Stability of steel shell structures

Journal of Constructional Steel Research 55 (2000) 159–181 www.elsevier.com/locate/jcsr

Stability of steel shell structures General Report Herbert 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 structures is discussed. Specific stress is put on the various approaches to a numerically based stability design. 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 ’99 he was informed that he had to write a state-of-the-art report on the “Stability and Dynamic of Shells”. However, because he is not really an expert on dynamics of shells, he decided to concentrate on the stability of shells. Accidentally this coincides with the fact that no papers on the dynamics of shells have been submitted for the Colloquium. Furthermore, the author has added into the title of this General Report the words “steel” and “structures”. The first addition signals, for example, that all the interesting shells constructed of other metals and of new materials such as fibrereinforced laminated plastics, are excluded; and the second addition signals that the report confines itself to those steel constructions for which civil engineers are primarily responsible, e.g. tanks, silos, processing containments, chimneys and towers. The simple reason is that only for these shells does the author feel competent enough to write a state-of-the-art report. Although the history of shell stability research is rather short—Section 2 of this report presents a very short retrospective view—the 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 0 1 4 3 - 9 7 4 X ( 9 9 ) 0 0 0 8 4 - X

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decades, drawn the attention of a great number of qualified applied mechanics researchers all over the world. Schulz [1] counted 20 years ago as many as ca. 2300 publications with test results; the number of theoretical publications at that time was certainly larger by a considerable factor. Since the 1970s a number of international conferences have been held dealing with steel shells, either specifically or in substantial sessions within a wider conference subject. For many of them, full proceedings have been issued [2–8]. A rather comprehensive review paper about buckling of shells has recently been published by Teng [9]; it contains more than 300 references. Furthermore, some technical journals have recently brought out special issues on metal shell structures [10,11]. Considering this huge amount of scientific material published in the last few years about steel shell stability, the author has decided not to try to review the whole of it within the limited space of this report. Instead, he will describe, from his personal subjective point of view as a researching, teaching, designing and controlling structural engineer and a member of some national and international standardization committees, the state-of-the-art of available knowledge about the stability design of steel shell structures. Unavoidably, the treatise will mainly be based on a Western European’s point-of-view.

2. Short historical retrospective view on shell stability research The stability of shells drew academics’ attention considerably later than the stability of plates. That was possibly to do with the fact that the technological skill to produce thin-walled curved surfaces for structural purposes developed later than the skill to produce thin-walled planar surfaces. Therefore, there may have been no research need. On the other hand, the lateness will also have been a consequence of the many times more difficult theory. After all, first empirical tests on axially compressed steel cylinders had been carried out—in connection with the planned Britannia and Conway bridges—as early as 1849 by Fairbairn [12]; and the Firthof-Forth Bridge had been built using riveted tubular members of large dimensions in 1890, about 20 years before Lorenz [13] and Timoshenko [14] found the theoretical solution for the critical axial load of a circular cylindrical shell. Further fundamental shell buckling cases were then theoretically solved in quick succession: a circular cylindrical shell under external pressure by v. Mises [15] and under torsional shear 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 on linear shell buckling theory followed in the 1930s. It is mainly connected with the names Flu¨gge, Timoshenko and Donnell. The first two academics published their famous monographies almost simultaneously [18,19], the latter created the “shallow shell theory” [20,21]. It turned out that linear shell buckling theory may be pursued on different theoretical precision levels—contrary to column, beam and plate buckling theories. Furthermore, it also turned out in this period that, no matter how precise the applied linear theory was, its results were much less suitable as a direct basis for practical design purposes than those of linear column/beam/plate buckling

