Engineering Structures 33 (2011) 3570–3578

Contents lists available at SciVerse ScienceDirect

Engineering Structures

journal homepage: www.elsevier.com/locate/engstruct

Buckling of cylindrical shells with stepwise variable wall thickness underuniform external pressureLei Chen, J. Michael Rotter ∗, Cornelia DoerichInstitute for Infrastructure and Environment, University of Edinburgh, Edinburgh, EH9 3JL, UK

a r t i c l e i n f o

Article history:Received 21 February 2011Received in revised form11 June 2011Accepted 1 July 2011Available online 16 August 2011

Keywords:Cylindrical shellsTanksSilosStepwise variable wall thicknessExternal pressureBuckling

a b s t r a c t

Cylindrical shells of stepwise variable wall thickness are widely used for cylindrical containmentstructures, such as vertical-axis tanks and silos. The thickness is changed because the stress resultantsare much larger at lower levels. The increase of internal pressure and axial compression in the shell isaddressed by increasing the wall thickness. Each shell is built up from a number of individual strakes ofconstant thickness. The thickness of the wall increases progressively from top to bottom.

Whilst the buckling behaviour of a uniform thickness cylinder under external pressure is welldefined, that of a stepped wall cylinder is difficult to determine. In the European standard EN 1993-1-6 (2007) and Recommendations ECCS EDR5 (2008), stepped wall cylinders under circumferentialcompression are transformed, first into a three-stage cylinder and thence into an equivalent uniformthickness cylinder. This two-stage process leads to a complicated calculation that depends on a chart thatrequires interpolation and is not easy to use, where the mechanics is somewhat hidden, which cannot beprogrammed into a spreadsheet leading to difficulties in the practical design of silos and tanks.

This paper introduces a new ‘‘weighted smearedwallmethod’’, which is proposed as a simplermethodto deal with stepped-wall cylinders of short or medium length with any thickness variation. Bucklingpredictions are made for a wide range of geometries of silos and tanks (unanchored and anchored) usingthe new hand calculation method and compared both with accurate predictions from finite elementcalculations using ABAQUS andwith the current Eurocode rules. The comparison shows that theweightedsmeared wall method provides a close approximation to the external buckling strength of stepped wallcylinders for a wide range of short and medium-length shells, is easily programmed into a spreadsheetand is informative to the designer.

© 2011 Elsevier Ltd. All rights reserved.

1. Introduction

Cylindrical shells of stepwise variable wall thickness are widelyused for cylindrical containment structures, such as vertical-axis tanks and silos. The increase of internal pressure and axialcompression in the shell is addressed by increasing the wallthickness [1]. Each shell is built up from a number of individualstrakes of constant thickness. The thickness of the wall increasesprogressively from top to bottom. Fig. 1 shows a typical exampleof a tank structure taken from ECCS EDR5 [2].

The thickness of thewall of these cylindrical vessels is generallyvery thin and the strength is often controlled by elastic bucklingfailure. The buckling behaviour of a cylindrical shell with uniformwall thickness under uniform external pressure has been exten-sively studied bymany researchers [3–7]. However, themechanicsof buckling in cylinders of stepwise variable wall thickness has not

∗ Corresponding author. Tel.: +44 131 667 3576; fax: +44 131 650 6781.E-mail address:m.rotter@ed.ac.uk (J. Michael Rotter).

0141-0296/$ – see front matter© 2011 Elsevier Ltd. All rights reserved.doi:10.1016/j.engstruct.2011.07.021

been captured well in current design procedures and has drawnlittle attention to date. This may be partly because the formulationof general rules for all possible patterns of thickness variation isdifficult.

In the European Standard for Shells [8] and European Recom-mendations on Shell Buckling [2], steppedwall cylinders under cir-cumferential compression are transformed, first into a three-stagecylinder and thence into an equivalent uniform thickness cylinderbased on the research of Resinger and Greiner [9–11]. This two-stage process leads to a complicated calculation that depends on achart that requires interpolation and is not easy to use, where themechanics is somewhat hidden, andwhich cannot be programmedinto a spreadsheet so that difficulties arise in the practical designof silos and tanks.

This paper introduces a new ‘‘weighted smeared wall method’’,which is proposed as a simpler method to deal with stepped-wallcylinders of short or medium-length with any thickness variation.The method is developed from an idea proposed by Trahairet al. [12] for stepped wall cylinders under external pressure.Buckling predictions are made for a wide range of geometriesfor silos and both unanchored and anchored tanks using the new

L. Chen et al. / Engineering Structures 33 (2011) 3570–3578 3571

Table 1Boundary condition terminology for cylindrical shells [6].

