Buckling and ultimate strength criteria of stiffened shells under combined loading for reliability analysis

Thin-Walled Structures 41 (2003) 69–88www.elsevier.com/locate/tws

Buckling and ultimate strength criteria ofstiffened shells under combined loading for

reliability analysis

P.K. Dasa,∗, A. Thavalingama, Y. Bai b

a Department of Naval Architecture and Marine Engineering, Universities of Glasgow and Strathclyde,Glasgow G4 0LZ, UK

b Offshore Technology Department, American Bureau of Shipping, Houston, TX 77066-6008, USA

Received 20 December 2001; received in revised form 21 October 2002; accepted 4 November 2002

Abstract

The work presented in this paper forms part of a broader task in establishing a guide toserve as technical documentation for buckling and ultimate strength assessment of varioustypes of marine structural components using the best state-of-the-art knowledge for extremeenvironmental loading. This paper concentrates on buckling and ultimate strength assessmentof ring stiffened shells and ring and stringer stiffened shells involving various modes of buck-ling and under various loading like axial compression, radial pressure and combined loading.Comparisons are made with screened test data, which have realistic imperfections and variousradius to thickness ratio values in the range generally used in offshore structures. The statisticaldata of model uncertainty factors in terms of bias and coefficient of variation (COV) arecalculated and may be used in a further reliability study. Comparisons are also made with thecodified rules, API BUL 2U and DNV buckling strength of shells. 2002 Published by Elsevier Science Ltd.

Keywords: Stiffened shells; Structural reliability; Offshore structure; Ultimate strength; Buckling

1. Introduction

Stiffened cylindrical shells are common structural components of various types offloating offshore structures and their rational design is desirable in order to achieve

∗ Corresponding author.E-mail address: [email protected] (P.K. Das).

0263-8231/03/$ - see front matter 2002 Published by Elsevier Science Ltd.doi:10.1016/S0263-8231(02)00093-9

70 P.K. Das et al. / Thin-Walled Structures 41 (2003) 69–88

not excessive but adequate levels of safety. The basic consideration for the establish-ment of strength criteria for stiffened shells in offshore applications differs consider-ably from those of other engineering disciplines primarily due to the greater stock-iness in configuration. In general the formulation must provide the means to accountfor the yield (collapse) mode and the buckling mode and any possible post-bucklingreserve strength. A unified treatment is required in order to couple these failuremodes and all likely failure modes need to be considered. In addition to these, effectsof imperfection, common to all shell-type structures, and residual stresses, especiallypronounced in fabricated steel structures, should also be included. While non-linearfinite element methods can cater to satisfy most of these considerations, they aretime-consuming and expensive and therefore not suitable for initial design. Thenumerical simplifications in the analytical formulations need checking/calibrated withtest results, before they can be used with confidence. A measure of closeness canonly be obtained by comparing its model uncertainty values with experimental ornumerical test data.

The main objective of the work is to establish a set of design equations for buck-ling strength assessment of ring and stringer stiffened shells related to marine struc-tures such as TLPs and Spars. The ABS rules for building and classing offshoreinstallations [1], mobile offshore drilling units [2], and steel vessels [3] together withthe ABS guide for floating production, storage and offloading systems [4], requirethat buckling strength be provided for the structure as a whole and for each structuralmember. For some design cases, however, the rules only provide limited informationon how to perform the buckling strength assessment and supplementary codes/guidesissued by, for example, API [5] and other recognised codes [6–8] has been used bydesigners. In the early 1980s, a joint industry project (JIP) was conducted by ABS,Conoco and the University of Glasgow on buckling and ultimate strength of stiffenedshells, in order to develop a model code for design of TLPs. The JIP consists offull-scale laboratory tests and theoretical work to derive ultimate strength formu-lation, called RCC formulation [9]. The JIP results were then adopted by API BUL2U [5]. In the early 1990s, a JIP was conducted by the University of Glasgowresulting in updating of the RCC formulation [10].

