Schmidt

Journal of Constructional Steel Research 55 (2000) 159181www.elsevier.com/locate/jcsr

Stability of steel shell structuresGeneral ReportHerbert Schmidt *

Department of Civil Engineering, University of Essen, D-45117 Essen, Germany

Abstract

The state-of-the-art of available knowledge about the stability design of steel shell structuresis discussed. Specific stress is put on the various approaches to a numerically based stabilitydesign. The European Prestandard ENV 1993-1-6 (the Shell Eurocode) is briefly described. 2000 Elsevier Science Ltd. All rights reserved.

Keywords: Shells; Stability; Buckling; Design; Numerical modelling

1. Introduction

When the author was invited to be General Reporter for Session 8 of SDSS 99he was informed that he had to write a state-of-the-art report on the Stability andDynamic of Shells. However, because he is not really an expert on dynamics ofshells, he decided to concentrate on the stability of shells. Accidentally this coincideswith the fact that no papers on the dynamics of shells have been submitted for theColloquium. Furthermore, the author has added into the title of this General Reportthe words steel and structures. The first addition signals, for example, that allthe interesting shells constructed of other metals and of new materials such as fibre-reinforced laminated plastics, are excluded; and the second addition signals that thereport confines itself to those steel constructions for which civil engineers are prim-arily responsible, e.g. tanks, silos, processing containments, chimneys and towers.The simple reason is that only for these shells does the author feel competent enoughto write a state-of-the-art report.

Although the history of shell stability research is rather shortSection 2 of thisreport presents a very short retrospective viewthe subject has, in the last three

* Tel.: +49-201-183-2766; fax: +49-201-183-2710.

0143-974X/00/$ - see front matter 2000 Elsevier Science Ltd. All rights reserved.PII: S 01 43 -974X(99)00 08 4- X

160 H. Schmidt / Journal of Constructional Steel Research 55 (2000) 159181

decades, drawn the attention of a great number of qualified applied mechanicsresearchers all over the world. Schulz [1] counted 20 years ago as many as ca. 2300publications with test results; the number of theoretical publications at that time wascertainly larger by a considerable factor. Since the 1970s a number of internationalconferences have been held dealing with steel shells, either specifically or in substan-tial sessions within a wider conference subject. For many of them, full proceedingshave been issued [28]. A rather comprehensive review paper about buckling ofshells has recently been published by Teng [9]; it contains more than 300 references.Furthermore, some technical journals have recently brought out special issues onmetal shell structures [10,11].

Considering this huge amount of scientific material published in the last few yearsabout steel shell stability, the author has decided not to try to review the whole ofit within the limited space of this report. Instead, he will describe, from his personalsubjective point of view as a researching, teaching, designing and controlling struc-tural engineer and a member of some national and international standardization com-mittees, the state-of-the-art of available knowledge about the stability design of steelshell structures. Unavoidably, the treatise will mainly be based on a Western Europe-ans point-of-view.

2. Short historical retrospective view on shell stability research

The stability of shells drew academics attention considerably later than the stab-ility of plates. That was possibly to do with the fact that the technological skill toproduce thin-walled curved surfaces for structural purposes developed later than theskill to produce thin-walled planar surfaces. Therefore, there may have been noresearch need. On the other hand, the lateness will also have been a consequenceof the many times more difficult theory. After all, first empirical tests on axiallycompressed steel cylinders had been carried outin connection with the plannedBritannia and Conway bridgesas early as 1849 by Fairbairn [12]; and the Firth-of-Forth Bridge had been built using riveted tubular members of large dimensionsin 1890, about 20 years before Lorenz [13] and Timoshenko [14] found the theoreti-cal solution for the critical axial load of a circular cylindrical shell. Further funda-mental shell buckling cases were then theoretically solved in quick succession: acircular cylindrical shell under external pressure by v. Mises [15] and under torsionalshear by Schwerin [16]; and a spherical shell under external pressure by Zoelly [17].

After these initiative treatises about 80 years ago, a period of intensive work onlinear shell buckling theory followed in the 1930s. It is mainly connected with thenames Flugge, Timoshenko and Donnell. The first two academics published theirfamous monographies almost simultaneously [18,19], the latter created the shallowshell theory [20,21]. It turned out that linear shell buckling theory may be pursuedon different theoretical precision levelscontrary to column, beam and plate buck-ling theories. Furthermore, it also turned out in this period that, no matter how precisethe applied linear theory was, its results were much less suitable as a direct basisfor practical design purposes than those of linear column/beam/plate buckling

161H. Schmidt / Journal of Constructional Steel Research 55 (2000) 159181

theories. Even safety factors as large as four, applied to the critical axial load of acylinder, proved to yield unsafe structures.