Boundary condition δu (Meridional displacement) δv (Circumferential displacement) δw (Radial displacement) δβ (Meridional rotation)

S1 r r r fS3 f r r fC1 r r r rC3 f r r r

Fig. 1. Typical example of tank structure (after [17]).

hand calculation method. These predictions are compared withboth accurate predictions from finite element calculations usingABAQUS and those of the current Eurocode rules. The comparisonshows that the weighted smeared wall method provides a closeapproximation to the external buckling strength of stepped wallcylinders for a wide range of short and medium-length shells. Itis easily programmed into a spreadsheet and is informative to thedesigner.

2. Buckling of cylinders of constant thickness under uniformexternal pressure

The buckling strength of a cylinder under uniform externalpressure is governed by the geometry of the cylinder. It dropsrapidly as the buckle becomes longer, so longer cylinders havea lower critical buckling pressure. The buckling strength is alsoaffected by the thickness of the cylinder: thinner cylinders havelower critical buckling pressures.

The theoretical external buckling pressure of a constant thick-ness cylinder can be obtained by minimising [3,13,14]:

pcr = Eπω

2 tr

2 α +

1α

2

×

1121 − ν2

+

ωπ

4 α

α +1α

4 (1)

where α =πmω

rt , r is the radius of the shell, t is the thickness

of the shell wall, E is Young’s modulus, ν is Poisson’s ratio,m is thenumber of complete buckles around the circumference andω is thedimensionless length parameter which is defined as:

ω =ℓ

√rt

(2)

in which ℓ is the half wave-height of the buckle, which is equal tothe whole height of a cylinder of uniform thickness.

Medium-length cylinders are defined in EN 1993-1-6 [8] asthose in the geometric range:

20 <ω

Cθ< 1.63

rt

(3)

in which Cθ is a buckling pressure factor to account for differentboundary conditions. Minimising Eq. (1) for this case, the criticalbuckling wave number may be closely approximated by [7]:

m2cr =

π

ω

rt

4361 − ν2

(4)

Table 2External pressure factors for medium-length cylinders Cθ (EN 1993-1-6, 2007).

Case Boundary conditions External pressure factors Cθ

1 S1 base, S1 top 1.52 S1 base, S3 top 1.253 S3 base, S3 top 1.04 S1 base, F (free) top 0.6

and the circumferential buckling pressure for medium-lengthcylinders is then [2] given by the classical equation as [8]:

pcr,D = 0.92E(t/r)2/ω (5)

which was derived by Ebner [4], based on classical linear Donnellshell buckling theory [14].

Using the boundary condition terminology of Singer andYamaki [6], Eqs. (1)–(5) all use classical simply-supported bound-aries S3 at both ends. The main boundary conditions used in prac-tice are defined in Table 1, where r means restrained and f meansfree. The structural consequences of different boundary conditionswere explored by Schnell [15] who showed that theymay be takeninto account by modifying the buckling external pressure by thefactors Cθ , as indicated in Table 2 for medium-length cylinders [8].

Restraint ofmeridional rotations at the boundaries has no effecton the buckling strength of medium-length cylinders, so S1 andC1, S3 and C3 are not differentiated in Table 2. For quite shortcylinders, the pressure factor Cθ is not constant but depends onthe dimensionless length parameter ω, which can be found in EN1993-1-6 [8].

3. Buckling of cylinders of stepwise variable wall thicknessunder uniform external pressure

For cylinders of stepwise variable wall thickness under uniformexternal pressure, the top thinnest parts are always involved in thebucklingmode. The buckling strengthmay reducewhen the lower,thicker strakes are also involved because the increasing lengthcan reduce the strength of the potential buckle, but at the sametime the increased thickness of the lower strakes may increase thepotential buckling pressure. The simultaneous change of lengthand thickness leads to two opposite trends, so it is not easy todetermine how large the critical buckle should be in a stepwisevarying thickness wall cylinder. A change in the wall thicknessdistribution can easily result in a different critical buckling mode.This is the reason why the current codified method [8] is rathercomplicated.

In the experimental study of Fakhim et al. [16], buckling modeswith buckles extending over the whole height of the shell wereobserved in shorter shells, while in longer shells, buckling modeswith buckle waves only in the top parts were observed.