2. Modelling criteria

The importance of good strength models has been illustrated in a number of earlierpapers [11–13] in which it has been shown that one of the most convenient waysto deal with model uncertainty is to compare experimental values as in the theoreticalpredictions. This model uncertainty may be associated with both the load and thestrength model and, when calculated from a reasonable number of population rep-resenting a broad range, will enable this parameter to be treated as a random variablelike other design parameters in the structural reliability analyses. The modelling para-meters are incorporated in the failure surface equation of each structural componentas follows:

Z � Xm1R�Xm2Q (1)

71P.K. Das et al. / Thin-Walled Structures 41 (2003) 69–88

in which

Xm1 � Xm2 �experimental value

predicted value, (2)

Xm1 and Xm2 are associated respectively with the strength effect R and load effectQ. Z is usually referred as g1(.) function in the First Order Second Moment reliabilityanalysis and represents the safety margin in the structural component. R and Q areagain a function of many design variables and the mean and coefficient of variation(COV) of these can be calculated using linear strip theory. Given a large populationof test data, Xm1 and Xm2 can be treated statistically and its mean and COV can becalculated. For a good strength model, bias for mean should tend to unity and theCOV should be as small as possible. Also Xm should show correlation with any basicvariables, i.e. no skewness should be inherent in the model.

3. Buckling of stiffened shells

The ring-stringer stiffened cylinder also known as orthogonally stiffened cylinderis a typical component of marine structures. It is widely used in offshore platformsas main structural elements especially in the legs of buoyant offshore platforms suchas semi-submersibles and tension leg platforms. This component consists of a fabri-cated cylinder with a stiffening system composed of longitudinal stringer stiffenerssupported laterally by more widely spaced ring frames. This type of component isparticularly suited to resist high axial loads and bending moments in combinationwith external pressure. These components have large dimensions (over 10 m indiameter) due to their application in the legs of the semi-submersible and TLPs.Because of that they are produced by butt welding cold or hot-formed plates. Thestiffeners are also welded to the cylinder in such a way that the structural continuityof the stringers is guaranteed. This type of fabrication introduces geometrical imper-fections as well as residual stresses. There are basically two ways in which a structuremay lose its stability, snap-through buckling and the other one is known as classicalor bifurcation buckling. Some possible failure modes are shown in Fig. 1.

Two types of formulations can be considered — lower bound and mean value.The first category can be characterised by the under prediction of the strength for aspecific percentile over all the test specimens representative of the modelled phenom-ena. The DNV formulations, the ECCS formulations and the API orthotropic formu-lations fall in this category. The mean value formulation is characterised by theaverage prediction of the strength for al test specimens, in this category we can findthe Rule Case Committee formulations and the API-discrete formulations. The meanvalue formulations are used to predict the strength while the lower bound formu-lations are intended for the design of the cylinders. Behind these formulations, severalseries of experiments were conducted everywhere, especially in the UK under theDepartment of Energy sponsorship and in the USA, sponsored mainly by ConocoInc and ABS.


Fig. 1. Modes of failure.

4. Strength modelling of ring stiffened shells

The design strength criteria for ring stiffened shells under axial compression exter-nal hydrostatic pressure and radial pressure is as follows. The paper deals with shellfailure [14] and for the stiffener design, separate consideration should be givenagainst general stability or torsional instability modes.

4.1. Axial compression

The expected elastic buckling stress of an imperfect ring-stiffened cylinder subjectto axial compression is assumed to be,

se � BrnCscr (3)

where σcr � 0.605Et /R is the classical buckling stress for a perfect thin cylinder;


C (length � dependent coefficient) � 1.0, if z�2.85

� 1.425 /z � 0.175z, if z � 2.85

ρn (nominal or lower bound knock-down factor to allow for shape imperfections)

� 0.75 � 0.003z(1�R/300t) for z � 1

� 0.75�0.142 (z�1)0.4 � 0.003z (1�R/300t), for 1�z � 20

� 0.35�0.0002 R/ t, if z�20

B (a mean bias factor assessed from elastic test data which compensates for thelower bound nature of ρn

� 1.2, ln�1

� 1 � 0.2ln , ln � 1

ln � �sy / (snCscrnominal slenderness parameter

z � 0.954L2 /Rt, the Batsdorf parameter.

