Thin-Walled Structures 40 (2002) 893–909 www.elsevier.com/locate/tws Stability of circular cylindrical steel shells under combined loading Th.A. Winterstetter a , H. Schmidt a,∗ a Department of Civil Engineering, University of Essen, D-45117 Essen, Germany Received 11 July 2001; accepted 11 January 2002 Abstract Circular cylindrical shells made of steel are used in a large variety of civil engineering structures, e.g. in off-shore platforms, chimneys, silos, tanks, pipelines, bridge arches or wind turbine towers. They are often subjected to combined loading inducing membrane compressive and/or shear stress states which endanger the local structural stability (shell buckling). A com- prehensive experimental and numerical investigation of cylindrical shells under combined load- ing has been performed which yielded a deeper insight into the real buckling behaviour under combined loading . Beyond that, it provided rules how to simulate numerically the realistic buckling behaviour by means of substitute geometric imperfections. A comparison with exist- ing design codes for interactive shell buckling reveals significant shortcomings. A proposal for improved design rules is put forward. 2002 Elsevier Science Ltd. All rights reserved. Keywords: Cylindrical steel shells; Shell buckling; Combined loading; Axial compression; Hoop com- pression; Torsion; Interaction 1. Introduction Thin-walled circular cylindrical shells in civil engineering applications are very often loaded in a way which produces not simply one of the three buckling relevant membrane forces — axial compression, circumferential compression and shear — but a combination of them. Thus, a design engineer has to consider not only the ∗ Corresponding author. Tel.: +49 201 183 2766; fax: +49 201 183 2710. E-mail address: [email protected] (H. Schmidt). 0263-8231/02/$ - see front matter 2002 Elsevier Science Ltd. All rights reserved. PII:S0263-8231(02)00006-X
Stability of circular cylindrical steel shellsunder combined loading
Th.A. Winterstettera, H. Schmidta,∗
a Department of Civil Engineering, University of Essen, D-45117 Essen, Germany
Received 11 July 2001; accepted 11 January 2002
Abstract
Circular cylindrical shells made of steel are used in a large variety of civil engineeringstructures, e.g. in off-shore platforms, chimneys, silos, tanks, pipelines, bridge arches or windturbine towers. They are often subjected to combined loading inducing membrane compressiveand/or shear stress states which endanger the local structural stability (shell buckling). A com-prehensive experimental and numerical investigation of cylindrical shells under combined load-ing has been performed which yielded a deeper insight into the real buckling behaviour undercombined loading . Beyond that, it provided rules how to simulate numerically the realisticbuckling behaviour by means of substitute geometric imperfections. A comparison with exist-ing design codes for interactive shell buckling reveals significant shortcomings. A proposalfor improved design rules is put forward. 2002 Elsevier Science Ltd. All rights reserved.
Thin-walled circular cylindrical shells in civil engineering applications are veryoften loaded in a way which produces not simply one of the three buckling relevantmembrane forces — axial compression, circumferential compression and shear —but a combination of them. Thus, a design engineer has to consider not only the
scientifically well-explored buckling of a cylindrical shell under fundamental stressstates, but also the interactive buckling.
As a further complication, the membrane stress distributions often differ consider-ably from those of the fundamental load cases “uniform axial compression” , “uniformcircumferential compression” and “ torsional shear” (Fig. 1). Even the seemingly cleardistributions of axial direct stresses and shear stresses from wind loading on cantil-evering cylindrical structures are not as simple as predicted by beam theory (see e.g.Peil & Nolle [25], Greiner & Derler [15]). However, in any case large parts of theshell wall would be deformed after buckling. Thus, it is generally not overconserv-ative to neglect — if no better method is available — the stress variations in circum-ferential and longitudinal direction and to perform a buckling design check with themaximum values of the individual membrane stress components, even in case theyoccur at different locations over the shell wall. This approach was adopted as so-called “Buckling Stress Design” in the recently issued “Shell Eurocode” [12]. Asbasis for the needed buckling interaction design formula, combinations of the threementioned fundamental load cases may be used (Fig. 1). The notation and termin-ology of the Shell Eurocode is used throughout this paper.
