&ffiffiffiffiffiffiffiffiffiffi&ffiffiffi%ffiffiWffiffiW%

Postbuckl ingTh eo ry

J. W. HUTCHINSONHARVA R D U N IVE RSI TY, CAM BR I DG E, MASSACHUSETTS

W. T. KOITERTECHNISCHE HOGESCHOOL, DELFT. THE NETHERLANDS

Introduction

A division of elast ic stabi l i ty theory and i ts appl icat ionsinto separate categories of buckl ing and postbuckl ing isnot entirely rat ional, but the creation of such an art i f i -cial dist inct ion is almost essential to bring either intomanageable proport ions. This point is brought home bymore than 1600 references on cyl indrical shel l buckl ingalone in the surveyby Grigol 'uk and Kabanov IRef. 1] .

Postbuckl ing aspects of stabi l i ty theory for f lat platesreceived prominence in the early 1930's, after Wagner

IRef. 2] had establ ished a sound theoretical foundationfor the load-carrying capacity of deeply wrinkled flatshear panels. The approximate discussion by Cox forplates in uniaxial compression IRef. 3] was fol lowed bya r igorous analysis by Marguerre and Trefftz [Ref. 4] .This work was continued f irst in Gernrany IRefs. 5, 6],and somewhat later in the United States [Refs. 7-15],Bri tain [Refs. 16-18] and The Netherlands IRefs.1.9-281 . Reviews of this work and subsequent dcvelop-ments are to be found in [Ref. 29] . The entirelydif ferent postbuckl ing behavior of thin shel l struc-tures came to l ight in the early 1940's when Karmanand Tsien IRefs. 30-331 showed that the large dis-crepancies bctween test and theory for the buckl ingof certain typcs of thin shel l structures was due to

the highly unstable postbuckl ing behavior of these

structures. At roughly the same t ime in war-t imeHolland, Koiter IRef. 34] developed a general theory ofstabi l i ty for elast ic systems subject to conservative load-ing which was published as his doctoral thesis in 1945.

Karman's and Tsien's papers spawned many more inthe fol lowing 30 years, and most of them have beendirected to obtaining more accurate results for thecyl indrical shel l under axial compression or to studyingthis structure under other loadings. Koiter 's work, on theother hand, attracted relatively little attention until theearly 1960's when interest in the general theory sprangup almost simultaneously in England and in the UnitedStates.

Our review will be centered mainly on the theory andapplications that have emerged from these two ap-proaches to postbucki ing theory. A major st imulus todevelopment in this subject continues to be generatedby the quest for predictive methods for the buckling ofshell structures, and a sizable portion of this review willconcern applications of the theory to this area, Astate-of-the-art discussion of shel l buckl ing as of 1958is given in the excel lent survey by Fung and Sechler

[Ref. 35], and that art icle provides a background forassessing the progress which has taken place in this sub-ject in just over a decade. Our lack of command of theRussian language prevents us from doing just ice to thecontr ibutions of Soviet scientists in this f ield. Referencemay be made, however, to Volmir 's outstanding treatiseon the stabi l i ty of deformable systems [Ref. 36], andto the survey on cyl indrical shel l buckl ing [Ref. 1].

Our discussion wil l not touch on research into stabi l-i ty of elast ic systems subject to nonconservative loadings,but fortunately we can cite a recent survey in AppliedMechanics Reuiews by Herrmann [Ref. 37] on thisaspect of stability theory, as well as a similar survey byNemat-Nasser IRef. 38] on thermoelastic stabi l i ty undergeneral loads. We also would cal l attention to anotherrecent review art icle by Thompson [Ref. 39] whichdoes cover some of the same ground as the presentsurvey, but the emphasis there is on elast ic systemscharacterized by a f ini te number of degrees of freedom,whereas most of the applications we will discuss will bewithin the realm of continuum theory, We also omit adetai led discussion of secondary branching IRefs. 29,40, 4ll of the initially stable postbuckling behavior off lat plates, even i f this phenomenon is of some im-portance for these and similar "completely symmetric"structures IRef. 42] .

Theory and Appl ications

The energy criterion of stability for elastic systemssubject to conservative loading is almost universally

i 353

adopted by workers in the field of structural stability

[Refs. 43-45] . A posit ive definite second variat ion ofthe potential energy about a stat ic equi l ibr ium state isaccepted as a sufficient condition for the stability ofthat state. Numerous attempts to undermine these twopillars of structural stability theory have been made, butconfidence in them remains undiminished. With a propershoring-up of certain aspects of these criteria, they willundoubtedly continue to serve as the foundation ofelastic stability theory. An account of the present statusof the fundamentals of stability theory is given in

IRef. 46] .In this review, we are concerned mainly with the

class of problems most frequently encountered in thefield of structural stability in which the loss of stabilityof one set of equi l ibr ium states of an ideal ized, or"perfect," elast ic structure is associated with bifurcationinto another set of equi l ibr ium states. The f irst set isreferred to as the prebuckl ing state and the bifurcatedstate as the buckled configuration. The bifurcation loadof the perfect structure is commonly called the classicalbuckl ing load and is denoted here by P-. This is by nomeans the only circumstance under which a structurecan become unstable. For example ) a very shal lowclamped spherical cap subject to a uniform pressureloading reaches a maximum, or limit, load at which pointi t becomes unstable under prescribed pressure with nooccurrence of bifurcation. However, due to the sym-metry of both the idealized structure and its loading,bifurcations frequently do occur at load levels below thelimit load.

The classical analysis of the stability of the prebuck-ling state of the perfect structure takes the form of aneigenvalue problem for the lowest load level for whichthe second variat ion of the potential energy is semi-definite. The Euler equations associated with this vari-at ional principle are l inear and the eigenmode (or modes)associated with the critical eigenvalue P. is termed thebuckl ing mode (or modes). While the classical analysisyields the load at which a perfect realization of a struc-ture wil l at least start to undergo buckl ing deflect ions, i tgives no indication of the character of the postbuckl ingbehavior, nor does it give any insight into the way a non-perfect real izat ion wil l behave.

Typical load-deflection curves characterizinq staticequilibrium configurations are shown in Fig. 1 for thecase of structures which have a unique buckling modeassociated with the classical buckling load. In each ofthe three cases shown the prebuckl ing state ofthe perfectstructure is stable for P I P, and is unstable fot P) P,where it is shown as a dotted curve. Case I illusrrares astructure with a stable postbuckling behavior which cansupport loads in excess of P, in the buckled state. Thebehavior of a slightly imperfect version of the samestructure (or perhaps for a slightly misaligned applicationof the load) is depicted by the dashed curve. Case II isan example of a structure which goes into a stable orunstable postbuckl ing behavior depending on whetherthe load increases or decreases fol lowine bifurcation. An

initial imperfection is all that is needed to prejudice thedeflection one way or the other. If an imperfection in-duces a positive buckling deflection, the load-deflectioncurve of Case II has a limit load P., the buckling load ofthe imperfect structure, which is- less than the bifurca-t ion load of the perfect structure P.. Case II is an ex-ample of asymmetric branching behavior, while Case III

illustrates a structure whose buckling behavior is sym-metric with respect to the buckl ing deflect ion and whoseinitial postbrrckling behavior is always unstable underprescribed loading condit ions.

The classical buckling load is a reasonably goodmeasure of the load level at which an imoerfect realiza-tion of the structure begins to undergo significant buck-ling deflections if the structure has a fully stable initialpostbuckl ing behavior. In the early days, rel iance on theclassical buckling load stemmed from the fact that allconfrontat ions between test and theory were for eithercolumns or plates, and both of these are stable in thepostbuckl ing regime. When tests were carr ied out on thinshell structures in the early 1900's, the classical theorybecame suspect because then actuai buckl ing loads werefrequently found to be as little as one-quarter of theclassical load.

Except for a very few structures, such as the columnunder axial compression [Ref. 47] and the circular r ingsubject to inward pressure [Ref. 43] , i r is not possibleto obtain closedform solut ions governingthe entire post-buckl ing behavior. Start ing with Karman's and Tsien'swork on spherical and cyl indrical shel ls, a large numberof numerical calculations in the large-deflection rangehave been made. Each of these calculat ions reDresenrsan attempt to obtain the load-deflection behavio-r of theperfect structure, and in part icular the minimum loadthe structure can support in the buckled state ( i .e., P*in Fig. 1). This load was held to be significant as a pos-sible design load on the grounds that the strucrure couldalways support at least this much load and that evenimperfections would not reduce the buckl ing load belowthis value. This concept could not be useful in any uni-versal sense since there are well-known examples of struc-tures with negative minimum postbuckl ing loads. An-other difficulty with this proposal is that the bucklingprocess of such a structure is a dynamic one and thebuckled structure may end up deformed quite differ-ently from what is predicted on the basis of stat icequil ibr ium considerations alone. In any case, efforts tovalidate the minimum load for design purposes for cylin-drical shell structures have not paid off, as we will dis-cuss in the next major section of this art icle; and thisidea perhaps should be abandoned.

Emphasis in the analysis of imperfection-sensitivestructures has shifted to the maximum load P, whichcan be supported before buckling is triggered and to re-lating P, to the magnitude and forms imperfectionsactually take. This was the point of view adopted in thegeneral nonlinear theory of stability [Ref. 34] and insubsequent investigations based on the general theory

IRefs. 49-55] .

1 354

The initial postbuckling analysis in its simplest form

yields an asymptotically exact relation between the load

parameter P and the buckling deflection 6, valid in theneighborhood of the bifurcation point of the perfectstructure. This expansion is in the form

P=P,( l+a6+b62+.. . ) (1)

where the coefficients d, b, ... determine the initial post-buckl ing behavior. A number of important problems

are characterized by multiple buckling modes associatedwith the classical buckling load. Two of these problemswil l be discussed in the next section. For such problems,the initial postbuckling analysis may be considerablymore complicated,

Asymptotically exact estimates of the buckling loadof the imperfect structure are obtained by including thefirst-order effects of small initial deflections. Only thecomponent of the initial deflection in the shape of theclassical buckling mode enters into the resultant formula.

, l

, l

Pl

CASE tr

CASE Itr

Gcn.rol izcd dof l6ct ion

FIG. 1. LOAD.DEFLECTION CUFVES FOR SINGLE

Buckl ing dcf lccl ion - 8

BIFURCATION BEHAVIOR

7-i

1 355

MODE

t.I f the ampli tude of this imperfection component is de-noted by 6-, then for Case II with 5 ) O, the generaltheory yields

ment for the effect of an imperfection - a slightly offsetload - on the buckling load is remarkably good.