For inelastic collapse several approaches are being examined. One such approachprovides the collapse stress from a quadratic interaction of λσy and σe:

sc � fsy (4)

where f � 1 /√(1 � l4e) and le � √sy /se, slenderness parameter.

This formulation tends to be slightly non-conservative for the stocky range andconservative for elastic collapse. These expressions for the knock-down factor are alittle more pessimistic than that resulting from the mathematical solution of aninitially imperfect cylinder and they are regarded as approximately a 95% confidencelower bound for knock-down calibrated from test data.

4.2. Hydrostatic pressure

For ring-framed cylinders subject to external hydrostatic pressure, formulation [14]is essentially identical with the approach in BS5500 [15]. It is noted that about 700model tests, with geometries in the range of 6�R/ t�250 and 0.04�L/R�50, lieabove the so-called ‘guaranteed’ shell collapse pressure predicted by this formulation.The bias of the mean strength for this lower bound curve is estimated to be 1.17and in the usual design range the COV is estimated to be 5% [16]. The guaranteedhydrostatic collapse pressure may be estimated by,

phc � � 0.5pm , if py � pm

py(1�0.5 py /pm) , if py � pm

(5)

where pm denotes the Von Mises buckling pressure (hydrostatic) and py denotes thepressure at which the maximum circumferential mean stresses reaches the yieldstress. The best known solution for elastic buckling of the unsupported cylinder isthat due to Von Mises [17] which is given by,


pm �

EtR

n2�1 �12�pRL �2 � 1

�n2� LpR�2

� 1�2�

t2

12R2(1�m2) �n2�1 (6)

� �pRL �2�2�minimised with respect to n (circumferential mode number). Windenburg [18] mini-mised the expression with respect to n, the number of complete circumferential wavesor lobes. By making further approximations, he obtained the following expressionfor the minimum buckling pressure,

pm �0.919 E(t /R)2

L / (Rt)1/2�0.636�

2.6E(t /2R)5/2

L /2R�0.45(t /2R)1/2 (7)

In the present study, pm is calculated using Eq. (7). This equation is invalid forvery small or very large values of L / (Rt)1 /2, but in the design range its accuracy isgood. The analysis assumes the cylinder is pinned at non-deflecting cylindrical sup-ports. More refined analyses are now available which, for example, consider theinfluence of the ring frames on the deformations before and during buckling. Theseshows that pm becomes inaccurate for closely spaced frames. Nevertheless, the VonMises expression is still widely used because it can be represented in a relativelysimple form and it is in most cases only slightly conservative. Moreover, great accu-racy is in any case not required since pm is used mainly as a parameter of lowinfluence which helps in the prediction of inelastic collapse pressures. Eq. (7) istherefore quite accurate for most design purposes and uniform framing. For mixedframing and/or uneven spacing a major buckling program such as BOSOR 4 couldbe used, but these require expert use. Von Sanden and Gunther carried out the firstcomplete analysis for elastic deformations of perfect uniformly framed cylinders [19].However, slightly more accurate formulae have been derived by Wilson [20] fromwhich relatively simple linear equations for the more important stresses have beenderived. The shell parameter py may be written as,

py �syt /R1�gG

(8)

where

g �A(1�m /2)