2. Existing design rules for fundamental loads
The buckling of cylindrical shells under the three fundamental loads shown inFig. 1 is well understood. Fig. 2 shows shell buckling reduction factors for thesefundamental loads as obtained from selected international design codes which maybe deemed to result from serious research efforts. The general agreement betweenthe various codes is quite good, though showing some differences due to differentexperimental data bases and/or different design approaches and/or different states ofknowledge at the date of codification.
Fig. 1. Fundamental load cases and combined loading.
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Fig. 2. Shell buckling reduction factor curves c � f(l) for the three fundamental load cases as given invarious design standards.
3. Experimental investigations
3.1. Former experimental studies
A comprehensive literature study has been performed by the first author collectingall accessible experimental data on interactive buckling tests [37]. The publishedresults cover nearly all relevant combinations and slenderness parameters for com-bined axial and radial loading, but they show gaps for other load combinations,including the practically important area of stocky cylinders under axial and shearforces, such as wind-loaded towers or pipelines (Fig. 3).
As a first direct conclusion from the experimental evidence, the shape of the inter-action function for cylinders buckling under combined loading proves to be slender-ness-dependent. Whereas the buckling load curves of stocky shells (small r/t-ratio)
Fig. 3. Collected and evaluated interactive buckling test data (the sources are cited in [37]).
approach the highly convex v. Mises-Huber-Hencky yield limit curve, slender shellswith high r/t-ratios show comparably straight-lined interaction curves. Furthermore,evaluation of the published experimental evidence shows that additional tensilestresses — circumferentially from internal pressure in case of axial compression, oraxially in case of circumferential compression, respectively — lower the compressivebuckling resistance of stocky shells because of their plastically-destabilizing effectdue to unfavourable biaxial material stressing, but raise the compressive bucklingresistance of very slender shells because providing an elastically-stabilizing effectdue to reduction of geometric imperfection influences.
3.2. Interactive buckling experiments at the University of Essen
Because of the abovementioned lack of test data, an own test series on stockysteel cylinders loaded by combinations of axial compression and torsion has been
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carried out by the first author [36,29]. Fig. 4a shows the test set-up. The specimenswere cut from mild steel plates and fabricated by cold-rolling and TIG welding ofthe longitudinal seam. The quasi-static yield strength fy,stat was determined by meansof a special material testing procedure and used as material reference value for com-parison analyses. The geometric imperfections were measured prior to testing, evalu-ated using Fourier decomposition and stored for comparison with design standardtolerances and for serving as input for geometrically realistic FE models of the testspecimens (see section 4.4). Table 1 shows the dimensions, material properties andexperimental buckling loads.
The experimental relative buckling stresses are plotted in Fig. 4b. They follownicely the yield limit curve and thus underline the abovementioned conclusionregarding interactive buckling of stocky shells.
4. Numerical investigations
Numerical investigations concerning shell instability problems are thinkable at avariety of different “calculation levels” . The new Shell Eurocode [12] tries to sys-tematize these levels in order to facilitate communication between engineers andresearchers. Its terminology is shown in Table 2, with gradually growing degree ofreality approximation.
Fig. 4. (a) Test set-up, (b) experimental buckling loads vs. yield limit.
Table 2Calculation levels for a shell buckling analysis according to ENV 1993-1-6 (“Shell Eurocode” )[12]
LA Linear analysis (linear elastic eigenvalue calculation, applied to the perfect structure)GNA Geometrically nonlinear analysis of the prebuckling state of the perfect structure (elastic
material), search for bifurcation or snap-through points)GMNA Geometrically and materially nonlinear analysis of the prebuckling state of the perfect
structure (nonelastic material), search for bifurcation or snap-through pointsGNIA Geometrically nonlinear analysis of the imperfect structure (elastic material),
introducing geometric substitute imperfectionsGMNIA Geometrically and materially nonlinear analysis of the imperfect structure (nonlelastic
Linear analyses (LA) of interactive buckling problems have been subject to numer-ous research papers, mainly due to their accessibility to analytical solutions. In ageneral combined loading case, all three fundamental membrane stress componentssx, sq and t (see Fig. 1) are acting on the structure. FE eigenvalue calculations atdifferent ratios of these components yield a “buckling interaction surface” as shownin Fig. 5, which includes also some selected buckling modes [37].