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theories. Even safety factors as large as four, applied to the critical axial load of a cylinder, proved to yield unsafe structures. The large discrepancies between theoretical solutions and the load carrying capacity of “real” shell structures had become obvious since the first systematic shell buckling test results had been compared to theoretical results [22]. It was not until the 1950s, however, that it was fully understood that the interaction between unavoidable imperfections and the ill-natured postbuckling behaviour of most of the instabilityendangered shell structures was the reason for the large discrepancies. The most important contributions to the development of this understanding were those of Karman and Tsien [23] and Donnell and Wan [24] who firstly calculated complete load– deformation curves of axially compressed cylinders with perfect and imperfect geometry, respectively, using nonlinear formulations of Donnell’s shell theory, and that of Koiter [25] who created the “initial postbuckling theory”. Immense efforts followed in order to calculate the postbuckling minima of perfect shells by means of nonlinear shell theory as precisely as possible. The aim was to use the postbuckling minima as conservative buckling load estimates for practical design purposes. The end of this research period is marked by the works of Hoff et al. [26] and Thielemann and Esslinger [27]. It ended with a clear understanding of the physical features that distinguish highly imperfection-sensitive shells with low postbuckling minimum from those of relative imperfection insensitivity: the different shares of membrane and bending energy in the resistance against initial buckling of a perfect shell structure. Simplified “membrane-reduced” shell buckling theories for estimating 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 the rapidly increasing power of computers in combination with the Finite Element Method (FEM). Researchers have concentrated more and more on trying to simulate numerically 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 carefully performed buckling tests. The conference proceedings mentioned earlier give evidence of this development.

3. Theory of shell stability The theory of shells has come to a state where the basic mechanics are concludingly laid down in good monographies, e.g. [30,31], and where the conversion into finite element formulations has reached such sophisticated levels that, from the steel shell designer’s standpoint, the present and further research has a certain tendency of only being of academical interest (exception: dynamic stability). With regard to a numerically based shell buckling design, using commercial FE computer programs, see Section 4. However, there is one area left concerning quasi-static shell stability where, in the author’s opinion, very promising theoretical research is still going on: that is the task of simulating quantitatively the imperfection influence on shell buckling phenomena.

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In the past, two separate policies of tackling this task have been pursued. The first policy is based on the fundamental theoretical hypothesis that it is possible to value the structural answer of an imperfect structure from certain properties of the perfectly idealized structure. Koiter’s [25] initial postbuckling theory belongs to this policy, as well as all the attempts to use postbuckling minima as estimates for realistic buckling resistances (starting with Karman and Tsien’s [23] pioneering work), as well 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 energy concept which will be briefly described further below. The second policy is based on the perception that the behaviour of an imperfect shell can only be simulated by analysing the imperfect shell itself. However, this logical perception leads inevitably to the question of how the imperfections of a real shell structure look. Without going into any detail, it may be stated that, no matter how sophisticated a numerical imitation of an imperfection field may be, it still represents merely a “substitute imperfection” because certain components of the real imperfections (e.g. residual stresses, inhomogenities, anisotropies, loading and boundary inaccuracies) are eventually not included and must therefore be “substituted” in the calculational imperfection model. Usually these substitute imperfections are introduced in the form of equivalent geometric imperfections. Some comments on this approach, using commercial FE computer programs, will be given in Section 4.4. Here, a recently published new theoretical concept to include the “definitely worst” equivalent geometric imperfection shape in the nonlinear buckling analysis of a shell in order to come up with a direct estimation of the real buckling resistance will be described further below. 3.1. Concept of minimum perturbation energy This concept was developed in the “Braunschweig School of Structural Analysis” [32–34] and is meanwhile capable of producing quantitative results which may be applied to design problems [35,36]. The concept refers to shell stability cases with at least one unstable decreasing postbuckling path in their load–deformation characteristics (Fig. 1). When a fundamental prebuckling equilibrium state of the perfect shell under a certain load level at which also a postbuckling equilibrium state exists is perturbed, it will snap through into the buckled shape if the introduced perturbation energy ⌸St is large enough. Of all possible perturbation fields, that one needing the smallest amount of energy min ⌸St to cause a snap-through is determined by means of solving a nonlinear eigenvalue problem. The latter passes over into the ordinary linear eigenvalue problem at the bifurcation load level where min ⌸St=0. The relation between the load–deformation curve and min ⌸St is illustrated in Fig. 1. The advantage (and potential for future development) of the perturbation energy concept is that the stability-reducing effect of imperfections is represented by a single quantity which can be determined at the perfect structure. Of course, the value min ⌸St which stands for a realistically imperfect structure has to be calibrated against known lower bound test results. Beyond that, it has to be normalized in a suitable manner before a direct extrapolation to unknown shell buckling cases is feasible.

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Fig. 1.

