NASA 98 Tp206290 Shell Stability

8/3/2019 NASA 98 Tp206290 Shell Stability

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NASA/ TP-1998-206290

The NASA Monographs on Shell Stability

Design Recommendations

A Review and Suggested Improvements

M ichael P. Nemeth and James H. Starnes, Jr.

Langley Research Center, Hampton, Virginia

January 1998


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NASA/ TP-1998-206290


Design Recommendations

A Review and Suggested Improvements

M ichael P. Nemeth and James H. Starnes, Jr.

Langley Research Center, Hampton, Virginia

January 1998


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Acknowledgments

The authors would like to express their thanks to Mrs. Vicki O. Britt, formerly of Langley Research Center, who con-ducted the experiments on the compression-loaded curved panels, and to Professor Johann rbocz of Delft Universityof Technology, for many useful discussions on the topic of this paper.


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Abstract

A summary of the existing NASA design criteria monographs for the design of

buckling-resistant thin-shell structures is presented. Subsequent improvements in the

analysis for nonlinear shell response are reviewed, and current issues in shell stabil-

ity analysis are discussed. Examples of nonlinear shell responses that are not

included in the existing shell design monographs are presented, and an approach for

including reliability-based analysis procedures in the shell design process isdiscussed. Suggestions for conducting future shell experiments are presented, and

proposed improvements to the NASA shell design criteria monographs are discussed.

Introduction

In the 1960s, the National Aeronautics and SpaceAdministration (NASA) experience with spacecraftdevelopment indicated a need for uniform design criteria.This need led to the development of a series of mono-graphs that provide design information and recommenda-tions in the areas of environment; material properties andprocesses; stability, guidance, and control; chemical pro-

pulsion; and structures. One of the structures mono-graphs, published in 1965 and revised in 1968, providesrecommendations for the design of buckling-resistantcircular cylindrical shell structures. This monograph isknown throughout the aerospace industry as NASASP-8007 (ref. 1). This monograph was followed in 1968by NASA SP-8019 (ref. 2), which gives recommenda-tions for the design of conical shells, and in 1969 byNASA SP-8032 (ref. 3), which gives recommendationsfor the design of doubly curved shells. These mono-graphs primarily emphasize the behavior of thin-walledmetallic shells subjected to axial compression, torsion,pressure, and bending loads, and to various combinations

of these loads. Prior to the publication of these mono-graphs, one of the most comprehensive collections ofshell stability information available was the series ofstructural stability handbooks written by Gerard andBecker (refs. 4 through 6). The NASA monographs usedand expanded the information provided in thesehandbooks.

The NASA structural stability monographs remainpopular among designers primarily because they addressone of the most important concerns associated withdesigning shells to satisfy stability requirements. Experi-ence has shown that large discrepancies often occur

between the classical shell stability analysis predictionsfor geometrically perfect shells and the correspondingresults from experiments. The NASA monographs pro-vide a reliable, but often overly conservative means ofdesigning shells by using simple, linear analytical mod-els and an empirical correction factor, referred to hereinas a knockdown factor. The format of the monographswas intended to satisfy the requirements of engineers andproject managers concerned with the preliminary designof spacecraft. However, the amount of information pre-

sented in the NASA monographs is somewhat limited,and as a result, their range of applicability to the designof high-performance shell structures, such as those madeof fiber-reinforced composite materials, is small.

Continued use of these NASA monographs by struc-tural designers and technical specialists, and recentNASA experience with the development of launch vehi-cles and aircraft structures have indicated that the mono-

graphs on shell stability need to be updated andexpanded. For example, the original NASA monographscontain practically no design information for lightweight,high-strength laminated composite shells subjected tomechanical or thermal loads. Such information could beused in the preliminary design of a high-speed civil trans-port aircraft or a single-stage-to-orbit reusable launchvehicle. The interest in updating the monographs is alsoinfluenced by the many advances in the state of the artof shell stability analysis that have taken place sincethe original monographs were published. Significantadvances in computer technology and computationalanalysis tools since the late 1960s have made it possible

to use much more sophisticated analytical models of non-linear shell response. These tools have also enabledin-depth investigations of the effects of complicatingstructural details such as cutouts and other discontinuitieson the buckling of shells and on their nonlinear behavior.In addition to advancements in analytical tools, manyadvancements have been made in experimental methodsand techniques. For example, technology is now avail-able to measure accurately the initial geometric imper-fections of shell test specimens, and new combined-loadtest capabilities have been developed and used to providemore carefully controlled experiments and higher fidelitytest results. Because of these technological advances andthe large body of experimental data that has beenamassed since the late 1960s, the development of mod-ern versions of the shell stability monographs is beingconsidered at Langley Research Center.

The present paper begins with a discussion of theapproach commonly used to design buckling-resistant,thin-walled shells and describes how the approachevolved. Then, an overview of the NASA monographson shell stability is given. Next, a discussion of some


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important issues that are presently confronting designersis presented, and two examples that illustrate some ofthese issues are described. The first example is the SpaceShuttle superlightweight external liquid-oxygen (LO2)tank. This contemporary thin-walled spacecraft structurewas partially designed by using NASA SP-8007. Thesecond example is a basic example that illustrates the

effect of cutout size on the buckling behavior of acompression-loaded curved panel. Both examples illus-trate shell behavior that is not addressed in the NASAmonographs. The present paper includes a brief discus-sion of a state-of-the-art nonlinear shell analysis codeand explains how it could be used to obtain a wide rangeof design information. In addition, a discussion of how toaddress design uncertainties and reliability in shelldesign is presented, and some suggestions for conductingfuture high-fidelity experiments are given. Finally,potential improvements to the NASA monographs onshell stability are discussed.

Common Approach to Stability Design

Prior to the late 1970s, the use of sophisticated ana-lytical methods, such as the finite-element method, wasnot widespread, and shell stability calculations weredone primarily with simple, specialized analytical mod-els. These analytical models were typically formulatedfor regular geometries with uniform properties, uniformloading conditions, and uniform boundary conditions,and certain aspects of the response were neglected inorder to obtain linear partial differential equations thatcould be solved readily. The simple analytical modelstypically neglected nonlinear prebuckling deformations,

and simply supported boundary conditions were oftenused to reduce the computational effort needed to con-duct parametric studies. This linearmodeling approach,referred to more accurately as a linear bifurcation buck-ling analysis, came into use not only because of the com-putational considerations mentioned above, but also asthe natural extension of the linear bifurcation bucklingapproach that had been used successfully for modelingcolumns and plates. Gradually, scientists and engineerslearned that the buckling behavior of shells is fundamen-tally different from that of columns and plates.