Essential conclusions from linear eigenvalue analysis are the following ones:
� The bifurcation interaction between axial compression sx and circumferentialcompression sθ (e.g. from external pressure) is nearly linear; circumferential ten-
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Fig. 5. Linear buckling analysis (LA): Three-dimensional interaction surface and two-dimensional inter-action curves for combined loading. Geometry: r/t=100, l/r=1. Material: E=210 kN/mm2, n=0.3. BCs:Simply supported, axially free.
sile stresses do not raise the axial bifurcation compressive buckling stress signifi-cantly (Flugge [13]).
� The bifurcation interaction between axial compression and torsion is nearly linear,too (Kromm [23], Batdorf et al. [5]).
� The parabolic shape of the bifurcation interaction curve between external pressureand torsion agrees with the results of Ho & Cheng [19] and Simitses [33].
� Axial tensile stresses raise the circumferential compression and torsion bifurcationbuckling stresses.
A refinement of the linear analysis (LA) is achieved with a geometrically nonlinearanalysis (GNA). However, for elastic cylindrical shells, these both analysis typesyield, as far as the shape of bifurcation interaction surfaces and curves is concerned,similar results (Almroth [1], Tennyson et al. [34], Yamaki [38]).
4.3. Geometrically and materially nonlinear analysis (GMNA)
In the case of stocky shells, the effects of plasticity must be included in the calcu-lation by performing a geometrically and materially nonlinear analysis (GMNA).Theoretical interaction curves using GMNA have been reported by Galletly & Pems-ing [14] and Krysik [24]. Fig. 6 shows some results from [37] using the linearly-elastic, ideally-plastic material law of structural steel with two different yieldstrength values.
The GMNA interaction curves are quite convex in their shapes, asymptoticallyapproaching either the LA/GNA elastic buckling interaction curves (when displayingbifurcation buckling) or the yield limit curve (when displaying snap-throughbuckling), respectively. The lower one of these two types of calculational limits isan upper bound for GMNA interactive buckling of perfect cylinders.
4.4. Geometrically and materially nonlinear imperfect analysis (GMNIA)
In a GMNIA analysis, additionally to the preceding theoretical levels, equivalentgeometric imperfections have to be introduced into the simulation model in order to
Fig. 6. Geometrically and materially nonlinear buckling analysis (GMNA): Interaction curves for (a)axial and circumferential buckling stresses, (b) axial and shear buckling stresses. Geometry: r/t=100 and200, l/r=1, perfect shell. Material: E=210 kN/mm2, fy=240 N/mm2 and 360 N/mm2, n=0.3. BCs: Clamped,axially restrained.
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Table 3Different approaches for the numerical simulation of geometrically imperfect shell structures
“Realistic” geometric Introduction of the “ real” geometric imperfections of the structure,imperfections determined e.g. by measurement“Worst” geometric imperfections Introduction of a mathematically determined “worst possible”
imperfection pattern“Stimulating” geometric Introduction of a simple substitute geometric imperfection pattern toimperfections “stimulate” the characteristic physical shell buckling behaviour (has
to be calibrated)
cover the effects of real imperfections (shape deviations, weld depressions, residualstresses, etc.). The second author tried to systematize the possible approaches [30],as summarized in Table 3.
4.4.1. “Realistic” geometric imperfectionsThe most realistic thinkable approach is to analyse the actual, i.e. “ real” imperfect
shell structure as generated from measurements after fabrication and erection (seee.g. Arbocz [3]). Following these thoughts, the imperfection data recorded from theEssen test specimens were directly introduced into FE models to re-calculate theexperiments described in section 3.2. Fig. 7 illustrates for one of the specimens theresults of this calculation. It shows a) the measured imperfection pattern, b) themeasured permanent experimental deflections after buckling and c) the numericaldeflections according to the geometrically imperfect FE model using the measuredimperfection pattern and material properties.
The experimental and numerical buckling modes in Fig. 7 show fairly good agree-ment; however, the remaining slight differences indicate uncovered influences likeresidual stresses or realistic (non-mathematical) boundary conditions. Unfortunately,the principally successful simulation of the experiments can not conceal the fact thatthis approach is not applicable in practical cases where the actual imperfect shapeof a civil engineering shell structure is not known at the design stage.
