The Design of Shells and Tanks in the Aerospace Industry, Some Practical Aspects

7/28/2019 The Design of Shells and Tanks in the Aerospace Industry, Some Practical Aspects

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The design of shells and tanksin the aerospace industry:some practicaHOry, H-G Reimerdes, J G6mez GarciaRWTH Aachen, Germany

.IaspectsSummaryThis review highlights some practical aspects of thedesign of thin-walled shells for aerospaceapplications. This type of shell must comply wi th themission profile. It s therefore necessary to find anoptimum structural concept w ith low weight, highstrength, high buckling load and a low imperfectionsensitivity. In an opt imum design, structuralinstabilityoccurs slightly below the material

strength or yield strength. In general, and bycontrast with other structural elements such asbeam and plates, a th in-walled cylindrical shell showsa high imperfectionsensitivity. Hence,recommendations are given concerning the designof shells and approximate stability analyses arepresented for different mechanical oadingconditions.

Progress in Structu ral Engineeringand Materials I998 Voi I(4): 404-4 I 4

Metallic and non-metallic shell structures of differentshapes and types of construction have widespreadapplications in aerospace, mechanical, civil andstructural engineering concepts. They are mainly usedas fuselages, rocket stages, habitation modules inspace stations, tank structures, junction elements,energy absorbers, silos or pipelines. All these differentapplications demand both low weight and highstrength.

Different aerospace technology programmesworldwide are all aiming to achieve a mass reductionand optimal design. These programmes investigate,among other things, the static and dynamic behaviourand the buckling strength of shell structures.Currentresearch projects include the ARIANE Programme,the European FESTIP (Future European SpaceTransportation Investigation Programme) Project, theresearch and development project for theInternational Space Station Alpha, the Europeanresearch and development project CRYOPLANE, the

Fig. I Internat ion al developm ent and aerospace researchprogrammesdevelopment project UHCA (Ultra High CapacityAircraft), several hypersonic, and other spacetechnology programs from different industrial andpublic institutions.

Abbravirlons TarmlnolonBD = bi-directional D = extensional t i h e s s in axial J,.L = smeared second momentsCFRP = carbon-flbre-reinforcedFESTIP = Future EuropeanSpaceplastics

TransportationInvestigationProgrammeMS = marginofsafetyQi = quasistatic

SS I = simply supported restrainofthe radial displacement553 = simpb supported restrain

ofthe circumferentialandradial displacementUD = uni-directiwlUHCA = ultra high capacity aircraft

E =EsecEtrn =Ered =EX .5 =E =I =G =1 1 '

direction of area inaxial.Youngs modulus circumferential directionsecant-modulus J = smeared torsionalsecondtangent-modulus moments of areareducedmodulus iu = ultimate oad actorYoung'smodulus inaxial. IT = test factorcircumferentialdirection k = numericallyor empiricallystrain tensor deflncd pomrsaccelerationdue togravity K, = bending stiffness nshearmodulus circumferential directionratiobetween he chess- I = critical wavelength n axialboard buckling orceflowandthe axisyrnmetric buckling shellsforce flow

direction or imropic

Ix.iy=L =i n =N Ny=

N =v =P =e =

critical wavelength naxial.circumferentialdirectionfor orchotmpic shellslength ofthe shellwave-number ncircumferentialdiredonaxial force per unitcircumference.circumferential orce perunit heightshear stress resultant perunit circumferencePoisson's ratioexternal pressurehelixangle

R =Rx,I$ =P =p m =s =sx,5 =a =z =

radiusradius naxial.circumferentialdirection(van derNeut's method)densityKnock4own factor(9nrpmbability)shell thicknesssmeared shell thickness naxial. circumferentialdirections m s s tensorratioof he 'critical load ncombination' o criticalload actingalone'

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SHELLS ND TANKS IN THE EROSPACE INDUSTRY 405

ARL4NEVFig. 2 Space tran sp or t systems

The shell structures used in aerospace applicationsmust meet special requirements. In spaceapplications, these shells are exposed to mechanicalloads (axial compression, bending, external andinternal pressure) and thermal loads from thecryogenic fuel and aerodynamic heating of thestructure. All these loads affect both thethermomechanical stress and strain behaviour and thestability of the shell structure (Fig.2). A successfuldesign must rely on an accurate definition of theloads, local stresses and load-carrying capacities, all ofwhich depend on precise evaluation of the boundaryconditions.

The buckling of thin-walled shells often limits theirload-carrying capacity. Even where this is not the case(eg framed thin-walled panels), structural instabilitiesmust be avoided if an interaction with theaerodynamics of the craft can occur. Furthermore,buckling of these panels must be prevented if fatiguecan occur under repeated buckling and relieving.final design analyses of these shell structures is doneusing commercial finite element (FE) codes. Usingthese programs, the analysis is often very time-consuming, particularly where many parametriccalculations are needed to find the optimum design(considering low weight, high strength and high loadcapacity). The physical influence of differentparameters can often be hard to understand. For thisreason, an analytical investigation is recommended asa first simplified step. However, a final analysis mustuse a sophisticated FE code like MARC, NASTRAN orANSYS.