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Load–deformation and load–perturbation energy curves (after [37]).

Scha¨fer [37] established interaction surfaces for the elastic buckling of cylinders under combined axial, radial and torsional loads (Fig. 2). Spohr [36] improved the concept with regard to the definition of the reference energy for the normalization of the perturbation energy and extended the concept to elastoplastic buckling. 3.2. Concept of “definitely worst” geometric imperfection shape This concept [38–40] includes the specific search for the “worst” geometric imperfection shape—within a given imperfection amplitude limit—implicitly in the nonlinear finite element analysis. The shell element is amended by three imperfection degrees 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].

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that has to be handled at each load step refers to a variable reference geometry. The developed algorithms lead automatically to the “definitely worst” imperfection shape which causes the lowest possible limit load. One of the advantages of this new concept seems to be that only stable equilibrium states have to be considered. No unstable or hypothetic states need to be relied on. Uncertainties from an eventual adulteration of physical instabilities by numerical instabilities in the vicinity of eigenvalue clusters are avoided. A further advantage is that, by introducing amplitude limits, the numerical imperfection model is directly coupled to the fabrication quality of the shell structure. The author of the present report thinks further development of this concept would be good. In particular, an even closer linking of fabrication reality and numerical imperfection model, e.g. by excluding from the “worst” shape search procedure certain imperfection patterns as unrealistic because of the method of fabrication/manufacture or erection, seems to be thinkable.

4. Numerical simulation of shell buckling Within the last decade, since Galambos [41] reviewed the finite element analysis of stability problems, the development of powerful computers and highly efficient numerical techniques has come to a state where any given shell structure can be calculated—no matter how complicated the geometry, how dominant the imperfection influence and how nonlinear the load carrying behaviour is. The necessary numerical tools are no longer only in the hands of researching academics, but are also available—in the form of commercial FEM packages [42,43]—for ordinary structural design 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 the characteristic buckling strength of the “real” shell which is needed for an equally safe and economic design [44]. Some efforts have been made to include relevant guidance in the draft of the new European Prestandard ENV 1993-1-6 for steel shell structures (the “Shell Eurocode”); it will be discussed in Section 5 of this report. The following comments are concerned with different thinkable numerical approaches 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 resistance Rcr for the simplest numerically based design approach. It is generally no problem to produce this eigenvalue for a given FE model. However, according to the author’s experience as a proof engineer, rather elementary mistakes are made again and again when defining the FE model of a given shell buckling case. Some simple illustrative examples follow hereafter.

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4.1.1. Cylindrical tower section under centric axial compression—number of elements Fig. 3 shows the eigenvalue results for a cylinder section under centric axial compression, calculated with four different discretization levels (nel=ca. 500, 1000, 2000, 4000 elements). Although the eigenmode for nel=476 looks quite reasonable, with the buckling deformations displaying a rather continuous shape (Fig. 3a), this model overestimates the correct analytical solution [45] by as much as 20%. It should be mentioned that the usual standard formula t sx,cr⫽0.605·E· r

(1)

based on Donnell’s shallow shell theory [21], is for this relatively long cylinder on the 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 bending—number of elements The 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 the correct analytical solution [46] by about 20%, and again eight times more elements are 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.

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Fig. 4. Linear bifurcation FE analysis for a cylinder under pure bending: eigenmode with (a) too small number 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 shortwaved eigenmode (Fig. 4b). 4.1.3. Cylindrical tower section under centric axial compression—membrane boundary conditions For shell problems the membrane boundary conditions are often more important than the bending boundary conditions. That is not self-evident for many an engineer who usually designs ordinary steel structures. Fig. 5 shows the eigenvalue results for the cylinder of Fig. 3 when being calculated—instead of using the “classical” boundary condition S3 with its axially free but circumferentially restrained in-plane edge displacements—the other three possible combinations of membrane boundary conditions (called S1, S2 and S4). For this example, in the case of restraining the edges not only circumferentially but also axially (S1), the eigenmode changes completely (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 changes the eigenmode completely (Fig. 5b) and reduces the critical buckling stress drastically by ca. 50%—independent of the axial displacement being restrained (S2) or free (S4). Every design engineer when modelling his shell case into a FE model in