4.4.2. “Worst” geometric imperfectionsThis approach means to find the mathematically determined “worst possible”
imperfection pattern. Many researchers have dealt with this theoretically highlydemanding topic (e.g. Koiter [22], Hutchinson [20], Wagenhuber & Duddeck [35],Deml & Wunderlich [7]).
However, it is doubtful whether these approaches provide imperfection patternswhich are close enough to real imperfections occurring in fabricated steel shellsand thus controlling the behaviour in practise. Moreover, no commercial or semi-commercial software implementations accessible to design engineers are at hand atpresent or in the near future.
4.4.3. “Stimulating” geometric imperfectionsAs already mentioned, the imperfection pattern of the structure after fabrication
and erection is not known to the design engineer. To obtain nevertheless a reliablebuckling resistance prediction by means of GMNIA, i.e. without the need of time-consuming and expensive tests, a feasible approach is to use common commercialFE packages and to introduce a simple equivalent geometric imperfection patternwhich stimulates properly the characteristic physical shell buckling behaviour. Thisis a well-explored approach. The Shell Eurocode proposes, for example, eigenmode-affine patterns or patterns adapted to the constructional detailing for use within aGMNIA analysis. However, the prescribed imperfection amplitudes are sometimesoverconservative and therefore tend to yield uneconomical designs [27,37].
Prior to thinking of a purely numerically based GMNIA buckling design for anew structural shell type, the amplitude of the considered “stimulating” imperfectionpattern has to be calibrated against existing experimental data bases for known shellbuckling cases. A very well-explored example for such an unfavourable “stimulating”imperfection pattern is a single axisymmetric inward predeformation in the cylinderwall. On the one hand, it corresponds to a real circumferential weld depression andon the other hand, it is suitable to reproduce the established design reduction factorsfor axial compression buckling (Hutchinson [20], Bornscheuer et al. [6], Rotter &Teng [26], Hautala [18]).
Of course, such an axisymmetric predeformation does not lower the circumfer-ential compression or torsion buckling strengths which are part of the general com-bined loading problem studied in this paper (Jurcke et al [21]). For these cases alongitudinal depression corresponding to the longitudinal seam position of a realwelded steel cylinder has to be introduced into the numerical model (Guggenberger[16]). The first author of this paper has used a combination of a circumferential anda longitudinal depression as “cross weld” equivalent imperfection pattern [37], seeFig. 8a) for a comprehensive parametric study using a large variety of shell geo-metries. The aim was to determine proper imperfection amplitudes to match therespective reduction factor curves (Fig. 9). As an important result, the necessaryamplitudes w0 proved to be approximately inversely connected to the respective lin-ear bifurcation buckling stresses, as expressed in Eq. (1). Thus, the “cross weld”equivalent imperfection pattern with amplitudes according to Eq. (1) may be lookedat as a consistent imperfection system which is able to cover all possible membrane
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Fig. 8. (a) “Cross weld” equivalent imperfection pattern, schematically representing circumferential andlongitudinal weld depressions in real structures. (b) Numerical buckling modes for axial and circumfer-ential compression, respectively. Geometry: r/t=500, l/r=0.65, imperfection pattern acc. to Fig. 8a.Material: E=210 kN/mm2, n= 0.3, fy=240 N/mm2.
Fig. 9. Calibration of “cross weld” equivalent imperfection amplitudes using shell buckling reductionfactors of established design codes.
stress states in a cylindrical shell with regard to a numerical GMNIA buckling designprocedure as codified in the Shell Eurocode [12].
w0
t�
kbifurcation stress
(1)
with kx�250, kq�100, kt�300 and the bifurcation stresses sxRc, sθRc and sRc
[N/mm2] acc. to ENV 1993–1–6, Annex D.Numerical GMNIA interaction curves based on this approach are shown in Fig.
10. They are markedly slenderness-dependent in their shapes. This agrees with simi-lar curves calculated by the authors using eigenmode-affine equivalent imperfectionsand amplitudes according to ENV 1993–1–6 [31]. On principle, the following majorfeatures known from the experimental evidence may be observed here again:
� Very slender shells with high r/t-ratios behave, in terms of the curve shapes,similarly to LA, with significant stability raising influence of additional tensilestresses on the buckling resistance, but with rather straight-lined biaxial com-pression interaction.