The fundamental difference between the buckling

behavior of columns and plates and the buckling behav-ior of shells was identified by von Krmn and Tsien(ref. 7) and was clarified by Donnell and Wan (ref. 8) andby Koiter (ref. 9). These references show that a majorreason for the large discrepancy between the analyticalpredictions of shell buckling behavior and the corre-sponding experimental results is a sensitivity of shellbuckling to initial geometric imperfections. Thissensitivity was shown to be a consequence of the fact thatshells are typically unstable at load levels equal to the

bifurcation load. Because of the practical limitations ofthe analytical models and the sensitivity of shells to geo-metric imperfections, a stability design process evolvedin which empirical knockdown factors were introducedto compensate for the differences observed between theresults of theory and experiments. As part of this designprocess, a designer was faced with the need to conduct

expensive experiments.


By 1960, many buckling tests of isotropic cylindersand curved panels had been conducted (e.g., see refs. 4,5, and 6) as part of an effort by the technical communityto establish a rational, practical approach for designingbuckling-resistant shells. At that time, NASA conceivedthe shell stability monographs to make the results ofthese tests and many proposed tests for other shellgeometries available to the aerospace structural designcommunity and to establish practical design recommen-

dations. The development of these monographs was acombined effort by members of industry, academia, andLangley Research Center. Much of the information givenin these monographs is based on the research conductedby Seide, Weingarten, and Morgan (ref. 10). The initialemphasis on cylinders and cones and the format of themonographs were originally intended to satisfy the needsof engineers and project managers concerned with thepreliminary design of launch vehicles and spacecraft.However, over time, it became evident that the mono-graphs were also of great interest to structural stabilityspecialists. The use of NASA SP-8007 was recentlydemonstrated in the shell analysis textbook by Vinson

(ref. 11).

The NASA monographs provide design informationin the form of empirical knockdown factors (referred toin the monographs as correction factors) and designrecommendations for isotropic, orthotropic, ring- andstringer-stiffened, and sandwich shells. The importantcharacteristics of various shell design problems, thesources of the design recommendations and their limita-tions, and discussions of how to proceed for cases withlittle known analytical and experimental data are alsopresented. In most cases, the knockdown factors aredefined as empirical corrections to linear bifurcation

buckling solutions for primarily elastic, simply supportedshells. The knockdown factors are lower bounds toexperimental data that were available at that time and areused to account for the large amount of scatter in thedata. The knockdown factors consist of corrections thatprimarily account for initial geometric imperfections,nonlinear prebuckling effects associated with edgesupports, and plasticity in some cases. The effects ofedge boundary restraints (e.g., a simply supported versusa clamped boundary condition) are included in the


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knockdown factors so that edge restraints are treated as arandom effect, in addition to the initial geometric imper-fections. Plasticity correction factors are given only forcases in which there was a sufficient amount of data tocharacterize the behavior in a conservative manner. Thebasic recommendation given in the monographs is thatany knockdown factor used for a design be substantiated

by experiments. This recommendation applies for shelldesigns in which the restraint or boundary conditions areto be accounted for more accurately, or for designs withunusual surface geometries, modal interaction associatedwith optimization, cutouts, joints, or other irregularities,or where there are little or no test data and analyticalresults. A brief overview of the contents of each mono-graph follows.

NASA SP-8007 (1968 Revision)

The 1968 revision of NASA SP-8007 consists pri-marily of discussions of research studies and design rec-ommendations for elastic, isotropic, cylindrical shells.However, some information is provided for orthotropicand sandwich cylinders. Design recommendations arepresented for isotropic cylinders subjected to axial com-pression, pure bending, uniform lateral pressure, uniformhydrostatic pressure, torsion, and combined loading con-ditions. The uniform lateral pressure loading conditiondoes not include the compressive axial load caused bypressure acting at the ends of a cylinder. In contrast, theuniform hydrostatic pressure loading condition includesthe lateral pressure load and the compressive axial load.Design recommendations for cylinders that are subjected

to combined loading conditions are limited almostentirely to isotropic shells. The combined loading condi-tions consist of axial compression and pure bending;axial compression and lateral pressure or hydrostaticpressure; axial compression and torsion; internal pressureand axial compression; internal pressure and pure bend-ing; and internal pressure, axial compression, and purebending loads.

Design recommendations and buckling formulas thatare lower bounds to experimental data for a wide rangeof radius-to-thickness ratios are given for isotropic cylin-ders subjected to axial compression or pure bending

loads. For cylinders loaded by lateral or hydrostatic pres-sure, a single knockdown factor, which is a lower boundto the corresponding experimental data, is given forshells that buckle with more than two circumferentialwaves. An additional empirical knockdown factor isgiven for long shells that buckle into a one-half-waveoval shape. For torsion loads, a single knockdown factorthat is a lower bound to the corresponding experimentaldata is given for moderately long cylinders. Because oflimited experimental verification, design recommenda-

tions are given in the form of conservative, linear buck-ling interaction equations for shells subjected tocombined axial compression and pure bending loads,combined axial compression and lateral pressure loads orhydrostatic pressure loads, and combined axial compres-sion and torsion loads. For shells subjected to combinedinternal pressure and axial compression or combined

internal pressure and pure bending loads, the bucklingload is expressed as a combination of the load caused bythe internal pressure, the buckling load for the unpressur-ized shell (including the appropriate knockdown factor),and an increase in the buckling load associated with thereduction in imperfection sensitivity caused by the inter-nal pressure. Empirically determined increases in thebuckling load, which are associated with the reducedimperfection sensitivity, are given for moderate ranges ofinternal pressures and radius-to-thickness ratios. Conser-vative, linear buckling interaction equations are alsogiven for shells subjected to combined internal pressure,axial compression, and pure bending loads.

Results are also presented in NASA SP-8007 forelastic, orthotropic cylindrical shells subjected to axialcompression, pure bending, uniform hydrostatic pres-sure, uniform lateral pressure, or torsion loads, and tocombined axial compression and bending loads. Theterm orthotropic is used to indicate single-layer andmultilayer composite monocoque shell wall construc-tions and stiffened shell wall constructions for which therings and stringers are perpendicular. These results con-sist primarily of design recommendations because of thesmall amount of experimental data for orthotropic cylin-ders that was available at the time. Formulas for comput-ing homogenized (smeared) elastic, orthotropicstiffnesses for multilayered stiffened cylinders, isotropicstiffened cylinders, and ring-stiffened corrugated cylin-ders are presented.