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

order to determine its linear eigenvalue should be aware of the great influence of the membrane boundary conditions. 4.1.4. Open cylindrical tank under wind loading—symmetry conditions It is common practice to utilize symmetry when creating a FE model of a symmetric structure under a symmetric load arrangement (see the example in Fig. 6). However, one should keep in mind that such systems have symmetric and antimetric eigenmodes. Therefore, as a matter of principle two effective half systems have to be generated, one with symmetric (Fig. 6c) and one with antimetric boundary conditions in the plane of symmetry. Only in this way is it guaranteed that the real critical 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 a reliable critical buckling resistance Rcr as the numerical basis for a traditional reduction factor shell buckling design. However, in some cases a geometrically nonlinear elastic analysis (i.e. large deflection theory) should be used to calculate the prebuckling path of the perfect shell configuration to which the eigenvalue search is applied. Unfortunately, only a few of the standard FE programs that ordinary

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Fig. 6. Linear bifurcation FE analysis for an open cylindrical tank under wind loading: (a) wind pressure distribution; (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, such a nonlinear bifurcation FE analysis requires considerable experience of the user. Typical examples of shell cases where a linear bifurcation analysis would overestimate the critical elastic buckling resistance of the perfect shell by amounts which are not neglectable are: 앫 conical and spherical caps that are shallow; 앫 conical and spherical caps that rest on supports that can displace radially; 앫 assemblies of cylindrical and conical shell segments without ring stiffeners at the meridional junctions and which are loaded meridionally by centric axial compression and/or global bending. Fig. 7 illustrates the latter case by means of two actual examples [47,48]. For the double cone shell (Fig. 7a) the nonlinear bifurcation load (GNA) is only 40% (!)

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Fig. 7. Critical buckling modes of (a) a cone/cone and (b) a cylinder/cone/cylinder shell assembly under axial 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 the cylinder/cone/cylinder shell (Fig. 7b) the relevant ratio is 65%, in this case with a significant difference in the buckling modes where the critical region even switches from the concave cylinder/cone junction (LA) to the convex cone/cylinder junction (GNA). 4.3. Fully nonlinear analysis of the perfect shell An “exact” FE analysis of the nominal shell configuration, i.e. including all nonlinear geometry and material influences, but with perfect geometry, has become quite popular among researchers. The reasons are in the author’s opinion: it is a challenge to master all the sophisticated nonlinear techniques; the computational power to achieve this is available these days; and the unpleasant problems of realistic imperfections are avoided. However, real shell structures are imperfect—unfortunately. Fig. 8 illustrates the situation for a cylindrical test specimen made of austenitic steel 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 nonlinear material influences are supposed to be dominant, none of the numerical models were able to simulate the experimental behaviour properly. As can be seen, the imperfection influences soften the prebuckling behaviour and lower the buckling load, compared to the numerical predictions. Furthermore, none of the numerical models found the actual periodic buckling mode of the specimen; all of them predicted an axisymmetric “elephant’s foot” collapse mechanism. From the foregoing explanations it may be concluded that, for a specific design case, a fully nonlinear analysis of the perfect shell is physically reasonable only for a relatively thick-walled shell for which a pronouncedly yield-induced collapse

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

mechanism is expected. Eventually the lacking imperfection influence has to be added by engineer’s estimation. Besides that, this “elastic–plastic buckling analysis” has, in recent years, been successfully used for systematic parametric studies aiming at fundamental insights into the buckling behaviour of a specific shell case in the elastic–plastic interaction slenderness region. Examples are the studies of Wunderlich et 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 terminology: the characteristic) buckling resistance should be determined from a numerical FE simulation, without adding any additional reduction, the imperfections must imperatively be included. The fact that any imperfection modelling, however sophisticated, represents merely a “substitute imperfection” has already been stressed in Section 3. The obvious straightforward way of building substitute imperfections into a FE model when using a commercial program are equivalent geometric imperfections. Three different approaches to this basic idea are thinkable and have been pursued in the research efforts of recent years. They are shortly discussed hereafter. 4.4.1. “Realistic” geometric imperfections It is understandable that a design engineer may ardently wish to include the “real” geometric imperfections of his structure, eventually amplified by a proper factor to take care of the material imperfections. But there is definitely no way of complying with this wish in a deterministic manner—except perhaps with regard to systematically fabrication-induced imperfections in series products. The only way of simulating “realistic” imperfections seems to be to model them stochastically [53,54],