Another reliable, fast and economical way todetermine the response for shells of revolution is ananalytical solutionof the governing differentialequation. The mathematical description of shellbehaviour is based on a complete shell bending theoryusing the Kirchhoff-Lovehypothesis. The resultingsystem of partial differential equations may bereduced to a system of ordinary differential equations

Nowadays, the greater part of both preliminary and

by expansion of the circumferential variation ofgeometrical and mechanical variables using theFourier series. The integration of this system ofordinary differential equations leads to the elementstiffness matrix via the transfer matrix. Thisformulation allows user elements to be implementedin commercial FE codes. The complete theory of thismethod has been documented by Rittweger et alui,Rittwegerizi, Albusr~i,Miermeistemi, Diekerrs,61, ndGdmez Garcia et ali7,si.

Load assumptionsTo comply with their mission, the shells of aerospacevehicles must meet appropriate design criteria withthe lowest possible structural mass. These lightweightand hence thin-walled shells are subjected to manyexternal loads (axial compression, external pressure,internal pressure, ovalizing, bending, torsion, etc.)and become very sensitive to several failure modes(buckling, fatigue or crack propagation undervibration, creep under long duration stationary loads,possibly at high temperature). To validate the design,the load levels are defined in the following terms.0 Limit loads are real, physical, maximum loads

which will be encountered by the vehicle with agiven probability (=1%) uring its lifetime. At alimit load level, the elastic deformations of thevehicle must not initiate a harmful degradation ofperformance (clearance, gaping, leakage, linear ornon-linear, static or dynamic instability, etc.). Areasonable safety margin (up to 10%)should beconsidered against permanent deformations (=o0,J.

0 Ultimate loads are limit loads multiplied by theultimate load factor , (j, =1.25for unmanned and1 .4 or 1.5for manned vehicles, respectively). Forvery sensitive satellites the flight vehicle must besubjected to an extensive acceptance test program.In general , =1.5will be adopted.At the ultimate load level, the load-carryingcapacity can just be exhausted. If the rupture load(related to the limit load level) is given by a testfactor (jT), the non-mandatory margin above theultimate level (MS, margin of safety) will be definedas follows:

The MS must be equal to or greater than zero. AnMS of 4-10% is reasonable. Higher margins willunnecessarily increase the structural mass, unlessother design criteria justify this. Such design criteriacould be stiffness or frequency limitations, thermalcontrol, manufacturing reasons (eg minimumfabrication sheet gauges), prices, etc.If the limit level stresses include thermalcomponents, the ultimate stress level will be definedwithout factoring the thermal stresses. However, the

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406 SHELLS ND CONTAINMENT STRUCTURESthermal stresses should be defined with both aconservative and an optimistic estimation, and bothpossibilities must be checked in the structuralanalysis. Indeed, thermal stresses can either increaseor alleviate mechanical stresses. Where the stressanalysis is at elevated temperature, thermaldegradation of the material characteristics must beincluded. These apply equally to all otherenvironmental effects, such as corrosive media, dust,humidity, frost, radiation, etc.

Structural failures due to fatigue and/or crackpropagation must be avoided during the whole life ofthe vehicle. Fracture control investigations mustprove that, due to a 'fail safe' and/or 'fail slow' designand with a reasonable probability, the mission will notfail or will not be curtailed in a unacceptable manner.This definition must be validated for the periodbetween two regular inspections in the aircraftindustry.axial loads at different temperatures can occur as:0 material rupture, tank leakage,0 material yielding or creep,0 structural instability.

Other structural failures under uni-axial or multi-

Manufacturing lightweight structures implies the useof materials of high performance, which are, ingeneral, very sensitive to cracks, stress concentrationsand cyclic loads, but high stress levels can besustained under other load cases. Fracture controlbecomes mandatory for structural elements whensuch materials are used.

The complexity and variety of different loadassumptions makes it impossible to present here allthe design criteria and failure modes of shell and tankstructures in the aerospace industry. The article istherefore restricted to the load capacity of thin-walledshells due to buckling under axial compression,external pressure and pure shear or torsion.Material selectionGENERALEMARKSThe environmental conditions of aerospace structuresneed materials that resist very high (up to2000 "C)and/or very low (cryogenic) emperatures. Thematerials used are metallic or non-metallic,homogeneous or composite, isotropic or anisotropic,etc. The materials must withstand both load cyclesand sustained stresses at these different temperatures.Often, they must fulfil other thermal requirements,such as conductivity, capacity, expansion andisolation. These matters are covered by numerouspublications beyond the scope of this article.