An important characteristic of the external pressure bucklingmode is that the bottom of a buckle always lies close to a changeof plate thickness because the buckling pressure falls as the lengthincreases. Since the bottomof the buckle is not normally pinned, itslocation is not precisely identifiable, but the approximation that itsbottom lies at a change of plate thickness is relatively good becausethe complete buckle is slightly longer (continuing slightly intothe restraining underlying course) and longer buckles have lower

3572 L. Chen et al. / Engineering Structures 33 (2011) 3570–3578

Fig. 2. Buckling modes of cylindrical tanks of 40 m diameter and different wallheights under uniform external pressure [7,17].

strengths, but the restraint of the underlying course increases thebuckling pressure again, so these two effects partially counteracteach other. The later part of this paper provides an effective andquite accurate correction to account for the restraining effects ofthe underlying course beneath the buckle.

The illustrative example in Chapter 11 of ECCS EDR5 [17],shown in Fig. 2, is quite helpful in giving an initial understandingof the buckling behaviour of step-wise variable wall thicknesscylinders under external pressure. It shows the buckling modesfor three cylindrical tanks with identical upper strakes, but withdifferent total heights, under uniform external pressure. Themodes indicate that the variation of the wall thickness results inbuckles developing only in the same group of upper strakes, whilethe lower ones provide considerable restraint against both radialand axial displacements at the boundary between two strakes [17].The height of the buckle does not depend on the thicknesses usedthroughout the whole wall if the buckling mode in the top parts isalready critical.

So for a given cylinder composed of several parts with strakesof stepwise varying thickness, determination of the critical buckleheight is an initial step in studying the buckling behaviour. Thecritical buckling mode may extend over one or more strakesbelow the thinnest one or may extend over the whole wall. Thiscorresponds physically to the fact that the thinner upper partsare restrained by the thicker lower parts. The design methodin Eurocode [8,17], in Trahair’s method [12], and the weightedsmeared wall method derived in this paper all rely on this concept.

4. Greiner’s method (1972) for stepped wall cylinders underuniform external pressure

Greiner’smethod [11] for steppedwall cylinders under uniformcircumferential compression was developed based on linearbifurcation analyses of cylinders under uniform external pressure,which was adopted into DIN 18 800 [18] and EN 1993-1-6 [8]. Themethod assumes no axial restraint at the base, corresponding toCθ = 1.0.

For cylinders consisting of more than three sections of differentwall thickness (Fig. 3a), the design method first replaces themultiple sections by an equivalent three-section-cylinder (Fig. 3b).The equivalent lengths ℓa, ℓb, ℓc and equivalent thicknessesta, tb, tc are determined by categorisation rules. Then the three-section-cylinder is replaced by an equivalent single cylinder of

Fig. 3. Transformation of stepped cylinder to equivalent cylinder [17]. (a) Cylinderof stepwise variable wall thickness. (b) Equivalent cylinder comprising threesections. (c) Equivalent single cylinder with uniform wall thickness.

effective length ℓeff and of uniform wall thickness t = ta (Fig. 3c)using a complicated set of charts (Fig. 4) for discrete valuesof the ratios of thickness between the three sections, makinginterpolation necessary. The effective length of the cylinder ℓeff isdetermined by:

ℓeff = ℓa/κ (6)

in which κ is a dimensionless factor indicating the effect of thestiffness of the lower parts of the cylinder and can be extractedfrom Fig. 4.

However, the curves in Fig. 4 present some challenges. Noequations are known to describe these curves, so they cannot beprogrammed into a spreadsheet. The published figure is small so itis not easy to obtain an accurate value of the parameter κ . It is clearthat thismethod in EN1993-1-6 [8] is rather tricky for practical usein the design of stepped wall cylinders, and accurate estimates ofbuckling pressures can only be obtained with difficulty.

After transformation, the general buckling formula equation(7) [17] for cylinders of uniform thickness under uniform externalpressure is then used to calculate the buckling pressure for theequivalent medium-length cylinder:

qRcr = 0.92CθErℓeff

tar

2.5

. (7)

When the cylinder has uniform thickness, the value of Cθ isgiven in Table 2. However, the axial restraint of the lower bound-ary may not have a significant effect on a cylinder with varyingthickness strakes [17], so the value of Cq must always be taken as1.0 [17].

5. The weighted smeared wall method for non-uniform walls

For a shell wall with both circumferentially and axially vary-ing stiffness, Trahair et al. [12] adopted a buckling mode withsinusoidal deformation in both directions and evaluated the cor-responding ‘‘effective thickness’’ of the components within thebuckle. This equivalent thickness is approximated by ‘‘smearingout’’ the thickness changes, replacing them with an equivalentthickness in the bucklingmode. Potential bucklingmodes can thenbe checked using the equivalent thickness to find the critical buck-ling mode with lowest buckling strength. The weighted smearedwall method derived in this paper is based on this concept.