� Stocky shells with low r/t-ratios show strong effects of biaxial yielding, i.e. convexbuckling curves, asymptotically approaching the biaxial yield limit curves.
4.5. Summary of numerical investigations
The calculated buckling resistance is reduced with growing calculation realitylevel: LA→GNA→GMNA→GMNIA (Fig. 11). Analogously to the slenderness-dependent behaviour under fundamental loads, a cylindrical shell structure undercombined loading responds to the ratio of bifurcation resistance over yield limit or,in other words, to the mutual positions of the LA and yield interaction curves,respectively. The lower one of these two curves is an upper bound for the “character-istic” buckling resistance of the real cylinder.
Coming from the classical linear eigenvalue calculation, the changes in interactivebuckling behaviour are, in case of a stocky shell, controlled by biaxial nonelasticmaterial behaviour, whereas the behaviour of very slender shells is governed bygeometric imperfections.
Existing design rules for interactive bucklingIn Fig. 12, an evaluation of some existing interactive buckling design rules for
steel cylinders is presented. The r/t-ratio, indicating different shell slendernesses, is
Fig. 10. Geometrically and materially nonlinear imperfect analysis (GMNIA): Interaction curves for (a)axial and circumferential stresses and (b) axial and shear stresses. Geometry: r/t varied, l=15√r/t. Material:E=210 kN/mm2, n=0.3, fy=240 N/mm2. BCs: Simply supported.
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Fig. 11. Schematic effect of different calculation levels and of different slenderness parameters on theshape of buckling interaction curves (axial and circumferential compression): (a) Moderately thin-walledcylinder (medium relative slenderness), (b) highly thin-walled cylinder (large relative slenderness)
Fig. 12. Existing codified design rules for buckling under combined loading: (a) axial and circumferentialbuckling stresses, (b) axial and shear buckling stresses. Geometry: r/t varied, l/r=1. Material: E=210kN/mm2, ν=0.3, fy=240 N/mm2. BCs: Simply supported, axially free.
used as a curve parameter. The graphs reveal significant deficits for interactive buck-ling: Some stress combinations are not covered at all, and sharp breaks in somecurves indicate physical inconsistencies.
For the practically important load combination of axial compression and internalpressure (left quadrant of the graph in Fig. 12a), an improved design proposal byRotter [27] has been used in the Shell Eurocode [12]; it contains still some minorinconsistencies concerning the adjustment of the introduced new reduction factors[37]. Another, similarly straightforward design proposal has been published by Kry-sik [24].
Of course, there are some special cases of non-uniform membrane stress states incylinders for which enough knowledge is available to formulate specific shell buck-
2 GNIA and tests---imperfect elastic shell Elastically-stabilizing effect of internalpressure on axial compression buckling
3 GMNIA---imperfect shell made of structural steel Nonelastic material behaviour underbiaxial strain
ling design rules instead of using a — sometimes overconservative — interactionequation. Such cases are silos on discrete supports [17], or purely transversely loadedtubular shells [32], or silo walls under unsymmetric load conditions [28]. However,if the stress gradients are small, or if there are some uncertainties concerning thestress peak magnitude or the area of shell wall over which the high stresses extend,an interactive buckling formula based on the combination of the three fundamentalload cases should be used when performing a “Buckling Stress Design” .
5. New proposal for design rules
A new consistent three step design procedure on the basis of existing fundamentalload codification is proposed [37], see Table 4, Eqs. (2)–(12) and Fig. 13. Majorfeatures are slenderness-dependency, the coverage of elastically-stabilizing tensilestresses and of all relevant influences of plasticity as well as consistent treatment ofall relevant stress combinations.
Step 1: Calculate the basic critical buckling stresses s(.)Rc,0 and tRc,0 for pure mem-
Fig. 13. Graphical evaluation of the new design concept. Geometry: l/r=0.5, r/t=50…2000. Material:E=210 kN/mm2, fy=240 N/mm2, ν=0.3. BCs: Simply supported.