Here, the main properties of current standardmaterials are outlined, with appropriate selectioncriteria. These criteria are strongly dependent on theload cases and failure modes to be considered. Somecharacteristic coefficients for the classic load cases,

structural responses and failure modes may be noted.0 Tension loads; burst pressure (o/p)

Stiffness,eigenirequency ( ~ / p ) ; G p )0 Elastic column buckling0 Elastic panel buckling0 Plastic buckling (od.-d;

(bcrip /P)Storage of strain energy

%;;(o/pP

BEHAVIOURF COMMON METALLIC MATERIALSIn this context, the material behaviour of metals in theplastic range is of special interest. The plasticbehaviour of the most common metallic materials canbe described very accurately using the Ramberg-Osgood formulation (Fig.3):

In(17/7) (2)=-[07 +(---I 6 '1 where k =1+E 00.7 7 6 0 . 7 1n(a0.7O0.85)This formulation ismost useful as it can predict the

secant modulus (EJ and tangent modulus (EtJ asfunctions of the equivalent stress as:

A(J

E

EFig. 3 Characteristic stress-strain curves for common aerospacemetallic materials

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SHELLSND TANKS IN THEAEROSPACE NDUSTRY 407

7060

so4030

20

lo

(3) --------~-

----

11 =7

Using these reduced moduli, the buckling stress inthe plastic range can be estimated by the Engesser-Kdrman procedurers**i.PROPERTIESF COMMON COMPOSITE MATERIALSSome physical properties of the commonestcomposite materials are given in the box below.

The new development of hollow fibres could be ofmajor significance.Hollow glass fibres are now fullydeveloped and can be used alone, as a sandwich coreor uniformly mixed with carbon fibres. Thesematerials give an important weight reduction. Hollowcarbon fibres are an even more attractive propositionbecause the skin of the fibres is harder. However,hollow carbon fibres are not yet in commercial usebecause they are currently too brittle. For highertemperature applications, hollow boron fibres in atitanium matrix could be very valuable.Some physical properties ofcomposite materials

Composite materials are commonly:0 High performance thin organic fibres embedded in a

polymerized duroplastic o r thermoplastic resin, wherethe fibres comprise about60% of the volume.0 Unidirectional(UD) ike a veneer, or woven like a fabricor cloth (bi-directional, BD), both being orthotropic.

Lamination of different layers can give a quasi-isotropic(QI) behaviour parallel to their plane, such asO0 /+6Oowiththree UD-layers, or 0 / ~ 4 5 " / 9 0 0 ith four UD-layers, oralternativelytwo fabric layersO"145" or filament-wound.The high strength T800 or T300 carbon fibres are the mostused, but also used are high modulus carbon fibres (whichare relatively brittle) and aramid fibres (which are difficultto machine as they are very tough and have a lowcompression strength). Boron fibres are relatively thickand hard, butare not easy to machine or to adapt to curvedsurfaces.The strength to density ratio (aly) =(alpg) for suchcomposites cannot be compared directly with that for anisotropic metallic material. Fig. 4 shows that these valuesare very different for UD,BD and QI laminates. Becausethese composite materials have a high performance, thestrength of smooth thin-walled structures is only limited bypanel or shell buckling.The stresslstrain behaviourofcompositeUD layers inthefibre direction andQIaminates n all directions remainselastic until rupture occurs. By contrast with thisbehaviour, BD aminates show plastic behaviour (eginthediagonal direction), because the matrix (resin) is the mainstress-carrying element inthisdirection.

80 ,7\UD

0 10 5000 10000 lSo00 ZDOOO

(KM)Fig. 4 Characteristic tensile stressesofcomposites

Design featuresSeveral structural forms are used for aerospacevehicles. Shell wall forms (Fig.5 )such as isotropic,sandwich, corrugated, ring-stringer, isogrid andwaffle grid are used to find optimum structuraldesigns with low weight including high strength, highbuckling loads and a low imperfection sensitivity.Optimum design requires that structural instabilityshould occur only slightly below the material strength(or yield limit).Isotropic shellThe simplest and cheapest shell is isotropic,characterized by a tremendous weight penalty (Fig.6) .This type of shell has a very high imperfection

Fig. 5 Examples of different design features0 ONSTRUCTIONESEARCHOMMUNICATIONSIMITED998 SSN 1365-0556


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408 SHELLS AND CONTA iNMENT STRUCTURESSK

10-2

-310

1u -610 a510Fig.6 Comparison of the shell weight of four different designfeatures (s =mass equivalent hic kness)

sensitivity and is only acceptable for compressionloaded aerospace applications like a pressurized shell(fuel tank) or as a very short shell.Sandwic h shellSandwich shells are the lightest structures whensecondary difficulties are ignored. These secondarydifficulties are the damage tolerance, the design ofinterface sections, the introduction of concentratedloads and the fixing of heavy equipment (these itemsneed supplementary rings and inserts), theuncontrolled amount of adhesive, potting resin forinserts, and the interfaces of honeycomb core panels.Corrug ated shellCorrugated cylinders with centric rings are slightlyheavier than sandwich shells. Using external ringstiffeners, they would become even lighter. This wallform does not display the secondary difficultiesenumerated for the sandwich shell. Using rivetedfittings on trapezoidal corrugations and rings, heavyequipment can be simply attached. By contrast,corrugated sheets have a low shear stiffness. Thecontrolling design criterion is therefore sometimes thelateral eigenfrequency rather than the axial loadcapacity, especially for short shells (eg satellite centraltubes). This disadvantage can be eliminated with asupplementary smooth sheet riveted to thecorrugation, increasing the specific weight andfabrication cost. Corrugated sheets are generallyinsensitive to imperfections.

