exp. perspective of press vessels

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J. BłachutInstitute of Physics,

Cracow University of Technology,

ul. Podchora _zych 1,

Krakow, 30-085 Poland

Experimental Perspectiveon the Buckling of PressureVessel Components

This review aims to complement a milestone monograph by Singer et al. (2002, Buckling Experiments—Experimental Methods in Buckling of Thin-Walled Structures, Wiley, NewYork). Practical aspects of load bearing capacity are discussed under the general um-brella of “buckling.” Plastic loads and burst pressures are included in addition to bifur-cation and snap-through/collapse. The review concentrates on single and combined static stability of conical shells, cylinders, and their bowed out counterpart (axial compression and/or external pressure). Closed toroidal shells and domed ends onto pressurevessels subjected to internal and/or external pressures are also discussed. Domed endsinclude: torispheres, toricones, spherical caps, hemispheres, and ellipsoids. Most experi-ments have been carried in metals (mild steel, stainless steel, aluminum); however,details about hybrids (copper-steel-copper) and shells manufactured from carbon/glass fibers are included in the review. The existing concerns about geometric imperfections,uneven wall thickness, and influence of boundary conditions feature in reviewed research. They are supplemented by topics like imperfections in axial length of cylinders,imperfect load application, or erosion of the wall thickness. The latter topic tends to bemore and more relevant due to ageing of vessels. While most experimentation has taken

place on laboratory models, a small number of tests on full-scale models are also refer-enced. [DOI: 10.1115/1.4026067]

Keywords: cones, ellipsoids, hemispheres, toricones, torispheres, toroids, external, inter-nal pressure, buckling, plastic load, burst pressure

1 Introduction

Buckling of thin-walled shell-like components, frequentlyfound in pressure vessels, has been the subject of numerous stud-ies over many decades. A substantial wealth of accumulatedknowledge exists in the form of books, conference proceedings,reviews, and other published material. To this end, books coveringexperimental aspects of buckling of pressure vessels or their com-ponents include: [1 – 6]. Reference [7] can be regarded as a mile-stone piece of work entirely devoted to experimentalmethodologies associated with, per se, buckling. It contains acomprehensive list of references. Conference proceedings dealingentirely with buckling include Refs. [8 – 13]. A large number of conference papers in the proceedings are explicitly devoted tobuckling experiments. Summary of the current set of design rec-ommendations on buckling prone, thin-walled shell components,e.g., cylinders, cones, and/or doubly-curved shells can be found inRef. [14]. The authors list some existing shell stability technicalissues—many of which are not addressed in the NASA recom-mendations (Refs. [15 – 17]). Initial geometric imperfections, non-linear prebuckling deformations, boundary conditions, loadintroduction effects, combined loads, and variation in material

properties are just a sample of topics still awaiting further investi-gations. The reliance on arbitrarily chosen knock-down factors isalso echoed in Refs. [18,19]. There have also been a number of published reviews of research on buckling, e.g., Refs. [20 – 22]. Inmany instances, experimental data was used to develop designstandards. A good example would be the predecessor of the cur-rent code [23], where design of externally pressurized hemispheri-cal/torispherical domes is entirely empirical. It is based on thelower bound to known experimental results being further reducedby a safety factor [24]. Authors of a review Ref. [25] poses a ques-

tion: why, despite a great research effort, has the scientific com-munity failed to develop procedures for shell design that are notbased on empirical data, i.e., on lower-bound followed by aknock-down factor? They do not provide a definitive answer tothis dilemma which still exists more than ten years on. Insteadthey propose a number of procedures aiming at improvement of

over-conservativeness of the lower-bound design philosophy for the case of axially compressed cylinders (including FRP cylin-ders). In a review of the buckling resistance of thin and slender structures typically found in nuclear industry, Ref. [26], back in1984, categorized vessel related components, prone to buckling,as: (i) stiff (buckling in plastic range), (ii) medium (elastic/plasticbuckling), and (iii) soft (elastic buckling). The criterion used herewas the ratio, REY , of elastic bifurcation-to-first yield load. For REY 5 the component was deemed to be stiff. For REY 0:2,structure was classified as soft. In this context, the results of 42experimental buckling tests have been compared with computedpredictions of buckling. The tests were on metallic torisphericaland elliptical heads, cylinders under: shear, axial compression or external pressure, spherical caps, spheres, and stiffened/unstiff-ened baffles. Scatter of results is provided here as Table 1 (from

