ARTICLE IN PRESS

Thin-Walled Structures 47 (2009) 661–667

Contents lists available at ScienceDirect

Thin-Walled Structures

0263-82

doi:10.1

� Corr

E-m

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

On the buckling strength of stiffened elliptic paraboloidal steel panels

Alphose Zingoni a,�, Victor Balden b

a Department of Civil Engineering, University of Cape Town, Rondebosch 7701, Cape Town, South Africab Centre for Research in Computational and Applied Mechanics, University of Cape Town, Rondebosch 7701, Cape Town, South Africa

a r t i c l e i n f o

Article history:

Received 30 July 2008

Received in revised form

19 November 2008

Accepted 20 November 2008Available online 21 January 2009

Keywords:

Shell buckling

Elliptic paraboloidal shell

Stiffened steel panel

Stiffened shell

Steel shutter

31/$ - see front matter & 2008 Elsevier Ltd. A

016/j.tws.2008.11.007

esponding author. Tel.: +27 21650 2601; fax:

ail address: alphose.zingoni@uct.ac.za (A. Zin

a b s t r a c t

This paper reports the results of a numerical study undertaken on the buckling behaviour of lightly

stiffened elliptic paraboloidal steel panels intended for use as long-span shuttering for lightweight

concrete bridge decks, walkways and floors. Steel panel shutters of double curvature may allow the

casting of concrete over relatively large spans while avoiding the use of supporting scaffolding and other

intermediate props. Such long-span shutters are desirable when the ground below the deck cannot

adequately support scaffolding, or the space below the deck carries traffic carriageways which should

not be obstructed by scaffolding. In this study, the effect of the wet concrete is simulated as a uniform

pressure normal to the shell surface (limiting case), while the dead weight of the hardening concrete is

simulated as a uniformly distributed loading on the horizontal projection of the shell surface. The

results show that panel rise and aspect ratio have a considerable influence on the buckling strength of

the shutter. Tentative viability limits are established.

& 2008 Elsevier Ltd. All rights reserved.

1. Introduction

The strength and stability behaviour of stiffened steel plateshas been extensively studied in the past, particularly with regardto loading in axial compression [1–4]. The results of studies of thistype readily find application in the ship-building industry (deck,hull and partitioner elements), offshore construction (platformdecks) and bridge construction (decks, box girders), whereconsiderations of strength-to-weight performance are of utmostimportance. Owing to the thin-ness of these panels and therelatively high in-plane or axial loads, the behaviour is dominatedby buckling. Many papers have been published on bucklingphenomena of stiffened steel plates. Some investigators have usedFEM-based parametric studies to examine the effect on load-carrying capacity and failure mode of such factors as plate aspectratio, stiffener size, initial imperfections of the plate and residualstresses (see, for example, Ref. [5]). Others have concernedthemselves with theoretical formulations predicting the ultimatestrength of stiffened steel panels (for example, Paik et al. [6]).

Studies of stiffened steel shells of single curvature, while stillnumerous, have been relatively fewer, with most of these havingbeen concerned with cylindrical panels and cylindrical shells,which readily find application in naval architecture and theaerospace industry, as well as in containment-vessel construction.For example, Bushnell and Bushnell [7] have considered the

ll rights reserved.

+27 21650 5864.

goni).

minimum-weight design of stiffened cylindrical panels and shellswith various imperfections, while Olowokere [8] has investigatedthe behaviour of stiffened steel liners for reactor containmentvessels, under wet-concrete loading during construction.

Stiffened steel shells of double curvature have not been studiedas extensively as stiffened steel plates and stiffened cylindricalpanels and shells. (The many stiffened shells of revolution foundin the aerospace industry are of laminated composite constructionrather than steel plate.) If one considers shells of double curvaturethat are not shells of revolution, such as elliptic and hyperbolicparaboloids, studies on stiffened steel shells of this form are evenrarer. Nayak and Bandyopadhyay [9] have studied the stiffenedshallow elliptic and hyperbolic paraboloids (among others),combining the doubly curved isoparametric thin shell elementwith the curved isoparametric beam element in their FEMmodelling to simulate the stiffened shell. However, their studyis confined to free-vibration behaviour. Wong and Teng [10] havedeveloped a permanent shuttering system for concrete shells, inthe form of an orthogonally stiffened thin steel shell assembled bybolting together modular units. The side of the shell with thestiffeners is the surface in contact with the wet concrete, and afterthe concrete has hardened, the steel shell and the concrete resistservice loads compositely. Experimental results of the bucklingbehaviour of the stiffened thin-shell shuttering system under wet-concrete loading have been reported.