The first application of initial postbuckling theorywas to the monocoque cylindrical shell under axial com-pression [Refs. 34, 50] which wil l be discussed in thenext section. The second was a study of a narrow cylin-drical panel under axial compression [Ref. 49] whichdisplays a transition, depending on its width, from astable postbuckling behavior typical of a flat plate to theunstable behavior which characterizes the cylindricalshel l . Beaty [Ref. 94] and Thompson [Ref. 63] appl iedthe general theory to the axisymmetric buckling of a

t'

I ( deg rees )

FIG. 2, COMPARISON OF THEORYON A SIMPLE TWO-BAR FRAME

to/'. t /2

AND EXPERIMENTS

(2)[' +]"=

[' ]J" =.rsrf,

_P-o6"

PC

P1__t

: t ' .

' \

- (ad) ' ' -

where a is a constant which depends on propert ies of the

structure. For the symmetr ic branching point of Case I I I

(a= o,b ( 0) , the analogous formula is

1 - P'

- (a l6- l ) tuPc

In both instances, very small values of q6 have a sizableeffect on PrlP- and it is this feature which accounts for

the extreme imperfection-sensit ivi ty of many structureswhich have an unstable postbuckling behavior.

Interest in the init ial postbuckl ing approach mush-roomed in England in the early 1960's [Refs. 56-93] .

Some f ine experimentation is included in this work, and

an extensive series of theoretical investigations have

centered around conservative systems characterized by af inite number of degrees of freedom. A great deal of this

work has been carr ied out at University College, London,

by Chilver, Thompson, Walker and their students.Sewell , at the University of Reading, also has concen-

trated on the postbuckl ing behavior of discrete systems.He has part icularly emphasized the systematics of the

perturbation procedure and has used the terminology

"the stat ic perturbation method" in referr ing to thismethod of analysis. Both Thompson and Sewell haveexplored a variety of bifurcation possibi l i t ies for suchsystems. Recent studies have been made with an eye to

providing a framework for postbuckling calculations viaf inite element methods. Some of this work is summarizedin [Ref. 39] .

Perhaps the best direct experimental val idation of

the init ial postbuckl ing approach was obtained in a

series of tests by Roorda [Refs. 67, 68]. The simpletwo-bar frame shown in Fig. 2 was loaded in a testingmachine which, in effect, prescribed displacement rather

than dead load and in this way the postbuckling equili-brium states were recorded. Experimental points and

theoretical predict ions IRefs. 53, 85] shown as a sol idl ine for the load, P, versus buckl ing rotat ion,0, for the

"perfect" structure are displayed in the upper half of

the f igure. The agreement between theoty and experi-

(3)

Effecl of loodeccenlr ic i l ies

1 356

complete spherical shel l . Their analysis yielded the ex-pected result that the sphere has an unstable postbuck-l ing behavior. However, i t has been shown recently

IRef. 55] that the init ial postbuckl ing analysis in i tssimplest form leads to results that are of little practicalconsequence for the complete sphere problem. We defera discussion of this matter also to the next section.

The general theory IRef. 34] has been applied to avariety of shel l structures by Budiansky and Hutchinsonand their students at Harvard University IRefs. 95-110] .Included among them are toroidal shel l segments undervarious loadings, cyl indrical shel ls subject to torsion andspheroidal shel ls loaded by external pressure.

Results shown in Fig. 3 due to Budiansky andAmazigo IRef. 103] very nicely i l lustrate the outcomeof such an analysis and i ts interpretat ion. In the upperhalf of the f igure the classical buckl ing pressure p. innondimensional form is olotted as a function of thelength param etet Z appropriate for either a simply-supported cyl inder of length L or a segment of length Lof an inf inite cyl inder reinforced by r ings which permitno lateral deflect ion but al low rotat ion. The init ial post-

, .2.t = (+) Jt-vz

FIG. 3. COMPARISON BETWEEN TEST AND CLASSICALTHEORY WITH INITIAL POSTBUCKLING PREDICTIONSFOR A CYLINDRICAL SHELL UNDER EXTERNAL PRES-SU RE

buckl ing behavior is symmetric with respect to thebuckl ing ampli tude 6, and therefore the pressure-deflect ion relat ion takes the form

where b is plotted in the lower half of Fig. 3. In thefigure, D is the bending st i f fness, z is the Poisson rat ioand t is the shel l thickness. In this case, the asymptoticrelat ionship between the buckl ing pressure and the im-perfect ion is

12 l5( -b) l ;

I t

n

p

where 6- is the ampli tude of the component of the im-perfect ion in the shape of the classical buckl ing mode.A wide range of test data, col lected by Dow IRef. 111],is also included in the figure. Measurements of initial de-f lect ions were not made in any of these tests, so i t is notpossible to make a direct comparison of test and theory.On the other hand, the coincidence of the large dis-crepancy between test and classical predict ions withinthe Z-range in which b is most negative bears out the im-perfect ion-sensit ivi ty pre dicted.

The complementary character of an init ial postbuck-ling analysis and a large deflection analysis is brought outby studies of long oval cyl indrical shel ls under axial com-pression. Kempner 's and Chen's IRefs. 112-114] numer-ical results for the advanced postbuckling behavior ofshel ls with suff iciently eccentr ic oval cross sectionshowed that loads above the classical bucklins loadcould be supported. Complete col lapse occurs onl l i whenthe buckling deflections engulf the high curvature endsof the shel l . According to an init ial postbuckl ing analysis

IRef. 105] , ini t ial buckl ing wil l be just about as sensit iveto imperfections as for the circular cylindrical shell aswould indeed be expected because of the shal low buck-l ing behavior i .r .ach case. The composite picture is thatinitial buckiing will be imperfection-sensitive but notcatastrophic, and loads above the classical values shouldbe possible. Recent tests [Ref. 115] have confirmedboth these features.

One aspect of shell buckling which has attracted agreat deal of theoretical and experimental attention inthe last few years is the role which stiffening plays instrengthening shel l structures. Some rather unexpectedeffects have turned up. One of the most interesting wasthe observation by van der Neut [Ref. 1 16] over 20 yearsago that the axial buci<ling load of a longitudinallystiffened cylindrical shell can be increased significantlyby attaching the stiffeners to the outer surface of theshell rather than the inner surface. This advantage hasbeen clear ly demonstrated by tests IRef. 117].

Init ial postbuckl ing results IRefs. 102, 109, 118] in-dicate that stiffened cylindrical shells tend to be less

f,, = ,- r(+l .

f, -\f" - 3\/tLo,J2

(4)

pcRLz

r2D-

1357

imperfection-sensitive than their unstiffened counter-parts, but the level of sensitivity can vary widely. Forexample, while the classical buckling load of an outside-stiffened cylinder may be much higher (a factor of twois not untypical) than an inside-stiffened one, rhe im-perfection-sensitivity of the outside-stiffened shell maybe much higher as well. Only a few experiments areavai lable to put these predict ions to a test but thosewhich have been reported [Refs. 119, I20l show pre-cisely this trend.

An extensive series of buckl ins tests on st i f fenedshell structures has been underwiy fot a number ofyears under the direct ion of Singer at the Technion inIsrael [Refs. 1.21., L22l. This program has producedample evidence that the classical buckling load is a morereliable measure of actual buckline loads for stiffenedshells than i t is for unsti f fened ,h"l l . . N"u"rtheless, theexperimental scatter and the discrepancies from theclassical predictions in these tests are indicative of thefact thai st i f fening does not el iminate the problem ofimperfection-sensit ivi ty. An up-to,date discussion of thismatter is given in IRef. 123] .

Other recent work in the area of postbuckl ing theoryincludes an investigation of the interaction of localbuckl ing and column fai lure for thin-walled compressionmembers [Refs. 124, 125]. This work may help toeradicate the naive approach to optimal design of struc-tures l iable to buckl ing by an attempt to equalize thelocal and overal l buckl ing stresses.

There have been important advances in both theo-ret ical and calculat ional aspects of shel l buckl ing in thelast decade. Buckl ing equations have been proposed

[Refs. 52, 126] which are exact within the context off irst-order shel l theory. Computer programs are nowavailable for accurate computation of classical bucklingloads for a wide class of shel ls of revolut ion subiect toaxial ly symmetric loads [Refs. 1,27, 1281. These pro-grams incorporate effects of nonl inear prebuckl ing be-havior and discrete r ings. When the prebuckl ing deforma-t ion of the perfect shel l is a purely membrane one withno bending, the init ial postbuckl ing analysis is general lysimpler than when bending, and nonlinear prebuckl ingeffects must be taken into account. Most appl icat ions ofpostbuckl ing theory to date have been in cases in whichthe prebucki ing response is exactly a purely membranestate or could be reasonably approximated by one.Within the last three years there have been several ap-pl icat ions of the general theory to problems in which i tis essential to include nonlinear prebuckl ing effects

IRefs. 104, 109, 110, 1291, and a general purpose com-puter program for shel ls of revolut ion subject to axisym-metric loads has been put together IRef. 130] .

Status of the Postbuckl ing Theory of the Cyl indricalShell under Axial Compression and the Spherical Shell

under Uniform Pressure

The cyl indrical shel l under axial compression hasserved as the prototype in studies of shell buckling

IRefs. 131-161], but i ts geometric simplici ty is decep-t ive, and in many respects this structure, together withthe sphere subject to pressure, has the most compli-cated postbuckling behavior of all. In large part, thisstems from the fact that a large number of bucklingmodes are associated with the classical buckl ing load ineach of these problems, and consequently these shel lsare susceptible to a wide spectrum of imperfectionshaoes.

i t is quite possible that a paper by Hoff, Madsen andMayers [Ref. 152] has put an end to the quest for theminimum load which the buckled shel l can support. Asequence of large deflect ion calculat ions of the minimumpostbucki ing load have been reported, each calculat ionmore accurate than those which preceded i t . Hoff, et al. ,give a convincing argument that a completely accuratecalculat ion, based on the procedure which had been em-ployed in al l the previous investigations, would lead to avalue for the minimum postbuckl ing load which wouldtend to zero for a vanishing thickness to radius rat io. Ofcourse, a shel l with a f ini te thickness to radius rat iowould actual ly have a nonzero minimum load, but therenow appears to be general agreement that this minimumis not nearly as relevant as had been thought.