It should be noted that the original idea of Trahair et al. was onlypublished in a small sub-section of a silos design guide in Australia,and without a derivation. It has not been widely used to date. Itwas written for corrugated silo walls with vertical stiffeners. It hasnever been verified either by classical equations or finite elementstudies. It assumes a half sine wave for the buckle shape in the

L. Chen et al. / Engineering Structures 33 (2011) 3570–3578 3573

Fig. 4. Factor κ for the determination of leff [8,17].

vertical direction with an S3 base boundary condition, but this isnot a precise representation of the true buckle shape (Fig. 2).

The equivalent thickness in this paper is evaluated as anappropriate weighted value based on energy principles [13]. It isobtained by considering the contribution of each part of the wallto the energy involved in the buckling mode, and this can thenbe equated to the energy of the destabilising pressures. This isequivalent to a single term Galerkin procedure or a first estimateof Rayleigh–Ritz evaluation. A strict smearing of thewall requires amixture of linear and cubic thickness terms to capture the relativeeffects of stretching and bending, but this leads to a process that istoo complicated for simple design use. The decision was thereforemade to use a single power of thickness throughout, and thevalue was chosen to provide a close, but lower bound, estimate ofthe computationally evaluated buckling pressure. This was foundto be best using only the bending stiffness, which still leads toreasonably accurate estimates, as is shownby the results presentedlater.

The equivalent uniform thickness teq is then found as:

t3eq =

4mπℓ

∫ l

0

∫ π/m

0t3z sin2 (mθ) sin2

πzℓ

dθdz. (8)

The thickness tz varies up the meridian, so the integral withrespect to z must be performed. For the reasons given above, thepower of the chosen thickness differs from Trahair’s correspondingequation, which was expressed as:

teq =

4mπℓ

∫ l

0

∫ π/m

0tz sin2 (mθ) sin2

πzℓ

dθdz. (9)

The buckling pressure predictions using Eq. (8) were verified toyield closer but conservative approximations to the accurate finiteelement predictions than those using t, t2, or t2.5, so Eq. (8) wasadopted throughout the following calculations.

As the thickness of the cylinder is constant around the circum-ference, the equivalent thickness needs no integration in the cir-cumferential direction and Eq. (8) reduces to:

t3eq =

2ℓ

∫ l

0t3z sin2

πzℓ

dz. (10)

For a buckling mode that extends over n strakes of constantthickness, Eq. (10) reduces to the sum of the integrals for eachindividual strake and becomes a simple summation:

t3eq =

1ℓ

n−i=1

t3i (χi − χi−1)

, i = 1, 2, . . . , n (11)

Fig. 5. The definition of parameter hi .

in which χi =

hi −

ℓ2π sin 2πhi

ℓ

and the sin term provides a

weighted thickness measure. The height of the potential bucklebeing considered is ℓ, corresponding to an integer number ofstrakes. The distance from the top of the cylinder to the bottomof the ith strake is defined as hi, as indicated in Fig. 5 (when i = 1the additional values h0 = 0 and χ0 = 0 are needed).

The chief difficulty in determining the buckling strength ofa stepped wall cylinder is to find the critical buckling mode:the mode with the lowest buckling pressure. Using the weightedsmeared wall method, this problem is solved in a different andeasier way.

First, different buckle heights for possible buckling modes areexamined. The equivalent thickness in each potential bucklingmode is found using Eq. (11). If the equivalent cylinder is ofmedium-length according to Eq. (3), Eq. (5) may be used directlyto assess the buckling pressure. The critical buckling wave numbermcr may be found using Eq. (4). If instead the potential bucklingmode corresponds to a short cylinder according to Eq. (3),Eq. (1) must be minimised with respect to m to find the bucklingpressure and mode. However, this situation seems to be relativelyrare in typical designs. Silo and tank structures of varying wallthickness appear never to produce ‘‘long’’ cylinders. All the abovecalculations are easily set out in a spreadsheet.

3574 L. Chen et al. / Engineering Structures 33 (2011) 3570–3578

Fig. 6. Three wall thickness distributions for Model 1.

6. Weighted smearedwallmethod applied to a varied thicknessdistribution

A verification of the weighted smeared wall method wasundertaken using four models, which led to a total of 821 separatetank designs. The verification begins with Model 1, which hasreference geometry of a defined set of wall thicknesses taken fromAnnex A of the API 650 standard [1]. Different distributions ofwall thickness (here termed ‘‘designs’’) were devised by scaling theratios of the thickness changes from one strake to another in thereference geometry. It may first be noted that the top strakes havea high radius to thickness ratio, which can be retained throughoutall the modified designs derived for Model 1.