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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 approach to a civil engineering shell structure. 4.4.2. “Worst” geometric imperfections The 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 old as the discovery of the detrimental influence of imperfections. It was common practice from the beginning, supposedly taken over from column and plate buckling experience, to consider that imperfection pattern to be the worst which is affine to the lowest eigenmode. For shell cases with clustered eigenvalues the question arose as to which one of the multiple eigenmodes was to be taken. Looking at the axially compressed cylinder as the prototype for such a shell case, the historical development went from checkerboard linear eigenmodes [24] via the axisymmetric linear eigenmode [56–58] to the fundamental conclusion of Ho [59]: that one of all linear combinations of the clustered eigenmodes which produces the steepest decreasing postbuckling path represents the worst imperfection pattern. Today we know that for a number of shell cases of which the prebuckling behaviour is significantly nonlinear, the search for the worst imperfection pattern should include nonlinear eigenmodes as well [60–62]. And to make it even more complicated, the results from new theoretical methods which search directly for the lowest possible limit load (see Section 3) suggest that single dimple imperfections may be worse than eigenmode-affine patterns covering larger areas of the relevant shell. The subject cannot be discussed in detail here. Numerous researchers have dealt with it. For a design engineer, the actual task is even more difficult when he tries to assume an imperfection pattern which, though being “nearly worst”, is not too far away from having at least a small probability of occurring in practice. The only advice that the author could give him is that he should keep in mind that it is only a “substitute” imperfection which he assumes, that the choice of the amplitude value is at least as important as the “worst” pattern, and that he must calibrate or verify his result anyway (see Section 5.2). 4.4.3. “Stimulating” geometric imperfections From 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 imperfection pattern as possible, deliberately abstaining from making it “realistic” or “worst”. Its function is to “stimulate” the characteristic physical shell buckling behaviour. For that purpose it must only have a certain geometric similarity to one of the critical eigenmodes. Its amplitude value has in any case to be calibrated somehow; it has no concrete meaning. The numerical investigations of Hautala [49] about axially compressed cylinders made of austenitic steels and loaded at various elevated temperatures are a typical example of using the technique of “stimulating” geometric imperfections. A single axisymmetric inward predeformation with a sinusoidal meridional shape located in

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the middle of the cylinder length was chosen. It provided the possibility of using a special shell-of-revolution computer program [63] which consumed only 10% computer time compared to a conventional FE package; that was important in view of the 6000 fully nonlinear shell analyses needed. Fig. 9 shows buckling curves which were simulated using different depth/radius ratios w0/r of the “stimulating” predeformation. The one for w0=r/500 correlates very well with the design buckling curve in ECCS [64]. It was therefore taken as the basis for the directly comparative nonlinear calculations using alternatively the bilinear elastic–plastic and the strain hardening stress–strain curves of structural steels and stainless steels, respectively. Similar single axisymmetric predeformations, instead of the full axisymmetric eigenmode, have already been used by several researchers when investigating axially compressed cylinders [65–68]. However, usually this predeformation was considered as a “realistic” geometric imperfection at circumferential welded junctions, rather than 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 a more generic type, i.e. without referring to special applications, are focussed on beam/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 relevant application 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 members thought it might be reasonable to develop a set of basic design rules for steel shell structures, independent of specific applications. Meanwhile the whole Eurocode 3 system has been restructured in a way that all general design rules are collected in the form of sub-parts of Part 1. Thus, a Part 1.6 “General Rules—Supplementary