Ring-strlnger shellStringer and ring stiffened sheets (aircraft-type)make a very reliable and simple structure behaving asan orthotropic shell. Due to the large number ofcomponents, this form for the wall is usually moreexpensive and slightly heavier than a corrugated shellwith centric ring stiffeners (Fig.6). It is, however,common for aircraft fuselages.lsogrld or w aw e grid shel lThe isogrid shell has some practical advantages andinvolves a smaller weight penalty at high loads. Atlower loads, other integrally stiffened shells (crossgrid or waffle grid) may be lighter. Integrally milledsheets display a flaw sensitivity which is reduced bysandblasting. This design concept is ideal for largerwelded pressure vessels.Composlte m ater lalsIf CFRP composites are used instead of aluminiumalloys, there is little influence on the design.Composites can give a mass reduction of 20% or 30%.Price is the main disadvantage of CFRP structures,together with damage tolerance due to the lack of theplastic deformation capacity of the fibres in somecases.

Structural instability n shellsThin-walled light constructions, like shell structures,are sensitive to vibration and their failure modesgenerally involve structural instability. If the loadcapacity is limited by elastic buckling, it usuallyoccurs far below the material strength. Stiffened thin-walled shells of revolution have higher buckling loadsthan unstiffened ones. Reasonable stiffening allowsthe yield stress(ieoo,2roccripptress) to be reached incompression.

Aerospace structures are often built from stiffenedthin-walled shells or curved panels. Unstiffenedisotropic shells (eg propellant tanks for launchingrockets), are stiffened by internal pressure, andbecome more resistant to buckling. Classic stiffenedforms used in the aerospace industry (eg fuselage of alarge transport aircraft or launch vehicles) are alsopartly stiffened by internal pressure. Stringers andrings are usually of thin-walled open or closed cross-section.BUCKLINGF PERFECT AND IMPERFECT STRUCTURESColumn buckling under compression is a classicalbifurcation stability problem. Indeed, the compressedperfect elastic column, free of imperfections and loadeccentricity, is in equilibrium at all load levels. Thecondition is, however, stable for load levels below theEuler load, unstable for loads above it and neutral atthe Euler critical load itself (Fig.7).

Plates and shells are also stable at loads below thecritical value. However, the post-bucklingbehaviour

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SHELLSND TANKS IN TH E AEROSPACE INDUSTRY 40 9BEHAVIOUR OF DIFFERENT ELASTIC STRUCTURES UNDE R COMPRESSION

B1 - p. . . . . . . .PLATE SHELLi'

. . .. . . ,a

Pw< PP: THEORETICAL BIFURCATION LOAD OF THE PERFECT SYS TEM

Fig. 7Stability behaviourofdifferent elastic structures under axialcompression

of these elements is quite different. The difference canbe illustrated at the critical load (neutral equilibrium)where a small displacement from the perfect axis stillprovides an equilibrium state. Under increasingdeflections, the load is approximately constant for acolumn, increases for a plate and decreases for acylindrical shell.higher loads than the bifurcation point, due to thegrowing transverse support of the stretched neutralplane. In experiments, it is often difficult to recognizethe bifurcation point.cylindrical shell drops dramatically. This behaviourcan be explained by the sinusoidal deformation of thecircular shape. Without deformation all points in thecross-sectionhave the same positive curvature. Thiscurvature increases for points which move outwardsand decreases for points moving inwards. For thelatter, the curvature can even change sign and 'snap-through' into a weaker equilibrium state.

The post-buckling behaviour of the three types ofstructural element explains the different compressionload-carrying characteristics. Real structural elementsare never perfect. In general, the behaviour of animperfect structure is not a stability problem any

In the post-buckling phase, the plate is able to carry

The post-buckling curve for a thin isotropic

Fig.8 The different load cases studied

more. With increasing deflection, the combinedstresses from compression and bending lead tostrength failure in columns and plates. Withoutmaterial failure, the deflection approaches the post-buckling curve asymptotically. In shells, snap-through usually occurs at a load far below bifurcation.