An empirical formula for knockdown factors is pre-sented for monocoque orthotropic cylinders loaded byaxial compression. This formula is based on a smallamount of experimental data and has a very limited rangeof validity. A similar formula is given for cylindersloaded by pure bending. A single knockdown factor,which is based on a small amount of experimental data,

is given for cylinders that are subjected to axial compres-sion or pure bending loads and that have closely spaced,moderately large stiffeners. A single knockdown factorthat is also based on a small amount of experimental datais suggested for cylinders loaded by lateral or hydrostaticpressure or by torsion loads. In addition, because of asmall amount of experimental data, a conservative, linearbuckling interaction formula is suggested for use withcylinders loaded by combined axial compression andpure bending loads.


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Design recommendations for sandwich cylinderswith isotropic face sheets and with either an isotropic oran orthotropic core are also presented in NASA SP-8007.Design recommendations are given for shells loaded byaxial compression, pure bending, uniform lateral pres-sure, or torsion loads. Knockdown factors are given onlyfor shells with cores that have high transverse shear stiff-

ness, and practically no experimental validation isdescribed.

Analytical results and design recommendations arealso presented in NASA SP-8007 for isotropic cylindri-cal shells that have an elastic core and that are subjectedto axial compression, uniform lateral pressure, or torsionloads, or to combined axial compression and lateral pres-sure loads. Based on experimental data, the knockdown-factor formula given for compression-loaded cylinderswithout an elastic core is recommended for use withcylinders that have an elastic core. For cylinders loadedby lateral pressure, a single knockdown factor is given

that is a lower bound to the corresponding experimentaldata. For the cylinders loaded by torsion, only design rec-ommendations are given. Similarly, a conservative linearbuckling interaction formula is recommended for cylin-ders loaded by combined axial compression and lateralpressure loads.

NASA SP-8019

NASA SP-8019 consists primarily of design recom-mendations for elastic, isotropic, conical shells subjectedto axial compression, pure bending, uniform hydrostaticpressure, torsion, or combined loads. The design recom-

mendations for cones subjected to combined loads aregiven for isotropic shells only. The combined loads con-sist of internal pressure and axial compression; internalpressure and pure bending; axial compression and purebending; internal pressure, axial compression, and purebending; uniform hydrostatic pressure and axial com-pression; torsion and uniform hydrostatic pressure; andtorsion and axial compression.

Design recommendations and a single empiricalknockdown factor that is a lower bound to experimentaldata are given for each of the single-component loadingconditions. Only conservative design recommendationsbased on rational arguments are given for loading condi-tions that consist of combined internal pressure and axialcompression and combined internal pressure and purebending because of the very small amount of experimen-tal data and the lack of analytical results that were avail-able at the time. Conservative, linear buckling interactionequations based on experimental results are given for allother combined load conditions.

Results are also presented in NASA SP-8019 forelastic, orthotropic conical shells (constant-thickness

orthotropic material and stiffened shells) subjected touniform hydrostatic pressure or to torsion loads. Theseresults consist primarily of design recommendationsbecause of the very small amount of experimental datathat was available at the time. Similarly, only design rec-ommendations are given for sandwich cones with isotro-pic or orthotropic face sheets and with either an isotropic

or orthotropic core.

NASA SP-8032

NASA SP-8032 consists primarily of discussions ofresearch studies and results for elastic, isotropic, doublycurved shells. Design recommendations are given forspherical caps that are loaded by uniform external pres-sure, by a concentrated load at the apex, or by a combina-tion of these loads. Buckling formulas that are lowerbounds to experimental data are given for clamped spher-ical caps that are loaded by uniform external pressure orby a concentrated load at the apex. A lower-bound,

empirical buckling formula is given for spherical capsthat are loaded by a concentrated load at the apex andthat have edges that are free to rotate and to expand in thedirection perpendicular to the axis of revolution. No con-clusive experimental results are given for spherical capsthat are loaded by combined uniform external pressureand a concentrated load at the apex.

Design recommendations are also discussed forcomplete prolate and oblate spheroidal shells subjectedto uniform external pressure and for complete oblatespheroidal shells subjected to uniform internal pressure.A single knockdown factor is given for the prolate sphe-

roidal shells, and a lower-bound, empirical buckling for-mula is given for the oblate spheroidal shells. Noexperimental validation is given for the results for theoblate spheroidal shells subjected to uniform internalpressure. Design recommendations are also discussed foroblate spheroidal and torispherical bulkheads that haveclamped edges and that are subjected to uniform internalpressure. An empirical knockdown factor is given for thetorispherical bulkheads; however, no experimental vali-dation is given for the oblate spheroidal bulkhead.

Design recommendations are discussed, and resultsare given for complete circular toroidal shells subjected

to uniform external pressure, and for shallow, equatorialsegments of complete toroidal shells. The toroidal shellsegments, which consist of barrel-shaped shells that arebowed outward from the axis of revolution (positiveGaussian curvature) and waisted shells that are bowedinward (negative Gaussian curvature), are subjected toaxial tension, to uniform lateral pressure, or to uniformhydrostatic pressure loads. Experimentally verified ana-lytical results are given for complete circular toroidalshells for a small range of geometric parameters.


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Similarly, an experimentally verified knockdown factoris given only for equatorial segments of toroidal shellsthat are loaded by axial tension and that are truncatedhemispheres.

Essentially no experimentally validated design infor-mation is given for orthotropic shells or for sandwich

shells that are doubly curved. Rational arguments areused to present design recommendations for speciallyorthotropic shells due to the absence of experimentaldata. No design recommendations are given for sandwichshells.

Shell Stability Issues

To adequately design a lightweight, buckling resis-tant, thin-walled shell structure, designers must under-stand several important shell stability issues, most ofwhich are not addressed in the NASA monographs.Some of these issues are listed as follows, and a few are

discussed subsequently.Initial geometric imperfectionsNonlinear prebuckling deformationsCutouts and jointsBoundary conditionsLoad introduction effectsThickness variationsVariation in material propertiesStiffener spacingLocal reinforcementCombined loadsVariation of loads with time

Small vibrationsLaminate constructionTransverse shear deformationSandwich constructionInelasticity and damageLocal eccentricities

Initial Geometric Imperfections

Sensitivity to initial geometric imperfections and theeffects of nonlinear prebuckling deformations are twomajor issues in the design of isotropic shells. Experiencehas shown that initial geometric imperfections with a

maximum amplitude on the order of one wall thicknesscan cause a reduction in the buckling load of a shell thatis on the order of 60 percent of the buckling load calcu-lated for the corresponding geometrically perfect shell.Thus, designing a minimum mass shell structure to bebuckling resistant is a difficult task because a designerusually does not know the initial geometric imperfectionshape and amplitude in advance. Because of this lack ofknowledge, an assumed imperfection shape must be usedto determine analytically a knockdown factor, or the

design must be based on a knockdown factor thatcorresponds to the lower bound to the known relevantexperimental data. Often, these data do not exist. In somecases, however, the shell manufacturing process mayconsistently produce a known imperfection shape with aknown maximum amplitude. If so, this information canbe used to determine a knockdown factor analytically.