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

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Rules for the Strength and Stability of Shell Structures” has been worked out and will 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 [69– 71]. Its scope and some selected features of its shell stability approach will be commented 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 Prestandard ENV 1993-1-6 covers not only the stability strength of shells (although often dominating the design of steel shell structures), but also their strength in terms of plastic collapse/tensile rupture (called “excessive yielding”), cyclic plasticity and fatigue. All these design aspects are formulated as ultimate limit states (ULS) for which characteristic design resistance quantities have to be determined. Three optional design approaches are offered for each limit state: 앫 design by means of standard expressions if available (called “direct design”); 앫 the classical approach where design stresses under acting loads are calculated and compared to design stress resistance values (called “stress design”); 앫 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 Element Method) concerning the whole structure (called “design by global numerical analysis”). The latter approach (or more precisely, the rules for the latter approach) will certainly be the object of one or the other criticism. Some engineers in “high-tech” shell application areas, e.g. space structures or nuclear power containments, could argue that sophisticated computer analyses need not be codified. However, especially for shell buckling, it is necessary to specify certain requirements for the numerical modelling, as may be realized from the explanations in the preceding section. As to the different theoretical levels and different levels of modelling when numerically analysing a shell (see Section 4), the relevant terminology needed clear definitions because between practising engineers of different countries and different technical communities quite often a considerable confusion had been observed. Table 1 summarizes the seven types of shell analysis as defined in ENV 1993-1-6, together with their abbreviations. In order to provide the user of the Prestandard with available algebraic informations about the strength and stability of shells, four annexes have been added to the main document of ENV 1993-1-6: 앫 앫 앫 앫

Annex Annex Annex Annex

A: B: C: D:

membrane theory stresses in shells; additional expressions for plastic collapse resistances; expressions for linear elastic membrane and bending stresses; and expressions for buckling design.

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

Type of analysis

Shell theory

÷

Membrane theory of shells

LA

Linear elastic shell analysis

Perfect

GNA

Geometrically nonlinear elastic analysis Materially nonlinear analysis Geometrically and materially nonlinear analysis Geometrically nonlinear elastic analysis with imperfections Geometrically and materially nonlinear analysis with imperfections

Membrane Not applicable equilibrium Linear bending Linear and stretching Nonlinear Linear Linear Nonlinear

Nonlinear Nonlinear

Perfect Perfect

Nonlinear

Linear

Imperfect

Nonlinear

Nonlinear

Imperfect

MNA GMNA GNIA GMNIA

Material law

Shell geometry Perfect

Perfect

5.2. Shell buckling limit state according to ENV 1993-1-6 5.2.1. Fabrication tolerances, quality classes As repeatedly emphasized in this report, the buckling strength of thin shells— contrary to the other abovementioned strength cases—is strongly dependent on the geometrical accurateness of the realized structure. Moreover, residual stresses, e.g. from welding, may in certain cases have a further deteriorating influence. Vicariously for the whole of possible imperfections, three essential and measurable geometric imperfection types have been chosen to define “buckling-relevant” fabrication tolerance limits. They are: 앫 the out-of-roundness of the shell: U r⫽

(dmax−dmin) dnom

(2)

앫 the accidental eccentricity ea at joints perpendicular to membrane compressive forces, related to the wall thickness t: ea Ue ⫽ t

(3)

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앫 the depth ⌬w0 of initial dimples in the shell wall, related to certain critical gauge lengths lg: U0⫽

⌬w0 lg

(4)

The values of the tolerance limit parameters U, as specified in ENV 1993-1-6 for three fabrication tolerance quality classes, are listed in Table 2. The introduction of quality classes reflects the sensitivity of the buckling design strength to fabrication accurateness and should encourage the fabricator to control it during fabrication. Of course, all buckling strength parameters have been made dependent on the quality class in such a manner that a higher quality class is rewarded with a higher design strength. The idea to class shells into strength groups according to the quality of construction was first proposed by Odland [73] and introduced into the Eurocode drafting work by Rotter. 5.2.2. Buckling stress design The 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 nondimensional shell slenderness l=(fy,k/sRc)0.5 with sRc being the critical buckling stress of the perfect shell. This format is compatible with the treatment of other stability cases 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 “buckling parameters”, is adaptable to any lower bound information of a shell buckling case and thus is open for future development: c⫽1 when lⱕl0

冋 册

c⫽1⫺b

l−l0 lp−l0

h

(5) when l0⬍l⬍lp

(6)

Table 2 Values for tolerance limit and equivalent imperfection parameters, respectively, according to ENV 19931-6 Fabrication tolerance quality class