The roie of boundary condit ions was extensively ex-plored in the 1960's and is now fair ly well sett led. Ac-curate numerical calculat ions of classical buckl ins loadswhich account for nonl incar prebuckl ing effects

", ' rd uo.-

ious boundary condi t ions have been made IRcf. 162].The classical load for cyl indrical shel l of moderatelength which is clamped at both ends is about 93 percent of the load predicted by the well-known "classical"formula based on a calculat ion which isnores end con,dit ions altogether. So-cal led *e^k bo,r, ldrry condit ionsfor which no tangential shear stress is exertcd on the endsof the shel l reduce the buckl ing load by a factor of abouttwo IRefs. 163,164]. More reccnt ly i t has been shownthat relaxed tangential restraint along a small fract ion ofthe edge has an almost equally detr imental effect

IRef. 165].Near perfect cyl indrical shel l specimens have been

produced by Tennyson IRef. 166] and Babcock andSechler IRef. 144], and these shel ls buckle very c lose rothe classical buckl ing load. High-speed photography hasresolved the apparent discrepancy betwecn observedbuckl ing patterns and the classical mode shapes IRefs.167-1711. Buckle wavelengths associated with the col-iapsed shel l are much longer than the classical buckl ingmode wavelengths, but the wavelengths associated withdeformation patterns photographed just after buckl inghas been tr iggered are in good agreement with the pre-dict ions of ini t ial postbuckl ing theory.

The long cyl indrical shel l subject to axial compres-sion is one of the examples used in IRef. 34] to i l iustratc

the general theory of elast ic stabi l i ty. Boundary condi-t ions are neglected, and consideration is restr icted tomode shapes which are periodic in both the axial andcircumferential coordinates. Due to the iarse number of

1 358

simultaneous buckl ing modes, there are many possible

stat ic equi l ibr ium branches which emanate from the bi-

furcation point of a perfect cyl inder, and al l of them are

unstable. Remarkably, the initial curvature of the curve

of load versus end shortening (Pversus e) is the same for

all of them and just after bifurcation

Effects of certain imperfections were also studied.

Among the shape possibi l i t ies represented by al l the

classical buckl ing modes, a geometric imperfection inthe shape of the axisymmetric buckl ing mode results inthe largest relat ive reduction in the buckl ing load IRef.51]. The rat io of the buckl ing load to the classical loadis related to the ampli tude of such an axisymmetricimperfection 6 by

where c = l ,6\-n and z is Poisson rat io. When themaximum load { is attained, the shel l has undergonelateral deflect ions which are only a fract ion of the shel lthickness, and an imperfection ampli tude of only one-f i f th a shel l thickness causes almost a 50 oer cent reduc-t ion in the buckl ing load.

fq. (7) is an asymptotic formula; and l ike al l ini t ialpostbuckl ing predict ions of this type, i t is accurate onlyfor suff iciently small imperfection ampli tudes. In general,

it is difficult to assess the range of validity of asymptoticresults. For this part icular imperfection shape, i t ispossible to obtain an independent and more accurateestimate of the P, uemns 5 relat ion which is based on af inite deflect ion calculat ion IRef. 51]. The asymptoricpredict ions ("q. 7) are plotted with the more accurateresults in Fig. 4, and two predict ions compare well overa substantial part of the range of interest. The calcula-t ion of IRef. 51] was repeated by Almroth [Ref. 172]with the aim of gett ing better accuracy. He found thatthe buckl ing load predict ion may be somewhat lowerthan those shown in Fig. 4 i f the parameter Z (Fig. 3)is very large.

Some experimental veri f icat ion of these buckl ingload calculat ions has been obtained by Tennyson andMuggeridge IRef. 173] with tests of cyl indrical shel lspecimens with axisymmetric sinusoidal imperfectionswhich were careful ly machined into them. Points repre-senting the experimental loads for f ive shel ls are plottedwith two theoretical curves in Fig, 5. The solid curve isthe result of the f ini te deflect ion predict ion of IRef. 51]for the imperfection wavelength coincident with thoseof the specimens. The dashed curve is a plot of the re-sults of a numerical caiculat ion which takes into accountnonlinear prebuckl ing deformations associated with theclamped end condit ions and the thickness variat ionspresent in the test cyl inders IRef. 173] . At extremelysmall imperfection levels, end condit ions dominate.Once the imperfection ampli tude is just a few per centof the shel l thickness, the end effect is negl igible and thetwo predict ions are virtual ly identical.

There have been some efforts to translate and adaotthe init ial postbuckl ing predict ions into a form direct iyuseful for engincering design IRefs. t29, 1"30, 1611 .

.#,-1] ' (6)

f 1, .12 3c l6- l pl t - - l=r l ' l ; e)L P,J 2lr l l ,

Gcn6rol fhcoryEq. (7)

-a.tEt

FIG. 4. EFFECT OF AXISYMMETRIC IMPERFECTIONS INTHE SHAPE OF THE AXISYMMETRIC BUCKLING MODEOF A PERFECT CYLINDRICAL SHELL UNDER AXIALCOMPRESSION

o.o2 o.o4 oo6

FIG. 5. COMPARISON BETWEEN TEST AND THEORY FORBUCKLING OF CYLINDRICAL SHELLS WITH AXISYM-METRIC IMPEBFECTIONS LOADED IN AXIAL COMPRES-SION

1 359

Reinforcing this trend is recent work by Arbocz andBabcock IRefs. 17 4, 17 5] in which imperfections

present in a number of test specimens were careful lymeasured and mapped. Special techniques have beenused to calculate buckl ing loads for certain imperfectionshapes which are quite dif ferent from the classical buck-l ing modes or for imperfection shapes which are onlycharacterized statistically. Some of these will be touchedon in the next section.

Since Karman's and Tsien's [Ref. 301 f irst large de-f lect ion calculat ion for the axisymmetric postbuckl ingbehavior of a complete sphericai shel l under uniformpressure, a fair ly large number of papers on the buckl ingof caps and spheres haue been published. Many of theseare referenced in a recent papcr on the subject IRef.55].Here, we wil l only mention some of the sal ient features

peculiar to the sphere problem which have recentlycome to l ight.

Like the cyl indrical shel l under axial comprcssion, a

complete sphcrical shel l under external pressure hasmany buckl ing modes associatcd with the classical buck-l ing pressure, and onc of these is axisymmetric. Manyunstable equi l ibr ium branchcs emanate from the bifurca-t ion point of the perfect shel l . A shal low shel l analysis ofmult imode buckl ing of the sphere IRef. 101] indicatesthat this shel l is highly sensit ive to both axisymmetricand nonaxisymmctric imperfections, again just as for thecyl inder. While this shal low shel l analysis is not r igor-

ously appl icable to thc complete sphere, i ts relat ive

simplici ty reveals the mechanics of the buckl ing phenom-ena of the spherc vcry clearly. In part icular, the shal lowbuckl ing modes are in str iking similari ty with expcri-mental modes .eported in IRef. 1761. An al tcrnat iveapproach to this problem by a straightforward pcrturba-t ion teclrnique is detclopcd in IRefs. 177,1781 .

Init ial postbuckl ing analyses for axisymmetric buck-l ing have been given by Beaty [Ref. 94] , Thompson

lRef.63l , Walker [Ref. 80 1, and most reccnt ly by

Koiter IRcf. 55] . A mult imode analysis in which

the axisymmetr ic mode coLlples wi th nonaxisymmetr icmodes, analogor.rs to cyl indrical shel l behavior, also wasgiven in IRef. 55] . Impcrfcct ion-sensi t iv i ty was found tobe highe r for mult imode buckl ing than for axisymrnetr icbuckl ing.

A most important discovery of 1nef. 55] was thatthe axisynlmetric rcsults just mentioned, which wereobtained by a perturbat ion expansion about the bi furca-t ion point, havc an cxtremely small range of val idi ty

which :LctLral ly vanishes as the thickncss to radius rat io

of thc shel l goes to zero. The principal reason for this

unusual l imitat ion on the init ial postbuckl ing results is

thc occurrence of modes, associatcd wi th eigenvaiueswhich are only vcry sl ightly largcr than the cri t icaleigenvalr.res, whosc ampli tudes become comparable in

magnitude to the ampli tudes of the classical buckl ingmodes outside of a very small neighborhood of the bi,furcation point. For al l practical purposes, the init iaipostbuckl ing analysis in i ts standard form breaks downunder such cr 'rcumstances. A more powerful form of

asymptotic expansion, also f irst detai led in [Ref. 34] ,with an extended range of val idi ty was applied to theaxisymmetric problem IRef. 55]. The outcome of thisspecial analysis is that the spherical shel l is highly im-perfect ion-sensit ive, more or less to the same degree asthe cyl indrical shel l under axial compression, even whenthe deformations are constrained to be axisymmetric.An approximate analysis IRef. 179] and several numer-ical calculat ions IRefs. 180-1831 back up this conclusion.

New Research Directions

It seems safe to predict that much of the impetus topostbuckl ing theory as a developing subject wi l l continueto come from questions which arise in the analysis ofshei l structures. Efforts are being directed to interpretingpostbuckl ing predict ions and rendering them useful forengineering purposes. I f this is to be accomplished,further calculat ions wil l be needed for more real ist icimperfections than those which are readi ly accommo-dated by the init ial postbuckl ing analysis. Approacheswhich incorporate stat ist ical descript ions of imperfec-t ions are also l ikely to receive growing attention in thecoming years.

Very reccntly, Sewell IRef. 184] has explored variousramif icat ions of the more powerful expansion methodof [Ref. 34] (which was employed to get around rhediff icult ies in the sphere problem referred to above) asit appl ies to conservative systems with a f ini te number ofdegrees of f reedom. Thompson IRcf. 185] has proposeda somewhat dif ferent variat ion of the method, also fordiscrete rather than continuous systems. The aim ineach case is to develop a uniformly val id asymptotic ex-pansion which gives an extended range of val idity forthose rathcr exceptional problems in which the standardexpansion breaks down outside the immediate vicinityof the bi furcat iorr poinc.

Effects of local ized dimple imperfections on thebuckl ing of a bcam on a nonl inear elast ic foundationhave been explored usir ig a variety of techniques, andasymptotic formulas analogous to eqs. (2) and (3) havebeen uncovered IRef. 186] . Appl icat ion of these specialtechniques has been made to study the effcct of localnonaxisymmetric imperfections on buckl ing of cyl indri-cal shcl ls under external pressure IRef. 187] . In addit ion,both theoretical and exocrimentai studies have beenmade of cyi indrical and conical shel is under axial com-pression in the presence of axisymmetric dimple im-pcrfect ions IRefs. 188, 1891. With a cont inuing rapidgrowth of numerical methods of shel l analysis, i t shouldsoon be possible to make reasonably accurate calcula-t ions of the buckl ing load of an arbitrary shel l structurein the presence of almost any form of imperfection. Thepotential of such methods has already been demonstrated

IRefs. 190-1931 for problems in which i t is necessary toconvert the governirrg nonl inear part ial dif ferential equa-t ions to an algcbraic system by spreading a two-dinensior.ral f ini te-dif fcrcnce srid over the shel l middlesurface. No doubt calcul l t ionl of this sort wi l l not be

I 360

inexpensive in terms of computation t ime for manyyears to come; but i f thcy are careful ly selected, suchcalculat ions should be most usefui for imperfection-sensi t iv i ty asscssmcnt.