Nonlinear Prebuckling Deformations

Nonlinear prebuckling deformations of shells aregenerally caused by the interaction between the compres-sive stresses in a shell and any localized bending defor-mations that arise, for example, from support conditionsor from discontinuities in stiffness that are caused byabrupt changes in thickness or joints. The significance ofthe nonlinear prebuckling deformations was first identi-fied by Stein for compression-loaded isotropic cylinders(refs. 12 and 13). As an isotropic cylindrical shell is com-pressed axially, it expands outward radially. At the sup-

ported edges, however, the radial expansion is restrained,which produces local bending deformations whose extentalong a generator depends on the cylinder radius andthickness. A similar condition exists for compression-loaded isotropic truncated conical shells where the extentof local bending deformations along a generator alsodepends on the vertex angle. Generally, as a cone getsflatter, the extent of the boundary bending deformationsgrows. The local bending deformations that occur arounda relatively large cutout in a compression-loaded cylinderor curved panel are another example of nonlinearprebuckling deformations. These bending deformationsare manifested by the coupling between the in-plane and

out-of-plane displacements in the strain-displacementrelations for curved panels or shells.

A very important consequence of substantial nonlin-ear prebuckling deformations is that a linear bifurcationsolution and a knockdown factor may be inadequate anduncharacteristic of the actual nonlinear response. Onesimple example of this deficiency is illustrated by thebehavior of a ring-stiffened cylindrical shell loaded byaxial compression or by external pressure (refs. 14and 15). For these shells, a linear bifurcation analysismay not only overpredict the buckling load, but may alsopredict an incorrect buckling mode. Another, more

complicated example is presented in reference 16 for theSpace Shuttle superlightweight LO2 tank shown infigure 1 and is discussed in the Examples section.

Cutouts

The effects of a cutout on the buckling behavior of ashell are another important shell stability issue fordesigners. The presence of a cutout may significantlyalter the prebuckling stress distribution in a shell,


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depending on the type of loading and the cutout size, andmay reduce its buckling load significantly. In addition,nonlinear prebuckling deformations that are local bend-ing deformations near the cutout, may be present and cansignificantly affect the characteristics of the bucklingbehavior. A cutout may also have a significant effect onthe imperfection sensitivity of a shell because as the cut-

out size increases, the amount of material removed by thecutout region, where imperfections may be very impor-tant, is reduced. Some effects of cutouts on the behaviorof compression-loaded curved panels are also discussedin the Examples section.

Laminate Construction

Approximately 25 years ago, researchers realizedthat there is a great potential for reducing structuralweight by using fiber-reinforced composite materials forstructures. The increased use of composite materials for

shell structures has led to additional shell stability issuesfor designers. For example, the effects of laminate con-struction (including sandwich construction) and trans-verse shear deformations on imperfection sensitivity arenot well understood. Transverse shear flexibility tends toreduce the effective stiffness of a structure and canreduce its buckling load. Similarly, knowing that lami-nated shell wall construction can greatly affect the atten-uation length of bending deformations implies that theeffects of nonlinear prebuckling deformations may besevere for some laminate constructions.

Examples

The common approach to stability design describedpreviously in the present paper is often used by industryin the preliminary design of shell structures. However, insome cases, the results of a linearized stability problemmay not adequately represent the underyling physics ofthe actual response. Two examples that illustrate thispotential pitfall are presented in this section. The firstexample is the Space Shuttle superlightweight LO2 tank.This example of a contemporary thin shell structure thatis subjected to combined loads illustrates complexnonlinear behavior that is dominated by local bending

deformations. The second example is a much simplersubcomponent-level example, that is, a compression-loaded curved panel with a cutout. Because cutoutsappear in nearly every kind of aerospace vehicle struc-ture, designing properly for their effects on the bucklingresistance of shells is very important. These two exam-ples illustrate some physical behaviors that are notcommonly understood and that are representative ofproblems that are dominated by effects that are currentlynot addressed in the NASA monographs.

Space Shuttle Superlightweight LO2 Tank

The Space Shuttle consists of the orbiter, two solidrocket boosters (SRBs), and the external tank (ET), asshown in figure 1. The external tank consists of a LO2tank, a liquid hydrogen (LH2) tank, and an intermediatestructure called the intertank (fig. 1). Currently, NASA is

engaged in the flight certification of a newly designedLO2 tank that is referred to as the superlightweight LO2tank. This new LO2 tank is significantly lighter than theone presently in service, and its buckling behavior is asignificant concern in its design. The superlightweightLO2 tank is a thin-walled monocoque shell that is madeprimarily of 2195 aluminum-lithium alloy. It consists ofa nose cone, a forward ogive section, an aft ogive sec-tion, a cylindrical barrel section, and an aft ellipticaldome section, as shown in figure 1. The intertank (fig. 1)is a right circular cylinder that is made from 2090 and7075 aluminum alloys. Details and dimensions of theLO2 tank and the intertank are given in reference 16.

An important loading condition that is illustrated bythis example is the prelaunch loading condition for whichthe LH2 and LO2 tanks are full. Compressive stresses arepresent in the ogive sections of the (monocoque) LO2tank directly above the solid rocket booster attachmentpoints for this loading condition. These compressivestresses are caused by the weight of the filled LH2 andLO2 tanks that is reacted at the two SRB attachmentpoints. Both linear bifurcation and nonlinear analyses arepresented in detail in reference 16. These results, whichwere obtained by using the Structural Analysis ofGeneral Shells (STAGS) nonlinear structural analysis

code (ref. 17), are described briefly as follows.The linear bifurcation solution yields a critical buck-

ling load factor ofpa = 3.78, where a value ofpa = 1.0corresponds to the magnitude of the operational loads.The corresponding buckling mode is shown in figure 2and consists of a short-wavelength buckle in the forwardpart of the aft ogive that is essentially a wrinkle in theskin. The shortness of the wavelength is caused by thehoop tension that resists the LO2 pressure.