Ref. [26]). Table 1 shows the difference between tests and numeri-cal predictions of buckling together with the number of tests car-ried out. It is seen here that the range of errors varies from 30%to þ50%. The discrepancies between experimental and computedvalues have been attributed to: (i) specimen geometry, (ii) bound-ary conditions, and (iii) material data. It was also noted that insome cases the buckling was difficult to observe experimentallyand it was subjective (internally pressurized domes; for example).Progress has been made in assessing the issues leading to big dis-crepancies between test data and theory since the publication of Ref. [26]. Testing methods, for example, have improved and theyhave been more focused [27 – 30]. It is true to say that bucklingexperiments are now better instrumented than in the past; for example, in Refs. [27,28,30 – 33]. Equally, more rigorous

Manuscript received August 18, 2013; final manuscript received November 14,2013; published online December 30, 2013. Editor: Harry Dankowicz.

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computational models have been developed, e.g., Ref. [7],[34 – 39]. References [7] and [30], for example, provide a thoroughreview of experimental techniques and results obtained in a wider context of structures prone to buckling. Recent review papers,Refs. [20,21], address structural behavior of domed ends and addto the accumulated know-how base. Design methodology is avail-able in the form of various codes [15 – 17,23,40 – 42], and are alsoavailable in a specialized stability handbook [43] (see also Ref.[44]). As mentioned earlier, one of unresolved issues within struc-tures prone to buckling is the effect of initial shape imperfectionson the magnitude of buckling load. Despite efforts aiming at find-ing a universal answer to the dilemma, it still remains a subject of active research. It is true to say that the detrimental effect of initialshape imperfections has been quantified for a range of structuralcomponents and loading conditions. However, it is, by-and-large,still based on a component by component basis. An attempt of aunified approach to the design of imperfection sensitive structuralcomponents can be found in Ref. [42], where a better manufactur-ing quality of a member results in the attainment of higher buck-ling loads. However, research into the derivation of less restrictiveknock-down factors still continues, e.g., Ref. [45]. A differentway of reducing imperfection sensitivity of axially compressedcylinders is explored in Ref. [46]. Numerical results suggest thatcylinders filled with, and/or surrounded by a compliant core, canbe less imperfection sensitive. Finally, it is worth mentioning a

better access to the specialized buckling-related, and web basedresources, e.g., Refs. [47,48].

2 Cylindrical Shells

Cylindrical shells constitute a backbone of pressure vessels andof other load bearing components in variety of on land, in the sea,and in the air applications. As mentioned earlier, the comprehen-sive review of cylindrical shells under buckling conditions isavailable in Ref. [7]. Historical background into early shell buck-ling tests is available in Ref. [49]. The earliest shell buckling testson thin-walled tubes under axial compression and bending were in1845–1850. These tests were followed by experimental bucklingtests on tubes under external and/or internal pressure. Reference[49] lists milestones in experimental buckling tests, and motiva-

tion behind them, for cylindrical shells for up to the 1970s. Tubu-lar bridges motivated the very first buckling tests on tubes in themid-19th century. The next demand for experimental data camefrom shipbuilders at the end of the 19th century. The needs arisingfrom the design of tunnel linings and submarines dominatedexperimentation early in the 20th century. The latter designs spe-cifically stimulated experimentation on stiffened cylinders. Air-craft structures were associated with tests on much thinner cylinders. By then, the range covered the radius-to-wall thickness, R / t , between 35 and 1440. The concept of “knock down factor”appears to have been founded at that time. The next big impetusin buckling of cylindrical shells took place in the 1960s and it wasassociated with careful studies of buckling and post-bucklingpatterns via high speed photography and the use of photoelasticity.

Further details about these milestones together with informationabout sources are given in Ref. [49]. Review of research into buck-ling of cylinders and domed closures onto cylindrical vessels isavailable in Ref. [50]. Plain cylinders, cylinders reinforced by: (i)rings, (ii) stringers, and (iii) rings and stringers are included. Load-ing includes: (i) axial compression, (ii) external pressure, and (iii)axial compression and external pressure. Nonaxial compression isalso mentioned. Some of more recent research into static stabilityof cylindrical shells is reviewed in what follows.