The large discrepancies between theoretical solutions and the load carryingcapacity of real shell structures had become obvious since the first systematic shellbuckling test results had been compared to theoretical results [22]. It was not until the1950s, however, that it was fully understood that the interaction between unavoidableimperfections and the ill-natured postbuckling behaviour of most of the instability-endangered shell structures was the reason for the large discrepancies. The mostimportant contributions to the development of this understanding were those of Kar-man and Tsien [23] and Donnell and Wan [24] who firstly calculated complete loaddeformation curves of axially compressed cylinders with perfect and imperfectgeometry, respectively, using nonlinear formulations of Donnells shell theory, andthat of Koiter [25] who created the initial postbuckling theory.

Immense efforts followed in order to calculate the postbuckling minima of perfectshells by means of nonlinear shell theory as precisely as possible. The aim was touse the postbuckling minima as conservative buckling load estimates for practicaldesign purposes. The end of this research period is marked by the works of Hoff etal. [26] and Thielemann and Esslinger [27]. It ended with a clear understanding ofthe physical features that distinguish highly imperfection-sensitive shells with lowpostbuckling minimum from those of relative imperfection insensitivity: the differentshares of membrane and bending energy in the resistance against initial buckling ofa perfect shell structure. Simplified membrane-reduced shell buckling theories forestimating buckling loads for design purposes have been derived from this finding[28,29].

The last three decades of shell stability research have been dominated by therapidly increasing power of computers in combination with the Finite ElementMethod (FEM). Researchers have concentrated more and more on trying to simulatenumerically the real, i.e. fully nonlinear, load carrying behaviour of realistically,i.e. imperfectly modelled shell structures, and to verify the results by means of care-fully performed buckling tests. The conference proceedings mentioned earlier giveevidence of this development.

3. Theory of shell stability

The theory of shells has come to a state where the basic mechanics are conclud-ingly laid down in good monographies, e.g. [30,31], and where the conversion intofinite element formulations has reached such sophisticated levels that, from the steelshell designers standpoint, the present and further research has a certain tendencyof only being of academical interest (exception: dynamic stability). With regard toa numerically based shell buckling design, using commercial FE computer programs,see Section 4. However, there is one area left concerning quasi-static shell stabilitywhere, in the authors opinion, very promising theoretical research is still goingon: that is the task of simulating quantitatively the imperfection influence on shellbuckling phenomena.


In the past, two separate policies of tackling this task have been pursued. The firstpolicy is based on the fundamental theoretical hypothesis that it is possible to valuethe structural answer of an imperfect structure from certain properties of the perfectlyidealized structure. Koiters [25] initial postbuckling theory belongs to this policy,as well as all the attempts to use postbuckling minima as estimates for realisticbuckling resistances (starting with Karman and Tsiens [23] pioneering work), aswell as all variants of lower bound or membrane-reduced shell buckling theories[28,29]. A recent research trend in this context is the minimum perturbation energyconcept which will be briefly described further below.

The second policy is based on the perception that the behaviour of an imperfectshell can only be simulated by analysing the imperfect shell itself. However, thislogical perception leads inevitably to the question of how the imperfections of a realshell structure look. Without going into any detail, it may be stated that, no matterhow sophisticated a numerical imitation of an imperfection field may be, it stillrepresents merely a substitute imperfection because certain components of the realimperfections (e.g. residual stresses, inhomogenities, anisotropies, loading andboundary inaccuracies) are eventually not included and must therefore be substi-tuted in the calculational imperfection model. Usually these substitute imperfectionsare introduced in the form of equivalent geometric imperfections. Some commentson this approach, using commercial FE computer programs, will be given in Section4.4. Here, a recently published new theoretical concept to include the definitelyworst equivalent geometric imperfection shape in the nonlinear buckling analysisof a shell in order to come up with a direct estimation of the real buckling resistancewill be described further below.

3.1. Concept of minimum perturbation energy

This concept was developed in the Braunschweig School of Structural Analysis[3234] and is meanwhile capable of producing quantitative results which may beapplied to design problems [35,36]. The concept refers to shell stability cases withat least one unstable decreasing postbuckling path in their loaddeformation charac-teristics (Fig. 1). When a fundamental prebuckling equilibrium state of the perfectshell under a certain load level at which also a postbuckling equilibrium state existsis perturbed, it will snap through into the buckled shape if the introduced perturbationenergy PSt is large enough. Of all possible perturbation fields, that one needing thesmallest amount of energy min PSt to cause a snap-through is determined by meansof solving a nonlinear eigenvalue problem. The latter passes over into the ordinarylinear eigenvalue problem at the bifurcation load level where min PSt=0. The relationbetween the loaddeformation curve and min PSt is illustrated in Fig. 1.

