other hand, for eachfiber direction the relative neutralsurfaceposition is characterizedby a single parameter[,,*. In fact, from eqns (13), replacingrelations (21),we obtain: cm = c,,(e = 00) = - ‘OJ i w”‘px = ’ +r’px = const. x,x cw = c,,(e = 900) = - ‘OS iyy’pY = ’ + Fw’pY = con&

FINITE

ELEMENT

Pw GW

APPROXIMATION

In this sectiona finite elementmodel of problem(17)is developed.It canbe observed that the regularity degreeof the unknown functions uo, v. , w. , I,v~,wyis lower than the regularitydegreeof the unknown displacementfunctionsin Kirchhoff theory.Therefore, the assumeddisplacementfield not only accountsfor the transversesheareffects,but also leadsto lower-orderequationsthat facilitate the developmentof the CO-element. The middle surfaceSzof the shellis partitionedinto a finite numberof isoparametric elements.The unknown displacementfunctionsareexpressedby the following interpolation relations:

whereN is the number of nodesin the mesh,fi(a,, ay) are the interpolation functions, and u;, vi, wk, vi and v- are the nodal values of the unknown displacements.By substitutingeqns (25) into eqn (17), the shell equilibrium equationscan be put in the discreteform: K(U)U = F,

(26)

whereK(U) is the stiffnessmatrix of the shell,U collectsthe nodalvaluesof thegeneralized displacementsand F is the nodal force vector.

426

D. BRUNO et al.

The numerical algorithm implementedto solve eqn (26) requires knowledgeof displacements,both to determinethe neutral-surfaceposition [eqn (12)] and the geometrical nonlinearstiffnessterms dueto von Karm&nstrains.So, an iterative procedure accounting for the nonlinear constitutive model and the geometrical nonlinearities introducedby the von K&man model must be used. The global procedureappliedto our specificproblem consistsof the following steps: (1) Assumethat the elasticmoduli of the layersare all in tensionor in compression. (2) Determine,by a standarditerative procedure,the displacementfield. (3) By displacementsobtainedat step2, determinethe neutral-surfaceposition &,# [eqn (12)]at eachGausspoint of any elementof the mesh. (4) Determinewhich zonesare in tensionor in compressionand updatethe stiffness matrix K accordingto Bert’s model. (5) With the stiffnessmatrix K it is possibleto calculatethe new displacements. (6) If the differencebetweentwo consecutiveneutral-surfacepositionsis lessthan an admissiblevalue6 stop the iteration, otherwisego to step3 and continue. NUMERICAL

RESULTS

In this sectionsomenumerical resultsare given, togetherwith useful comparisons betweenclosed-formand finite elementsolutions. Moreover, the main mechanicaland geometrical.parametersare investigatedshowingthe influenceof boundaryconditions, radii of curvature,stackingsequence, angleof fiber orientationandmaterialproperties,etc. Shells subjectedboth to uniform and sinusoidallydistributed transverseload are analyzed. The finite elementanalysisis developedusinga 5 x 5 nine-nodeshell elementmesh and reducedintegrationof shearterms. In numerical applicationstwo schemesof shellsare examined.Theseschemesare shownin Fig. 3, and correspondto cylindrical and sphericalshells. The elasticmaterial propertiesconsideredin the analysisare givenin Table 1.

Fig. 3. Schematics of example problems: (a) cylindrical shell; (b) spherical shell. Table 1. Material properties Aramid-rubber

Graphite-epoxy

Properties

Tensile

Compressive

Tensile

Compressive

4 GPa) 4 @Pa) % @Pa) G,, @Pa) G3 KM V12

3.5842 0.00909 0.0037 0.0037 0.0029 0.416

0.0120 0.0120 0.0037 0.0037 0.00499 0.205

152.52 8.28 2.59 2.59 2.59 0.25

151.72 7.59 2.59 2.59 2.59 0.25

421

Nonlinear analysis of shells Table 2. Boundary conditions Side (x, 0)

ss HH

Side (0, y)

Side (x. b)

Side (a, y)

uo,woeh

uoswo9wy

uo,wo,wx

00, woe K

UOl woe V”

00. was vx

uo* wo9 v, uo, wo, K

In addition, the unimodular material (UM) adoptedin the computationsis characterizedby the following mechanicalparameters: El/E,

= 25;

v12 = 0.25;

G12/EZ = G13/EZ = 0.5;

Gz3/E2 = 0.2.