In the present study, the buckling strength of doubly curvedlightly stiffened steel panels, in the form of shallow ellipticparaboloids of rectangular plan form, is investigated on thebasis of finite-element modelling. Such panels are intended as

ARTICLE IN PRESS

A. Zingoni, V. Balden / Thin-Walled Structures 47 (2009) 661–667662

long-span shuttering units for the casting of lightweight concretebridge decks, walkways and floors. The high initial costs offabricating such units may very well be offset by the benefits ofthe ensuing system. It is known that shells of double curvature,particularly elliptic paraboloids, have the ability to span overrelatively large distances without the need of intermediatesupports, in comparison with flat plates and cylindricalpanels of the same general proportions. Under the weightof the wet concrete, orthogonally stiffened shuttering in theform of a shallow elliptic paraboloid would, therefore, beexpected to be able to carry the concrete without the need forpropping scaffolding underneath, provided that the edges areappropriately supported.

What has motivated this investigation is the increasingnumber of collapses of scaffolding systems for bridge decks andbuilding floors in South Africa and elsewhere, as a result ofcompressive overloading of the supporting scaffolding, or thefailure in bearing of the ground supporting the scaffolding. Wherethe ground below the deck of a bridge or walkway is too weak tosupport scaffolding, or where the space below the deck carries atraffic carriageway which should not be obstructed by the erectionof scaffolding, long-span shutters of the type being proposed canprovide the solution.

2. Geometric details and numerical modelling

Shutter units were considered to comprise a thin elliptic-paraboloidal shell of rectangular plan form of length a and width b

Fig. 1. Discretisation of shell and stiffeners (partial view).

Table 1Critical buckling load intensities (in kN/m2) for pressure and gravity loading.

h/b ¼ 2.5% h/b ¼ 5.0%

pcrit qcrit pcrit

b/a ¼ 0.2 Sine 1.37 1.46 2.10

Elliptic 4.26 4.59 11.90

b/a ¼ 0.4 Sine 0.46 0.47 0.93

Elliptic 1.54 1.60 4.69

b/a ¼ 0.6 Sine 0.21 0.21 0.56

Elliptic 1.03 1.06 3.26

b/a ¼ 0.8 Sine 0.11 0.11 0.38

Elliptic 0.79 0.82 3.03

b/a ¼ 1.0 Sine 0.08 0.08 0.32

Elliptic 0.72 0.74 3.29

(aXb), the rise of the shell being h. The midsurface of the shell canbe described by the equation

z ¼ h�ðða=2Þ � xÞ2

ða=2Þ2þððb=2Þ � yÞ2

ðb=2Þ2

" #h (1)

where the coordinate x is measured in the direction of length a,the coordinate y is measured in the direction of the width b, theorigin of the xy coordinate system on the rectangular plan formcoinciding with one of the corners of the rectangle. Assuming thatthe panel lies with the x and y axes in the horizontal plane, thecoordinate z gives the elevation (i.e. vertical coordinate) of theshell midsurface, this being zero at the midpoints of the edges ofthe panel (datum level), h at the centre of the panel and �h atthe corners.

For comparison purposes in the early stages of the investiga-tion, a sinusoidal surface described by the equation

z ¼ h sinpx

a

� �sin

py

b

� �(2)

(with the datum level coinciding with the straight edges of thepanel, and all points now lying above the xy plane) was alsoconsidered.

With the panel length a fixed at 10 m, and the panel width b

varied from 2 to 10 m in steps of 2 m (to cover aspect ratios b/a of0.2, 0.4, 0.6, 0.8 and 1.0), and adopting a shell thickness t of 3 mm,metal strips of width 5 mm (i.e. plate thickness) and height 25 mmwere assumed to be welded onto the lower surface of the shell at aregular spacing of 1 m in each of the two directions parallel to theedges of the rectangular plan form, to form an orthogonal patternof stiffeners on the surface. The outermost stiffeners coincide withthe edges of the panel. The upper surface of the shell (which lieson the side of the shell midsurface away from the centres ofcurvature) will be the surface that comes into contact with thewet concrete during casting.