2.1 External Pressure. Corrosion damage to the wall thick-ness can affect the buckling strength of externally pressurized ves-sels. These effects have recently been researched for cylindricalshells and domed ends. Buckling strength of cylindrical pressurehulls with artificial damage introduced to the wall has beenresearched in Refs. [51 – 54]. Motivation for this comprehensiveexperimental program was linked to naval submarines, which, if not adequately protected, can suffer from corrosion damage. Oncethe surface corrosion is identified in the pressure hull, a range of repair possibilities exist. This includes anything from weld build-up of the lost wall thickness to the full replacement of affectedshell plates. However, all of these routes carry side effects, e.g.,appearance of residual stresses, geometrical distortions, or changeof material properties in heat affected zones. An alternativeapproach would be to allow operation of a corrosion affected ves-

sel, but within a modified envelope of operations. References[51,52] detail buckling tests on laboratory models of approxi-mately 220mm diameter, 2.5 mm nominal wall thickness, andmanufactured from 6082-T6 aluminum alloy. Buckling tests on20 ring stiffened shells subjected quasi-static external hydrostaticpressure were carried out. Eight models were “near-perfect” andthe rest had localized wall thinning in the form of rectangular patches positioned at half length. The wall thickness loss of up to25% was introduced on the outer surface, only. Corrosion to T-type rings was also considered. Here, the breadth of the localizedflange damage was up to a maximum of 50% (with the web beingintact). Comparison of test data with submarine design formulaeprediction of collapse pressure is provided for both intact and cor-roded models. It is seen here that the experimental collapse valuesare higher than those given by the existing design manual. How-

ever, at the same time, the reduction of collapse and yield pres-sures caused by the wall thinning was 20 and 40%, respectively,when compared to intact models. Tests on additional eight mod-els, three intact and five with simulated corrosion, are described inRef. [54]. This time a different aluminum alloy, AA-6082-F28,was used. The principal idea behind choosing this alloy was itselongation at break, which is comparable to steel plate used inconstruction of submarines. Uniaxial tests have shown anisotropyin material properties and the elongation at break ranging from 9to 20% was recorded. However, the profile of stress–strain curvesresembled that of true material used in submarine pressure hulls.Tests showed that collapse strength was related not only to theyield point of the material, but also to the plastic reserve of thematerial. In consequence, the full stress–strain curve would beneeded in the numerical predictions of collapse. Results based on

elastic perfectly plastic modeling would lead to conservative esti-mates of buckling strength. Although the tests were carried out onaluminum models, their pattern of failure resembled that of hullsmade from high-strength steel found in real submarine pressurehulls. Several proprietary finite element (FE) codes wereemployed in Ref. [53] to utilize the vast amount of experimentaldata collected (in Ref. [52]) in order to numerically estimate col-lapse pressures. Accuracy of the FE results is about 11%, whilethe conventional design manuals are accurate to within 20% for intact models, and 26% for corroded models. It has transpired thatthe fine details had to be modeled and Ref. [53] provides thesedetails. The sensitivity of buckling pressures to manufacturingdefects in pipes with the ratio of diameter-to-wall-thickness, D / t ,within 10 to 25 has been investigated in Ref. [55].

Table 1 Comparison of 42 experimental and computed resultsfrom Ref. [26] (P exp tl is experimental buckling load, and P c iscomputed buckling load)

Pexp tl Pc

Pexp tl Number of tests

30% to 20% 320% to 10% 510% to 0% 50% to 10% 7

10% to 20% 620% to 30% 630% to 40% 540% to 50% 5

010803-2 / Vol. 66, JANUARY 2014 Transactions of the ASME




2.2 Axial Compression. Buckling strength of axially com-pressed cylindrical shells still attracts sizeable amount of research.One specific topic has been devoted to buckling of variable lengthcylinders under axial compression, as sketched in Fig. 1. Whentwo or more cylindrical segments form a prime load bearing struc-ture then the interaction between two neighboring segmentsbecome critical when the load is axial compression. The possibil-ity of buckling of either one or both segments complicates the seg-ment-to-segment interaction. Typical application exists inaerospace where the gap between segments is filled by shimming[56]. Once axial compression is applied to two segments wherethere is a variable gap between them, then the uneven load results.