An imperfect column carries a little more, and anelastic plate much more, than the bifurcation load. Animperfect elastic isotropic shell (if not very short) failsby a snap-through far below the bifurcation load level.Real materials cause earlier failure due to yieldingand/or local rupture.as seen for example in an 'oil can'. A transverse loadmay initiate a deflection of the flat vault of the canwall and i t suffers 'snap-through' at a critical loadlevel.No equilibrium is possible above the maximumload (near this deformation state). If there is anotherequilibrium state with a different configuration, 'limitpoint' instability is connected with a 'snap-through'phenomenon.walled structures, this description is limited to linearbuckling analyses of cylindrical shells under axialcompression, external pressure and torsiodshear ,together with combinations of these loads.

Another type of instability is 'limit point' instability,

Due to the abundance of buckling problems in thin-

BUCKLINGF CYLINDRICAL SHELLS UNDER AXIAL LOADCylindrical shells can buckle under uniform axial loadin axisymmetric buckling and in a chess-boardbuckling mode.I so t rop i c shellsThe axisymmetric buckling of an isotropic cylindricalshell (Fig.9) can be represented mathematically by abeam on an elastic foundation. A generator of the shellof unit width may be treated as a beam, with thecircumferential extensional stiffness treated as an

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410 SHELLS ND CONTAINMENT STRUCTURES:a_

-1nit width!LN VN

Shellthickness s

IIN

Fig.9 Mod el of a beam on elastic foundation

Fig. I0Axis ymm etric and chess-board buckling of a cylindrical shellunder unifor m axial compressionelastic foundation.critical stress (the real buckling stress is much smaller)and the critical wavelength for axisymmetric bucklingare[9**,17**1:

(4)

The formulas used to calculate the theoretical

Es nRs$ i p jJX,C,th = 1= ( k )=R 31 vThese formulas are only valid for moderate lengths

and simply supported boundary conditions. The post-buckling behaviour of the axisymmetric buckling issimilar to that of the beam (ie without a dramaticreduction). By contrast, buckling into a chess-boardpattern is associated with a serious post-bucklingreduction. Chess-board pattern buckling (Fig.10)isonly observed in some cases. Large deformationsdevelop rapidly after buckling and the chess-boardpattern degenerates into diamond shapes. The criticalbifurcation stress of an isotropic cylindrical shell isidentical for the axisymmetric and chess-boardpatterns.To estimate the real buckling stress it is alsonecessary to consider the influence of initialimperfections. Different theoretical or empirical

models may be used, one of which is the efficientAlmroth formula:

-0.54qC r~ 9 9% with ~ 9 9%6.48(F) (5)The knock-down factor~ 9 9 %s commonly

defined. This factor arises from a statistical evaluationof several hundred laboratory tests and defines theload at which there is a 99%probability that a test willhave a higher buckling strength than this. Otherknock-down factors (eg~ 9 0 %nd ~50%) ay also beused for different probabilities of buckling:

-0.54 -0.54~ 9 0%8.76 ( ~ 5 0 % 11.86 ( (6 )

Other causes of a discrepancy between the real andtheoretical loads are as follows.0 Boundary conditions- re-buckling deformations: Poissons law (like- ccentric load application: beam-column effects- on-uniform load applicationplasticity (estimation)

imperfections)

0 Inelastic buckling, influence of the material

The influence of material plasticity can be included byusing the Engesser-Kdrmdn critical strain concept.Areduced modulus of elasticity is usually used:

Or tho t rop lc shellsA shell is orthotropic if the extensional and/orbending stiffnesses are different in the axial andcircumferential directions and there is no couplingbetween linear strains (E~, y )nd shear strains (E,~).Thin-walled cylinders which are densely stiffenedwith longitudinal stringers and flexible rings can beconsidered as orthotropic. They can be analysedassuming smearedstiffnesses.Symmetric compositelaminates are also orthotropic.

In aerospace, stiffened shells are often used.Generally, the shear-carrying-skin, the longitudinalstiffening stringers, and the circumferentialringstiffeners have centre lines which are not coincident:the orthotropy is thus eccentric.This effect has asubstantial influence on bifurcation loads for non-axisymmetric (chessboard)buckling load. Stiffenerslying outside the shear-carrying-skin are always moreefficient.characteristics which are listed in the box, next page.orthotropic cylindrical shell under axial compressioncan also be treated as a beam on elastic foundation.buckling stress and the critical wavelength inaxisymmetric buckling of an orthotropic cylindricalshell of moderate length and with simply supportedboundary conditions:

Orthotropic shellsshow some specialThe axisymmetric buckling analysis of an

The following formulas are found for the theoretical

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SHELLS ND TANKS IN TH E AEROSPACE INDUSTRY 41 ISome special features o f orth otrop ic shells

0 eccentric stiffeners: stiffeners outs ide the sheapcarrying skin are mor e effective

0 influence of boundary conditions: simply supportedshells(SS &SS3)- o infl uence for axially stiffened shells- ringstif fened shells: 50% strength reduction for

0 influence of t he boundary conditions: clamped shellsshear weak boundaries (eg SS I )

(C I &C3)- igher streng ths for axially stiffened shells- o influence for ring-stiffened shells

where s, , syare smeared shell thicknesses,Ex, Y reYoungs modulus in the axial and circumferentialdirection and J, is the smeared second moment of areain the axial direction.