The large set of designs devised in this way can produce thefull range of buckling modes, from buckling over the full height tobuckling only in the top strakes, simply by changing the ratio of theplate thicknesses at each step. In each altered design, the thicknessof the thinnest strakes tb1 is used as the reference thickness. Thethickness of the next strake is taken as the thickness of the thinneststrake plus k times the difference between its reference value andthat of the thinnest strake so that k = 1 corresponds to the API 650reference design, k = 0 corresponds to a uniform wall and large kgives large lower strake thicknesses causing the buckles to be onlyin the thinnest strake. The altered thickness of the ith strake in anygiven altered design is determined as:

ti = tb1 + k (tbi − tb1) (12)

where i is the number of the strake and the reference designrelative thickness is tbi. This device systematically transforms thereference design fromAPI 650 [1] into a large range of designs, all ofwhich have quite different buckling modes. The different designsdepend only on the value of k, so this parameter should be notedfor the discussion which follows.

The relative thicknesses of the strakes in three example designsare shown in Fig. 6. The thickness of each strake has been madedimensionless relative to the thinnest top strakes.

7. Buckling predictions using the weighted smeared wallmethod

The weighted smeared wall method was used to determinethe buckling behaviour for many cylinders with different patternsof wall thickness, all termed Model 1. The calculated equivalentthickness teq for each potential buckle height is shown in Fig. 7.The equivalent thickness always increases as the assumed buckleheight rises because the lower strakes are always thicker thanthose above. A low value of k in Eq. (12) can be seen to give onlysmall changes in the equivalent thickness, but high values producelarge changes.

Fig. 7. Equivalent thickness teq against the potential buckling height for differentwall thickness distributions.

Fig. 8. Critical circumferential buckling wave number mcr against the potentialbuckling height for different wall thickness distributions.

The critical circumferential buckling mode mcr for a set ofpossible axial buckling lengths is shown in Fig. 8. Either usingEq. (4) or minimising Eq. (1) produces the same result (acomparison is shown for k = 0.1 in Fig. 8). For all patterns ofwall thickness, the critical circumferential buckling wave numbermcr decreases as the buckling height rises. This is natural becausethe complete buckling mode tends to be approximately square, sotaller buckles also have longer circumferential wavelengths (lowmodes). The wave number or circumferential mode mcr is alwayslowest for a full height buckle. Larger values of k, with largereffective thicknesses, naturally lead to lower modes (Eq. (4)).

The smeared wall buckling pressures pcr,S for all potentialbuckling heights were evaluated using Eq. (5) (Fig. 9). The bucklingpressures were all made dimensionless by using a reference valuepcr,S0 which is the calculated value when the buckle is confinedto the top thinnest strakes and is invariant for all patterns of

L. Chen et al. / Engineering Structures 33 (2011) 3570–3578 3575

Fig. 9. Evaluated dimensionless critical buckling pressure against the potentialbuckling height with different wall thickness distributions.

wall thickness distribution. The minimum of each curve in Fig. 9therefore gives the critical buckle height or critical axial mode.

When k = 0.1, the critical axial buckling mode is found toextend over the whole shell height. When k = 10, the criticalaxial mode is found to occur only in the top thinnest strakes.Whenthe value of k is moderate, the critical axial mode occurs in strakesof mixed thickness, sometimes with several buckle heights havingsimilar critical pressures.

Not all possible buckle heights need to be investigated. Since thebuckling pressure varies smoothly, decreasing from the shortestheight and passing through aminimum before rising again (Fig. 9),the calculations only need to proceed until the minimum has beenreached.

8. Comparison of theweighted smearedwallmethod, Greiner’smethod in EN 1993-1-6 (2007) and finite element predictions

8.1. Comparison of the buckling modes using the weighted smearedwall method and finite element predictions

To establish the validity of the weighted smeared wall method,accurate predictions of the buckling behaviour of stepped wallcylinders were performed using the finite element softwareABAQUS. Thewall thickness patterns shown in Fig. 6 were adoptedfor computational simulation. Exploiting the symmetry of thebuckling modes, a 180◦ half cylinder model was used to obtainaccurate results. At the base, the first calculation was performedfor S1 and a second for S3 boundary conditions (Table 2 [6])(radial translations restrained but meridional rotation free, witheither axial translations restrained S1 or free S3). S1 correspondsto typical silos and anchored tanks whilst S3 corresponds to

unanchored tanks [8]. A ring stiffener with sufficient flexuralstiffness was used at the top to maintain circularity. The ringstiffener effectively provided an S3 boundary. The shell was madeof isotropicmaterial with Young’smodulus E = 2.0×105 MPa andPoisson’s ratio ν = 0.3.