907T.A. Winterstetter, H. Schmidt / Thin-Walled Structures 40 (2002) 893–909
brane stress states acc. to ENV 1993–1–6, Annex D. If relevant, increase them forthe influence of tensile stresses by means of Eq. (2) to (4).
sqRc � sqRc,0·�1 � 0,5·| sxd
sxRc,0|�, in case of sxd being tensile (2)
tRc � tRc,0·�1 � 0,5·| sqdsqRc,0|�, in case of sqd being tensile, (3)
tRc � tRc,0·�1 � | sxd
sxRc,0|�, in case of sxd being tensile. (4)
Calculate the slenderness parameters l(.) and the reduction factors c(.) acc. to ENV1993–1–6.
Step 2: Take the elastically — stabilizing effect of internal pressure into accountby means of Eq. (5) to (7). The empirical formulas from DIN 18800–4 [10] are usedhere; however, the axpe-formula of ENV 1993–1–6 could be taken as well.
cxq � cx, if lx�0,7, (5)
cxq � cx�1 � 1,2lx�qi
E�rt�2�0,38lx�0,7
0,3 �, if 1,0�lx�0,7, (6)
cxq � cx�1 � 1,2lx�qi
E�rt�2�0,38�, if lx�1,0. (7)
Step 3: Take the nonelastic material behaviour under biaxial stress states intoaccount by means of Eqs. (8) to (12).
If one or two of the stress components are not present or if they are tensile,then the respective reduction factors are c=1,0. Tensile stresses are introduced withnegative signs.
As a result of comprehensive experimental and numerical investigations, a newproposal for a consistent “Buckling Stress Design” for interactive buckling of circularcylindrical steel shells has been put forward. In the context of the work presentedin this paper, two topics should — in the authors’ opinion — be target to furtherresearch efforts: Firstly, the use of a “cross weld” substitute imperfection patternwith bifurcation-stress-dependent amplitudes as presented in section 4.4 should bechecked by means of numerical GMNIA calculations for conical and spherical shells,too. Secondly, more efforts are needed to establish generally applicable, economical,safe and unmistakable guidelines for a purely numerical buckling design.
References
[1] Almroth BO. Influence of edge conditions on the stability of axially compressed cylindrical shells.AIAA J 1966;4(1):134–40.
[2] API. Bulletin 2U on Stability Design of Cylindrical Shells, 1st Ed. American Petroleum Institute,Washington, 1987.
[3] Arbocz J. The imperfection data bank, a means to obtain realistic buckling loads. In Ramm E.Buckling of shells. Stuttgart, Germany, 1982.
[4] ASME Section III Code Case N-284. Metal containment shell buckling design methodes. Am. Soc.of Mech. Eng., New York, USA, 1980.
[5] Batdorf SB, Stein M, Schildcrout M. Critical combinations of torsion and direct axial stress for thin-walled cylinders. NACA TN 1345, Washington D.C., 1947.
[6] Bornscheuer BF, Hafner L, Ramm E. Zur Stabilitat eines Kreiszylinders mit einer Rundschweißnahtunter Axialbelastung. Stahlbau 1983;10:313–8.
[7] Deml M, Wunderlich W. Direct evaluation of the “worst” imperfection shape in shell buckling.Comp. Meth. Appl. Mech. Eng. 1997;149:201–22.
[8] Det norske Veritas. DnV Rules for the design, construction and inspection of offshore platforms,Appendix C — Steel Structures. Hovik, Norway, 1992.
[11] ECCS-R. Buckling of steel shells — European recommendations, 4th ed. Brussels, Belgium: Gen.Sec. ECCS, 1988.
[12] ENV 1993–1–6. Eurocode 3: Design of steel structures, Part 1–6: General Rules — SupplementaryRules for Shell Structures. Brussels, Belgium: CEN, 1999.
[13] Flugge W. Stresses in shells, 2nd ed. New York/Heidelberg/Berlin: Springer, 1973.[14] Galletly GD, Pemsing K. On design procedures for the buckling of cylinders under combined axial
compression and external pressure. Trans. ASME, J. Press. Vess. Techn. 1984;106(5):134–42.[15] Greiner R, Derler P. Effect of imperfections on wind-loaded cylindrical shells. Thin-Walled Struc-
tures 1995;23:271–81.[16] Guggenberger W. Buckling and postbuckling of imperfect cylindrical shells under external pressure.