Under axial compression, an isotropic cylindricalshell always has the same minimum bifurcation load,independent of the buckling mode (whetheraxisymmetric or chess-board). Orthotropic shellsgenerally have many different bifurcation loads fordifferent modes, and the critical load is usually foundby systematic study of potential modes. The chess-board minimum is often lower than the axisymmetricbifurcation load.approximate methods may be used, such as those ofvan der Neut[is**i,Dicksonci2*1 nd Tennyson[isl.Themethod of van der Neut is reviewed here. It is simpleand quick, but in this form ignores boundaryconditions.

Van der Neuts method uses smearedstiffnesses,which more careful analysis shows is valid if thebuckle half wavelength covers more than 1.6stiffeners(stringers or rings). However, the simple methodyields only the ratio of wavelengths in the twodirections, and if the actual wavelength must bechecked, a more sophisticated method is needed.the axisymmetric buckling force:

To estimate the lowest critical load, different

The orthotropic shell buckling load is found from

It is then possible to find the buckling load for thechess-board buckling mode:N,cr=qNx,cr,syrn

(9)

,/(Y +A +$)( +B + +)+F2 -Fwith q = (10)Y + A + Y

The terms in this expression can be written as:

(11)

The value of Y depends on the ratio of bucklingwavelengths (ly/lx), o different values must beexplored, but a first approximation for qmin an befound using Y =dC , following which minimization ofN,,,, with respect to (ly/lx) s simple.

As for an isotropic shell, the real buckling load isreduced by effects such as:0 influence of initial geometrical imperfections- maller influence for stiffened shells: use forexample Almroths formula0 boundary conditions- revention of pre-buckling deformations:- ccentric force introduction: beam column- on-uniform load applicationplasticity

Poisson effectseffects

0 inelastic buckling: influence of the material

If the ring spacing in the orthotropic stiffening is notvery dense, the stringers may buckle between rings,and this should be explored too. Aircraft fuselageshells must have a smooth outer surface,so thestiffeners are inside the skin. This requirement leads toa more complicated design and reduced bifurcationloads (outward eccentricities are more efficient).Outside stringers are allowed on launch rockets andhypersonic vehicles: however, outside rings areforbidden as they could cause shock waves and highlocal temperatures.BFig. I I Eccentrically stiffened ortho tropic cylindrical shell underuniform axial load

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4 t 2 SHELLS ND CONTAINMENT STRUCTURESThe single straight trapezoidal corrugation with

outside ring stiffeners is the most efficient orthotropicshell type, but it must be designed with care. The platesides of the trapezoidal corrugation must not bucklebelow the shell crippling load. This requirementapplies to all thin-walled stringers. One failure modefor these ring- and stringer-stiffened shells is cripplingunder stresses from combined primary and secondaryloads.

Primary axial loads are caused by axial force andbeam-bending moment together with membrane-warping stresses. Secondary stresses result from localbending of the stringers where loads are introducedeccentrically or from thermal stresses. Radialtemperature gradients bend stringers, but this isprevented by rings or the skin. Axial local bendingthermal stresses occur of the order of

For ultimate strength evaluation, only mechanicalstresses will be factored:

(13)The thermal gradient can be reduced by local thermalisolation of the top of the stringers (or corrugations), ifthey are exposed eg to kinetic heating.

The general instability of the shell is calculated, as afirst approximation, using smeared stiffnesses neither van der Neuts or Dicksons method. In general,the buckling chess-board-like pattern deforms therings as well as the stringers. The shell can bucklebetween the rings, remaining circular. This panelinstability corresponds to column buckling of thestringers, with an effective width of skin, between thecircular frames. Column buckling is calculated usingthe Euler-Johnsonprocedure for thin-walledcolumns[s**~,djusted for trapezoidal corrugatedsections. Three failure modes must be considered.0 General instability with stringer and ring0 Panel instability: buckling of the stringers between0 Local failure of the stringers due to crippling

deformationthe rings. The rings remain circular

In strict mathematical terms, the mass-optimalstructure could have the same instability load forgeneral buckling and panel-buckling modes. Inpractice, however, they should be separated toprevent a reduction of capacity through interactionbetween buckling modes, triggered by unavoidableimperfections and eccentricities.

As already mentioned, orthotropic stiffening notonly increases the bifurcation load, but also rendersthe shell less sensitive to initial imperfections. Wherestrong ring stiffeners are used, the axisymmetricbuckling strength of the perfect shell and the cripplingstrength of the stringers can be reached.

BUCKLINGF CYLINDRICAL SHELLS UNDER EXTERNALPRESSURELateral pressure alone means that only the cylindricalenvelope is under pressure and no axial compressionacts on the shell (this contrasts withhydrostaticloading).