The dcvelopment of numerical methods for carry ingout the init ial postbuckl ing analysis seems l ikely to pro-grcss along the l ines of both the f ini te dif ference pro-ccdure IRef. 1301 and the f in i te element method

IRefs. 80, 194]. An interest ing scheme has been putforward for approximately accounting for nonl inear pre-buckl ing behavior in an init ial postbucki ing analysis in away which exploits the computational advantages of thef in i te element procedure IRefs. 195, 196] . The idea be-hind the scheme is the treatment of the prebuckl ingnonlincari ty as a general ized imperfection.

Prel iminary attempts have been made to come togrips with somc of the stat ist ical aspects of imperfection-sensi t iv i ty IRefs. 197-199]. A rather complete analysis

IRcf. 200] of an inf inite beam with random init ial de-f lect ions resting on a nonl inear elast ic foundation yieldsa relat ionship between the buckl ing load of the beam andthe root mean square of the imperfection ampli tudeswith an implici t dependence on the correlat jon func-t ion for the imperfection. Calculat ions for the buckl ingload of an inf inite cyl indrical shel l under axiai compres-sion with random axisymmetric imperfections have beenmade IRefs. 20I, 202]. Here again, there emergeasymptotic formulas similar to eqs. (2) and (3) whichnow relate the buckl ing load to a value of the imperfec-t ion power spectral density at a part icular frequencywhich corresponds to the frequcncy of the classicalaxisymme tr ic mode IRef. 201] . Thcre is st i l l a long wayto go before results such as thcsc wil l be useful to thestructural cngineer, but an encouraging f irst stcp in thisdirect ion has been taken IRef. 2031.

Important extensions of postbucki ing theory remainto be made to include buckl ing under dynamicai ly ap-pl ied loads and buckl ing in the plast ic range. Dynamicbuckl ing calculat ions are not new, but no unifyingtheory is avai lable which is at al l as comprehcnsive asthat for stat ic conservative loadings. The init ial post,buckl ing analysis has been broadened in an approximatefashion to include dynamic ef fects IRefs. 95-97], andsome simple results with general implications havebeen found.

Buckl ing in the plast ic range is a phenomenon that isnot yet ful ly understood. Nonuniqueness of incrementalsolut ions in plast ic branching problems starts at thelowest bifurcation load, usually without loss of stabi l i ty,and a further load increase is required init ial ly in thepostbuckl ing range [Rcfs. 204, 205). A detai led dis-clrssion of the famous column controversy amongConsiderc, Engesser, von Karman and Shanley IRefs.206-2711 is given in [Ref.2121. ln the case ofplatesand shel ls the problem is evcn more complicated becausethe physical iy unacceptable deformation theory ofplast ici ty yields results in better agreement with exper-iments than thc simplest f low theory. Even i f f low theoryis ccrtainly more acceptable from the physical point of

view, i t is somewhat doubtful , however, that the s implest

J 2 theory would be adequare in bucki ing problems. Tosome extent th is discrepancy also has been explained byOnat and Drucker IRef. 213] , by proper al lowance forin i t ia l imperfect ions in their s imple example. Even moresigni f icant, perhaps, are Hutchinson's recent resul ts onthe postbuckl ing behavior in the plast ic range of struc-tures which are highly imperfect ion-sensi t ive in theelast ic domain [Ref. 2141 . l t is found in his examples(a s imple structural model and a spher ical shel l underexternal pressure) that the sensi t iv i ty to imperfect ionsis equal ly marked in the plast ic range, in spire of theinitial rise in the load at the lowest bifurcation point ofthe perfect structure. A s imi lar phenomenon has beenobserved by Leckie IRef. 215] in the postcoi lapse be-havior of a r ig id-plast ic spher icai shel l cap. Hutchinson,sanalysis also conf i rms the ear l ier f indings IRef. 213]that the discrepancy betweer.r the predict ions of deforma-t ion theory and f low theory largely disappears in thepresence of imperfect ions.

References

[1] Gr igol 'uk, E. I . , and Kabanov, V. V., "Stabi l i ry of Cyl in-dr ical Shel ls," ( in Russian), Ser ies Mechanics 8, Nauki , Moscow,L969.

l ,2 l Wagner, H., "Ebene blechwandtrager mit sehr dunnemstegblcch," Zeitschr. f. I:lugtechn. u. Motorluftschiffahrt 2O,200, 227 , 256 , 27 9 , 306 , t929 .

L3] Cox, H. L. , "Buckl ing of th in plares in compression,"Acron. Res. Comm., R. and M. 7554,1,934.

l,4l Marguerre, K., and Trcfftz, E., "Ubcr die tragfahigkcite ines langsbclasteten plat tenstrei fens nach. Uberschrei ten derbeul last ," Zei tschr. fur Angew. Math u, Mech. 17. 85-1OO, 1.937.

t5] Marguerre, K. , "Die mit t ragcndc brei te der gcdrucktenplatte," Lu ftf ahr t for s chung 1 4, 121-1 28, 19 37 .

t6] Kromnt, A. , and Marguerre, K. , "Verhal ten eines vonschub- und druckkraf ten beansoruchten olat tenstrei fens ober-halb der beulgrenzc," Luf t fahrt f i rschung 14, 627-639, 1937.

I7l Fr iedr ichs, K. O., and Stoker, J. J. , "The nonl inearboundary value problem of the buckled plate," American J. ofMath. 63, 839-888, 1 941.

t8l Fr icdr ichs, K. O., and Stoker, J. J. , "Buckl ing of thccircular plate beyond the critical thrust," l. Appl, Mech. g,A7-414, t942.

t 9] Bodner, S. R., "The postbuckl ing behavior of a c lampcdcircular plate," Quart . Appl . Math. 12, 397-401.,1955; AMR 8(79 55), P.ev. 2677 .

[10] Kuhn, P., " Invest igat ions on the incompletely dc-veloped plane diagonal tension f ie ld," NACA Rcport 697,1940.

[11] Kuhn, P. and Peterson, J. P. , "strength analysis ofst i f fened bcam webs," NACA Tech. Notc 1364, 1947; AMR 1( 1948), Rev. 634.

l12l Levy, S., "Large def lect ion theory for rectangularplates," Proc. First Syrnp. Appl . Math. , Am. Math. Soc.,197-210,1949; AMR 3 (1950), Rev. 1067.

[13] Levy, S., "Bending of rectangular plates wi th largedef lect ions," NACA Rcport 737, 1942.

114l Levy, S., Fienup, K. L. and Wool ley, R. M., "Analysisof square shear web above buckl ing load," NACA Tech, Note962,1.945.

[15] Lcvy, S., Wool ley, R. M., and Corr ick, J. N., "Analysisof deep rectangular shear web above buckl ing load," NACATech. Note 1009,1946.

[16] Cox, H. L. , "The buckl ing of a f lat rectangular platcunder axial comprcssion and i ts behavior af tcr buckl ing," Aeron.Rcs. Counci l , R. and M. 2041,,1945.

1361

t.[17] Hemp, W. S., "Theory of f lat panels buckled in com-

pression," Aeron. Res. Counci l , R. and M, 2778, 1,945.

[18] Leggett , D. M. A., "The stresses in a f lat panel undershear when the buckl ing load has been exceeded," Aeron. Res.Counci l , R. and M. 243O, t95O; AMR 4 (1951), Rev. 2455.

[19] Benthem, J. P. , "The reduct ion in st i f fness of com-binations of rectangular plates in compression after exceedingthe buckl ing 1oad," NLL Report S 539, Reports and Trans-actions, Nationaal Luchwaartlaboratorium 23, 1959:, AMR 15(1962), Rev.6976.

[20] Botman, M., and Bessel ing, J. F. , "The ef fect ive widthin the plast ic range of f lat p lates under compression," NLL Re-port S 445, Reports and Transactions, Nationaal Luchtvaart-laborator ium 19, S27-560, I954; AMR I (1956), F.ev. 2774.

[21] Botman, M., "The effective width in the plastic rangeof f lat p latcs under compression (part 3) ," NLL Report S 465,1955; AMR 10 (1957), Rev. 680.

[22] F1oor, W. K. G., and Burgerhout, T. J. , "Evaluat ion ofthe theory on the postbuckling behavior of stiffened, flat, rect-angular plates subjected to shear and normal 1oads," NLL Re-por l S 370, Reports and Transact ions, Nat ionaal Luchtvaart-laborator ium 16, S9-S23, 1951; AMR 6 (1953), Rev. 3375.

[23] Floor, W. K. G., " Invest igat ion of the postbuckl ingeffcctive strain distribution in stiffened, flat, rectangular platessubjected to shear and normal loads," NLL Report S 427,Fte-ports and Transactions, Nationaal Luchwaartlaboratorium 19,S1-S26, 1953; AMR 9 (1956), Rev. 716.

[24] Floor, W. K. G., and Plantema, F. J. , "Dc mecdragcndebreedte van vlakke platen; verbetering van de theorie vanMarguerre" ("The effective width of flat plates; improvement ofMarguerre 's thcory") , NLL Report S 262, 1942.

l25l Koi ter , W. T. , "De mecdragcnde breedte bi j groteoverschrijding der knikspanning voor verschillende inklemmingder olaatranden" ("The effective width at loads far in excess ofthe cr i t ical load for var ious boundary condi t ions") , NLL Reports 287. 1943.

[26] Koi ter , W. T. , "Het schui fp looiveld bi j grote ovcr-schrijdingen van de knikspanning" ("The incomplete tensionfield at stresses far in excess of the critical shear stress"), NLLReport S 295,1,944.

[27] Neut, A. van der, "Expcr imental invest igat ion of thepostbuckling behavior of flat plates loaded in shear and com-pression," Proc. 7th Int . Congr. App1. Mech., London 1,174-1,86,1948; AMR 2 (1949), Rev. 995 and 3 (1950), Rev. 1671.

[28] Kuijpers, W. J. J., "Postbuckling behavior of a plarestrip with clamped edges under longitudinal compression" (inDutch), Report 231, Laboratory of Appl . Mech., Del f t , 1962.