In this particular study, the edges of the panel were assumed tobe fully restrained against translation, but free to rotate in thevertical section (i.e. simply supported or fully pinned boundaryconditions). In practice, the shutter would only be supported inthis fashion along the two opposite shorter edges and unsup-ported along the two longer sides, but since this study was aninitial feasibility investigation of maximum possible bucklingstrengths, all four edges were given full lateral restraint.

Two loading cases were considered, namely: (i) a uniformpressure p (force per unit area of the actual curved surface, acting

h/b ¼ 7.5% h/b ¼ 10.0%

qcrit pcrit qcrit pcrit qcrit

2.29 3.43 3.77 5.00 5.57

12.99 20.36 22.65 31.03 35.10

0.97 1.52 1.77 2.16 2.32

5.04 9.05 9.92 13.27 15.27

0.57 0.91 0.93 1.28 1.34

3.48 6.54 7.25 9.65 11.44

0.39 0.62 0.65 0.88 0.93

3.23 6.23 7.00 9.07 11.06

0.33 0.52 0.54 0.72 0.77

3.53 6.56 7.57 9.36 11.88

ARTICLE IN PRESS

Fig. 2. LPF plots for the elliptic 10 m�4 m panel: (a) pressure, 5% rise; (b) gravity, 5% rise; (c) pressure, 7.5% rise; (d) gravity, 7.5% rise.

A. Zingoni, V. Balden / Thin-Walled Structures 47 (2009) 661–667 663

normal to the shell surface at any given point); (ii) a verticalimposed loading of intensity q (force per unit area of thehorizontal projection of the shell surface, acting vertically down-wards at all points). The first loading would simulate the pressureof the wet concrete, while the latter would simulate the deadweight of the hardening concrete. Strictly speaking, the effect ofnormal wet concrete is closer to that of dead weight thanhydrostatic pressure, but the pressure-loading case provides auseful bound to the actual loading conditions, particularly in thecase of concretes of high fluidity and certain types of slurries.

The shell and its stiffeners were discretised into a regular meshof finite elements (30 elements per metre in each coordinatedirection — see Fig. 1) using the finite element programmeABAQUS [11]. For both the shell and its stiffeners, the fullyintegrated finite-membrane-strain 4-node shell element of ABA-QUS (designated S4) was employed. This element allows for finitemembrane strains, transverse shear deformation and arbitrarilylarge rotations, and is therefore suitable for large-strain analysis.As steel is the material of construction for both shell andstiffeners, the material properties assumed in the analysis wereE ¼ 200 GPa (Young’s modulus), n ¼ 0.3 (Poisson’s ratio) andr ¼ 7800 kg/m3 (mass density).

For each of the five aspect ratios adopted (b/a ¼ 0.2, 0.4, 0.6,0.8, 1.0), four rise ratios (h/b ¼ 2.5%, 5.0%, 7.5%, 10%) wereconsidered, and a linear buckling analysis carried out using the

programme ABAQUS, to determine the lowest buckling loads (firstmode solution) for the pressure loading, pcrit, and for the gravityloading, qcrit. To account for geometric non-linearities andmaterial plasticity, a non-linear analysis employing a modifiedversion of Riks’ method [12] was also carried out. Themodified Riks procedure allows the load-displacement responseto be traced in the post-buckling range of negative stiffness. Thenon-linear analysis is evidently more general and moreaccurate than the eigenvalue linear buckling analysis, but as iswell-known, the latter is computationally much faster. Thepurpose of running the two in parallel in this study was toestablish, for all ranges of parameters of interest, the reliability ofthe linear eigenvalue analysis (with the non-linear analysisserving as the benchmark) in predicting the critical bucklingloads of the panels.

3. Numerical results

3.1. Linear buckling analysis

The buckling applied-load intensities for pressure loading (pi)and gravity loading (qi) were computed as

pi ¼ lipapplied ¼ lið1 kN=m2Þ ¼ li (3a)

ARTICLE IN PRESS

Fig. 3. Buckled panels at the end of simulation sequence for the elliptic 10 m�4 m panel: (a) pressure, 5% rise; (b) gravity, 5% rise; (c) pressure, 7.5% rise; (d) gravity, 7.5%

rise.

A. Zingoni, V. Balden / Thin-Walled Structures 47 (2009) 661–667664

qi ¼ liqself (3b)

where the li are the calculated buckling eigenvalues, papplied is thepressure applied during the simulation (taken as 1 kN/m2), andqself is the self-weight of the shutter (vertical load per unit area ofthe shell surface). Note that the above load intensities are due tothe applied loading (from the wet concrete) only; the self-weightof the steel shutter (which is taken into account by the analysis)has been deducted at the end. Thus pi is numerically equal to thecorresponding li (with units of kN/m2), while qi is equivalent tothe self-weight of the shutter amplified by the corresponding li.