The calculated critical buckling modes (Fig. 10) are very similarto those of the weighted smeared wall method (Fig. 9), but somesmall differences are also evident. Based on the assumption of S3boundaries at both ends, the weighted smeared wall method givesthe shape of the buckle as a half sine vertical wave. By contrast, thefinite element predicted shape includes the restraining effect of thestrakes below the buckle and, for whole wall buckles, the restraintof axial deformations at the base. This difference will later be seento cause a minor difference in the buckling strength predictionsdepending on the specific wall thickness distribution. It should benoted that Greiner’s method and the weighted smeared methodassume S3 boundaries at both ends.

8.2. Comparison of the buckling pressure predictions using the threemethods

The buckling pressures pcr derived from the three methodswere next compared: the weighted smeared wall method,Greiner’s method and finite element predictions. The bucklingpressures were all made dimensionless using Eq. (5) with anadopted thickness of the thinnest strakes, but with the length atthe critical buckle height in each case. This ratio permits rapid pre-cise assessments of accuracy over a wide range of geometries. Thebuckling pressure ratioφ (Eq. (13)) is plotted against thewall thick-ness distribution factor k in Fig. 11.

φ = pcr/pcr,D. (13)

The buckling pressure ratio φ is always larger than 1.0 becausethe reference pressure pcr,D has been calculated using the thicknessof the thinnest strakes t1.

When k is large (k ≥ 5), local buckling occurs only in thethinnest strakes. The buckling pressure from the weighted smea-redwall method is then constant as k changes, because the flexuralrestraint of the strake below the thinnest one is ignored. The baseboundary condition has no effect. However, the actual critical pres-sure steadily rises slightly as the restraint from below increases, soan increase in k causes a steady slight increase in the critical pres-sure.

When the buckling mode encompasses the whole wall (k ≤

0.3), finite element predictions with the classical S3 boundarycondition closely match both Greiner’s method and the weightedsmeared wall method, but the S1 boundary yields higher strengths

(a) k = 10,n = 2.

(b) k = 5,n = 3.

(c) k = 2,n = 4.

(d) k = 1,n = 5.

(e) k = 0.8,n = 6.

(f) k = 0.5,n = 8.

(g) k = 0.1,n = 9.

(h) k = 0,n = 9.

Fig. 10. Finite element predictions of the critical buckling mode with S1 base boundary (n is the number of the strakes in each buckle).

3576 L. Chen et al. / Engineering Structures 33 (2011) 3570–3578

(a) Wide range of distribution factors k. (b) Detail at low values of distribution factor k.

Fig. 11. Relationship between the buckling pressure ratio φ and the wall thickness distribution factor k (Fig. 11b shows the detail for small values of k).

Table 3The relative thickness of the basic case for different geometric models.

Basic case tb1 tb2 tb3 tb4 tb5 tb6 tb7 tb8 tb9 tb10 tb11 tb12 tb13

Model 1 1 1 1.125 1.25 1.55 1.8575 2.2 2.525 2.85 – – – –Model 2 1 1 1 1 1.125 1.25 1.5 2 2.5 3 3.5 4 5Model 3 1 1 1.125 1.25 1.55 1.875 2.2 2.525 2.85 – – – –Model 4 1 1.063 1.125 1.25 1.55 1.875 2.2 2.525 2.85 – – - –

due to the axial restraint at the base. For a uniform wall (k = 0),the increase is 25%, as expected (Cθ = 1.25 in Table 2), but the S1boundary has less effect at increasing k and disappears completelyat k = 0.5. The final recommendations given in this paper permitthe S1 strength gain to be exploited.

When local buckling occurs in strakes ofmixed thickness (0.3 <k < 5), all three calculations are very close in the range 0.3 <k ≤ 1.5. For k > 1.5, the wall thicknesses are outside the scope ofGreiner’smethod. Theweighted smearedmethod can then be usedfor a wider range of wall thicknesses and its predictions are closeto accurate numerical values.

9. Buckling pressure predictions for differentmodels using theweighted smeared wall method

The above results were all obtained for Model 1 (Fig. 6). Toobtain a more comprehensive check of the weighted smearedmethod, three more models with different wall thickness patternswere explored. The key geometric parameters for each model areshown in Tables 3 and 4: these are the relative thickness tbi, thelargest radius to thickness ratio, the reference height of each strake,the number of strakes, the number of stakes at the top and theadjusted thicknesses. All three models were modified using thefactor k in the same way to produce a range of designs.