[291 Koi ter , W. T. , " Introduct ion to the postbuckl ingbehavior of f lat p lates" in "Comportement postcr i t ique desplaques ut i l isces en construct ion metal l ique," Memoircs de laSociete Royale des Sciences de Licge, 5me ser ie, tome VII I ,fasciculc 5, 1963.

[30] Karman, Th. von, and Tsicn, H. S., "The buckl ing ofspherical shells by external prcssure," J. Aero. Sci- 7, 43-50,L939.

t31] Karman, Th. von, Dunn, L. G., and Tsicn, H. S., "Theinfluenie of curvature on thc buckling characteristics of struc-

tures," / . Aero. Sci . 7,276,1940.

l32l Karman, Th. von, and Tsien, H. S., "Thc buckl ing of

th in cyl indr ical she1ls under axial compression," J. Aero' Sci .8,

303, 1941.

t33l Tsien, H. S., "A thcory for the buckl ing of th in shcl ls,"

J. Aero. 5ci .9,373, 1,942.

1.341 Koi tcr , W. T. , "Over dc stabi l i te i t van het elast isch

cvenwicht" ("On the stability of elastic equilibrium"), Thcsis,

Del f t , H. J. Par is, Amstcrdam, 1945. Engl ish t ranslat ion issued

as NASA TT F-10, 833,1967.

t35] Fung, Y. C. and Sechler, E. E. , " Instabi l i ty of th in

elrs i ic shcl ls," Structural Mechanics, Proc' First Symp. Naval

Struct . Mech., edi ted by J. N. Goodier and N' J. Hoff , Pergamon

Prcss, 115, 1960; AMR 14 11961) ' Rev. 2486'

[36] Volmir , A. S. , "Stabi l i ty of Dcformable Systenrs,"( in Russian), Nauka, Moscow, 1967.

l37l Herrmann, G., "Stabi l i ty of equi l ibr ium of elast ic sys-tems subjected to nonconseruat ive forces," AMR 20. 103-108,1967.

[38] Ncmat-Nasser, S. , "Thermoelast ic stabi l i ty undcr gcn-eral loads," AMR 23, 615-625,1970.

[39] Thompson, J. M. T. , "A gcncral thcory for thc eqrr ilibrium and stability of discrete conservative systems," Zeit, furMath. und Phys. 2O,797-846,1969.

[40] Stein, M., "Thc phenomenon of change in bucklepattern in elast ic structures," NASA TR R-39, 1959; AMR 13(1960), Rev.6268.

t41] Skaloud, M., "Cr i tere de l 'etat l imi te dcs plaqucs etdes systemes de plaqucs" in "Comportenrcnt postcr i t ique desplaques ut i l isees en construct ion metal l iquc," Mcmoires de laSociete Royale des Sciences de Liege, 5n1e ser ic, tomc VII I ,fascicule 5,1963.

l42l Masur, E. F. , "Buckl ing, postbuckl ing and l imi t anal-ysis of completc ly symmetr ic elast ic structuras," Int . J. Sol idsS truc t. 6, 587 -604, I97 0.

t43l Koi ter , W. T. , "On the concept of stabi l i ty of equi-l ibr ium for cont inuous bodies," Proc. Kon. Ned. Ak. Wct, B66,1.73-1.77,1963; AMR 18 (1965), Rev. 3966.

[44] Koi ter , W. T. , "On thc instabi l i ty of cqui l ibr ium in theabse nce of a minimum of thc potcnt ia l cnergy," Proc. Kon. Ncd.Ak. Wct, 868, 107-113, 1965.

[45] Koi ter , W. T. , "Thc cncrgy cr i ter ion of stabi l i ty forcont inuous elast ic bodies," Proc. Kon. Ned. Ak. Wct,868,778-202, t965.

146l Koi ter , W. T. , "Purpose artd achicvcments of researchin c last ic stabi l i ty ," Proc. Fourth Technical Confercnce, Soc, forEngng. Sci . , North Carol ina State Univcrsi ty, Raleigh, N. C.,1966.

l47l Southwcl l , R. V., "Theory of c last ic i ty," Sccond Ed.,Oxford Univ. Press, 1941.

[48] Carr ier , G. F. , "On the buckl ing of e last ic r i r rgs,"

J. Math. Phys. 26,94-103,1947; AMR 1 (1948), Rev. 256.

[49) Koi tcr , W. T. , "Buckl ing and postbuckl ing behavior ofa cyl indr ical panel under a-xial comprcssion," NLR Rcport S 476,Rcports and Transact ions Nat ional Aero. Rcs. Inst . 20, Amster-dam, 1956;AMR 11 (1958), Rev.819.

[50] Koi ter , W. T. , "Elast ic stabi l i ty and postbuckl ingbchavior," Proc, Synrp. Nonl inear Problcms,257-275, Univ. ofWisconsin Press, Madison, 1963.

[51] Koi ter , W. T. , "The ef fcct of u isymmctr ic imperfec-t ions on the buckl ing of cyl indr ical shel ls under axial compres-sion," Proc. Kon. Ncd. Ak. Wct,866,265, 1963; AMR 18(1965), Rev.3387.

I52l Koi ter , W. T. , "General equat ions of e last ic stabi l i tyfor th in shel ls," Proc. Symp. in honor of L loyd H. Donnel l ,edi ted by D. Muster, Univ. of Houston, 1966.

[53] Koi tcr , W. T. , "Postbuckl ing analysis of a s imple two'bar f rame," Rccent Progrcss in Appl icd Mcchanics, Almqr.r ist andWikscl l . Stockholn. 1966.

[54] Koi ter , W. T. , "A suff ic ient condi t ion for the stabi l i tyof shal low shcl ls," Proc. Kon. Ncd. Ak. Wct, 870, 367-375,1.967.

[55] Koi tcr , W. T. , "The nonl inear buckl ing problem of acompletc spher ical shcl l under uni form external pressure," I , I I ,I I I and lV, Proc.-Kon. Ned. Ak. Wet,B72, 40-1.23, L969.

[56] Mansf ie ld, E. H., "Some ident i t ies on structural f lex,ib i l i ty af ter buckl ing," Aero. Quart . 9, 300-304, 1958;AMR 12(1959), Rev.3312.

l57l Thompson, J. M. T. , "Stabi l i ty of e last ic structuresand tlreir loading deviccs," J. Mech. l|ngineer. Sci. 3, 753-162,196 1 ; AMR 15 (7962), Rev. 5189.

[58] Thompson, J. M. T. , "Basic pr inciplcs in the generaltheory of elastic stability," J. Mech. Phys. Solids 11, 73-20,1963;AMR 16 (1963), Rcv. 5731.

1362

t59] Br iwec, S. J. , and Chi lver, A. H., , ,Elast ic buckl ing ofrigidly-jointed braced framcs," J. E"g. Mech. Diu.,4SCE 89,21.7, 1963;AMR 17 (1964), Rev. 5667.

160] Britvec, S. J., "Overall stability of pin-jointed frame-works after thc onset of elastic buckling," Ing, Arch, 32,443-452, 1963 ; AMR 17 (1.964), P.ev. 7 047.

[61] Br i tvec, S. J. , "Elast ic buckl ing of p in- jo inted frames,"Int . J. Mech. Sci . 5,447-46O, 1963;AMR 17 (1964), Rev. 5666.

162l Thompson, J. M. T. , "Eigenvalue branching conf igura-tions and the Rayleigh-Ritz proccdure," QuarL AppL Math. 22,244-251.,1964;AMR 18 (1965), Rev. 1447,

[63] Thompson, J. M. T. , "Thc rotat ional ly-symmetr icbranching bchavior of a cornplcte spher ical shel l , " Proc, Kon.Ncd. Ak. Wet, 867, 295-311,1,964.

t641 Thompson, J. M. T. , "Discrete branching points inthe gencral theory of elastic stability," J. Mech. Phys. Solids 13,295-370,1965; AMR 19 (1966), Rev. 6213.

[65] Thompson, J. M. T. , "Dynamic buckl ing under steploading. Dynamic stabi l i ty of structures," Proc. of an Inter-nat ional Confercncc, Northwestern Univ. , Evanston, I l l inois,Pergamon Press, 1965.

[66] Sewcl l , M. J. , "The stat ic perturbat ion technique inbuckl ing problenis," / . Mech. Phys. Sol ids 13,247-265,7965.

167l Roorda, J. , "The buckl ing behavior of impcrfect struc-tural systems," J. Mech. Phys. Sol ids 13,267-28O, '1,965.

[68] Roorda, J. , "Srabi l i ty of structurcs wi th smal l im-perfcct ions," J. I lng. Mech. Div. , ASCI| 91,87-1.06,1965;AMR18 (1965), Rcv.4035.

[69] Sewel l , M. J. , "On the connect ion bctween stabi l i tyand tlre shape of the equilibrium surfacc,"/. Mech. Phys. Solids4, 203-230, 1966 ; AMR 20 (1967 ) , Rev. 3978.

[70] Chi lver, A. H., "Coupled modes of e last ic buckl ing,"J. Mech. Phys. Sol ids 15, 15-28, 1967; AMR 2O (1967), Rev.77 55.

I71l Supple, W. J., "Coupledthe elast ic buckl ing of symtnetr icNlech. Sci . 9,97-112, 1967.

branchirrg conf igurat ions in

structural systems," 1nl . /

1.721 Godley, M. H. R., and Chi lver, A. H., "The clast icpostbuckl ing bchavior ofunbraced plane frames," Int , J. Mech.Sci . 9, 323-333, 1967; AMR 21 (1968), Rcv. 968.

173l Supplc, W. J. , and Chi lver, A. H., "Elast ic postbuckl ingof compressed rectangular plates, contr ibut ion to th in-wa11edstructures," cdi ted by A. H. Chi lvcr, Chatto and Windus, 1967.

174l Thompson, J. M. T. , "The est i rnat ion of c last ic cr i t icalloads," / . |v lech. l>hys. Sol ids 15, 311-31.7, 1967; AMR 21( i968), Rev.4813.

[75] Popc, G. G., "On the bi furcat ional buckl ing of e last icbeanrs, p lates and shal low shel ls," ,4ero Quart , 19,20-30,7968;AMR 22 (1969), Rcv.3192.

176l Pope, G. G., "The buckl ing behavior in axial corn-pression of s l ight ly-curved panels including the ef fcct of sheardeformabi l i ty ," Int . J. Sol ids Struct . 4, 323-340, 1968; AMR21 (1.968), Rev. 5611.