Diminishing axial gaps result in a variable length of hoop contactbetween two cylinders and in localized plastic deformations.These local effects at the imperfect end of the cylinder can propa-gate along the shell’s length. They in turn can trigger asymmetricbifurcation buckling or collapse, and as such they can pose designlimitations. One aspect of this problem was studied in Ref. [57].Cylinders with sinusoidal waviness of axial length were subjectedto axial compression by a rigid disk moving vertically. Numericalresults have highlighted a complicated nature of the interactionbetween the plate and imperfect cylinder. A big drop in bucklingstrength has been obtained for relatively small amplitude of wavi-ness in length. In Ref. [58], 18 mild steel cylinders with thelength-to-radius ratio, L / R 2.4 and with the radius-to-wall thick-ness ratio, R / t 185 were collapsed by axial compression. Cylin-ders had variable length at one end of the sinusoidal profile. The

amplitude-of-axial-waviness-to-wall thickness ratio, 2 A / t , wasvaried between 0.05 and 1.0. Experimental results show that buck-ling strength strongly depends on the axial amplitude of imperfec-tion (see Fig. 2). Average imperfect cylinders, with 2 A / t ¼ 1.0, areable to support 49% of experimental buckling load obtained for geometrically perfect model. The largest sensitivity of bucklingstrength was associated with small amplitudes of axial length. For example, for axial length imperfection amounting to 25% of wallthickness the buckling strength was reduced by 40%. It appearsthat the number of sinusoidal waves in the imperfection profileplays a secondary role, i.e., its role in reducing the bucklingstrength is not a dominant factor. The paper provides experimentaldetails and comparisons with numerical results based on the FEanalyses.

2.3 Combined Loading

2.3.1 Straight Cylinders. Experimental program aiming atreassessing NASA Space Vehicle Design Criteria guide SP-8007(Ref. [15]) is reported in Refs. [59 – 61]. Metallic cylinders withthe radius-wall-thickness ratio, R/t, varying between 250 and1500 were buckled under static loading. Experimental modelswere from copper, aluminum or stainless steel. Their diameter was 135 mm. The whole program consisted from 150 carefullyconducted buckling tests. The role of geometrical imperfectionswas of particular interest and it was closely monitored duringtests. It was noted, for example, that internal pressure reduced the

influence of imperfections on buckling, resulting in higher buck-ling strength. A stable post-buckling behavior of copper model ( R / t ¼ 1350) was obtained for the case of simultaneous action of in-ternal pressure and bending. But for other cases, internal pressuretriggered local yielding and this, in turn, accelerated elastic–plas-tic buckling. Some models were retested under a different set of

Fig. 1 Cylinders with uneven length at the top end

Fig. 2 Buckling response for axially compressed cylinderswith various waviness of length

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loads. While these tests were within elastic domain, the concernswere raised about the retest data. In view of this, Ref. [61] pro-vides two sets of test data, i.e., “single test” and “retest” bucklingresults. In a separate study, the influence of the thermal insulationlayer onto buckling performance of cantilevered cylindrical shellis reported in Ref. [62]. This is both an experimental and numeri-cal study. Vacuum induced buckling tests of small steel cylindersare reported in Refs. [63,64]. Models were mass manufacturedindustrial containers for storage of paint. Initial geometry of cylin-der’s generators was carefully measured and the loadingamounted to hydrostatic external pressure. The content of Ref.[65] is in many respects unique. It provides insights into load car-rying capacity of on land vertical storage tanks, usually found in

the petroleum industry, under buckling conditions. Four steeltanks with volume capacity ranging from 1000 m3 to 65,000 m3

(with radii between 5 m and 35 m) were in situ measured for ge-ometry (geometrical imperfections), and then buckled by externalpressure (through the application of internal vacuum). The paper provides a wealth of practical information including: design codeestimates, implications of quality of manufacturing, knock downfactors, and FE analyses (including axial compression generatedby roof-loading).

2.3.2 Bowed-out Cylinders. Buckling strength of axially com-pressed bowed out cylindrical shell can be larger than bucklingload of mass equivalent cylinder, as shown in Fig. 3 (with detailsin Refs [66,67]). At the same time, barrels are more efficient insupporting external pressure. This idea has been explored in Refs.

[68 – 73] for underwater applications. Earlier background to theabove idea can be found in Refs [74,75]. It is seen in Fig. 4 thatexternal hydrostatic pressure increases with the amount of out-ward barrelling; it reaches maximum and then it drops when theshell becomes the outer half of a toroid. A number of mild steelmodels have been machined to test numerical predictions. Allmodels had the same mass as the reference cylinder and they hadintegral flanges at both ends. Each shell after over machining wasstress relieved. Wall thickness and shape have been measuredprior to testing. The diameter of the tested shells was about200 mm, their length varied from 75 to 100 mm in order to securethe constant mass while the wall thickness was nominally 3.0 mm.Thick flat plates were attached to tested models, which were thenfilled with oil and vented to the atmosphere. Models were

immersed in 350 mm 1000 mm vessel and single incrementalquasi-static pressure was applied while the amount of expelled oilwas measured. All shells failed suddenly with a loud bang and