For the three models, the buckling pressure ratio φ varies inthe same manner as was seen for Model 1 (Fig. 11) with thewall thickness factor k (Figs. 12–14). The buckling modes can bedivided into the same three categories, with the chief differencethat the value of k at the boundary between categories changesfrom one model to another. The difference is caused by manyfactors including the plate thickness ratios and the geometry of thetop thinnest strakes. The S1 base boundary condition is again seento affect only the whole wall buckling mode, with a slight spill-over into the first mixed wall modes with k below 0.5. For all otherconditions, the S1 base boundary condition has no influence on thebuckling strength.

10. Exploitation and enhancement of the weighted smearedwall method based on accurate finite element predictions

The analyses described above used three different methods topredict the buckling pressures of stepped wall cylinders under

Fig. 12. Relationship between the buckling pressure ratio φ and the wall thicknessdistribution factor k (Model 2).

Fig. 13. Relationship between the buckling pressure ratio φ and the wall thicknessdistribution factor k (Model 3).

uniform external pressure. The weighted smeared wall methodsometimes slightly underestimates or overestimates the bucklingpressure. Corrections for these effects are described here. The

L. Chen et al. / Engineering Structures 33 (2011) 3570–3578 3577

Table 4The geometric parameters of different models based on the basic case.

Model name Reference largest radius tothickness ratio r/t1

Reference height of eachstrake

Total number of strakes The number of top strakes ofuniform thickness

The adjusted thickness ofeach strake (ti)

Model 1 2000 1800 9 2ti = tb1 + k(tbi − tb1)(For all models)

Model 2 2000 1500 13 4Model 3 1000 1000 9 2Model 4 500 300 9 1

Fig. 14. Relationship between the buckling pressure ratio φ and the wall thicknessdistribution factor k (Model 4).

buckling modes were first divided into the three categories ofwhole wall, mixed strakes and top strake buckling. The ratio of thefinite element buckling pressure pcr,FE to the smeared wall valuepcr,S was characterised by the parameter ψ = pcr,FE/pcr,S .

For whole wall buckling, the variation of ψ with the ratio ofthe equivalent plate thickness teq to the bottom strake thicknesstb is shown in Fig. 15 for all four models. For the S3 base boundarycondition, the results need no adjustment, but the strengthmay beunderestimated by up to 6%. For the S1 base boundary, the ratioψreaches Cθ = 1.25 in all models when teq/tb = 1, correspondingto a uniform thickness cylinder. For axially restrained bases (S1),a more accurate estimate of the true buckling pressure, pcr,M , canthen be obtained as:

pcr,M/pcr,S = −1.08 + 4.4teqtb

− 2.07

teqtb

2

(14)

which is shown as the dashed line in Fig. 15. Eq. (14) gives aconservative close estimate of the buckling strength within 1% formost results, except for a few for Model 2.

For local buckling modes involving strakes of mixed thickness(Fig. 16), the weighted smeared wall method slightly underesti-mates the buckling pressure in most cases. The results are plot-ted against tn/tn+1, where tn is the thickness of the nth (lowest)strake in the buckle and tn+1 is that of the strake below it. Wherethe change in thickness is very small (tn/tn+1 > 0.9), the result canbe seen to be affected by the S1 or S3 boundary condition and cor-responds to the part of Fig. 11b just after mixed thickness modesbegin, where all but the short bottom strake are involved. A few S3boundary predictions are here unconservatively evaluated. Whenthe buckling mode involves the top zone and only one more strakeinModels 2 and 4, a slightly larger error occurs, but it is less than 7%for most results. For Model 1, based on the practical design of API650, the largest error is around 3%. Thus for practical designs, theweighted smeared wall method does give direct good estimates ofthe buckling pressure.

For local buckling modes only in the top thinnest strakes, thebuckling pressure ratio ψ = pcr,FE/pcr,S steadily rises as thethickness t2 of the second strake immediately below it increases

Fig. 15. Whole wall buckling: Discrepancy between FE and smeared wall bucklingpressures for all models (S1 and S3 base boundaries).

(Fig. 17). This is because the strake below it progressively adds axialrestraint to the top strake until it is so thick that it acts as an S1boundary condition, leading to Cθ = 1.25.