The advantage (and potential for future development) of the perturbation energyconcept is that the stability-reducing effect of imperfections is represented by a singlequantity which can be determined at the perfect structure. Of course, the value minPSt which stands for a realistically imperfect structure has to be calibrated againstknown lower bound test results. Beyond that, it has to be normalized in a suitablemanner before a direct extrapolation to unknown shell buckling cases is feasible.


Fig. 1. Loaddeformation and loadperturbation energy curves (after [37]).

Schafer [37] established interaction surfaces for the elastic buckling of cylindersunder combined axial, radial and torsional loads (Fig. 2). Spohr [36] improved theconcept with regard to the definition of the reference energy for the normalizationof the perturbation energy and extended the concept to elastoplastic buckling.

3.2. Concept of definitely worst geometric imperfection shape

This concept [3840] includes the specific search for the worst geometric imper-fection shapewithin a given imperfection amplitude limitimplicitly in the nonlin-ear finite element analysis. The shell element is amended by three imperfectiondegrees of freedom at each of its nodes. Thus, the resulting set of nonlinear equations

Fig. 2. Interaction surfaces for a cylinder with r/t=100 under combined axial, radial and torsional loading(after [35]): calculated with perturbation energy concept, calibrated against DIN axial load buckling; DIN [72].


that has to be handled at each load step refers to a variable reference geometry. Thedeveloped algorithms lead automatically to the definitely worst imperfection shapewhich causes the lowest possible limit load.

One of the advantages of this new concept seems to be that only stable equilibriumstates have to be considered. No unstable or hypothetic states need to be relied on.Uncertainties from an eventual adulteration of physical instabilities by numericalinstabilities in the vicinity of eigenvalue clusters are avoided. A further advantageis that, by introducing amplitude limits, the numerical imperfection model is directlycoupled to the fabrication quality of the shell structure. The author of the presentreport thinks further development of this concept would be good. In particular, aneven closer linking of fabrication reality and numerical imperfection model, e.g. byexcluding from the worst shape search procedure certain imperfection patterns asunrealistic because of the method of fabrication/manufacture or erection, seems tobe thinkable.

4. Numerical simulation of shell buckling

Within the last decade, since Galambos [41] reviewed the finite element analysisof stability problems, the development of powerful computers and highly efficientnumerical techniques has come to a state where any given shell structure can becalculatedno matter how complicated the geometry, how dominant the imperfec-tion influence and how nonlinear the load carrying behaviour is. The necessarynumerical tools are no longer only in the hands of researching academics, but are alsoavailablein the form of commercial FEM packages [42,43]for ordinary structuraldesign engineers. However, the main task of the design engineer is, more than ever,to model his shell problem properly and to convert the numerical output into thecharacteristic buckling strength of the real shell which is needed for an equallysafe and economic design [44]. Some efforts have been made to include relevantguidance in the draft of the new European Prestandard ENV 1993-1-6 for steel shellstructures (the Shell Eurocode); it will be discussed in Section 5 of this report.

The following comments are concerned with different thinkable numericalapproaches to shell buckling when using a commercial FEM package.

4.1. Linear bifurcation analysis

The lowest eigenvalue load of the perfect shell is needed as critical buckling resist-ance Rcr for the simplest numerically based design approach. It is generally no prob-lem to produce this eigenvalue for a given FE model. However, according to theauthors experience as a proof engineer, rather elementary mistakes are made againand again when defining the FE model of a given shell buckling case. Some simpleillustrative examples follow hereafter.


4.1.1. Cylindrical tower section under centric axial compressionnumber ofelements

Fig. 3 shows the eigenvalue results for a cylinder section under centric axial com-pression, calculated with four different discretization levels (nel=ca. 500, 1000, 2000,4000 elements). Although the eigenmode for nel=476 looks quite reasonable, withthe buckling deformations displaying a rather continuous shape (Fig. 3a), this modeloverestimates the correct analytical solution [45] by as much as 20%. It should bementioned that the usual standard formula

sx,cr50.605Etr

(1)

based on Donnells shallow shell theory [21], is for this relatively long cylinder onthe unsafe side, too, by 12.5%. An eight times larger number of elements (Fig. 3b)is needed to obtain a suitable approximation (Fig. 3c).