The boundaryconditionsexaminedin the computationsare shownin Table 2. In Tables3a,b, c a simply supported(SS)cross-plysphericalshellmadeof unimodular material(UM) with aspectratio a/b = 1, andsubjectedto a sinusoidallydistributedtransverseload, is analyzed. Two valuesof the side-to-thicknessratio (a/h = 10;a/h = 100)wereconsideredfor three stacking sequencetypes (O”/900; 0”/!30”/0”; O”/!900/900/Oo).Moreover, several valuesof the geometricparameterradius-to-thicknessratio p/a are examined. Numerical results were obtained using both the closed-form solution and FEM procedure. The analysis of these shows a good agreementbetween analytical and numericalresults. Table 3. Nondimensional center deflection P = (wch3E,/qou4) - ld versus radius-to-thickness ratio p/a, for a cross-ply spherical shell (u/b = 1) under sinusoidally distributed transverse load. (a) Cross-ply O”/90”; (b) cross-ply 0°/900/Oo; (c) cross-ply O”/900/900/Oo

(4

a/h = 10 P/a : 3 4 5 10 1030

0.))

1 2 3 4 5 19 1030

6)

=

100

Closed form

FEM

Closed form

FEM

8.156 4.023 10.062 10.958 11.428 12.122 12.373

8.147 4.013 10.057 10.954 11.426 12.122 12.373

0.2111 0.0536 0.4634 0.7%9 1.1948 3.5760 10.653

0.2105 0.0534 0.4623 0.7951 1.1923 3.5703 10.650

o/h

P/a

a/h

=

10

a/h

=

100

Closed form

FEM

Closed form

FEM

3.2588 5.3047 5.9964 6.2826 6.4245 6.6238 6.6930

3.2529 5.3021 5.9959 6.2831 6.4255 6.6255 6.6950

0.0536 0.2068 0.4391 0.7237 1.0337 2.4109 4.3370

0.0534 0.2063 0.4382 0.7222 1.0318 2.4081 4.3360

u/h

=

10

u/h

=

100

P/a

Closed form

FEM

Closed form

FEM

: 3 4 5 10 lore

5.2537 3.2289 5.9380 6.2212 6.3615 6.5587 6.6271

5.2511 3.2229 5.9374 6.2215 6.3624 6.5603 6.6290

0.2054 0.0532 0.4362 0.7192 1.0279 2.4030 4.3368

0.2049 0.0531 0.4353 0.7178 1.0260 2.4002 4.3351

D. BRUNO et al.

428

Table 4. Dimensionless maximum displacements for cross-ply (O”/90”) cylindrical panels made of aramid-rubber bimodulus material, b/h = 10; SS-sinusoidally distributed load. f+ = (w,mUE,,h3/qob4)x 10’; P = (uF”Ezch3/q,,b4)x 10’; ii = (uo”“E2ch3/qob4)x 10

Thick-shelltheory

Thin-shell theory

a/b

ii,

1s

B

iiJ

ii

I7

43.9849 185.3851 284.4248 335.9739

0.2562 4.6931 8.9444 10.9476

6.7779 36.6201 56.8794 67.3947

0.2390 1.0227 1.4295 1.5887

2.0142 7.4611 10.0832 11.1475

0.4069 1.I759 2.5264 2.8415

2.5313 12.3637 17.8149 20.1848

p,/h = 10 0.5 1.0 1.5 2.0

39.9775 174.5531 262.5691 306.6569

0.2421 3.5384 6.6718 8.1080

6.1930 32.2518 49.0138 57.4118

p,/h = 20 0.5 1.0 1.5 2.0

46.7015 194.4911 267.6268 297.5952

0.2945 1.1897 1.6331 1.8053

0.8728 3.8487 5.3544 5.9752

53.1660 213.6214 294.7210 329.1962

p/h = 50 0.5 1.0 ::i

45.6675 188.2312 261.6269 292.8734

0.4310 1.7951 2.5195 2.8221

3.3083 14.2131 19.9254 22.3775

53.2172 209.7143 290.1484 324.2099

In Table 4 the dimensionlessdisplacementsof a simply supported(SS) cylindrical cross-plytwo-layershellaregivenfor somevaluesof p/h and a/b ratios. The resultswere obtainedusing closed-formsolutions and the distributed load is of the sinusoidaltype. Theseresultswere obtained by using both thin and thick closed-formsolutions. More precisely,the thin solution was obtainedby applying the penalty functional method to eqn (22), and assuminglarge valuesfor the penalty parametersAi and AZin eqns(23), while for the thick solution, the penalty quantitiesA: and Af which appearin eqns(23) assumethe actualvaluesof the shearstiffnesscoefficientsAU nd ASS.The increasein the maximum transversedisplacementwhen passingfrom thin to thick theory is evident.On the contrary, this is nevertrue for maximum in-planedisplacementsand the differences increasewith p/h ratio. The FEM analysis of a sphericalshell (p, = pv = p) made of UM material and subjectedto hingedboundaryconditions(HH) and uniformly distributedload wascarried out, to show the effects of geometricalnonlinearies;seeFig. 4. Both for linear and

5 Fig. 4. Spherical shell (a/b = 1; p/a = 10) under uniform transverse load: diiensionless transverse central deflection I? = w,/h versus h/p ratio and load level 4 = qop2/E, h2 (UM material; HH conditions).