Tolerance parameters

Class

Description

Ur,max

Ue,max

U0,max

Equivalent imperfection parameter U0,eff

A B C

Excellent High Normal

0.007 0.010 0.015

0.14 0.20 0.30

0.006 0.010 0.016

0.010 0.016 0.025

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

c⫽

a when lpⱕl l2

(7)

where (see Fig. 10): l0 is the squash limit slenderness defining the “plastic plateau” of the buckling curve; lp is the plastic limit slenderness above which purely elastic buckling is assumed: lp=[a/(1⫺b)]0.5; b describes the relative stress level at lp; a is the well-known elastic imperfection reduction factor (“knock-down factor”); and h describes the shape of the elastic–plastic buckling interaction between lp and l0. The buckling parameters a, b, h and l0 are for basic shell buckling cases given in Annex D of ENV 1993-1-6, e.g. for externally pressurized cylinders of fabrication tolerance quality class B: a⫽0.65, b⫽0.6, h⫽1, l0⫽0.4

5.2.3. Buckling design by global numerical analysis Rules 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 1—are given in ENV 1993-1-6. In both cases, proper imperfection reduction factors have to 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 deduction based on comparable buckling cases), ENV 1993-1-6 prescribes conservatively to take the reduction factor values for an axially compressed cylinder. Rules for a straightforward shell buckling assessment by a fully nonlinear GMNIA analysis (see Section 4.4) are also given in ENV 1993-1-6. The imperfection influence may be simulated by equivalent geometric imperfections. Their pattern “should be chosen in such a form that it has the most unfavourable effect on the buckling behaviour of the shell”. The difficulties inherent to this banal sentence have been

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discussed in Section 4.4. The maximum deviation of the equivalently imperfect geometry from the perfect shape, i.e. the imperfection amplitude, should be assumed not smaller than ⌬w0,eff⫽U0,eff·lg and ⌬w0,eff⫽100U0,eff·t

(8)

with the parameter U0,eff according to Table 2. As can be seen, the assumed geometric imperfections are about 60% larger than the allowed tolerance limits for measurable initial dimples. This is because the equivalent geometric imperfections are “substitute” imperfections (see Section 3) and have to cover eventual non-measurable imperfections (e.g. residual stresses) as well. An additional rule recommends to check if a 10% smaller amplitude than the one from Eq. (8) possibly delivers a smaller ultimate load. If that is the case (which happens in some shell configurations), the minimum has to be determined. Prior to using the numerical outcome Rcr,GMNIA of the analysis (see Fig. 11) as a direct basis for the design, it has eventually to be calibrated. For this purpose, according to ENV 1993-1-6, other shell buckling cases, for which either test results or reliable characteristic buckling resistance values are available, and which are comparable in their buckling controlling parameters, have to be calculated with the same numerical tool using similar imperfection assumptions and similar modelling techniques.

6. Concluding remarks This “general report” could only address a few selected aspects within the wide field of steel shell stability. Much progress has been achieved in the last years, in terms of better physical insight into the complex behaviour of shells and better numerical tools for its handling when designing a shell structure, as well as of general design rules for fundamental shell stability cases. However, if the reader of this report 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 according to ENV 1993-1-6.

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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 structures, there is a mismatch between the needs for structural design and current codified provisions”. Future theoretical shell stability research should, from the view point of structural design engineers, concentrate on applying the high theoretical knowledge and the powerful numerical tools to unsolved (or hitherto unsatisfactorily solved) application problems, rather than turning the basic shell cases (e.g. the cylinder under uniform axial compression) round and round and coming up with another alleged 0.x% error in Eq. (1) or Eq. (6). Such application problems are: any type of local loads and supports, stiffenings, openings and cutouts, shell combinations, incomplete shells, non-uniform stress states in shells; the list is far from being complete. An excellent example for this type of sophisticated and purposeful numerical research, as the author has it in mind, are the Graz investigations on locally supported cylindrical steel silos or containments [74,75]. A last point that the author would like to emphasize is the continuing need for shell buckling tests. Of course, the basic philosophy of experimental programs has changed in view of today’s numerical facilities. No longer is it the statistics-focussed quantity of tests what matters, but their quality as physical verification benchmarks for numerical models. By the way, tests have an important educational secondary effect on young researchers: they learn that the numerical model is not the reality.

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