Results of nonlinear analyses presented in refer-ence 16 are reproduced in figures 3 and 4. The solid linesshown in figure 3 represent the normal displacements

along the length of the aft ogive shell wall for values ofthe applied load factor pa approximately equal to 3.0,4.0, and 5.0. Overall, negative values of the normal dis-placements are indicated by the left-hand-side ordinatefor these three lines because of contraction of the aftogive that is caused primarily by the LO2 thermal load.The linear bifurcation mode is represented in the figureby the dashed line with the normalized amplitude givenby the right-hand ordinate of the figure. The solid linesshown in figure 3 indicate a short-wavelength bending


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response in the aft ogive over the SRB attachment point(fig. 2) that is similar in shape to the corresponding linearbifurcation mode shape. The overall slope of the solidlines (obtained by fitting a straight line to each curve) is aresult of outward displacements of the shell wall (indi-cated by less negative values) that are caused by theinternal pressure and that are represented by a nonlinear

analysis. This effect is not represented in the prebucklingstress state that is used in a linear bifurcation bucklinganalysis and, as a result, does not affect the overall slopeof the dashed line.

The results presented in figure 3 predict a stable non-linear response at load levels greater than the bucklingload predicted by a linear bifurcation analysis. As theapplied load increases, substantial bending deformations(indicated by the waviness of the curves) develop andgrow in the shell wall. These bending deformationsreduce the apparent meridional stiffness of the aft ogive.The nonuniformity of the bending deformations is caused

by thickness variations in the ogive and the presence ofcircumferential weld lands. Similar results are presentedin reference 16 which indicate that a geometric imperfec-tion with a small negative amplitude and with the shapeof the linear bifurcation mode greatly increases the sever-ity of the stable bending deformations. This imperfectioncauses the growth of the bending deformations to beginat much lower load levels than the linear bifurcationbuckling load.

The reduction in the apparent meridional stiffness ofthe aft ogive is shown more explicitly in figure 4. In thisfigure, the intensities of the largest bending deformations

(indicated by the largest magnitude of the normal dis-placement amplitude) for the geometrically perfect shelland a geometically imperfect shell are given as a functionof the load factorpa. The amplitude w shown in figure 4is the distance from the maximum value of the shell-walldisplacement to the adjacent minimum value andrepresents the intensity of the local bending deformationin the response. The filled circles in the figure corre-spond to results for a geometrically perfect shell, and theunfilled squares correspond to results for geometricallyimperfect shells with an imperfection-amplitude-to-wall-thickness ratio ofA/t= 0.3 (t= 2.540 mm (0.100 in.)).The horizontal dashed line in the figure represents thelinear bifurcation buckling load level.

The results presented in figure 4 indicate that theamplitude of the greatest local bending deformationgrows with increasing load and that the amount ofgrowth increases substantially with increasing geometricimperfection amplitude. The results predict that the shellcan support loads greater than the critical buckling loadpredicted by the linear bifurcation analysis. Mostimportantly, the results show that the linear bifurcation

analysis does not represent accurately the mechanics ofthe actual shell response. Moreover, a design based onthe linear bifurcation analysis and a knockdown factorthat was determined by using an intuitive approach likelywould be overly conservative.

Compression-Loaded Curved Panel With a

Cutout

Several tests of compression-loaded 6061-T6 alumi-num singly curved panels with a central circular cutoutwere conducted at Langley Research Center. The panelshad a nominal radius of curvature ofR = 152.4 cm(60 in.) and a nominal thickness of t = 2.54 mm(0.10 in.). The length and arc-width of the panels wereapproximately 37.47 cm (14.75 in.) and 36.83 cm(14.5 in.), respectively. The panels were loaded slowly inaxial compression by uniformly displacing the two oppo-site curved edges with a 1334-kN (300-kip)-capacityhydraulic testing machine. The loaded ends of a panel

were clamped, and the unloaded edges were simply sup-ported by a test fixture. The length and arc-width of thepanels between the inside edges of the test fixture(unsupported area) were both 35.56 cm (14.0 in.). Elec-trical resistance strain gauges were used to measurestrains, and direct current differential transformers wereused to measure axial displacements and displacementsnormal to the panel surface. Shadow moir interferome-try was also used to monitor displacements normal to thepanel surface.

Experimental results for load versus end shorteningare presented in figure 5. The load is nondimensionalized

by the linear bifurcation buckling load for a panelwithout a cutout = 62,988 N (14,161 lb) thatwas obtained from STAGS. This buckling load is basedon a length L = 35.56 cm (14.0 in.), an arc-widthW= 35.56 cm (14.0 in.), a nominal thickness oft= 2.54mm (0.1 in.), a Youngs modulus ofE = 72.4 GPa(10.5 106 psi), and a Poissons ratio of = 0.33. Theend-shortening is nondimensionalized by the nominalpanel thickness t. The dashed line in the figure corre-sponds to a panel without a cutout, and the solid linescorrespond to panels with cutout-diameter-to-panel-width ratios d/W= 0.3, 0.4, and 0.5.

The experimental results presented in figure 5 indi-cate that the character of the nonlinear response of apanel changes significantly as the cutout size increases.For example, the results indicate that the panels withd/W= 0 and 0.3 exhibit buckling behavior that involves adynamic change from one stable equilibrium configura-tion to another. Similar results, not shown in the figure,were obtained for panels with d/W = 0.1 and 0.2.The results in figure 5 also indicate that the panelswith d/W = 0.4 and 0.5 do not exhibit this type of

Pbi fo


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behavior but exhibit stable, monotonically increasingnonlinear responses. The results show that the intensityof the dynamic buckling process decreases substantiallyas d/Wincreases from a value of 0 to 0.3. The intensity ofthe dynamic buckling response is indicated by the differ-ence between the buckling load and the lowest stablepostbuckling load.

The results presented in figures 6 through 9 provideadditional insight into the effect of cutout size on thecharacter of the nonlinear response. The results in thesefigures are shadow moir patterns on the convex or outersurface of the panels. The shadow moir patterns for thepanel without a cutout are shown in figure 6 for values of

= 0.86 ( just before buckling) and 0.57 ( just afterbuckling). The top pattern in figure 6 indicates that nosignificant nonlinear prebuckling deformations arepresent. This finding is consistent with the straightness ofthe initial portion of the dashed line shown in figure 5.The bottom pattern in figure 6 indicates that the stable

postbuckling mode shape consists of a single half-wavealong the panel length and across the panel width. Theradial displacements of this postbuckling mode areinward.

Shadow moir patterns for the panel with a cutoutwith d/W = 0.3 are shown in figure 7 for values of

= 0.72 (just before buckling) and 0.67 (just afterbuckling). The top pattern in figure 7 indicates that sig-nificant nonlinear prebuckling deformations occuraround the cutout, which is consistent with the deviationfrom straightness of the initial portion of the solid lineshown in figure 5 for d/W= 0.3. The radial deformations

around the cutout are outward. The bottom pattern in fig-ure 7 indicates that the stable postbuckling mode shapeconsists of an outward deformation pattern on the left-hand side of the cutout, similar to the nonlinear prebuck-ling deformation pattern shown on the left side of the toppattern in the figure, and an inward buckle on the right-hand side of the cutout. This buckle consists of approxi-mately a single half-wave along the panel length andacross the panel half-width.