4.1.2. Cylindrical tower section under pure bendingnumber of elementsThe same cylinder section as used for the preceding example has been calculated

under pure bending (Fig. 4). Again the FE model with nel=476 overestimates thecorrect analytical solution [46] by about 20%, and again eight times more elementsare needed for a sufficiently good approximation (Fig. 4c). It might be noteworthy

Fig. 3. Linear bifurcation FE analysis for a cylinder under centric axial compression: eigenmode with(a) too small number of elements; (b) sufficient number of elements; (c) critical buckling stresses.


Fig. 4. Linear bifurcation FE analysis for a cylinder under pure bending: eigenmode with (a) too smallnumber of elements; (b) sufficient number of elements; (c) critical buckling stresses.

that for pure bending Eq. (1) represents asymptotically the correct analytical solution,although originally derived for centric axial compression. The reason is the short-waved eigenmode (Fig. 4b).

4.1.3. Cylindrical tower section under centric axial compressionmembraneboundary conditions

For shell problems the membrane boundary conditions are often more importantthan the bending boundary conditions. That is not self-evident for many an engineerwho usually designs ordinary steel structures. Fig. 5 shows the eigenvalue resultsfor the cylinder of Fig. 3 when being calculatedinstead of using the classicalboundary condition S3 with its axially free but circumferentially restrained in-planeedge displacementsthe other three possible combinations of membrane boundaryconditions (called S1, S2 and S4). For this example, in the case of restraining theedges not only circumferentially but also axially (S1), the eigenmode changes com-pletely (compare Fig. 3b and Fig. 5a) and the critical buckling stress increases (Fig.5c). On the other hand, allowing the edges to deform circumferentially again changesthe eigenmode completely (Fig. 5b) and reduces the critical buckling stress drasti-cally by ca. 50%independent of the axial displacement being restrained (S2) orfree (S4). Every design engineer when modelling his shell case into a FE model in


Fig. 5. Linear bifurcation FE analysis for a cylinder under centric axial compression: (a) eigenmodewith S1 boundary conditions; (b) eigenmode with S2 boundary conditions; (c) critical buckling stressesfor all four S boundary conditions.

order to determine its linear eigenvalue should be aware of the great influence ofthe membrane boundary conditions.

4.1.4. Open cylindrical tank under wind loadingsymmetry conditionsIt is common practice to utilize symmetry when creating a FE model of a sym-

metric structure under a symmetric load arrangement (see the example in Fig. 6).However, one should keep in mind that such systems have symmetric and antimetriceigenmodes. Therefore, as a matter of principle two effective half systems have tobe generated, one with symmetric (Fig. 6c) and one with antimetric boundary con-ditions in the plane of symmetry. Only in this way is it guaranteed that the realcritical eigenmode leading to the lowest eigenvalue is found (Fig. 6b).

4.2. Geometrically nonlinear bifurcation analysis

Usually the linear bifurcation analysis discussed above is sufficient to produce areliable critical buckling resistance Rcr as the numerical basis for a traditionalreduction factor shell buckling design. However, in some cases a geometrically non-linear elastic analysis (i.e. large deflection theory) should be used to calculate theprebuckling path of the perfect shell configuration to which the eigenvalue searchis applied. Unfortunately, only a few of the standard FE programs that ordinary


Fig. 6. Linear bifurcation FE analysis for an open cylindrical tank under wind loading: (a) wind pressuredistribution; (b) correct eigenmode (antimetric); (c) eigenmode of half system with symmetry conditions.

design engineers have access to provide the latter option. And, on top of that, sucha nonlinear bifurcation FE analysis requires considerable experience of the user.

Typical examples of shell cases where a linear bifurcation analysis would overesti-mate the critical elastic buckling resistance of the perfect shell by amounts whichare not neglectable are:

O conical and spherical caps that are shallow;O conical and spherical caps that rest on supports that can displace radially;O assemblies of cylindrical and conical shell segments without ring stiffeners at the

meridional junctions and which are loaded meridionally by centric axial com-pression and/or global bending.

Fig. 7 illustrates the latter case by means of two actual examples [47,48]. For thedouble cone shell (Fig. 7a) the nonlinear bifurcation load (GNA) is only 40% (!)


Fig. 7. Critical buckling modes of (a) a cone/cone and (b) a cylinder/cone/cylinder shell assembly underaxial load, according to linear (LA) and geometrically nonlinear (GNA) bifurcation analysis (after [47]).

of the linear one (LA)although the buckling modes look rather similar. For thecylinder/cone/cylinder shell (Fig. 7b) the relevant ratio is 65%, in this case with asignificant difference in the buckling modes where the critical region even switchesfrom the concave cylinder/cone junction (LA) to the convex cone/cylinder junc-tion (GNA).