Thin-Walled Structures 1995;23:351–66.[17] Guggenberger W. Proposal for design rules of axially loaded steel cylinders on local supports. In:
Proc. of the Int. Conf. on Struct. Steelwork, Hong Kong. 1996. p. 1225–30.[18] Hautala KT. Buckling of axially compressed cylindrical shells made of austenitic stainless steels at
ambient and elevated temperature. University of Essen, Dr. thesis., 1999.[19] Ho BPC, Cheng S. Some problems in stability of heterogenous aelotropic cylindrical shells under
combined loading. AIAA J 1963;1(7):1603–7.
909T.A. Winterstetter, H. Schmidt / Thin-Walled Structures 40 (2002) 893–909
[21] Jurcke RK, Kratzig WB, Wittek U. Kreiszylinderschalen mit wulstartigen Imperfektionen. Stahlbau1983;52:241–4.
[22] Koiter WT. The effect of axisymmetric imperfections on the buckling of cylindrical shells underaxial compression. Proc. Kon. Ned. Akad. Wet. B. 1963;66:265–79.
[23] Kromm A. Die Stabilitatsgrenze der Kreiszylinderschale bei Beanspruchung durch Schub- und Lang-skrafte. Jahrbuch der deutschen Luftfahrtforschung 1942;:602–16.
[24] Krysik, R. Stabilitat stahlerner Kegelstumpf- und Kreiszylinderschalen unter Axial- und Innendruck.University of Essen, Dr. thesis., 1994.
[25] Peil U, Nolle H. Zur Frage der Schalenwirkung bei dunnwandigen, zylindrischen Stahlschornsteinen.Bauingenieur 1988;63:51–6.
[26] Rotter JM, Teng JG. Elastic stability of cylindrical shells with weld depressions. J. Struct. Eng.1989;15:1244–64.
[27] Rotter JM. Design standards and calculations for imperfect, pressurized axially compressed cylinders.In: Krupka V, Schneider P, editors. Proc. of the 2nd Int. Conf. on Carrying Capacity of ShellStructures, Brno. 1997. p. 354–60.
[28] Rotter JM. The analysis of steel bins subject to eccentric discharge. In: Proc. of the 2nd Int. Conf.on Bulk Materials — Storage, Handling and Transportation. IEAust., Wollongong. 1986. p. 264–71.
[29] Schmidt H, Winterstetter ThA. In: Chan SL, Teng JG, editors. Proceedings of the ICASS’99, HongKong, China. Buckling interaction strength of cylindrical steel shells under axial compression andtorsion. 1999.
[30] Schmidt H. Stability of steel shell structures. Gen. Rep. for TS 8 of the SDSS ‘99 (Timisoara,Romania) Journ. Construct. Steel Res. 2000;55:159–81.
[31] Schmidt H, Winterstetter ThA. Cylindrical shells under combined loading: Axial compression, exter-nal pressure and membrane shear. In: Teng JG, Rotter JM, editors. Buckling of Thin Metal Shells.(to be published in 2001).
[32] Schmidt H. Winterstetter ThA. 2000. Cylindrical shells under shear and torsion. In: Teng JG, RotterJM, editors. Buckling of Thin Metal Shells, (to be published in 2001).
[33] Simitses GJ. Instability of orthotropic cylindrical shells under combined torsion and hydrostaticpressure. AIAA J 1967;5(8):1463–9.
[34] Tennyson RC, Booton M, Chan KH. Buckling of short cylinders under combined loading. Trans.ASME, J. Appl. Mech. 1978;45:574–8.
[35] Wagenhuber W, Duddeck H. Numerischer Stabilitatsnachweis dunner Schalen mit dem Konzept derStorenergie. Arch. App. Mech 1991;61:327–43.
[36] Winterstetter ThA, Schmidt H. Beulversuche an langsnahtgeschweißten stahlernen Kreiszylindersch-alen im elastisch-plastischen Bereich unter Axialdruck, Innendruck und Torsionsschub. Res. Rep.82, Dept. Civil Eng., University of Essen, 1999.
[37] Winterstetter ThA. Stabilitat von Kreiszylinderschalen aus Stahl unter kombinierter Beanspruchung.University of Essen, Dr. thesis., 2000.
[38] Yamaki N. Elastic stability of cylindrical shells. Amsterdam/New York/Oxford: North Holland,1984.