Shadow moir patterns for the panel with a cutoutwith d/W = 0.4 are shown in figure 8 for values of

= 0.46 and 0.71. The patterns in figure 8 and thecorresponding curve in figure 5 indicate that significantoutward nonlinear prebuckling deformations around thecutout dominate the response. There is no dynamic buck-ling response for this panel. Similarly, the shadow moirpatterns for the panel with a cutout with d/W= 0.5 thatare shown in figure 9 for values of = 0.50 and0.70, and the corresponding curve in figure 5 indicatethe same type of response.

In summary, this simple example illustrates aresponse for compression-loaded curved panels that is

typically not well understood, is not considered bydesigners, and is not addressed in the NASA mono-graphs. The response trends change with loading, bound-ary conditions, and material systems, such as a laminatedcomposite system. How these trends affect the cutoutsize at which the response changes its character is gener-ally unknown. Information of this kind would be a valu-

able contribution to an updated shell design monograph.

Concept for New Design Recommendations

Development of new, expanded versions of theNASA monographs is now possible because of signifi-cant technological advances and advances in the under-standing of shell stability. In particular, advances incomputers and analysis tools have increased greatly theability to solve complex shell stability problems. Thus, abrief description of the capabilities of an advanced, state-of-the-art analysis tool that could be used to obtain awide range of analytical results that could be included in

expanded versions of the NASA monographs is pre-sented in this section.

Before embarking on an endeavor to revise theNASA monographs, a two-part question remains to beaddressed; that is, What kind of an approach to stabilitydesign should be used, and how should problem uncer-tainties be addressed? A basic, first-approximationanswer to this question is suggested later in this section.The approach is based on the premise that many of theshell response parameters are not necessarily probabilis-tic in nature and that a completely probabilistic approachmay tend to obscure the physical understanding of

behavior. Thus, a hybrid approach to shell stabilitydesign is under consideration and will be discussedbriefly in this section.

Another major consideration in the formulation ofnew design recommendations for a revised set of NASAmonographs is experimental testing. With shell bucklingbehavioral trends established analytically, selectiveexperiments can be identified and conducted to establishcredible design recommendations. This selective testingapproach, made possible by advanced analysis tools, isparticularly important when considering the costs of con-ducting experiments and the costs of test specimens such

as those made of fiber-reinforced composite materials.Moreover, to establish the best possible design recom-mendations, it is imperative to use high-fidelity experi-mental results. This step is necessary to prevent theintroduction of excessive conservatism through the useof poor-quality experimental results. Some suggestionson how to obtain high-fidelity experimental results arealso given in this section. Finally, some specific sugges-tions for improving the NASA monographs arepresented.

P/Pbi fo

P/Pbi fo

P/Pbi fo

P/Pbi fo


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Capabilities of an Advanced Analysis Tool

Advances in the finite-element method during thelast 15 years have improved the capability for analyzingcomplex nonlinear shell problems and for obtainingaccurate buckling and nonlinear response predictions.For example, an advanced, state-of-the-art structural

analysis code has been used to conduct in-depth nonlin-ear analyses of the Space Shuttle superlightweight LO2tank (refs. 16 and 17). This code was chosen for analyz-ing this problem because of its robust state-of-the-artnonlinear-equation solution algorithms and its generaluser-input capability that is convenient for modelingbranched shells typically used for launch vehicles. Thecode uses both the full and modified Newton methods toobtain an accurate nonlinear solution, and large rotationsin a shell are represented by a co-rotational algorithm atthe element level. The Riks arc-length projection methodis used to continue a solution past limit points, and theThurston (ref. 18) equivalence transformation processor

is used for solution-branch switching in the vicinity of abifurcation point. The code also permits complex geome-tries, loading conditions, boundary conditions, and initialgeometric imperfections to be modeled in a direct man-ner by using user-written subroutines. These subroutinesare essentially independent of the mesh discretizationand provide analysts with a great deal of flexibility formodeling complex structural configurations (e.g., seeref. 16) and conducting mesh refinement studies.

Advanced analysis tools with the capabilities men-tioned above make it possible to determine accurate ana-lytical estimates of the sensitivity of a shell buckling loadto initial geometric imperfections or other destabilizingirregularities. Thus, state-of-the-art nonlinear shell analy-sis codes can be used to establish shell buckling behav-ioral trends deterministically for a wide range of systemparameters and to identify any unusual, possibly unex-pected nonlinear behavior that designers should consider.

Basic Approach to Stability Design

Modern, high-fidelity nonlinear shell analysis codes,such as STAGS, have enabled accurate predictions of thenonlinear response and buckling loads of thin-shell struc-

tures. The response of a shell can be determined accu-rately when its dimensions and properties are known tosufficient precision. For example, the effects of initialgeometric imperfections can be dealt with by measuringthe true shape of the shell and by modifying the shellanalysis model to represent the true measured geometry.Such deterministic analyses are valuable for identifyingand isolating important contributions to the nonlinearresponse and for systematically quantifying the effects ofchanges in structural and material design parameters.

The reliability of current shell design procedures canbe improved by using these more accurate deterministictools, provided that accurate information on the dimen-sions and material properties is available. If some dimen-sions and properties are not well known, however, itshould be possible to modify the design process toinclude such uncertainties. By coupling a probabilistic

representation of uncertain dimensions, tolerances, andmaterial properties with a deterministic analysis thatincorporates the better-known parts of the design prob-lem, a hybrid design process could be developed. A typi-cal result of the process might be a stiffened shell with aprescribed buckling load, complete with a rationallyobtained confidence interval. The hybrid approach couldalso serve as the basis for a reliability-based designprocedure.

Suggestions for Future Experiments

The determination of meaningful knockdown factors

for shell buckling depends greatly on high-fidelity exper-imental results. Some of the scatter in the post-1930stest data for buckling loads of isotropic cylindrical shellscan be attributed to nonuniform load introduction or to apoor simulation of the boundary conditions by the testfixture. When questionable test results are used to deter-mine knockdown factors from lower bound curve fitapproximations to the test data, the knockdown factor islikely to be overly conservative. Thus, it is very impor-tant to know the pedigree of a given set of test data.