rapid outflow of oil from inside. Hence, there has been no diffi-culty in identifying buckling pressure. Experimental bucklingpressures varied from 8 to 22 MPa. Figure 4 indicates four config-urations “a”, “b”, “c,” and “d,” which were subject of experimen-tation. Bifurcation buckling was predicted for points “a” and “b.”Photographs in the figure show the lobar mode of failure at “a”,and “b.” At point “c,” both the machined barrel and tested modelare seen in the insert. Photographs of collapsed barrels at points“c” and “d” are also depicted in Fig. 4. Good agreement has beenobtained between experimental results and numerical predictions.The ratio of pnum= pexp tl varied from 0.98 to 0.99 for reference cyl-inders, and 0:90 pnum= pexp tl 1:02 for mass equivalent barrels.The sensitivity of buckling/collapse pressures to the initial, eigen-mode type geometric imperfections has been assessed for both cy-lindrical and bowed out geometries. It appears that barrelling does

not necessarily increase the sensitivity of buckling pressure toshape deviations from perfect geometry. Reference [68] providesfurther details. Buckling tests carried out in Ref. [68] proved that,on a like for like basis, barrels were able to support pressure of nearly 90% higher than mass equivalent cylinders. A more practi-cal configuration of bowed out shell would be associated with abarrel having the same length and top/bottom radii as the master cylinder. This would require the readjustment of the wall thick-ness in the barrel in order to have both shells of the same mass. Inaddition, a shell’s generator could be searched in a different classof profiles than circular arcs. This approach was adopted in Ref.[76]. The shape of the generator was assumed to be defined by ageneralized ellipse [77]

x

Ro þD

n1

þ

y

0:5 Lo þ n3

n2

¼1 (1)

where Ro is a barrel’s radius at top/bottom ends; D is the amountof barrelling at the equator.

Parameters n1, n2, and n3, which strongly influencing the bar-rel’s meridian, were chosen as optimization variables. Simulatedannealing, SA, was utilized in the search for the optimal shapeleading to the maximum of buckling pressure under the equalitycondition imposed on masses, m, of the cylinder and barrel, i.e.,mcyl¼ mbarrel.

A section through the design space is plotted in Fig. 5. It is seenhere that a significant increase in buckling pressure above the cyl-inder’s one was predicted. Detailed calculations were performedfor the reference cylindrical geometry given by Lo / Ro¼ 1.0 and

Fig. 3 Combined stability plot for cylindrical shell (axial com-

pression, F , versus external pressure, p ) [67]

Fig. 4 Load carrying capacity of equivalent barrels as a func-tion of barrelling, D / R o . Photographs of tested models at a, b, c,and d.





Ro / t o¼ 33.33. The reference cylinder under consideration, failsthrough the bifurcation buckling at pbif ¼ 10.5 MPa. The eigen-mode corresponds to n

¼7 circumferential waves. The SA algo-

rithm has found the global optimum at popt¼ 14.71 MPa, and itcorresponded to the optimal design vector (n1, n2, 2n3 / Lo)opt ¼ (2.2, 2.0, 1.0). The predicted failure mechanism at the optimumwas through the axisymmetric collapse. A section through thedesign space is shown in Fig. 6, where it is seen that feasible do-main is not convex. Two nominally identical barrels were col-lapsed experimentally and they failed at 16.97 MPa and16.83 MPa, respectively. After removal from the test tank bothbarrels were photographed and these are seen as inserts in Fig. 6.It is worth noting that the experimental buckling pressure for twoequivalent cylinders was 11.66 MPa and 11.58 MPa, respectively.For barrels, this gives an increase of 45% in magnitude of externalhydrostatic pressure above the master cylinder’s bucklingstrength. Depending on numerical modeling the ratio, pnum / pexp tl,varied from 0.90 to 1.02. Additional calculations have shown that

the shape optimization has not created the solution which wouldbe dismissed from a practical point of view because of greatlyenhanced imperfection sensitivity of buckling load to initial geo-metric imperfections. It is seen from results given in Ref. [76] thatboth the initial (cylinder) and optimal (barrel) designs have a com-parable sensitivity to initial shape deviations from perfectgeometry.