To estimate the critical buckling pressure, severaltheoretical models can be used. One model, based onsemi-membrane theory solved using the Galerkinmethod[sv, leads to an expression for the theoreticalcritical pressure pcr,thand the circumferentialbuckling mode mcr,,, for an orthotropic shell withsimply supported ends (SS3) .In the axial direction,the buckling mode is usually a single half-wave:

(14)

For an isotropic shell the corresponding formulasare:

(15)Rm i r t h 7 c - 4L

These formulas are again valid for simplysupported boundaries and a moderate length shell.Different boundary conditions may be adopted usingDAST 013[111 or similar rules.

Under lateral loading, a cylindrical shell is muchless sensitive to imperfections than under axialcompression. According to Koiter[ic*~,he sensitivitydiminishes as the length decreases. The effect ofimperfections is usually estimated with a knock-down factor of 0.7.Several other effects, such as theboundary conditions and material plasticity alsoinfluence the buckling load.BUCKLINGF CYLINDRICAL SHELLS UNDERSHEA~TORSIONAs with the shell under external pressure, the stabilityof an orthotropic shell with simply SS3-supportedboundary conditions can be estimated using semi-membrane theory and the Galerkin method. Underpure torsion, the lowest buckling load has a bucklingmode with only one half-wave over the length butseveral circumferential waves. The phase angle of them circumferential waves changes with axialcoordinate, with buckle wave summits on a spiral linewith a helix angle8.This analysis leads to thetheoretical critical shear stress resultant, thecircumferential buckling mode, and the angle8 for anorthotropic cylindrical shell of moderate length:

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SHELLS ND TANKS IN THEAEROSPACE NDUSTRY 413Nxy,cr,th3.45DX -8J Fmcr,,,, =2.95 g/FL R*OD ; 'and for an isotropic shell:

(16)

(17)

Donne11 also provided semi-empiricalequations forthe theoretical critical shear stress resultant, thecircumferential buckling mode, and the angle 8 in anisotropic shell as follows:

1.25Nxy,cr,th =0'74Es t()

0.388 0.306m&, =2.36 (f) ( , (18)0.417 -0.208e o =62.38(") (

A 'knock-down-factor ' or initial imperfections of0.7 is again typically used.BUCKLINGF CYLINDRICAL SHELLS UNDER COMBINEDThe above covers the common buckling load cases foraerospace applications.A few comments are madehere on combined loads. Often the loads used above(axial compression, external pressure, torsion) do notoccur alone, but are combined. The combinations maylead to lower buckling loads than for each loadingalone.

Most existing numerical codes offer a simpleinteraction formula for the effect of combined loads.Where two loads interact, the formula could bewritten:

LOADS

Z? +zF=1 (19)whereZ represents the ratio of the 'critical load actingin combination' to the 'critical load acting alone' andk , kjare numerically or empirically defined powers.

Dunkerley proposed k i=ki=1,which is generallyconservative. Among others, the research group ledby Tennyson established analytical interaction curvesfor smooth thin-walled laminated orthotropic shells.They used a nonlinear elastic analysis and assumedsimply supported boundary conditions. Three simpleformulas are given here that offer a firstapproximation to the combined load interaction.

Lat eral pressure and tors io n:

where k could lie between 1.85 (according oTennysonris])and 2.5 using the semi-membranetheory and the Galerkin-method.Axia l compress ion and la tera l p ressure:

Ax i a l comp ress ion and tors ion:

(20)

(22)

ConclusionsThis article has presented an overview of differentdesign aspects of thin-walled shell structures foraerospace applications under different mechanicalloads.0 The main design criteria are low weight, high

buckling loads, high strength and low imperfectionsensitivity with damage tolerance and perhaps along service life.

0 Shells have a completely different post-bucklingresponse from beams or plates. The isotropic shellunder axial compression suffers a dramatic loss ofload capacity and ishighly sensitive to geometricimperfections.

0 Orthotropic stiffened shells are less sensitive togeometrical imperfections than isotropic shells.They permit designs in which the buckling load isclose to material strength of high performancematerials, even where the design is controlled bycompressive loads.

0 For orthotropic stiffened shells, general instability,panel instability and local failure (eg crippling)must all be investigated.

0 Simple formulas have been given to estimate boththe buckling load and buckling mode under axialcompression, external pressure, torsion andcombinations of these loads.

0 Final design analyses must be performed usinghigh performanceFEMcodes, such as NASTRAN,ANSYS, MARC, etc.

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414 SHELLS NDCONTAINMENT STRUCTURESReferences and recommended readingPapersofparticular interest have been marked* Special Interest+* Exceptlonal nterest

[ I] RlttwegmrA, Schermann Th, RelmerdesH-G&6ry H. Influenceof geometric imperfectionson the load capacityoforthotropic stlffened andcomposite shells of revolution with arbitrary merldlans and boundary conditions.ThiPWdkdS~IUCW*S I 9 9 5 23: 237-254.