A � twt � 2Nt /a, A � Ar(R /Rs)2 and aL � 1.285L /�Rt

Here Ar is the cross-sectional area of ring frame, R is the mean radius, Rs is the


radius to centroid of stiffener, tw is the stiffener web thickness, m is the Poisson’sratio and L is the unsupported length of shell. G and N are transcendental functionsof aL and are calculated as follows:

N �coshaL�cosaLsinhaL � sinaL

and G �

2 �sinhaL2

cosaL2

� coshaL2

sinaL2 �

sinhaL � sinaL(9)

4.3. Radial pressure, shell collapse

An acceptable approximation for the radial pressure which would cause elasticinstability of the shell can be obtained by assuming that the axial stress and radialpressure interact linearly. It can then be shown that the elastic collapse radial pressureis given by:

prm � pm / (1 � 0.5 pm (R / t) /se) (10)

in which σe is obtained from Eq. (3) and pm is the Von Mises hydrostatic collapsepressure. Treatment of inelastic collapse of the shell may follow the same approachas in the case of hydrostatic pressure by replacing the Von Mises pressure, pm, inEq. (5) with its counterpart prm in Eq. (10); i.e:

prc � {0.5prm , if py � prmpy (1 (11)

� 0.5 py /prm) , if py � prm

where py is the hydrostatic yield pressure for the shell as before. It is thought thatthe additional assumptions implied in this approach are somewhat conservative sincethe shell subject to radial pressure would experience less beam-column effect thanone under hydrostatic pressure.

4.4. Combined axial compression and pressure

As in the case of most design codes, the problem of shell collapse under thecombined action of axial compression and pressure is treated in this paper using aninteraction approach. The interaction relation is generally expressed in the form ofa polynomial:

(s /sc)m � (p /prc)n � 1 (12)

or segments of polynomial curves. Recommendations by various codes are founddiffering widely, ranging from the linear interaction (m � n � 1) recommended byECCS [7] to a circular one (m � n � 2) required by DNV [6,10]. The ASME CodeCase N-284 suggests a combination of straight lines and parabolas which appearsto agree quite well with test data. It is suggested that the parabola (m � 1, n � 2)offers the best fit to available data and is very close to the ASME recommendations.The Xm is defined as follows:

Xm � {(s /sc) � (p /prc)2} (13)


The bending compressive stress can be treated in a similar way to the axial com-pression. The stress in Eq.(12) is the sum of bending stress and the stress due topure compression.

5. Test results and comparison

Before validating any theoretical formulation or codified rules with experimentalresults, it is important to critically examine these data before incorporating into thedata bank. Only by following a systematic procedure can bogus test data be elimin-ated to leave reliable data. As the buckling collapse load depends on many parametersand is sensitive to residual stresses and initial imperfections, it is important that thesevalues, if available, should be recorded correctly. In general, a systematic classi-fication can be based on the following:

� Geometric properties, i.e. the length, internal radius, thickness, ring stiffenerdimensions and spacing, stringer dimensions and spacing, number of bays.

� Material properties, i.e. yield and tensile stress, whether tensile or compressive,and the rate of loading, etc.

� Method of production.� Geometric imperfection, contour of initial imperfections and maximum initial

imperfections.� Test conditions, i.e. boundary conditions, experimental collapse loads starting

from local failure, if any.

The formulation considers is generally applicable to shells with parameters of thefollowing ranges.

� The radius upon thickness ratio, R/t (100–500).� The stringer spacing upon thickness ratio, s/t (25–130).� The Batdorf width parameter, Zs (4–60).� The Batdorf length parameter, Zl (10–700).� The ring spacing upon radius ratio, L/R (0.2–1.6).� The ratio between the stiffeners and shell areas (0.1–0.6).