177) Sewel l , M. J. , "A general theory of equi l ibr ium pathsthrough cr i t ical points," Part I , Proc. Roy. Soc. 4306.201-223;Part 11, Proc. Roy. Soc. A306, 225-238, 1,968.

[78] Walkcr, A. C., and Hal l , D. G., "An analysis of thelarge de flections of beams using the Rayleigh-Ritz finite elementmethod," Aero Quart . 19,357-367,1968;AMR 22 (1969),P.ev.845 5.

179) Roorda, J. , "Or the buckl ing of symmetr ic structuralsystems with first and second order imperfections," Int, J,Sol ids Struct . 4, 1737-1148,1968;AMR 22 (1969), Rev. 7731.

180] Walker, A. C., "An analyt ical study of the rotat ionai ly-symrnetr ic nonl incar bucki ing of a complete spher ical shel l underexternal pressure," Int , J, Mech. Scl . 10, 695-710, 1968; AMR22 (1.969), Rev.9395.

[81] Thompson, J. M. T. , and Walker, A. C., "The non-l inear perturbat ion analysis of d iscrete structural systems," 1r.?t

J. Sol ids Struct . 4,757-768,1968; AMR 22 (1969), Rcv. 4254.

[82] Thompson, J. M. T. , and Hunt, G. W., "Comparat iveperturbation studics of the elastica," Int. J. Mech. Sci. 11,999-L01.4, t969.

[83] Thompson, J. M. T. , and Walkcr, A. C., "A gcneraltheory for the branching analysis ofdiscrete structural systens,"Int . J. Sol ids Struct .5,281,-288, 1969;AMR 22 (1969), Rcv.9384.

[84] Thompson, J. M. T. , and Hunt, G. W., "Pcrturbat ionpatterns in nonl inear branching theory," IUTAM Symp. on In-stabi l i ty of Cont inuous Systems, Hcrrenalb, Spr inger-Ver lag,1969.

[85] Roorda, J. , and Chi lver, A. H., "Frame buckl ing: anillustration of the perturbation techniquc," Int. J. NonlinearMech. 5,235-246, 1970.

[86] Supplc, W. J. , " In i t ia l postbuckl ing behavior of a c lassof elast ic structural systcms," Lnt, J. Nonl inear Mech. 4, 23-26,1969 AMR 23 (1970), Rcv. 2668.

[87] Thompson, J.M.T., "Thc branching analysis of pcr-fcct and imperfect discretc structural systems," J. Mech. Phys.Sol ids 17, 1-10,1,969 AMR 22 (1969), Rev. 9385.

[88] Walker, A. C., "A nonl inear f in i te elemcnt analysis ofshal low circular arches," Lnt . J. Sol ids Struct . 5.97-107, 1,969;AMR 22 (1969), Rev. 8507.

[89] Walker, A. C., "The postbuckl ing bchavior of s implysupported square plates," Aero Quart , 20, 203-222, 1969; AMR23 (1970), Rev.3434.

[90] Thompson, J. M. T. , "Basic theorems of e last ic stabi l -i ty ," Int . J. Engng. Sci . 8,307,797O.

[91] Huseyin, K. , "Fundamcntal pr inciplcs in the buckl ingof structurcs under combined loading," Int . J. Sol ids Struct . 5,479,1969.

l92l Huseyin, K. , "The convcxi ty of the stabi l i ty boundaryof symmetr ic structural systems," Acta Mech. 8, 188-204, 1,970.

[93] Huseyin, K. , "The elast ic stabi l i ty of structura- l sysremswith indcpcndent loading parameters," Int , J, Sol ids Struct , 6,677, 1970.

l94l Beaty, D., "A theoret ical and exper incntal study ofthe stability of thin elastic spherical shclls," Thesis, Univ. ofMichigan, 1962.

[95] Budiansky, 8. , and Hutchinson, J. W., "Dynamic buck-ling of imperfection-sensitive structures," Proc. Xl Internat.Congr. Appl . Mech., Munich, 636-651, Jul ius Spr inger-Ver lag,Bcr l in, 1964; AMR 19 (1966), Rev. 7709.

[96] Hutchinson, J. W., and Budiansky, B., "Dynamicbtrckl ing est imates," ,41,4 A J. 4, 525-530, 1966; AMR 20 (7967),Rev. 930.

197) Budiansky, B., "Dynamic buckiing of elastic struc-tures: cr i ter ia and est imates," "Dynanic stabi l i ty of structures,"Proc. of an Internat, Conf. , Northwestern Univ. , Evanston,I l l inois, Pergamon Press, 1965,

[98] Hutchinson, J. W., "Axial buckl ing of pressur izcd im-perfect cylindrical shells,",41,4 A J. 3, 1461-1466, 1965.

[99] Budiansky, B., and Hutchinson, J. W., "A survey ofsome buckl ing problems," AIAA l . 4, 1505-1510, 1966; AMR20 (1967) , F.ev. 237 3.

11001 Hutchinson, J. W., " In i t ia l postbuckl ing behavior oftoroidal shcl1 segments," Int . J. Sol ids Struct . 3, 97-115, 7967;AMR 20 (1967), Rev. 4687.

[101] Hutchinson, J. W., " Imperfect ion-sensi t iv i ty of ex-ternally pressurized spherical shells," /. Appl. Mech. 34. 49-55,1967; AMR 20 (1967\, Ftev. 7754.

11,021 Hutchinson, J. W., and Amazigo, J. C., " Imperfcc-tion-sensitivity of eccentrically stiffened cylindrical shells," AI AAJ.5,392-401.,1967; AMR 23 (1970), Rev. 6069.

11031 Budiansky, 8. , and Amazigo, J. C., " In i t ia l post-buckl ing behavior of cyl indr ical she1ls under external pressure,"J. Math. Phys. 47,223-235,1968; AMR 22 (1,969), Rev. 9399.

[ 104] Fi tch, J. R., "The buckl ing and postbuckl ing bchaviorof spherical caps under concentrated load," Int. J, Solids Struct.4,421-446,1968;AMR 22 (1,969), Rev. 4104. Sce also,41,4,4 / .7,7407,1.969.

1 363

[105] Hutchinson, J. W., "Buckl ing and in i t ia i postbuckl ingbehavior of oval cylindrical she1ls under axial compression," /,Appl . Mech. 35,66-72, 1968; AMR 21 (1.968), Rev. 8072.

[106] Amazigo, J. C., "Buckl ing under axial compressionof long cylindrical shells with random axisymmetric imperfec-t ions," Quart . Appl . Math. 26, 538-566, 1969; AMR 22 (1,969),Rev.6785.

[107] Budiansky, B., "Postbuckl ing behavior of cyl indersin tors ion," Proc. Second IUTAM Symp. on the Theory of ThinShel ls, Copenhagen, edi ted by F. I . Niordson, Spr inger-Vcr lag,1969.

[108] Danielson, D. A., "Buckl ing and in i t ia l postbuckl ingbchavior of spher iodal shel ls under pressure," AIAA J, 7,936-944,1969; AMR 23 (r970), Rev. 964.

[109] Hutchinson, J. W., and Frauenthal , J. C., "Elast icpostbuckl ing behavior of st i f fened and barreled cyl indr icalshel ls," / . Appl . L lech. 36, 7 84-790, 1,969.

[110] Fi tch, J. R., and Budiansky, 8. , "The buckl ing andpostbuckl ing of spher ical caps under axisymmetr ic load," AIAA

J. 8,686-692,7970.

[111] Dow, D. A., "Buckl ing and postbuckl ing tests of r ingst i f fcned cyl inders loaded by uni form pressute," NASA TND-31i 1, Novernber, 1965.

11121 Kcmpncr, J. , and Chen, Y.-N., "Large def lect ions ofan axial ly compresscd ovai cyl indr ical shcl l , " Proc. XI Internat.Congr. Appl . Mech., Munich, 299, Jul iLrs Spr inger-Ver lag, Bcr l in1964; AMR 20 (1967), Rcv. 9249.

[113] Kempner, J. , and Chen, Y.-N., "Buckl ing and post-buckl ing of an axial ly comprcssed oval cyl indr ical shel l , " Proc.Symp. on the Theory of Shcl ls to honor Lloyd H. I )onnel l , Univ.ofHouston,1967.

11141 Kempner, J. , and Chen, Y.-N., "Postbuckl ing of anaxial ly cornpressed oval cyl indr ical shel l , " Proc. Twelf th Int ,Congr. Appl . Mcch,, Stanford Univ. 1968, Spr inger-Ver lag, 1969.

[115] Tennyson, R. C., Booton, M., and Caswcl l , R. D.,"Buckl ing of imperfect e l l ipt ical cyl indr ical shel ls under axialcompression," to bc publ ished in AIAA J.

[116] Neut, A. van der, "Gcneral instabi l i ty of st i f fenedcyl indr ical shcl ls under axial compression," Report S 314,Nat ional Aeron. Rcs. Inst . (NLR), Amstcrdarn, 1947,

111.7) Card, M. F. , and Joncs, R. M., "Exper inrental andtheoret ical resul ts for buckl ing of eccentr ical ly st i f fened cyl in-ders," NASA TN D-3639, October, 1966;AMR 20 (1967), Rcv.922.

[118] Brush, D. O., " Impcrfcct ion-sensi t iv i ty of str inge rst i f fened cyl inders," ALAA J. 5, 391-401, 7967.

[119] Garkisch, H. D., "Exper imentel le untersuchung desbculvcrhal tcns von kreiszyl indcrschalen mit exzcntr ischen langs-vcrstc i fungen," Dcutschc Luft-u. Raumfahrt , Forsch., Ber, DLRFB 67-75,1967;AMR 22 (1969), Rcv. 3211.

[120] Thielernann, W. F. , and Essl inger, M. E., "Post-buckl ing equi l ibr iurn and stabi l i ty of th in wal led c i rcular cyl in-ders under axial compression," "Theory of th in shcl ls." Edi tor ,F. I . Niordson, Spr inger-Ver lag, 1969.

1121 I Singer, J. , Baruch, M., and Harar i , O., "On the stabi l -i ty of eccentr ical ly st i f fened cyl indr ical shel ls undcr axial com-prcssion," Int . J. Sol ids Struct .3,445-470, 1967; AMR 21(1968), Rcv. 169.

[122] Singer, J. , "The inf luence of st i f fencr geometry andspacing on the buckl ing of u ia l ly compressed cyl indr ical andconical shcl ls," Proc. Second IUTAM Symp. on the Theory ofThin Shel ls, Copenhagen, 1967, edi ted by F. I . Niordson,Spr ingcr.Vcr lag. I 969.