[ I ] RlttwegetrA. Statik, Stabilitflt und EigenschwingungenanisotroperRotatlons-schalen bellebigen Meridians mlt der Obervagunpmatrizen-Methode,Dlssertatlon, RW TH Aachen, 1992.Auslegung belicbigerZylinderschalen. Dissertatlon,RWTHAachen. 1994.Schalenstrukturen-Allgemelne Grundglclchungen In analytischer Darstellungundlhre Anwendung. Dissertatlon, nstltut fUr Lelchtbau. RWTH Aachen. 1995.

I S ] D k k r 5. Statik. Stabilltflt und Elgenschwingungen der Torusschale unterbelleblgen Randbedlnyngen. Dksertatlon. RWT H Aachen, 1986.161 Dl eb rS.Hybrid analysis: couplingofthe flnlte-elcmente-method withanalytical approaches. European orumonAedastics andSvuctuml Dynamics.

Aachen. 17-19Aprll 1989. Paper89-053.elliptical cylindricalshells under combined mechanlcaland thermal load. PtuceedingsInternoclonalESA Confirence on SpocecmpStructures,Materials and MechankalTestng.Noordwijk, The Netherlands, 1996.Energleaufnahmwerhalten metallischerZylinderschalenunter axialerStoObelastung. DGLR-J ahrestagungDresden, Band111 1996.** [9] 6ry H. Structural designofaerospace vehlcles 11. Proceedings SPACECOURSE. Aachen, t 99 I.

131 Albur J. Analytische und semi-analytische Bemchnungsverfahrcnzur[4] MlermehterM. Statik und Stabllitatnlchtrotatlonssymmetrischer

[ t ] G6mezCarcb J, Albur J &Relmerdes H-G. Statlc and stabllityof

[S] GlmezGarciaJ, Marsokk &RelmerdesH-C.

A comprehenslve and accurate overview ofthe urrent numericalaswellasanalyticalapproachesand methods Inthe structural deslgnof aerospacevehicles.

H 6 r y Prof Dr-lng D r hc (H)Professor Emeritus, Departmentof Aerospace Structures:Lightweight Construction, lnstitut fur Leichtbau,RWTHAachen, WullnerstraOe 7,52062 Aachen, GermanyHans-G Reimerdes Prof Dr-lngHead, Department of Aerospace Structures: LightweightConstruction, lnstitut fur Leichtbau, RWTH Aachen,WullnerstraOe7,52062 Aachen, Germany

[01 BasarY&KratzlgWB.Mechanikder Fl~chentmgwerke.[I ] DAST Rlchllnle013.8eulsiche~eirsnachweisir Scholen. K6ln:[ 21 DlcksonJN&Bmllhr RM. The general nstabilityofring-stiffenedAnalogous to the work done by van der New[ 19q.thlspubllcatlon provldesan excellent overvlewofthe stability ofring-stiffened shells. A fast andrellable methodtodetermlne the critlcal buckllng oad was Introduced nthlsreport[ 31 FlUggeW Stresses n shdk. 2nd edition. Berlin: Springer-Verlag. 1973.

Braunschwelfliesbaden: Friedr. Vleweg&Sohn. 1985.Stahlbauverlag.1980.*corrugated cylinders under axial compression.NASA TN-D-3089.1966.

** [41 KolterWT.O n the stability of elastic equillbrlum. Thesis, Delft.Amsterdam: HJParis. 1945. (in Dutch).

A n excellent thesis dealing with llnear and nonllnear stablllty behavlourofstructures undertheconsideration of geometrical Imperfectlons.[I I] anghaarHL.Energymethods inapp/ied medronlcs.New York J ohn

Wiley. 1962.[ 61 SchnellW &EschenauerH. Elostizirijtstheorie. ZUrich BI-

Wissenschafts Verlag. 1993.** [ 71 Tlmoshenko SP &GereJM.Theoryofekasticstability.New YorkMcGraw-Hill. I9 8 I.

A brilliant bookcoverlnga wide rangeofthe basic theoretical models inthestability analysisofthin-walled structures.[S] TennysonRC et al. Effectof axisymmetric shape Imperfectionson the

bucklingof aminated anisotropic circular cylinders. Tmnsaaions CanadianAeronautics and Spoce Institute. University of Toronto. Vol5. I9 7 I. 3 -1 39.** [I 91 van der Neu t A. General nstabilityofstiffenedcylindrical shells underaxial compression. N ot Luchtworlab.13,Repon 53 14.The Netherlands 13. 1947.

A n excellent report, being the Rrst dealing with the stabllltyof orthotroplcshells under axlal load and Introducinga simple and fast method ortheestlmatlon ofthe buckling oads.

JesusG6mez Garcla Dipl-lngResearch Fellow, DepartmentofAerospace Structures:Lightweight Construction, lnstitut fur Leichtbau, RWTHAachen, WullnerstraOe7,52062Aachen, GermanyE-mail: [email protected]

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The Design of Shells and Tanks in the Aerospace Industry, Some Practical Aspects

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