5.1. Ring stiffened shells

The test data considered in this study are taken from various references [e.g 21–25] for axial compression, hydrostatic pressure and combined axial and radial press-ure. Finite element models have been developed for non-linear analysis for stiffenedcylinders under various loading conditions and these can be seen in Refs. [26,27].The statistical results under axial, hydrostatic and combined loading using differentstrength models are given in Table 1. For axial loading of tests models made of highstrength materials, the model did not behave well and hence data for these modelsare not considered in the statistical analysis. Figs. 2–4 illustrate the strength analysis


Table 1Statistical results of theoretical models for ring-stiffened shells

DNV API Bul 2U Recommended model

Axial compression Mean 1.18 0.97 1.06COV 10.11% 12.47% 7.22%

Hydrostatic pressure Mean 1.11 1.20 1.09COV 12.32% 15.56% 12.06%

Combined loading Mean 1.54 0.80 1.21COV 17.11 18.67% 17.03%

Fig. 2. Strength analysis for DnV model under axial compression (ring-stiffened shells).

Fig. 3. Strength analysis for API model under axial compression (ring-stiffened shells).


Fig. 4. Strength analysis for proposed model under axial compression (ring-stiffened shells).

of test vs. theory under axial loading (population 21 models) and Figs. 5–7 illustratethe strength analysis of test vs theory under hydrostatic loading (population 49models) for the DNV and API and recommended models, respectively. The test vs.theory diagrams under combined loading (population 23 models) are shown in Figs.8–10 for the DNV, API and recommended models, respectively. The theoretical andinteraction curve for the proposed model under combined loading along with experi-ment results are illustrated in Fig. 11.

5.2. Ring and stringer stiffened shells

There is a large number of available experimental results from the mid-1960swhich are the outcome of an extensive aerospace research programme. These tests

Fig. 5. Strength analysis for DNV under hydrostatic pressure (ring-stiffened shells).


Fig. 6. Strength analysis for API model under hydrostatic pressure (ring-stiffened shells).

Fig. 7. Strength analysis for proposed model under hydrostatic pressure (ring-stiffened shells).

were mainly conducted in the elastic range and the majority of them used highstrength aluminium alloy models. The stringers in these models were large in numberand are thus closely spaced. These results are useful for providing information onelastic buckling strength. Also they are machine finished which means that the imper-fections and residual stresses are minimal. In the context of offshore design, thematerial to be used is steel and, due to the method of fabrication, they have consider-able inherent initial imperfections and residual stresses. These structures also gener-ally fail in the elasto-plastic range. In the 1970s there was a large test programmecarried out by UK universities and elsewhere in Europe and in the early 1980sadditional tests was conducted by Conoco/ABS. The test data were reviewed andthe tests passing the screening are incorporated here. The various test data that havebeen adopted have originated from various sources and represents both small-scale


Fig. 8. Strength analysis for DNV model under combined loading (ring-stiffened shells).

Fig. 9. Strength analysis for API model under combined loading (ring-stiffened shells).

model and large-scale cylinders. Their method of manufacture is also different andhence the amount of residual stress and initial imperfections vary from model tomodel but was reported to be within tolerable limits. However, these strength formu-lations do not take into account specific value of residual stress or initial imperfec-tions but allow for an average value depending on their method of fabrication, etc.

5.2.1. Axial compressionThe results of model uncertainty factor Xm for 48 steel models under axial loading

have been calculated. API Bul 2U (discrete analysis) and proposed formulation arebased on a similar approach except that the knockdown factor associated with shellterm is different. The statistical analyses were carried out using offshore data for allthe formulations. The statistical results using DNV, API and recommended modelsare as illustrated in Table 2. The strength analysis of test vs. theory under axial


Fig. 10. Strength analysis for proposed model under combined loading (ring-stiffened shells).

Fig. 11. Theoretical interaction curve along with experimental values for proposed model under com-bined loading (ring-stiffened shells).