4.3. Fully nonlinear analysis of the perfect shell

An exact FE analysis of the nominal shell configuration, i.e. including all nonlin-ear geometry and material influences, but with perfect geometry, has become quitepopular among researchers. The reasons are in the authors opinion: it is a challengeto master all the sophisticated nonlinear techniques; the computational power to achi-eve this is available these days; and the unpleasant problems of realistic imperfectionsare avoided. However, real shell structures are imperfectunfortunately.

Fig. 8 illustrates the situation for a cylindrical test specimen made of austeniticsteel AISI 304 (material-no. 1.4301) under axial load at 100 C temperature [49,50].Although the r/t-ratio of 150 classifies this shell as medium-thick where the nonlin-ear material influences are supposed to be dominant, none of the numerical modelswere able to simulate the experimental behaviour properly. As can be seen, theimperfection influences soften the prebuckling behaviour and lower the bucklingload, compared to the numerical predictions. Furthermore, none of the numericalmodels found the actual periodic buckling mode of the specimen; all of them pre-dicted an axisymmetric elephants foot collapse mechanism.

From the foregoing explanations it may be concluded that, for a specific designcase, a fully nonlinear analysis of the perfect shell is physically reasonable onlyfor a relatively thick-walled shell for which a pronouncedly yield-induced collapse


Fig. 8. Load-shortening curves of a perfect cylinder under axial load, according to various fully nonlinearFE analysis (GMNA) variants and compared to the experimental curve (after [49]).

mechanism is expected. Eventually the lacking imperfection influence has to beadded by engineers estimation. Besides that, this elasticplastic buckling analysishas, in recent years, been successfully used for systematic parametric studies aimingat fundamental insights into the buckling behaviour of a specific shell case in theelasticplastic interaction slenderness region. Examples are the studies of Wunderlichet al. [51] on externally pressurized toriconical shells and of Esslinger and v. Impe[52] on axially compressed conical shells.

4.4. Fully nonlinear analysis of the imperfect shell

If, for a stability-endangered shell structure, the realistic (in the Eurocode termin-ology: the characteristic) buckling resistance should be determined from a numericalFE simulation, without adding any additional reduction, the imperfections mustimperatively be included. The fact that any imperfection modelling, however sophis-ticated, represents merely a substitute imperfection has already been stressed inSection 3. The obvious straightforward way of building substitute imperfections intoa FE model when using a commercial program are equivalent geometric imperfec-tions. Three different approaches to this basic idea are thinkable and have been pur-sued in the research efforts of recent years. They are shortly discussed hereafter.

4.4.1. Realistic geometric imperfectionsIt is understandable that a design engineer may ardently wish to include the real

geometric imperfections of his structure, eventually amplified by a proper factor totake care of the material imperfections. But there is definitely no way of complyingwith this wish in a deterministic mannerexcept perhaps with regard to systemati-cally fabrication-induced imperfections in series products. The only way of simulat-ing realistic imperfections seems to be to model them stochastically [53,54],


eventually based on measurements at a great number of similar shell applications[55]. However, the author is not aware of any effected application of this approachto a civil engineering shell structure.

4.4.2. Worst geometric imperfectionsThe idea to find the worst possible geometric imperfection pattern for a given,

to-be-designed shell structure and to introduce it into the nonlinear analysis is as oldas the discovery of the detrimental influence of imperfections. It was common prac-tice from the beginning, supposedly taken over from column and plate bucklingexperience, to consider that imperfection pattern to be the worst which is affine tothe lowest eigenmode. For shell cases with clustered eigenvalues the question aroseas to which one of the multiple eigenmodes was to be taken. Looking at the axiallycompressed cylinder as the prototype for such a shell case, the historical developmentwent from checkerboard linear eigenmodes [24] via the axisymmetric linear eigen-mode [5658] to the fundamental conclusion of Ho [59]: that one of all linear combi-nations of the clustered eigenmodes which produces the steepest decreasing post-buckling path represents the worst imperfection pattern. Today we know that for anumber of shell cases of which the prebuckling behaviour is significantly nonlinear,the search for the worst imperfection pattern should include nonlinear eigenmodes aswell [6062]. And to make it even more complicated, the results from new theoreticalmethods which search directly for the lowest possible limit load (see Section 3)suggest that single dimple imperfections may be worse than eigenmode-affine pat-terns covering larger areas of the relevant shell.