11231 Petcrson. J. P. , "Buckl ing of st i f fcncd cyl inders inaxial compression and bending, a revicw of test data," NASATN D-5561, Dccember 1969; AMR 23 (1970), Rcv. 2663.

[124] Ncut, A. van der, "The interact ion of local buckl ingand column fai lure of th in-wal led compression members," Proc.Twelf th Int . Congr. Appl . Mcch., Stanford Univ. , 1968, Spr inger-Ver lag, 1969.

11251 Graves Sni th, T. R., "The ul t imate strength oflocal ly buckled columns of arbi t rary lcngth," in "Thin wal led

steel sfuctures," edi ted by K. C. Rockey and H. V. Hi11, CrosbyLockwood and Son, Ltd. , London, 1969.

[1,26] Budiansky, B., "Notes on non- l inear shel l theory,"

J. Appl . Mcch. 35,393-401,,1968; AMR 22 (1969), Rev. 5028.

11,271 Almroth, B. O., and Bushnel l , D., "Computer analysisof var ious shel ls of revolut ion," AIAA J. 6, 1848-1856, 1968;AMR 22 (1969), Rev. 5052.

[128) Stcin, M., "Recent advances in the invest igat ion ofshel l buckl ing," AIAA J. 6, 2339-2346, 1968; AMR 22 (1969),Rcv.6787.

[129] Cohen, G. A., "Ef fect of a nonl inear prcbuckl ingstate on the postbuckl ing behavior and in ipcrfect ion-scnsi t iv i tyof c last ic structures," AIAA J. 6, L616-1.620, 1968; AMR 22(1969), Rev. 2397. Sce also AIAA J. 7 , 1407-8, 1.969.

[130] Cohen, G. A., "Computer analysis of impcrfect ionsensi t iv i ty of r ing-st i f fened orthotropic shel ls of revolut ion," tobe publ ished ) ,nAIAA J.

11311 Lcggctt , l ) . M. A., and Jolcs, R. P. N., "The be-havior of a cyl indr ical shel l under axial cornprcssion when thebuckl ing load has been exceeded," Aero. Res. Counci l , London,R. and M., 2190,1,942.

l I32l Cicala, P. , " I l c i l indro in parete sot t i le compressoassial inente. Nuovo or ientamcnto del f indagine sul1a stabi l i taelast ica," L 'A erotec nica- 24, 3-t8, 19 44,

[133] Michielsen, H. F. , "The bcl iavior of th in cyl indr icalshe1ls af ter buckl ing under uial compression," J. Aero. Scl . 15,738,1948;AMR 2 (1949), Rev. 1377.

[134] Donnel l , L. H., and Wan, C. C., "Effcct of imperfec-t ions on buckl ing of th in cyl inders and columns under axialcompressions," J. AppL Mech. 17, 73, 1950t AMR 4 (1951),Rev.635.

[ 135] Cicala, P. , "The ef fect of in i t ia l deformat ions on thebehavior of a cyl indr ical shel1 undcr axial compression," Quart .Appl . Math. 9, 273-293, 1951; AMR 5 (1952), Rev. 1341.

[136] Lo, H., Crate, H., and Schwartz, E. 8. , "Buckl ing ofth in-wal lcd cyl inder undcr axial compression and intcrnal pres-sure," NACA Report 1027, 1951; AMR 5 (1952), Rcv. 2589.

[137] Kempner, J. , "Postbuckl ing bchavior of axial ly com-prcsscd circular cyl indr ical shel ls," / . Aero. Sci . 21,329,1954;AMR 7 (1954), Rev. 3171.

[ 138] Loo, T. T. , "Ef fccts of large def lect ions and imperfcc-t ions on the elast ic buckl ing of cyl inder under tors ion andaxial compression," Proc. Second U,S. Nat. Congr. Appl . Mech.,345-357,1954; AMR 9 (1956), Rcv. 1421.

[139] Kempner, J. , Pandalai , K. A. V., Patcl , S. A. , andCrouzet-Pascal , J. , "Postbuckl ing bchavior of c i rcular cyl indr icalshel ls under hydrostat ic pressure," J. Aero. Sci , 24, 253-263,1957; AMR 10 (1957), Rev. 3596.

[140] Nash, W. A., "Buckl ing of in i t ia l ly imperfect cyl in-drical shclls subjcct to torsion," J. Appl. itlech. 24, 125-130,1957; AMR 10 (1957), Rev. 2084.

1141] Thielemann, W. F. , "New developments in thc non-l inear theor ies of the buckl ing of th in cyl indr ical shel ls," Aeron.and Astro. 76-121, Pergamon Press, New York, 1960.

11,421 Lu, S. Y., and Nash, W. A., "Buckl ing of in i t ia l lyimperfect axial ly compressed cyl indr ical shel ls," NACA Tech.Note D-15 10, 1962.

11431 Kempner, J. , "Some resul ts on buckl ing and post-buckl ing of cyl indr ical shel ls," NASA TN D-1510, 173,1962.

[144] Babcock, C. D., and Sechler, E. E. , "The ef fect ofin i t ia l imperfect ions on the buckl ing stress ofcyl indr ical shel ls,"NASA TN D-2005, 1963; AMR 17 (1964), Rev. 2C05.

[145] Almroth, B. O., "Postbuckl ing behavior of u ia l lycompresscd circular cyl indcrs," AIAA J. 1, 630, 1,963; AMR 16(1963), Rev. 4480.

11461 Almroth, B. O., "Postbuckl ing behavior of or tho-tropic cylinders under axial compression," AIAA J, 2, 1795-1799,1964;AMR 18 (1965), Rev. 2054.

11471 Sobey, A. J. , "The buckl ing strength of a uni formcircular cyl inder loaded in axial compression," Aero. Res.Cotrnci l , London, R. and M., 3366,1,964; AMR 17 (1964), Rev.5032.

1364

11481 Sobcy,, A. J. , "The buckl ing of an axial ly loadedcircular cyl indcr wi th in i t ia l impcrfcct ions," Royal Aircr . Est .Tcch. Rcpt. 64016, 1-964.

11491 Weingartcn, V. i . , and Seide, P., "Elast ic stabi l i ty ofth in-wal led cyl indr ical and conical shel ls undcr cornbincd ex-tcrnal prcssurc and axial compression," AIAA J, 3, 913-920,1965; AMR 18 (1965), Rcv.6678.

[150] Weingarten, V. L, Morgan, E. J. , and Seide, P.,"Elastrc stabi l i ty of th in-wal led cyl indr ical and conical shel lsunder combined internal pressure and axial compression," ,41-4,4J. 3, 11.1,8-71.25, 1965; AMR 18 (1965), Rev. 7 342.

[ 151] Hoff , N. J. , "The perplexing bchavior of th in c i rcularcylindrical shclls in axial compression," Israel J. of 'I 'ech. 4,1-28, 1966\ AMR 21 (1968), Rev. 173.

[152] Hoff , N. J. , Madsen, W. A., and Maycrs,buckl ing equi l ibr ium of axial ly compressed circularshel ls," ,4. l ,4,4 J. 4, 1 26-133, 1966.

[153 ] Meyer-Pienine, H., "Zv bcrcchnung vonlasten dunnevandiger kreiszyl inder endl ichcr langc,"der WGLR, 377-383, 1967 .

[154] Thielanann. W. F. , and Essl ingcr, M., "Beul-undnachbeulver-hal ten isotroper zyl inder unter ausscndruck," S tahlbau 36,167-175,1967.

[155] Narasimhan, K. Y., and l lof f , N. J. , "Calculat ion ofthe load carry ing capaci ty of in i t ia l ly s l ight ly impcrfect th in-wal led c i rcular cyl indr ical shel ls of f in i tc length." to bc pub-l ishcd in/ . App. AIech.

f 156] Dym, C. L. , and Hoff , N. J. , "Pcrturbat ion solut ionsfor the buckl ing problcms of axial ly comprcssed thin cyl indr icalshel ls of inf in i te or f in i tc lcngth," / . Appl . L lech. 35,754, 1"968.

[157) Thielemann, W. F. , and Essl ingcr, M. E., "On thepostbuckl ing behavior of th in-wal lcd axial ly compressed circularcyl indcrs of f in i tc lcngth," Proc. S1mp. on thc Thcory of Tlr inShel ls to honor Lloyd H. Donncl l , Univ. of Houston, 1968.

[158] Hoff , N. J. , "Sornc rccent studies of the buckl ing ofth i r r shel ls," Aero. J. of the Royal Aero. Soc. 73, 1057-1070,L969.

1159j Khot, N. S., "Buckl ing and postbuckl ing behavior ofcomposi tc cyl indr ical shcl ls undcr axial comprcssion," AIAA J.8, 229-235, 1.970.

[160] Khot, N. S., "Postbuckl ing bchavior of gcomctr ical lyimpcrfect composi te cyl indr ical shcl ls under uial comprcssion,"ArAA J. A,579,1.970.

[161] Alrnroth, B. O., Burns, A. B., and Pi ther, E. V. ,"Dcsign cr i ter ia for axial ly loaded cyl indr ical shcl ls," AtAA/ASME 11th Structures, Struct . Dynarnics, and Mats. Conf. ,Dcnvcr, Colorado, Apr i l 1970.

11621 Almroth, B. O., " ln{ lucncc of cdge condi t ions on t}rcstabi l i ty of axial ly comprcsscd cyl indr ical shel \s," AIAA J. 4,134-1.40,1966.

[163] Hoff , N. J. , "Low buckl ing strcsses of u ia l ly com-pressed circular cyl indr ical shcl ls of f in i tc length," / . Appl . Nlech.32,542-s46,1965.

11,64) Simmonds, J. G., and Daniclson, D. A., "Ncw rcsul tsfor thc buckl ing lcads of axial ly compresscd cyl indr ical shcl lssubject to relaxed boundary condi t ions," l . AppL Mech. 37,93 1.00, 1970.

[165] Hoff , N. J. , and Soong, T-C, "Buckl ing of axial lycomprcssed circular cyl indr ica. l shel ls wi th non-uni form boundarycondi t ions," in "Thin wal lcd stccl structurcs," cdi tcd by K. C.Rockcy and H. V. Hi l l , Crosby Lockwood and Son, Ltd. ,London,1969.

1166] Tennyson, R. C., "Photoelast ic c i rcular cyl indcrs inaxial comprcssion," ASTM STP 419, 1967 .