The above results prompted the examination of multisegmentvessels made from bowed out cylinders (see Ref. [78]). It was

decided to investigate bowed out cylindrical shells which had thesame wall thickness as the reference cylinder. Although, per se,these were not equivalent models since the mass was not thesame. Nevertheless, large increases in external buckling pressureswere predicted by numerical calculations. One of the outstandingissues was the joining of two neighboring segments. One possibil-ity was to have the integral flanges of adjoining segments boltedtogether. Here, dimensions of flanges heavily influenced the loadcarrying capacity of the whole assembly as illustrated in Fig. 7.Two configurations were chosen for experimentation and they aredenoted in Fig. 7 as models DB1 and DB2. Numerical calculationsindicated that in both cases, the excessive yielding at the jointscontrolled the failure. Post-collapse pictures of models DB1 andDB2 are also seen in Fig. 7. One 4-segment vessel has also beenstudied. Its collapsed view of global nature is shown in Fig. 8 (see

Ref. [79] for discussion of inter-stiffener collapse and overall col-lapse of cylinders). Connecting segments by bolting externalflanges performed satisfactorily for given arrangements. However,not all possibilities, e.g., external versus internal flanges wereexplored. In addition, the wall thickness of barrels was kept con-stant. It might be beneficial if variation of the wall thickness isallowed in order to mitigate the edge effects. Numerical resultsobtained in Ref. [70] indicate that in barrels made from fiber rein-forced plastics (FRP), it is possible to reduce the edge effectsthrough the appropriate lamination stacking. In particular, free ra-dial displacements at the top and bottom edges do not automati-cally lead to an inferior performance when compared with an

Fig. 5 The magnitude of failure pressure for a shell generatordescribed by generalized ellipse [76]. View of barrel’s shape atbifurcation corresponding to p max.

Fig. 6 Plot of the cost function versus design vector compo-nents n15n2. Also, photograph of two tested barrels E1 andE1a [76].

Fig. 7 Buckling strength of two-segment vessel versus theflange thickness [67]

Fig. 8 Collapsed four-segment vessel [67]





equivalent cylinder. However, no experimental verification of thishas been carried out.

3 Conical Shells

Buckling of cones has closely been associated with bucklingperformance of cylindrical counterparts. Strong motivation for ex-perimental and theoretical research into buckling of cones hadbeen, and still is [80,81], rooted into their substantial role in mis-siles and in space launchers. Account of these efforts is providedin Ref. [82]. Past experiments on unstiffened conical shells arebriefly reviewed in Ref. [83]. Between 1958 and 2008, there havebeen 484 buckling tests on unstiffened cones. Reference [83]shows the number of tests per year, the type of applied loading, to-gether with the source of data and the type of material from whichthe tested shells were made. Apart from plain, i.e., not reinforcedcones, a range of stiffened cones has also been tested in the past:111 in the same period. Most tests were on cones reinforced byrings, and in most cases cones were made from steel. By far themost frequent buckling tests have been carried out on cones subjected to a single load. However, buckling tests were also carriedout when two or more loads were applied at the same time. Detailsabout these tests can be found in Refs. [83 – 85]. The above testshave been carried out in elastic range. The next sections summa-rize recent experimental research effort for cones subjected toexternal or internal pressure, axial compression, and cones sub-

jected to combined loading. Most of these papers are related toelastic–plastic buckling with very few experimental data availablehere.

3.1 Externally Pressurized Cones. It has to be mentionedthat studies into elastic–plastic buckling of cones have not been asubject of extensive research effort. Nevertheless some fresh ex-perimental results have been reported. For example, details aboutbuckling tests on ten mild steel cones subjected to quasi-staticexternal pressure are reported in Ref. [86]. Shells were manufac-tured by rolling flat sheets followed by longitudinal welding. Thebase diameter was about 300 mm and the wall thickness variedbetween 0.5 mm and 0.8 mm. The collapse was a gradual processbut it is not reported what was the capacity the vacuum sourcewhich served as means of loading. Measured geometry was uti-

lized in subsequent FE analyses. The ratio of pexp tl / pnumerical, var-ied between 0.40 and 0.92. Tests on 19 steel, seamless conesobtained by using the metal spinning, are described in Ref. [87].The base diameter of all models was 500 mm and the wall thick-ness was 0.635 mm. Four models were unstiffened while theremaining had external rings at various positions on the slant.Models were subjected to quasi-static external hydrostatic pres-sure. The experimental buckling pressures were established visu-ally once any bulge was visible on the cone surface. In this case,the ratio of pexp tl / pnumerical varied between 0.62 and 0.80. In a sep-arate research, with scattered results in Refs [88 – 90], a series of 15 mild steel cones reinforced by external rings have been testedunder quasi-static external hydrostatic pressure. It is reported thatthe failures were sudden with a loud bang accompanying failure.Once a cone failed the pressure instantly dropped (for example,