The subject cannot be discussed in detail here. Numerous researchers have dealtwith it. For a design engineer, the actual task is even more difficult when he triesto assume an imperfection pattern which, though being nearly worst, is not toofar away from having at least a small probability of occurring in practice. The onlyadvice that the author could give him is that he should keep in mind that it is onlya substitute imperfection which he assumes, that the choice of the amplitude valueis at least as important as the worst pattern, and that he must calibrate or verifyhis result anyway (see Section 5.2).

4.4.3. Stimulating geometric imperfectionsFrom the foregoing explanations it becomes clear that for extensive parametric

studies where a great number of fully nonlinear shell analyses have to be performed,it may be a good idea to choose an as simple equivalent geometric imperfectionpattern as possible, deliberately abstaining from making it realistic or worst. Itsfunction is to stimulate the characteristic physical shell buckling behaviour. Forthat purpose it must only have a certain geometric similarity to one of the criticaleigenmodes. Its amplitude value has in any case to be calibrated somehow; it hasno concrete meaning.

The numerical investigations of Hautala [49] about axially compressed cylindersmade of austenitic steels and loaded at various elevated temperatures are a typicalexample of using the technique of stimulating geometric imperfections. A singleaxisymmetric inward predeformation with a sinusoidal meridional shape located in


the middle of the cylinder length was chosen. It provided the possibility of using aspecial shell-of-revolution computer program [63] which consumed only 10% com-puter time compared to a conventional FE package; that was important in view ofthe 6000 fully nonlinear shell analyses needed. Fig. 9 shows buckling curves whichwere simulated using different depth/radius ratios w0/r of the stimulating predefor-mation. The one for w0=r/500 correlates very well with the design buckling curve inECCS [64]. It was therefore taken as the basis for the directly comparative nonlinearcalculations using alternatively the bilinear elasticplastic and the strain hardeningstressstrain curves of structural steels and stainless steels, respectively.

Similar single axisymmetric predeformations, instead of the full axisymmetric eig-enmode, have already been used by several researchers when investigating axiallycompressed cylinders [6568]. However, usually this predeformation was consideredas a realistic geometric imperfection at circumferential welded junctions, ratherthan a stimulating geometric imperfection in the beforementioned sense.

5. Standards and recommendations for the stability design of steel shells

Traditionally, nearly all national standards with structural steel design rules of amore generic type, i.e. without referring to special applications, are focussed onbeam/column/bar type structures. Shell-like elements would be included, if at all,only as tubular members. If shells are used in certain application fields (e.g. silos,offshore platforms), the necessary design information would be given in the relevantapplication standard. Though historically explicable, this situation is unsatisfactory.Therefore, when the drafting of Parts 3 and 4 of Eurocode 3 (chimneys, towers,masts, silos, tanks, pipelines) started a few years ago, the project team membersthought it might be reasonable to develop a set of basic design rules for steel shellstructures, independent of specific applications. Meanwhile the whole Eurocode 3system has been restructured in a way that all general design rules are collected inthe form of sub-parts of Part 1. Thus, a Part 1.6 General RulesSupplementary

Fig. 9. Numerically simulated buckling curves of axially compressed cylinders, using an axisymmetricsingle inward stimulating imperfection of depth w0 (after [49]).


Rules for the Strength and Stability of Shell Structures has been worked out andwill be officially issued as European Prestandard ENV 1993-1-6 by the end of 1999.It has been partially presented and discussed in some recent conference papers [6971]. Its scope and some selected features of its shell stability approach will be com-mented on hereafter.

5.1. Scope of ENV 1993-1-6

Unlike the well-known European Recommendations for buckling of steel shells[64] and the German Standard for the stability of steel shells [72], the PrestandardENV 1993-1-6 covers not only the stability strength of shells (although often domin-ating the design of steel shell structures), but also their strength in terms of plasticcollapse/tensile rupture (called excessive yielding), cyclic plasticity and fatigue.All these design aspects are formulated as ultimate limit states (ULS) for whichcharacteristic design resistance quantities have to be determined. Three optionaldesign approaches are offered for each limit state:

O design by means of standard expressions if available (called direct design);O the classical approach where design stresses under acting loads are calculated and

compared to design stress resistance values (called stress design);O the modern, strongly computer-oriented approach where the limit state is assessed

by means of a sophisticated numerical analysis (e.g. based on the Finite ElementMethod) concerning the whole structure (called design by global numericalanalysis).