A modified buckling pressure pcr,M can be obtained using thethickness ratio t1/t2 as:

pcr,M/pcr,S = 1.4 − 0.7 (t1/t2) (15)

where t1 is the top strake thickness and t2 the thickness of the nextstrake below it. The dashed line in Fig. 17 shows that Eq. (15) givesa slightly conservative buckling pressure estimate for all models,with a maximum error around 5% for Model 4. It should be notedthat this correction is necessary in Greiner’s method too, since itis also based on the assumption of S3 boundary conditions at bothends.

11. Conclusions

The buckling behaviour under uniform external pressure ofcylinders with stepwise variable wall thickness has been studiedwith both anchored and unanchored base boundaries. A newweighted smeared wall method has been developed. The mainconclusions are:

1. The weighted smeared wall method can determine both thecritical buckling mode and the buckling pressure with goodaccuracy.

2. Three categories of buckling mode have been identified: wholewall buckling, mixed thickness buckling, and top thinnest zonebuckling. The critical category of a given shell depends on thespecific wall thickness distribution.

3. A new design method has been presented and verified over alarge number of designs. It gives accurate results for a verywiderange of shell walls, is easier to use than that of Greiner [7] andcan easily be programmed into a spreadsheet. The direct pre-dictions can be enhanced by minor modifications that accountfor adjacent restraint effects.

3578 L. Chen et al. / Engineering Structures 33 (2011) 3570–3578

Fig. 16. Mixed thickness buckling: Discrepancy between FE and smeared wall buckling pressures for all models.

Fig. 17. Top zone buckling: Discrepancy between FE and smeared wall bucklingpressures for all models.

References

[1] API STD 650 Welded Tanks for Oil Storage. American Petroleum Institute;2007.

[2] Rotter JM, Schmidt H. European recommendations for steel construction:buckling of shells. 5th ed. Brussels: European Convention for ConstructionalSteelwork; 2008.

[3] Batdorf SB. A simplified method of elastic stability analysis for thin cylindricalshells. NACA TN 1947; No.1341.

[4] Ebner H. Theoretical and experimental investigations on buckling ofcylindrical tanks subjected to external pressure. Stahlbau 1952;21:153–9.

[5] Flügge W. Stress in shells. 2nd ed. Berlin: Spring Verlag; 1973.[6] Yamaki N. Elastic stability of circular cylindrical shells. North-Holland:

Amsterdam; 1984.[7] Greiner R. Cylindrical shells under uniform external pressure. In: Teng JG,

Rotter JM, editors. Buckling of thin metal shells. London: Spon; 2004.p. 154–74.

[8] EN 1993-1-6 Eurocode 3: design of steel structures, Part 1.6: strength andstability of shell structures. Brussels: CEN; 2007.

[9] Resinger F, Greiner R. Zum Beulverhalten von Kreiszylinderschalen mitabgestufter Wanddicke unter Manteldruck. Stahlbau 1974;43:182–7.

[10] Resinger F, Greiner R. Praktische Beulberechnung oberirdischer zylindrischerTankbauwerke für Unterdruck. Stahlbau 1976;45:10–5.

[11] Greiner R. Ein baustatisches losungsverfarhen zur beulberechnung dun-nwandiger Kreiszylinderschalen unter manteldruck, vol. 17. Verlag WilhelmErnst & Sohn, Heft: Bauingenieur-Praxis; 1972.

[12] Trahair NS, Abel A, Ansourian P, Irvine HM, Rotter JM. Structural design of steelbins for bulk solids. Sydney: Australian Inst Steel Constr 1983.

[13] Brush DO, Almroth BO. Buckling of bars. In: Plates and shells. New York:McGraw-Hill; 1975.

[14] Donnell LH. Beams, plates and shells. New York: McGraw Hill; 1976.[15] Schnell W. Einfluss der Randverwölbung auf die Beulwerte von Zylinder-

schalen unter Manteldruck. Stahlbau 1965;34:187–90.[16] Fakhim YG, Showkati H, Abedi K. Experimental study on the buckling

and post-buckling behaviour of thin-walled cylindrical shells with varyingthickness under hydrostatic pressure. Valencia, Spain. In: Proceedings of theInternational Association for Shell and Spatial Structures (IASS) Symposium2009; 2009.

[17] Greiner R, Rotter JM. Cylindrical shells of stepwise variable wall thickness.In: Rotter JM, Schmidt H, editors. Brussels: European convention for construc-tional steelwork. Stability of steel shells: European design recommendations,2008. p. 217–35.

[18] DIN18800 Steel structures: stability, buckling of shells. Berlin: DeutschesInstitut fuer Normung; 1990.