Nonlinear analysis of shells

429

Fig. 5. Spherical shell (u/b = 1; p/h = 500; a/h = 50) under-central point load: dimensionless central transverse deflection (iit = w,/h) versus load level P = Pu/E,h3 (UM material; HH boundary conditions).

nonlinear results, the increasein structural rigidity is seenas h/p increases.It can be observedthat with increasingload level, the nonlinearequilibrium pathscrossthe linear onesand then the structuresbecomestiffer in their nonlinearrange.This is producedby the changeof configuration. A spherical shell made of UM material was analyzedusing the FEM procedure, showingthe influenceof thelayeringandstackingsequence(0”/90”/. . .; 45”/-45”/. . .) on the geometricalnonlinearbehavior;seeFig. 5. The shellwassubjectedto hingedboundary conditions(HH) and to a centraltransversepoint-loadP. From this figure we can seethe tendencytoward snappingbehavior, especiallyfor two-layer angle-plylaminates.Obviously, by increasingthe layering,the structureincreasesits rigidity. The influenceof shear deformability is shownin Fig. 6, wherebya sphericalshell (p, = pY = p) subjectedto a sinusoidaltransverseloadandto simply supported(SS)boundaryconditionswasanalyzed. The ratio betweenthe centraltransversedeflectionobtainedby using Shear-Deformation Q/h = 100 e/u = 10 a/b = 1 UN

-----8

layore

-2

layere

Fig. 6. Cross-ply spherical shell (p/h = 100, p/a = 10; a/b = 1) under sinusoidal transverse load: influence of shear deformability on the central transverse deflection (UM material; SS boundary conditions).

430

D. BRUNOet 01.

Theory (wsnr) and that obtainedby usingThin ClassicalTheory (w& is shownversus the side-to-thicknessratios a/h. The differencesbetweenthe resultsobtainedby the two theoriesfor a/h ratios lessthan 20 are evident. Note that while for the plate structurethe ClassicalPlate Theory solution can be obtained by using the Shear-DeformationTheory in which large values of a/h are assumed,for the shell structure appropriate thin and thick theories must be used to analyzethe influenceof sheardeformability. In Figs 7a, b the influenceof fiber orientationon deformability is shownfor various numbersof layersgiving both thin and thick solutions.Shell made of quasi-unimodular graphite-epoxymaterial subjectedto uniform transverseloads were analyzedby the FEM procedure,taking only linear strain termsinto account.Numerical resultsshowthe strong influence of layering and also of the lamination angle of the structural rigidity. In fact, from Fig. 7a we can observethat a significant increasein flexural rigidity with respectto the orthotropic solution is obtainedfor four and eight layersat 0 = 45”. On the contrary when a thin shell is considered(p/h = 500; a/h = lOO),thesedifferences (a) ‘*’ Thiok mlution

-------_-----0

20

2 Layer8 4 k&pru 8 lagers 40

e/h = 60 a/h=10 80

,y

80

Fii. 7. Simply supported cylindrical shell under uniform transverseload: influence of fiber orientation angle B on dimensionlesscentral transversedeflection [W = (wCE2cR3/qOu’) x Id; db = 1; +e/-ei...].

ir’ -1

Nonlinear analysis of shells

(a) /

140

Awu)

/

RUBBIR

190 1o/s0

c M- - -,-,/. / // /

I

I7

lb)

/ /

,I

I

60

I I

I

/

/

-Bimodular ---_

ComprW#h

--------_-_-

q-H

Fig. 8. Cross-ply cylindrical shell under uniform transverse load: influence of aspect ratio a/b on the dimensionless central transverse deflection iir = (w&h3/q,,a4) x 10”.

disappearand an optimal valueof the angle0 = 45” is found with comparableincrement of flexural rigidity for all layering. Thin (p/h = 500; a/h = 100) and thick (p/h = 50; a/h = 10) cylindrical shells madeof bimodular material andsubjectedto uniform transverseload wereconsideredto investigatebimodularity effect; seeFigs 8 and 9. The analysiswascarried out using the FEM techniqueand including only linear strain terms. Two bimodular materials were considered:graphite-epoxyand aramid-rubber.