from 14.46 MPa to 7.32MPa for cone no. 10 in Ref. [88]). Straingauges on the inner surface, primarily used to indicate the number of lobes in the failure mode, recorded large plastic strains justprior to buckling. Thus, subjective identification of buckling loadappears to be removed during the latter tests. Comparisonsbetween experimental and numerical predictions measured by theratio, pexp tl / pnumerical, varied from 0.24 to 0.84 (Tables 4 and 5 of Ref. [88]; see also Table 6 of Ref. [89], and Table 3 of Ref. [90]).In an earlier series of buckling experiments on ring-reinforcedsteel and aluminum cones summarized in Ref. [91], the same ratiovaries from 0.56 to 0.87. Over the years, a number of attemptshave been made to find a simplified design equation which couldprovide an estimate of the buckling strength of cones subjected tohydrostatic pressure. One such equation has been proposed in Ref.

[92] as a result of extensive parametric studies. Reference [93]compares estimates given in Ref. [92] with numerical predictionsgiven by BOSOR5 code, Ref. [94] (see Fig. 9). It is seen herethat all Bosor5 predictions are higher than those given inRef. [92]. Comparisons have also been made between 17 test dataon steel/aluminum cones and the proposed equation. The ratio, pexp tl / pnumerical, was found to be between 0.57 and 1.41. Estimatesof buckling pressures for thicker cones were on a safe side whilefor thinner cones the above ratio was on unsafe side (between0.57 and 0.91).

3.2 Internally Pressurized Cones. Results of an experimen-tal study into the buckling of six scaled-down models of the inner vessel relevant to a liquid metal fast breeder nuclear reactor arereported in Ref. [95]. The models consisted of two cylindricalshells connected by cone and toroid (all from stainless steel).These compound shells were manufactured by rolling, forming,and welding, and they were subjected to internal pressure and/or concentrated loads applied to the stand-pipes. Diameters of cylin-ders were 420 mm and 810mm with the wall thickness being0.8 mm. The ratios of experimental buckling load to the FE esti-mates were between 0.95 and 2.02. Conical shells are frequentlyused in metallic silos and storage tanks where internal pressure isoften an important loading condition. References [96 – 98] discussinternally pressurized cone-cylinder intersections under uniform

internal pressure. Details about a single, mild steel test cone-cylinder fabricated by rolling and welding steel sheets of 1 mmthickness are given in Ref. [96]. Radius and length of the cylinder were both 500 mm. Cone apex half angle was 60 degrees. The FEresults indicated that buckling load of the cone-cylinder intersec-tions subjected to internal pressure appears not to be sensitive toinitial geometric imperfections. Test data on a further three cone-cylinder intersections is provided in Ref. [97]. Cones were madeby rolling and seam welding of 1 mm thick mild steel sheets.

Fig. 9 Comparison of buckling loads with Bosor5 predictions[93]





Their apex half-angle was 40 deg. Under single incremental load-ing, they failed by bifurcation with a number of hoop waves at thecone–cylinder junction. Additional cone–cylinder model, thistime with a horizontal ring of 20 mm depth and placed at the junc-tion, is tested in Ref. [98]. It appears that in all three references,an exact bifurcation buckling was difficult to establish experimen-tally and remained a fairly subjective process. The above threereferences provide detailed discussions around this subject.

3.3 Buckling Under Axial Compression. Recent tests are

reported in Refs. [99,100]. Five steel models fabricated from steelsheets by cold rolling and longitudinal seam-welding were col-lapsed by quasi-static axial compression. The base diameter of allmodels was 450 mm and the wall thickness varied between0.7mm and 0.9 mm. Heavy plates were attached at both ends tosimulate clamped boundary conditions. Models failed through theformation of axisymmetric bulge at the small-radius end. The testultimate loads were normalized by Rankine limit load, and the ra-tio varied between 0.98 and 1.11. In addition, 40 aluminum conesfabricated by spinning [100] were subjected to axial squashingbetween platens. The base diameter varied between 256 mm and304mm with the wall thickness being between 0.625mm and1.45 mm. Although the main objective was to evaluate the energyabsorption and folding mechanisms on the post-collapse path, thecollapse loads are also reported. Results of the FE analyses for

some models are given, and the ratio of experimental collapse forceto the FE estimate, Fexp tl / Fnumerical, was between 0.17 and 0.75.