Table 1 summarises the results obtained for the lowestbuckling load (corresponding to the first eigenmode), for thefive aspect ratios and four rise ratios of interest. In this simulation,both the elliptic paraboloid and the sinusoidal shell wereconsidered.

The results show that the shallow elliptic paraboloid is farmore efficient than the shallow sinusoidal shell in resistingbuckling, especially as the rise (h/b) is increased and/or as theaspect ratio (b/a) is increased. (The sinusoidal panel losescurvature towards the edges, compromising double-curvatureshell action, hence the much lower buckling strengths.) Thesinusoidal form has merely been used to benchmark the ellipticparaboloid, and will not be pursued any further.

3.2. Non-linear analysis

The modified Riks procedure solves for loads and displace-ments simultaneously, and uses the ‘‘load proportional factor’’(LPF) as a solution variable. In ABAQUS [11], the progress of thenon-linear simulation is measured with respect to the ‘‘arclength’’ along the static equilibrium path in load-displacementspace. The applied load intensities at any given instance forpressure loading (parc) and for gravity loading (qarc) were

computed as

parc ¼ ðLPFÞpapplied ¼ ðLPFÞð1 kN=m2Þ ¼ ðLPFÞ (4a)

qarc ¼ ðLPFÞqself (4b)

respectively, where papplied and qself are as previously defined, andpapplied is taken as 1 kN/m2 as before.

To illustrate the results, Fig. 2 shows the LPF plots (versus arclength) for the elliptic 10 m�4 m panel with selected rise ratios(5% and 7.5%), for both pressure and gravity loading. These plotsexhibit peaking, softening and recovery of strength, which istypical of local buckling of individual panel bays. Fig. 3 shows thebuckled panels at the end of the simulation sequence. Similarresults are depicted in Figs. 4 and 5 for the 10 m�6 m panel.

Focussing attention on the elliptic configuration, the results ofthe linear buckling analysis (lowest buckling values) have beencompared with those of the non-linear modified Riks procedure(as read off the LPF plots at the onset of first buckling). Thecomparisons are summarised in Table 2, where li (the firsteigenvalue of the linear buckling analysis) is shown versus the LPFvalue for first buckling (non-linear analysis). In the non-linearanalysis, first buckling is taken as the point at which local‘‘peaking’’ of the applied load first occurs, with some softeningfollowing immediately after that. It is not just the point where theLPF plot shows a sudden decrease in slope; peaking and softeningare necessary.

The comparisons show that the linear buckling analysispredicts the first buckling load reasonably well for low aspectratios (b/a ¼ 0.2 and b/a ¼ 0.4), but tends to considerably over-estimate the buckling strength in the case of higher aspect ratios(b/a ¼ 0.8 and b/a ¼ 1.0). As previously stated, these comparisonsare only meant to evaluate the performance of a standard linearbuckling analysis versus the more exact non-linear modified Riks

ARTICLE IN PRESS

Fig. 4. LPF plots for the elliptic 10 m�6 m panel: (a) pressure, 5% rise; (b) gravity, 5% rise; (c) pressure, 7.5% rise; (d) gravity, 7.5% rise.

A. Zingoni, V. Balden / Thin-Walled Structures 47 (2009) 661–667 665

analysis; the discussion and recommendations that follow will bebased on the results of the non-linear analysis.

Let us calculate the equivalent depth of wet concrete for thebuckling loads obtained, based on the gravity load cases. For thispurpose (and subsequent discussion), we use the results of thenon-linear analysis. Assuming the concrete weighs 24 kN/m3, theequivalent depth of wet concrete is simply qarc1st/24 (in m), whereqarc1st is the applied gravity-load intensity corresponding to thepoint of first buckling along the non-linear load-displacementpath. The results are shown in Table 3.

4. Discussion and recommendations

The elliptic paraboloidal panels that have been considered inthis paper are lightly stiffened, with stiffeners forming anorthogonal net on the shell surface of cell dimensions 1 m�1 m.They are primarily intended for the construction of lightweightconcrete bridge decks, walkways and floors. The discussion belowis based on the results of the non-linear analysis.