GRAPHITE-EPOXY

\ Y-0.6

,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,, 0 1 2

/ / // ad 3

4

Fig. 9. Cross-ply cylindrical shell under unifortu transverse load: neutral-surfaces position C = CJh. (a), (b) Neutral-surface positions at the central point of the shell [c, = de (e = 0’); hv = c,s (0 = 900]. (c), (d) Neutral-surface positions at x/a = 0.5 for aramid-rubber. (Continued overle4f)

432

BRUNO et al.

D.

a>

:'(b) 2:

I I I

ARAMID-RUBBER

\

-1:

/

----_

-20

a/h

=

a/h

= 100

/

/

/

/

to

a/b

,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,’, 1

2

t 1 (cl

3

-a/b -----a/b

4 = =

0.5 3

0

ARAMID-RUBBER -l-

-2-

o.r:f

0.2

ARAMID-RUBBER cr/h= 10 tnz

(d)

______-------

_I_-._--.-_I_-_-----.~~-----.-----------.

90”

-0.3 -

-w

I---r---------------a/b -----&

= =

0.5 3 v/b

,111, 0.0

,1,,,‘,,,,,,,,

I l,,,,,

0.2

l,,“”

0.4 Fig.

9-continued.

0.6

,,,,‘,‘,,,,,,

.I

0.8

1

Nonlinear

analysis of shells

433

The different behaviorsof the structures,obtainedby usingonly six compressiveor only six tensile,overall 12bimodular material properties,are shownin Fig. 8. The strong differencebetweenunimodular and bimodular behavior is well evidencedby aramidrubber (EIT/EIC = 298.7) as shown in Fig. 8a, in which the dimensionlesstransverse deflection ii, = (wCEzch3/qoa4) x lo3 is plotted versusthe aspectratio (a/b). The strong influenceof the aspectratio a/b on the bimodularity effectsis clarified by Fig. 9, in which the neutral surfacepositions (<,; &,,) are plotted versusthe aspectratio a/b, and shell coordinatesy/b at x = a/2. From thesefigureswecandeducethat for a/b equalto 0.5theneutral-surfaceposition
n

40: CRAPEITE-BPOXY a/b

o/so ss

=

100

,.-- *.-*.-*.-*.-.*.-) . /( ..-/ *.-- ) ’

0 ,““,,,,,,,i.,,,,,,

0. I

‘NH

.-

,,,-.,,,,,,,.,,,,

0.3

0.k

0.7

,.-- .I’

,

Oh

Fig. 10. Cross-ply cylindrical shell under uniform transverse load: influence of thickness ratio T = t,/t2 on the transverse deflection ii = (wc&h3/q0d) x ld.

D. BRUNO et al.

434

The influenceof the thicknessratio tl/tz is shownin Fig. lOa,b, wherebytwo-layer cross-ply(O”/!20”)cylindrical shells [(p/h = 50; a/h = 10); (p,./h = 500; u/h = lOO)] made of graphite-epoxybimodular material were examinedfor three valuesof aspect ratio a/b = (1, 3 and 5). In this casethe analysiswas also carried out using the FEM computationalproceduretaking only linear strain terms into account. The shellsweresubjectedto a uniform transverseload and SS boundaryconditions. It canbeobservedthat the centraldeflectionis,= wJ& h3/qoa4 increaseswith t/t2 ratio. This behavioris more marked for narrow shells(a/b = 3-5) in which the fiber direction of the upper layer is along the generatricesof the cylindrical shell. The qualitative behavioris similar for thin and thick shells.The effectsof geometricnonlinearitieswere analyzedfor a single-layercylindrical shell made of aramid-rubberbimodular material and subjectedto SS boundaryconditions and to a uniform distributed load. The geometric parameterswereasfollows: p,,/h = 50; a/b = 1;a/h = 10.Numericalresultswere obtainedusing FEM computationsand including sheardeformability. Theseresultsare shownin Fig. 11,in which both bimodular and unimodular shellsare examinedfor two valuesof the fiber orientation8 : 8 = 0“; 0 = 45”. The unimodular resultscorrespondto

-----

Tensile

2.0 :I

EimoduZar

; (b)

-----

TUtl#i&

ARAMID-RUBBER 1.0

45O SS

Fig. 11. Geometrically nonlinear analysis of single-layer biiodular cylindrical shell under uniform transverse load 4 = (qJE&(u/h)’ iir = w,/h; u/b = 1; u/h = 10; p,,/lr = 50; SS boundary conditions].