11671 Almroth, B. O., Holmcs, A. M. C., and Brush, D. O.,"An exper imental study of thc buckl ing of cyl indcrs undcraxial comprcssion," tsxper. Mach. 4,263, 1,964; AMR 18 (1965),Rcv.799.

[ ]68] Evcnscn, D. A., "High speed photographic obscrva-t ion of thc buckl ing of th in cyl inders," l ixper. Mech. 4, 110-117,1 964; AMR 17 (1964), Rcv. 5027.

[169] Tennyson, R. C., "Buckl ing modes of c i rcular cyl in 'dr ical shel l under axial compression," AIAA J.7,1481-1487,1969.

[170] Koi ter , W. T. , "Knik van schalen" (Buckl ing ofshcl ls) , de Ingenieur 76, 33,1964.

l17I l Essl inger, M., "Hochgeschwindigkei tsaufnahmenvombeulvorgang dunnwandiger, axialbelastcter zyl inder," Stahlbau39. 2-4, 1.970.

1L72) Almroth, B. O., " Inf luence of inrperfect ions andedge restraint on thc buckl ing of axial ly compresscd cyl inders,"NASA CR-432, Apr i l 1966;AMR 19 (1966),P.ev.4219.

[173] Tennyson, R. C., and Muggeridgc, D. B., "Buckl ingof axisymmetr ic imperfect c i rcular cyl indr ical shcl ls under axialcompression," AIAA J. 17, 2127-2131, 1969; AMR 23 (L970),Rev.3428.

11741 Arbocz, J. , and Babcock, C. D., "The ef fcct ofgen-eral impcrfections on the buckling of cylindrical shelIs," J. Appl.Mech. 36,28-38,1969.

11,751 Singcr, J. , Arbocz, J. , and Babcock, C. D., "Buck-l ing of impcrfect st i f fened cyl indr ical shel ls under axial com-pression," to bc publ ished inAIAA J.

l l76l Car lson, R. L. , Sendelbeck, R. L. , and Hoff , N. J. ,"Exper imcntal studies of the buckl ing of complete spher icalshel ls," Erper. Mech. 7, 28I-288, 1967; AMR 22 (1969), kcv.8508.

1177] Rcissner, E. , "On postbuckl ing behavior and impcr-fcct ion sensi t iv i ty of th in elast ic plates on a nonl inear c last icfoundat iorr ," Studies in Appl . Math. 49, 45-48, 1970.

1178] Reissncr, E. , "A note on postbuckl ing behavior ofprcssurizcd shallow spherical shclls," J. Appl. Mech. 37, 533-534,197 0.

1179] Thompson, J. M. T. , "The clast ic instabi l i ty of acomplete spher ical shel l , " Aero. Quart . 13, 189 201 , 1962; AMR15 (1962), Rev. 6977.

[180] Thurston, G. A., and Pcnning, F. A. , "Ef fcct ofaxisymmetr ic imperfect ions on the buckl ing of spher ical capsunder uni form prcssurc," AIAA J. 4,319-327,1966; AMR 20(1967), Rcv.6400.

[181] Bushnel l , D., "Nonl inearshcl ls of rcvolut ion," AIAA J, 5,(1967), Rcv.7083.

axisymmctr ic behavior of432-439, 1967; AMR 20

[182] Koga, T. , and Hoff , N. J. , "Thc axisymmerr ic snap-through buckl ing of th in-wal lcd sphcr ical shcl ls," lnf . J. Sol idsStruct . 5,679,7969 AMR 23 (1970), Rcv. 1807.

[183 I Kalnins, A. , and Bir ic ikoglu, V. , "On the stabi l i tyanalysis of impcrfcct spher ical shcl ls," to bc publ ishcd in thcJ. of Appl. Mech.

I 1841 Sewel l , M. J. , "A mcthod of postbuckl ing analysis,"J. 1\ '1ech. Phys. Sol ids 17,21.9-233, 1969.

[185j Thompson, J. M. T. , "A ncw approach to c last icbrarrching analysis," / . Mech. Phys. Sol ids 18,29-42,197O.

11861 Amazigo, J. C., Budiansky, 8. , and Carr icr , G. F. ,"Asymptot ic analyses of thc buckl ing of impcrfcct columns onnonlincar elastic for-rndations," Int. J, Solids and Struct. 6.1341.1356,1.970.

[187i Amazigo, J. C., and Frascr, W. B., "Buckl ing undcrcxtcrnal prcssurc of cyl indr ical shcl ls wi th dimplc shapcd in i t ia limperfcct ions," Harvard Universi ty Rcport SM-40, lu ly 1,970.

[188] Schi f fncr, K. , "Untersuchung dcs stabi l i tatsvcrhal tcnsdunnwandigcr kcgclschalcn untcr uialsymmctr ischcr bclastungmit tc ls c incr nicht l inearcn schalcntheor ic," Jahrbuch dcr WGI-R,448-453,1 965; AMR 20 (1967), Rcv. 8455.

[ ]89] Hutchinson, J. W., Tcnnyson, R. C., arrd Muggcr idgc,D. B., "Thc ef fcct of a local axisymmctr ic impcrfcct ion on thcbuckl ing bchavior of a c i rcular cyl indr icai shcl l undcr axial com-pression," to be publ ishcd in AIAA J.

[190] Famil i , J. , and Archcr, R. R., "Fini tc asymmetr icdeformat ion of sh:r l low sphcr ical shcl ls," ,4. f ,4,4 / . 3,506,510,1965; AMR 18 ( l 965), Rcv. 4648.

[191 I l ] rogan, F. , and Almroth, I ] . O., "Buckl ing of cyl in-dcrs wi t l r cutorr ts," AIAA J. 8,236 240,1970.

J. , , ,Posr-

cyl indr ical

nachbeul-

Jah rbuch

I 365

t-11,921 Almroth, B. O., Brogan, F. A. , and Marlowe, M. 8. ,

"Col lapse analysis for e l l ipt ic cones," to be publ ished in AIAA J.11931 Kao, R., and Perrone, N., "Asymmetr ic bucki ing of

spher ical caps with asymmetr ical imperfect ions," to be publ ishedinJ. Appl. Mech.

11"941 Thurston, G. A., "Cont inuat ion of Newton's methodthrough bi furcat ion point ," / . AppL Mech, 39, 425-430,1969.

[195] Haftka, R. T. , Mal let t , R. H., and Nachbar, W.,"Adapt ion of Koi ter 's method of f in i te element analysis ofsnap-through buckling behavior," to be published in Int. J.Solids and Struct.

[196] Haftka, R. T. , Mal let t , R. H., and Nachbar, W.,"Evaluat ion of a Koi ter- type f in i te element mcthod," to bcpubl ished.

11971 Bolot in, V. V. , "stat ist ical methods in thc nonl inearthcory of e last ic shel ls," NASA TT F-85, 1962.

[198] Thompson, J. M. T. , "Towards a gene ral star ist icaltheory of imperfect ion-scnsi t iv i ty in c last ic postbuckl ing," / .Mech. Phys. Sol ids 15, 413-417, i967; AMR 21 ( i968), Rev.407 5.

[199] Roorda, J. , "Somc stat ist ical aspccts of the buckl ingof imperfection-scnsitive structurcs," J. Mech. Phys, Solids 17,11.1-123,1969; AMR 23 (1970), Rcv. 954.

l2o0l Fraser, W. 8. , and Budiansky, B., "The buckl ing of acolunrn with random initial dcflections," /. Appl. Mech. 36,233 -240, 1 969 ; AMR 23 (197 O), Rev. 6054.

[2O1,] Amazigo, J. C., "Buckl ing under uial compression oflong cyl indr ical shel ls wi th random uisymmetr ic imperfect ions,"Quart . Appl . Math. 26, 537-566, 1969; AMR 22 (1969), Rev.6785.

12021 Fcrsht, R. S., "Almost sure stabi l i ty of long cyl in-dr ical shcl ls wi th random impcrfect ions," NASA CR-l161,August 1968.

[203] Tcnnyson, R. C., Muggcr idge, D. B., and Caswcl l ,R. D., "The ef fect of axisymmetr ic impcrfcct ion distr ibut ions

on the buckling of circuiar cylindrical shells under axial com-pression," AIAA/ASME 11th Structures, Structural Dynani ics,ard Materials Conference, Denver, Colorado, April 1970.

[204] Hi l l , R. , "A general theory ofuniqueness and stabi l -ity in elastic-plastic solids," J. Mech. physics of Solirls 6,236-249, 1958; AMR 11 ( i95s), Rev. 4966.

[205] Hi l l , R. , "Some basic pr inciples in the mechanics ofsol ids wi thout a narural t ime," / . Mech.-phys. Sot ic ls - l , 2Og-225,1959 AMR 14 (1961J,Rev. 1255.

[2061 Considere, "Resistance des pieces comprimecs,, ' Con-gres Internat ional des Procedes de Construct ion, 371-, Annexe,Librair ie Polytechniquc, Par is, 1891.

[207] Engesser, F. , "Ueber knickfragen,, , Schu,<, izeischeBauzei tung 26, 24, 1,895.

[208] Karman, Th. von, "Unrersuchunsen uber knick-festigkeit, mitteilungen uber forsch u ngsarbeitcn j' Vc rein Deu tsch-er Ingenieure 81, 1910.

l209l Karman, Th. von, "Discussion of paper , Inclast ic

colurrrn thcory ' by F. R. Shanley," J. Aero. Sci . 14-,267,1g47.12L0l Shanley, F. R., "The column paradox,, , / . Aero. Sci .

13,678,7946.

[21,1,1 Shanley, F. R., " Inelast ic column theory," J. Aero.Sci . 14, 261., 1"947; AMR 1 ( i948), Rev. 72.

121,21 Hoff, N. J., "Inelastic buckling of colurnns in thcconvcnt ional test ing machine," Proc, Symp. on the Theory ofShel ls to honor Lloyd H, Donncl l , Univ. of Houston, 7967.

12131 Onat, E. T. , and Druckcr, D. C., " Inelast ic instabi l i tyand incremcnta. l theor ies of p last ic i ty," / . Aero. Sci , 20, 181-186,1953; AMR 6 (1953), Rev. 3397.

l2 l4 l Hutchinson, J. W., "On the postbuckl ing bchavior ofimperfection-sensitive structures in the plastic rangc," HaruardUniversi ty Rcport SM-41, July 1970.

[215] Leckie, F. A. , "Plast ic instabi l i ty of a spl icr ical shcl l , "in "Theory of thin shells," editcd by F. I. Niordson, Springer-Vcr lag, 1969; AMR 22 (1969),Ptcv.7737.

1 366