The latter approach (or more precisely, the rules for the latter approach) will certainlybe the object of one or the other criticism. Some engineers in high-tech shellapplication areas, e.g. space structures or nuclear power containments, could arguethat sophisticated computer analyses need not be codified. However, especially forshell buckling, it is necessary to specify certain requirements for the numerical mod-elling, as may be realized from the explanations in the preceding section.

As to the different theoretical levels and different levels of modelling whennumerically analysing a shell (see Section 4), the relevant terminology needed cleardefinitions because between practising engineers of different countries and differenttechnical communities quite often a considerable confusion had been observed. Table1 summarizes the seven types of shell analysis as defined in ENV 1993-1-6, togetherwith their abbreviations.

In order to provide the user of the Prestandard with available algebraic infor-mations about the strength and stability of shells, four annexes have been added tothe main document of ENV 1993-1-6:

O Annex A: membrane theory stresses in shells;O Annex B: additional expressions for plastic collapse resistances;O Annex C: expressions for linear elastic membrane and bending stresses; andO Annex D: expressions for buckling design.


Table 1Types of shell analysis according to ENV 1993-1-6

Abbreviations Type of analysis Shell theory Material law Shell geometry

Membrane theory of shells Membrane Not applicable Perfectequilibrium

LA Linear elastic shell analysis Linear bending Linear Perfectand stretching

GNA Geometrically nonlinear elastic Nonlinear Linear Perfectanalysis

MNA Materially nonlinear analysis Linear Nonlinear PerfectGMNA Geometrically and materially Nonlinear Nonlinear Perfect

nonlinear analysisGNIA Geometrically nonlinear elastic Nonlinear Linear Imperfect

analysis with imperfectionsGMNIA Geometrically and materially Nonlinear Nonlinear Imperfect

nonlinear analysis withimperfections

5.2. Shell buckling limit state according to ENV 1993-1-6

5.2.1. Fabrication tolerances, quality classesAs repeatedly emphasized in this report, the buckling strength of thin shells

contrary to the other abovementioned strength casesis strongly dependent on thegeometrical accurateness of the realized structure. Moreover, residual stresses, e.g.from welding, may in certain cases have a further deteriorating influence. Vicariouslyfor the whole of possible imperfections, three essential and measurable geometricimperfection types have been chosen to define buckling-relevant fabrication toler-ance limits. They are:

O the out-of-roundness of the shell:

Ur5(dmax - dmin)

dnom(2)

O the accidental eccentricity ea at joints perpendicular to membrane compressiveforces, related to the wall thickness t:

Ue5ea

t(3)


O the depth Dw0 of initial dimples in the shell wall, related to certain critical gaugelengths lg:

U05Dw0

lg(4)

The values of the tolerance limit parameters U, as specified in ENV 1993-1-6 forthree fabrication tolerance quality classes, are listed in Table 2. The introduction ofquality classes reflects the sensitivity of the buckling design strength to fabricationaccurateness and should encourage the fabricator to control it during fabrication. Ofcourse, all buckling strength parameters have been made dependent on the qualityclass in such a manner that a higher quality class is rewarded with a higher designstrength. The idea to class shells into strength groups according to the quality ofconstruction was first proposed by Odland [73] and introduced into the Eurocodedrafting work by Rotter.

5.2.2. Buckling stress designThe stress design approach follows on principle the long-established reduction

factor approach as used in many national and international standards, e.g. [64,72].Its gist is the yield-stress related stability reduction factor c as function of the non-dimensional shell slenderness l=(fy,k/sRc)0.5 with sRc being the critical buckling stressof the perfect shell. This format is compatible with the treatment of other stabilitycases in Eurocode 3 (Fig. 10). For c an algebraic expression has been chosen(proposed by the author and modified by Rotter) which, by means of four free buck-ling parameters, is adaptable to any lower bound information of a shell bucklingcase and thus is open for future development:

c51 when l#l0 (5)

c512bF l- l0lp- l0Gh

when l0,l,lp (6)

Table 2Values for tolerance limit and equivalent imperfection parameters, respectively, according to ENV 1993-1-6

Fabrication tolerance quality Tolerance parameters Equivalentclass imperfection

parameterClass Description Ur,max Ue,max U0,max U0,eff

A Excellent 0.007 0.14 0.006 0.010B High 0.010 0.20 0.010 0.016C Normal 0.015 0.30 0.016 0.025


Fig. 10. Stability reduction factor c=f(l) in ENV 1993-1-6.

c5al2

when lp#l (7)

where (see Fig. 10): l0 is the squash limit slenderness defining the plastic plateauof the buckling curve; lp is the plastic limit slenderness above which purely elasticbuckling is assumed: lp=[a/(12b)]0.5; b describes the relative stress level at lp; ais the well-known elastic imperfection reduction factor (knock-down factor); andh describes the shape of the elasticplastic buckling interaction between lp and l0.