To obtain high-fidelity experimental results, severalissues must be addressed and several tasks must be per-

formed. Prior to conducting an experiment, initial geo-metric imperfections of the shell surface, the wallthickness distribution, unevenness of the loaded edges,and the material properties should be measured. Knowl-edge of these quantities is extremely important forobtaining good correlation between theory and experi-ment. The instrumentation for a test should be plannedadequately to facilitate the correlation between theoryand experiment and to provide enough data to help oneunderstand the expected behavior. The data samplingrate should be high enough to capture adequately theshell response. The instrumentation should include back-to-back strain gauges for monitoring bending strains and

local nonlinear deformations; direct-current differentialtransformers (DCDTs), or other similar devices, formonitoring displacements normal to the shell surface;and shadow moir interferometry for qualitatively moni-toring buckle patterns. In many cases, the amount andtype of instrumentation needed can be determined frompreliminary analyses. It is important to reiterate that forsome shell stability problems, a linear bifurcationanalysis may not adequately represent the shell behavior,and as a result, may be inadequate for planning


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instrumentation. For experiments that involve load intro-duction by displacing a platen of a loading machine,proper alignment of the platens should be verified, andDCDTs, or other similar devices, should be used todefine the plane of the loading platen and to detectany load introduction anomaly. The loaded edges ofcompression-loaded shells should be measured to ensure

that the edges are as close to flat and parallel as possible.A loading rate that is consistent with the goals of the testshould be selected. Details of the test fixture and its rela-tionship to the desired boundary conditions should beclearly defined when reporting test data; all instrumenta-tion locations that correspond to the reported resultsshould be indicated clearly.

For experiments that involve thermal loading orcombined mechanical and thermal loading, additionalissues must be considered. An in-depth discussion ofseveral of these issues has been presented by Blosser(ref. 19), and some of the information needed to charac-

terize experimental results adequately is summarizedas follows. First, the temperature distribution of thestructure and its test fixture, as well as the heat flux at allthe surfaces, needs to be recorded adequately to facilitatethe correlation between theory and experiment. In addi-tion, any difference in coefficient of thermal expansionof the specimen and the test fixture, any heating of theloading platens, and all locations of insulated surfacesand heat conduction paths should be recorded. Completedescriptions of the thermal test fixture components,including coolant passages and cavities, should be given,and any interaction of the thermal components with thecomponents used to introduce mechanical load should be

identified. Other important details that should berecorded are the air temperature in the area surroundingthe test specimen, the method of heating or cooling usedfor the specimen and test fixture, and changes in materialproperties of the specimen and test fixture withtemperature.

Potential Improvements to the NASA

Monographs

Certainly one of the most significant improvementsto the NASA monographs would be the inclusion ofdesign recommendations for laminated composite shells

that are based on the analytical and experimental studiesthat have been conducted over the past 25 years. Anotherimprovement would be to base knockdown factors onaccurate analytical models of nominally perfect shells(such as shells free of initial geometric imperfections andmaterial variances) that include the proper boundary con-ditions (as opposed to only simply supported boundaryconditions, which are used to a large extent in the currentmonographs) and possibly the effects of nonlinearprebuckling deformations. These tasks can be done for a

wide range of parameters by using specialized codessuch as BOSOR4 and DISDECO, which compute bifur-cation buckling loads that include the effects of nonlinearprebuckling deformations and various boundary condi-tions by solving a nonlinear eigenvalue problem (refs. 20and 21). Isolating the effects of nonlinear prebucklingand boundary conditions are essential steps to under-

standing the shell behavior and to obtaining reliableknockdown factors that are not overly conservative.

Another significant improvement to the NASAmonographs would be to establish practical nondimen-sional parameters that contain the appropriate geometricand material variables and that enable concise represen-tations of behavioral trends and sensitivity of theresponse to variations of the parameters (e.g., seeref. 22). Guidelines for including damage tolerance andthe sensitivity of a design to load introduction effectswould be valuable additions to the monographs. One ofthe most significant improvements that can be made

immediately is to provide insight into, and quantitativeresults for, the true nonlinear interaction of combinedloads that has been treated very conservatively in theNASA monographs as a linear interaction. Furthermore,providing design recommendations for thermal loads andfor combined mechanical and thermal loads would be asignificant improvement.

Another issue that must be addressed to obtain a newset of useful and practical design monographs is designuncertainties. A significant contribution to this area canbe made by providing guidelines for determining whichshell stability issues are more adequately handled in a

deterministic rather than in a probabilistic manner. Froma practical viewpoint, this information indicates approxi-mately the number of experiments and analyses neededto establish meaningful design recommendations andreliable, but not overly conservative, knockdown factors.Ultimately, the improvements to the NASA monographsshould be focused on the practical needs of industrystructural designers and chief engineers and shouldreflect the scientific advances that have been made overthe last 25 years. The end result of such an effort wouldbe a collection of scientifically based knockdown factorsand design recommendations.

Concluding Remarks

A summary of the existing National Aeronautics andSpace Administration (NASA) monographs for thedesign of buckling resistant thin-shell structures has beenpresented. Improvements in the analysis of nonlinearshell response have been reviewed, and current issues inshell stability analysis have been discussed. Examples ofnonlinear shell responses that are not included in theexisting NASA shell design monographs have been


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presented, and an approach for including reliability-based analysis procedures in the shell design process hasbeen discussed. Suggestions for conducting future shellexperiments to obtain high-fidelity results have been pre-sented, and proposed improvements to the NASA shelldesign criteria monographs have been discussed.

NASA Langley Research CenterHampton, VA 23681-2199November 3, 1997

References

1. Anon.: Buckling of Thin-Walled Circular Cylinders. NASASpace Vehicle Design Criteria, NASA SP-8007, 1965.

2. Anon.: Buckling of Thin-Walled Truncated Cones. NASASpace Vehicle Design Criteria, NASA SP-8019, 1968.

3. Anon.:Buckling of Thin-Walled Doubly Curved Shells. NASA

Space Vehicle Design Criteria/Structures. NASA SP-8032,1969.

4. Gerard, George; and Becker, Herbert:Handbook of StructuralStability: Part IIIBuckling of Curved Plates and Shells.NACA TN-3783, 1957.

5. Becker, Herbert:Handbook of Structural Stability: Part VIStrength of Stiffened Curved Plates and Shells. NACATN-3786, 1958.

6. Gerard, George:Handbook of Structural Stability: Supplementto Part IIIBuckling of Curved Plates and Shells. NASATN D-163, 1959.

7. Von Krmn, Theodore; and Tsien, Hsue-Shen: The Bucklingof Thin Cylindrical Shells Under Axial Compression.J. Aero-naut. Sci., vol. 8, no. 8, June 1941, pp. 303312.

8. Donnell, L. H.; and Wan, C. C.: Effect of Imperfections onBuckling of Thin Cylinders and Columns Under Axial Com-pression.J. Appl. Mech., vol. 17, no. 1, Mar. 1950, pp. 7383.