For shallow panels of the proportions in question, the load-carrying capacities between pressure and gravity loading are not

too different (the differences generally not exceeding 10%), withpressure (i.e. loading normal to the shell surface) causing firstbuckling at a slightly lower load intensity than gravity loading(vertical). Thus the initial conditions at the time of placement ofconcrete, when the concrete is at its most fluid and pressureeffects prevail, govern the design of the shuttering.

As expected, the load intensity (in kN/m2) required to causebuckling (i.e. the load-carrying capacity of the panels) increases asthe rise of the panel (i.e. the h/b ratio) increases. For instance, inmoving from an h/b ratio of 2.5% to an h/b ratio of 5%, the bucklingload intensity increases by a factor of 2.8 for the narrow10 m�2 m panel and by a factor of 2.2 for the relatively wider10 m�8 m panel. When the rise is as large as 10%, thesame panels become 6.5 and 7.0 times stronger in relation totheir h/b ¼ 2.5% counterparts. Thus the buckling strengthof the panels is strongly dependent upon the rise of the panels,the gain in strength being always proportionately greater than thegain in rise.

For the fixed panel length of a ¼ 10 m adopted in the presentstudy, as the panel width b is increased from the minimum valueof 2 m, there is a rapid decrease in the buckling strength ofthe panels. For instance, taking a panel of 5% rise, the buckling

ARTICLE IN PRESS

Fig. 5. Buckled panels at the end of simulation sequence for the elliptic 10 m�6 m panel: (a) pressure, 5% rise; (b) gravity, 5% rise; (c) pressure, 7.5% rise; (d) gravity, 7.5%

rise.

Table 2Comparison of linear and non-linear results for first buckling of the elliptic panel.

h/b ¼ 2.5% h/b ¼ 5.0% h/b ¼ 7.5% h/b ¼ 10.0%

l LPF l LPF l LPF l LPF

b/a ¼ 0.2 Pressure 4.26 4.46 11.90 13.00 20.36 20.50 31.03 27.30

Gravity 17.91 21.10 50.43 59.00 87.24 96.00 133.80 133.50

b/a ¼ 0.4 Pressure 1.54 1.57 4.69 4.13 9.05 7.96 13.27 11.45

Gravity 6.28 7.95 19.72 19.20 38.52 29.30 58.60 56.10

b/a ¼ 0.6 Pressure 1.03 0.85 3.26 2.33 6.54 4.18 9.65 7.20

Gravity 4.20 5.00 13.67 10.90 28.16 23.70 43.84 36.60

b/a ¼ 0.8 Pressure 0.79 0.62 3.03 – 6.23 3.64 9.07 5.75

Gravity 3.23 4.12 12.68 8.96 27.14 16.60 42.19 27.70

b/a ¼ 1.0 Pressure 0.72 0.53 3.29 1.53 6.56 – 44.99 –

Gravity 2.92 – 13.84 7.64 29.21 – 13.27 –

Table 3Equivalent depth (in mm) of wet concrete that can be carried by the elliptic panel.

10 m�2 m 10 m�4 m 10 m�6 m 10 m�8 m 10 m�10 m

h/b ¼ 2.5% 225 85 55 45 –

h/b ¼ 5.0% 630 205 115 95 80

h/b ¼ 7.5% 1040 315 255 180 –

h/b ¼ 10.0% 1460 610 400 305 –

A. Zingoni, V. Balden / Thin-Walled Structures 47 (2009) 661–667666

strength of the panel reduces to 32% (of the strength of the10 m�2 m panel) when b is increased to 4 m, and to only 15%when b is increased to 8 m. Similar trends are exhibited for the

other rise ratios. Thus, the buckling strength of the panels isstrongly dependent on the aspect ratio b/a of the panel, the loss ofstrength generally being proportionately greater than the gain inaspect ratio.

For the lightweight panels in question, let us define functionalviability as the capacity to carry an equivalent depth of wetconcrete of at least 200 mm. On this criterion, and by reference toTable 3, it is evident that the 10 m�10 m panel is not viable, forany chosen rise up to and including the maximum of 10%. The10 m�8 m panel is only viable if the rise is greater than 8% (valuesother than the ones studied have been obtained by interpolation),and the maximum load-carrying capacity associated with the

ARTICLE IN PRESS

A. Zingoni, V. Balden / Thin-Walled Structures 47 (2009) 661–667 667

maximum rise of 10% is a modest 305 mm. The 10 m�6 m panelbecomes viable at a rise of 6.5%, with a maximum load-carryingcapacity of 400 mm at a rise of 10%. The 10 m�4 m panel is viablefor a rise of 5% or greater, attaining a maximum capacity of610 mm at a rise of 10%. The 10 m�2 m panel is viable for allranges of rise (equal to or greater than the prescribed minimum of2.5%), attaining a maximum capacity of 1460 mm at theprescribed maximum rise of 10%.