The buckling parameters a, b, h and l0 are for basic shell buckling cases givenin Annex D of ENV 1993-1-6, e.g. for externally pressurized cylinders of fabricationtolerance quality class B:

a50.65, b50.6, h51, l050.4

5.2.3. Buckling design by global numerical analysisRules for the assessment of the ultimate buckling limit state of an arbitrary shell

of revolution by means of a numerical LA/GNA analysis (see Sections 4.1 and 4.2)or a GMNA analysis (see Section 4.3)abbreviations according to Table 1aregiven in ENV 1993-1-6. In both cases, proper imperfection reduction factors haveto be applied to the numerical result. Of course, the latter is the crux of this approach.If no specific factors are available (e.g. from specific tests or from analogy deductionbased on comparable buckling cases), ENV 1993-1-6 prescribes conservatively totake the reduction factor values for an axially compressed cylinder.

Rules for a straightforward shell buckling assessment by a fully nonlinear GMNIAanalysis (see Section 4.4) are also given in ENV 1993-1-6. The imperfection influ-ence may be simulated by equivalent geometric imperfections. Their pattern shouldbe chosen in such a form that it has the most unfavourable effect on the bucklingbehaviour of the shell. The difficulties inherent to this banal sentence have been


discussed in Section 4.4. The maximum deviation of the equivalently imperfectgeometry from the perfect shape, i.e. the imperfection amplitude, should be assumednot smaller than

Dw0,eff5U0,efflg and Dw0,eff5100U0,efft (8)with the parameter U0,eff according to Table 2. As can be seen, the assumed geometricimperfections are about 60% larger than the allowed tolerance limits for measurableinitial dimples. This is because the equivalent geometric imperfections are substi-tute imperfections (see Section 3) and have to cover eventual non-measurable imper-fections (e.g. residual stresses) as well. An additional rule recommends to check ifa 10% smaller amplitude than the one from Eq. (8) possibly delivers a smaller ulti-mate load. If that is the case (which happens in some shell configurations), the mini-mum has to be determined.

Prior to using the numerical outcome Rcr,GMNIA of the analysis (see Fig. 11) as adirect basis for the design, it has eventually to be calibrated. For this purpose, accord-ing to ENV 1993-1-6, other shell buckling cases, for which either test results orreliable characteristic buckling resistance values are available, and which are compa-rable in their buckling controlling parameters, have to be calculated with the samenumerical tool using similar imperfection assumptions and similar modelling tech-niques.

6. Concluding remarks

This general report could only address a few selected aspects within the widefield of steel shell stability. Much progress has been achieved in the last years, interms of better physical insight into the complex behaviour of shells and betternumerical tools for its handling when designing a shell structure, as well as of generaldesign rules for fundamental shell stability cases. However, if the reader of thisreport would now have the impression that no further research is needed, this

Fig. 11. Alternative definitions of buckling resistance from global GMNA and GMNIA analysis accord-ing to ENV 1993-1-6.


impression would be fatally misleading. The author agrees fully with Rotter [69]who stated in a key note lecture on shell structures: In many aspects of shell struc-tures, there is a mismatch between the needs for structural design and current codi-fied provisions.

Future theoretical shell stability research should, from the view point of structuraldesign engineers, concentrate on applying the high theoretical knowledge and thepowerful numerical tools to unsolved (or hitherto unsatisfactorily solved) applicationproblems, rather than turning the basic shell cases (e.g. the cylinder under uniformaxial compression) round and round and coming up with another alleged 0.x% errorin Eq. (1) or Eq. (6). Such application problems are: any type of local loads andsupports, stiffenings, openings and cutouts, shell combinations, incomplete shells,non-uniform stress states in shells; the list is far from being complete. An excellentexample for this type of sophisticated and purposeful numerical research, as theauthor has it in mind, are the Graz investigations on locally supported cylindricalsteel silos or containments [74,75].

A last point that the author would like to emphasize is the continuing need forshell buckling tests. Of course, the basic philosophy of experimental programs haschanged in view of todays numerical facilities. No longer is it the statistics-focussedquantity of tests what matters, but their quality as physical verification benchmarksfor numerical models. By the way, tests have an important educational secondaryeffect on young researchers: they learn that the numerical model is not the reality.

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