9. Koiter, Warner Tjardus: A Translation ofThe Stability of Elas-tic Equilibrium. AFFDL-TR-70-25, U.S. Air Force, Wright-Patterson Air Force Base, Feb. 1970.

10. Seide, P.; Weingarten, V. I.; and Morgan, E. J.: The Develop-ment of Design Criteria for Elastic Stability of Thin Shell

Structures. STL/TR-60-0000-19425, U.S. Air Force, Dec.1960.

11. Vinson, Jack R.: The Behavior of Shells Composed of Isotropicand Composite Materials. Kluwer Academic Publ., 1993.

12. Stein, Manuel: The Effect on the Buckling of Perfect Cylin-ders of Prebuckling Deformations and Stresses Induced byEdge Support. Collected Papers on Instability of Shell Struc-tures1962, NASA TN D-1510, 1962, pp. 217227.

13. Stein, Manuel.: The Influence of Prebuckling Deformationsand Stresses on the Buckling of Perfect Cylinders. NASATR R-190, 1964.

14. rbocz, J.: Comparison of Level-1 and Level-2 Buckling andPostbuckling Solutions. Faculty of Aerospace Engineering,Report LR-700, Delft University of Technology, Delft, TheNetherlands, Nov. 1992.

15. Sridharan, Srinivasan; and Alberts, Jessica: Numerical Model-ing of Buckling of Ring-Stiffened Cylinders.AIAA J., vol. 35,no. 1, Jan. 1997, pp. 187195.

16. Nemeth, Michael P.; Britt, Vicki O.; Collins, Timothy J.; andStarnes, James H., Jr.:Nonlinear Analysis of the Space ShuttleSuperlightweight External Fuel Tank. NASA TP-3616, 1996.

17. Brogan, F. A.; Rankin, C. C.; and Cabiness, H. D.: STAGSUser Manual. Lockheed Palo Alto Research LaboratoryReport, LMSC P032594, 1994.

18. Thurston, G. A.; Brogan, F. A.; and Stehlin, P.: PostbucklingAnalysis Using a General-Purpose Code. AIAA J., vol. 24,no. 6, June 1986, pp. 10131020.

19. Blosser, Max L.: Boundary Conditions for AerospaceThermal-Structural Tests. Aerospace Thermal Structures and

Materials for a New Era: Volume 168Progress in Astronau-

tics and Aeronautics, Earl A. Thornton, ed., AIAA, 1994,pp. 119144.

20. Bushnell, D.: Stress, Stability, and Vibration of ComplexBranched Shells of RevolutionAnalysis and Users Manual

for BOSOR4. NASA CR-2116, 1972.

21. rbocz, J.; and Hol, J. M. A. M.: Koiters Stability Theory ina Computer-Aided Engineering (CAE) Environment. Int. J.Solids & Struct., vol. 26, no. 910, 1990, pp. 945973.

22. Nemeth, M. P.: Nondimensional Parameters and Equations forBuckling of Anisotropic Shallow Shells. J. Appl. Mech.,vol. 61, Sept. 1994, pp. 664669.


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Figure 1. Space Shuttle external tank components.

USA

UnitedStates

LO2 tank

Intertank

LH2 tank

Nose coneExternal tank

Solid rocketbooster (SRB)

Orbiter

SRB beam

Aft orbiterattachmentfitting

Forward orbiterattachmentfitting

Aft SRBattachmentpoint

Thrust panel

Aft ellipticaldome section

Cylindricalbarrel section

Aft ogivesection

Forwardogivesection


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Figure 2. Linear bifurcation buckling mode for a 99000 degree-of-freedom model (load factorpa = 3.78).

Solid rocket booster(SRB) attachment point


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Figure 3. Predicted nondimensional normal displacement w/tof aft ogive of a geometrically perfect shell for increasingLH2 interface loads.

Figure 4. Predicted local nondimensional displacement amplitude w/tof aft ogive surface for increasing LH2 interfaceloads; geometrically perfect and geometrically imperfect shells.

Load factor, pa

= 4.02

Linearbifurcationbuckling mode

w/tof bucklingmode

(dashed line)

Aft ogive

Load factor, pa

pa = 3.0

pa

= 5.0

pa

= 3.78

xL

wt

5

3

4

2

1

0

1

2

w/t(solid lines)

8

12

10

14

16

18

200 .25 .50

Nondimensional aft ogive coordinate x/L

.75 1.00

6

5

4

3

2

1

0

Displacement amplitude, w/t

654321

Loadfactor,pa

Load factor, pa

w

Linear bifurcationbuckling load level

Imperfection-amplitude-to-wall-thicknessratioA/t= 0.3

Geometrically

perfect shell

Aft ogive

t


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Figure 5. Nondimensional load versus end-shortening curves for aluminum curved panels with a central circular cutout;is the analytical prediction of the linear bifurcation buckling load for the panel without a cutout.

Figure 6. Shadow moir patterns for aluminum curved panels without a cutout.

1.0

.8

.6

.4

.2

0 .1 .2

Nondimensional end-shortening, /t

P/Pobif

.3 .4

L/W= 10.3

0.4

0.5

R/t= 600d/W= 0

P

R

W

d

L

t

Pbifo

P/Pobif= 0.86 (before buckling) P/P

obif= 0.57 (after buckling)


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Figure 7. Shadow moir patterns for aluminum curved panels with a central circular cutout (d/W= 0.3).



P/Pobif= 0.72 (before buckling) P/P

obif= 0.67 (after buckling)

P/P

o

bif= 0.46 P/P

o

bif= 0.71

P/Pobif= 0.50 P/P

obif= 0.70


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January 1998 Technical Publication

The NASA Monographs on Shell Stability Design Recommendations

A Review and Suggested Improvements WU 522-11-41-02

Michael P. Nemeth and James H. Starnes, Jr.

L-17632

NASA/TP-1998-206290

Paper presented at the 38th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and MaterialsConference.

A summary of the existing NASA design criteria monographs for the design of buckling-resistant thin-shellstructures is presented. Subsequent improvements in the analysis for nonlinear shell response are reviewed, andcurrent issues in shell stability analysis are discussed. Examples of nonlinear shell responses that are not includedin the existing shell design monographs are presented, and an approach for including reliability-based analysisprocedures in the shell design process is discussed. Suggestions for conducting future shell experiments are pre-sented, and proposed improvements to the NASA shell design criteria monographs are discussed.

Shell stability; Design recommendations; Buckling 21

A03

NASA Langley Research CenterHampton, VA 23681-2199

National Aeronautics and Space AdministrationWashington, DC 20546-0001

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