Nonlinear analysis of shells

435

the sameshellmadeof a unimodular materialwhosemechanicalpropertieswereassumed to be equal to the tensilepropertiesof the aramid-rubberbimodular material. It can be observedthat the effectsof geometricalnonlinearitiesaremore pronouncedfor bimodular shellsthan unimodular ones.This is relatedto the high deformability of bimodular shell structures.On the contrary, if the unimodular model is adoptedthe structurebecomes stiffer andtheeffectsof geometricalnonlinearitiesdisappearfor theload levelsconsidered. CONCLUSIONS

In this paperthe nonlinearbehaviorof doubly curvedcompositeshellswas studied, taking both geometricand material nonlinearitiesinto account. Material nonlinearities involve the different elastic responsethat some advancedcompositematerials exhibit under tensile strains or compressivestrains. These last nonlinearities were modeled accordingto the fiber-governedschemeproposedby Bert. A shear-deformationversionof Sanders’theory was developed,including geometricnonlinearitiesin the senseof von K&rm&n.Moreover, the governingequationswerebasedon a variational formulation in which both thin and thick theories of shells were incorporatedusing an appropriate additional penalty functional term. Closed-formsolutionsweredevelopedfor both thin andthick bimodular shellsin the particular caseof a sinusoidallydistributedtransverseload action and simply supported cross-plylaminates,accountingfor linear strains only. Numerical FEM solutions were also given for generalloadings, general boundary conditions and arbitrary stacking sequences, including geometricnonlinearities.Somecomparisonsbetweenanalyticaland numerical FEM solutionsvalidate the latter. Numericalresultsshowthe stronginfluenceof materialandgeometricalnonlinearities on the overall behaviorof compositeshell structures,as well as the influence of other characteristicparameterssuchas boundaryconditions, layering, geometricalratios, etc. Further developmentsare in progressinvolving nonlinear dynamicsof bimodular shellsand the study of more refinednonlinearshellmodels,which will enablethe analysis of problemssuchas buckling, postbucklingand failure to be carried out. REFERENCES Ambartsumyan, S. A. (1%5). The axisymmetric problem of a circular cylindrical shell made of material with different stiffnesses in tension and compression. Izvestiya Akademiya Nauk SSSR. Mekhanika 4, 11-85. English translation: National Tech. Information Center Document AD. 675312 (1967). Bert, C. W. (1967). Structural theory of laminated anisotropic shells. J. Compos. Mater. 1, 414-423. Bert, C. W. (1977). Models for fibrous composites with different properties in tension and in compression. J. Engng Mat. Tech., Trans ASME 998(4), 344-349. Bert, C. W. and Reddy, V. S. (1982). Cylindrical shells of bimodulus composite material. J. Engng Mech. Div., ASCE 108, 675-688. Dong, S. B., Pister, K. S. and Taylor, R. L. (1962). On the theory of laminated anisotropic shells and plates. J. Aeronaut. Sci. 29, %9-975. Koiter, W. T. (1960). A consistent first approximation in the general theory of thin elastic shells. In Proc. IUTAM Symp. on The Theory of Thin Elmtic Shells, pp. 12-33. North-Holland, Amsterdam, and Interscience, New York. Mindlin, R. D. (1951). Influence of rotary inertia and shear on flexural motions of isotropic elastic plates. J. Appi. Mech. 18, 31-38. Ramana Murthy, P. V. and Rao, K. P. (1985). Analysis of laminated bimodulus composite thin shells of revolution using a doubly curved quadrilateral finite element. Int. J. Num. Meth. 21, 285-299. Reddy, J. N. (1984). Exact solutions of moderately thick laminated shells. J. Engng Mech. Div., ASCE 110, 794-m.

Reddy, J. N. and Chandrashekhara, K. (1985). Geometrically non-linear transient analysis of laminated, doubly curved shells. Int. J. Non-Lin. Mech. 20(2), 79-90. Reissner, E. (1945). The effect of transverse shear deformation on the bending of elastic plates. J. Appl. Mech. 12,69-77.

Reissner, E. (1949). Small bending and stretching of sandwich type shells. NACA TN-1832. Saint Venant, B. (1864). Notes to the 3rd Edn of Navier’s R&umddes Lecons de la Resistance des Corps Solides p. 175. Paris. Tiioshenko, S. (1941). Strength of Materials, Part 2: Advanced Theory and Problems, 2nd Edn. Van Nostrand, Princeton, NJ. Yang, T. Y. (1973). High order rectangular shallow shell finite element. J. Engng Mech. Div., ASCE EM1 99, 157-181.