Clearly, these load-carrying capacities can be enhanced if thepanel construction is modified (use of steel plate of thicknessgreater than the 3 mm adopted in this study; use of stiffeners ofdimensions greater than the assumed 25 mm�5 mm; adoption ofstiffener spacing smaller than the 1 m of present considerations),but the above results are sufficient to show that the proposedshell panel systems are viable as shuttering, particularly inlightweight concrete deck construction.

5. Summary and conclusions

This study has shown that edge-supported lightly stiffenedsteel panels in the form of shallow elliptic paraboloids are viableas free-spanning shuttering for lightweight concrete bridge decks,walkways and floors, provided that the correct aspect ratio andrise of the panel are chosen. The initial conditions at the time ofplacement of concrete, when the concrete is at its most fluid andpressure effects prevail, govern the design of the shuttering.

For b/a ratios ranging from 0.2 to 1.0, and h/b ratios rangingfrom 2.5% to 10%, this numerical investigation has quantified theactual buckling strengths for panels of 3 mm shell thickness,stiffened with 25 mm�5 mm welded-on steel strips forming anorthogonal pattern on the inner shell surface and spaced at 1 m inthe two directions parallel to the panel-projection edges. Thebuckling strength of the panels is strongly dependent on the riseof the panel (the higher the h/b ratio, the greater the load intensitythe panel can carry) and the aspect ratio of its rectangularprojection (the higher the b/a ratio, the smaller the load intensitythe panel can carry).

Cut-off shallowness limits for functional viability (in terms ofthe ability to carry a minimum depth of 200 mm of wet concrete)have been established for various panel aspect ratios, but clearly,the load-carrying capacity of these panels can be enhanced wellbeyond these limits by using thicker steel plate, more robuststiffeners and a denser network of stiffeners. These viabilityextensions, as well as experimental testing of prototype panels,and a study of the effect of imperfections (knife-edge dents andlocal ‘‘flats’’ on the upper shell surface), will form the subjectof future work.

References

[1] Murray NW. Buckling of stiffened panels loaded axially and in bending. TheStructural Engineer 1973;51(8):285–301.

[2] Sherbourne AN, Liaw CY, Marsh C. Stiffened plates in uniaxial compression.IABSE Proceedings 1971;31:145.

[3] Bonello MA, Chryssanthopoulos MK, Dowling PJ. Ultimate strength design ofstiffened plates under axial compression and bending. Marine Structures1993;6:533–52.

[4] Grondin GY, Chen Q, Elwi AE, Cheng JJR. Stiffened steel plates undercompression and bending. Journal of Constructional Steel Research 1998;45(2):125–48.

[5] Grondin GY, Elwi AE, Cheng JJR. Buckling of stiffened steel plates: a parametricstudy. Journal of Constructional Steel Research 1999;50(2):151–75.

[6] Paik JK, Thayamballi AK, Kim BJ. Large deflection orthotropic plate approachto develop ultimate strength formulations for stiffened panels undercombined bi-axial compression/tension and lateral pressure. Thin-WalledStructures 2001;39:215–46.

[7] Bushnell D, Bushnell WD. Approximate method for the optimum design ofring and stringer stiffened cylindrical panels and shells with local, inter-ring,and general buckling modal imperfections. Computers and Structures1996;59(3):489–527.

[8] Olowokere O. Failure of stiffened steel liners for nuclear reactor containmentstructures under construction. Computers and Structures 1984;19(4):669–72.

[9] Nayak AN, Bandyopadhyay JN. On the free vibration of stiffened shallowshells. Journal of Sound and Vibration 2002;255(2):357–82.

[10] Wong HT, Teng JG. Buckling behaviour of model steel base shells of thecomshell roof system. In: Zingoni A, editor. Progress in structural engineering,mechanics and computation. London: A.A. Balkema/Taylor and Francis; 2004.p. 185–9.

[11] ABAQUS Standard (version 5.8). Hibbit, Karlsson and Sorenson Inc., Newark,California, 1998.

[12] Riks E. An incremental approach to the solution of snapping and bucklingproblems. International Journal of Solids and Structures 1979;15:529–51.