Table 2Statistical results of theoretical models for ring and stringer stiffened shells under axial compression


Mean 0.967 1.025 1.008COV 21.26% 13.68% 13.7%


Fig. 12. Strength analysis for DNV model under axial compression (ring and stringer stiffened shells).

loading for the DNV, API and recommended models are shown in Figs. 12–14.While, for the DNV model, the bias is good, the spread of the model uncertaintyfactor is wide and this reflects in the COV values. The assumption of unstiffenedshell behaviour for modelling stringer stiffened shell, while it may hold for broadpanelled cylinder, is surely conservative while modelling narrow panelled cylinder.This was reflected in large COVs in DNV’s model.

5.2.2. Radial pressureThe model uncertainty values Xm under radial pressure for 11 models are calculated

for different strength formulations. It is to be noted that the statistical results for 11of the population is to be used in reliability analysis. One model from the population

Fig. 13. Strength analysis for API model under axial compression (ring and stringer stiffened shells).


Fig. 14. Strength analysis for proposed model under axial compression (ring and stringer stiffenedshells).

of 12 is discarded because the geometry of the model is not consistent with othermodels used in the population. The strength analysis of test vs. theory for the DNV,API and recommended models are shown in Figs. 15–17 and the statistical resultsusing DNV, API and recommended models are given in Table 3.

5.2.3. Combined loadingIn this calculation, among the stiffener sizes, the size of stringer is the only input.

In the absence of proper ring frame sizes, it is assumed the area of ring frame istwice that of stringer area. The model uncertainty values Xm are calculated for 35population and the statistical results are shown below. In assessing the statistics of

Fig. 15. Strength analysis for DNV model under radial pressure (ring and stringer stiffened shells).


Fig. 16. Strength analysis for API model under radial pressure (ring and stringer stiffened shells).

Fig. 17. Strength analysis for proposed model radial pressure (ring and stringer stiffened shells).

Table 3Statistical results of theoretical models for ring and stringer stiffened shells under radial pressure


Mean 1.39 1.21 1.14COV 39% 14.53% 13.4%


Fig. 18. Strength analysis for DNV model under combined loading (ring and stringer stiffened shells).

the modelling parameter, both hydrostatic and general combined loading test resultswere used. Axial compression and radial pressure results were not used, as this wouldproduce results identical to single load action models. The model predictions usethe experimental values of the load components at failure and thus calculates theproportions in which the load approaches the failure surface. The parameters in thecombined equation are calculated using the single load action of the appropriatestrength model. The strength analysis of test vs. theory for the DNV, API and rec-ommended models are shown in Figs. 18–20 and the statistics of the modellingparameter for various models are shown in Table 4.

Fig. 19. Strength analysis for API model under combined loading (ring and stringer stiffened shells).


Fig. 20. Strength analysis for DNV model under combined loading (ring and stringer stiffened shells).

Table 4Statistical results of theoretical models for ring and stringer stiffened shells under combined loading


Mean 1.705 1.137 1.109COV 25.1% 14.94% 19.79%

6. Conclusions

A set of design equations and its application for the assessment of bucklingstrength of ring and stringer stiffened shells related to marine structures such as TLPand SPARS are described. Comparisons are also made with screened test data. Thestatistical data of model uncertainty factors in terms of bias and coefficient of vari-ation are calculated and may be used in a further reliability study. Comparisons arealso made with the codified rules, API BUL 2U and DNV buckling strength of shells.The equations and model uncertainties derived from this paper will then be veryuseful to define (partial) safety factors. It may be noted that the population on testdata, under pressure loading of ring-stiffened shells, are not many and some furthertests are required to obtain realistic statistics of these theoretical models. This workmay be continued to develop strength criteria in both WSD and LRFD formats andto calibrate the new criteria against each other. Structural reliability methods will beapplied to calibrate design criteria. The equations and model uncertainties derivedfrom this paper will then be very useful to define (partial) safety factors. The newcriteria shall also be compared with current industry practice.


Acknowledgements

The authors would like to thank the American Bureau of Shipping for supportingthis project.

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Buckling and ultimate strength criteria of stiffened shells under combined loading for reliability analysis