3.4 Combined Loading. It appears that there has been verylimited research into elastic–plastic buckling performance of coni-cal shells, under the action of combined loads. The previousresearch effort has primarily been limited to buckling by externalhydrostatic pressure which in fact corresponds to combined load-ing due to axial compression resulting from pressure action ontop/bottom flanges, e.g., Refs. [86,88,89,91].

In order to gain a better understanding of buckling of conesunder combined loading, two series of buckling tests on labora-tory size steel cones have recently been carried out in Liverpool.Models were CNC-machined and all of them had integral top andbottom flanges. They were subjected to: (i) axial compression, (ii)

external radial pressure, or (iii) any combination of pressure andaxial compression. Nominal dimensions of the first set of models,13 in total with designated names C1,… C13, were as follows: thecone semiangle, b¼ 26 deg; the ratio of the larger radius, r 2, towall thickness, t , was r 2 / t ¼ 34.0; the wall thickness, t ¼ 3.0mm.Details about experimentation and related FE computations canbe found in Refs. [83,93,101 – 106]. The second set of ten modelshad the following geometry: b ¼ 14 deg; r 2 / t ¼ 54.0, and the wallthickness, t ¼ 2.0 mm. Here, details about the experiments and theFE results can be found in Refs. [107 – 111]. Figure 10 shows two,as manufactured models (b¼ 26 deg in Fig. 10(a), and b ¼ 14degin Fig. 10(b)). None of the models were stress relieved prior totesting. Combined loading was applied to them using arrangementshown in Fig. 10(c). Heavy top and bottom plates were attachedto each cone. Flanges were partially embedded into the plates in

order to secure clamped-clamped boundary conditions as realisticas possible. Axial compression was applied through a ramattached to the bottom plate and connected, through an internalbar and pivoted coupler, to the top flange (see Refs. [83,101,102).The whole arrangement seen in Fig. 10(c) was immersed in350 mm 1000 mm pressure tank. External pressure in the tankwas controlled manually. At the same time, through a separatepressure line, compressive force was applied via the ram. Variousloading scenarios were explored during experiments. Both sets of models developed plastic straining before buckling. Hence differ-ent loading paths were explored first, and this included: (i) pres-sure preloading followed by incremental axial force loading, (ii)axial force preloading followed by incremental radial pressure,and (iii) proportional loading. References [102,104,106] discuss

these in detail. It is shown in them that the magnitudes of bucklingload remain nearly the same for different loading paths; althoughthere were small differences in values of plastic strains at buck-

ling. Figure 11(a) depicts loading paths for the first set of cones(models C1,…, C13). All models were filled with oil and ventedto the atmosphere. During loading, the amount of expelled oil wasmeasured. Failure of all models was sudden with large outflow of oil and accompanied by a loud bang. As mentioned earlier, bothsets of cones were machined from two different billets of steeland tests were carried out in order to establish the exact materialproperties. Several round tensile specimens (10mm diameter,200mm long) were cut from billets in different directions andthey have not been stress relieved prior to uniaxial tensile tests.Full details about the material properties are available inRefs. [102,104] for the first set of cones (b¼ 26 deg). Tensilestress–strain curves confirmed mild steel characteristics of mate-rial, i.e., well defined upper/lower yield followed by horizontalplateau. Details about the evaluation of material properties for the

second set of cones are available in Refs. [108,111]. Under uniax-ial tension the second material exhibits continuous strain harden-ing without a clearly defined yield point. Hence, 0.2% proof stresswas assumed for the yield point. The FE computing related to theabove experimentation identified the first yield envelope in addi-tion to the collapse envelope. These are depicted in Fig. 11 for b¼ 26 deg and in Fig. 12 for b¼ 14 deg models. It is seen herethat the first yield envelope is of “by-and-large” bilinear format.For lower values of axial force, the plastic strains begin to growfrom larger-radius end of the cone. For higher values of axialforce, the plastic strains start to develop at the smaller-radius endof the cone. The hatched area in Fig. 11 indicates the elastic–plas-tic domain. Experimental results shown in Fig. 11 are cast in twodifferent formats. Experimental data points in Fig. 11(a) are nor-malized: by average experimental collapse force, Fexp tl

avg , and by

average experimental buckling pressure, pexp tl

avg (note that therewere two nominally identical models tested under pure axial com-pression and two models tested under pure radial pressure). Exper-imental data points shown in Fig. 11(a) are normalized by the FEpredicted collapse force, Fcoll

o , and by the FE predicted pure radialcollapse pressure, pcoll. Both interactive plots shown in Figs. 11(a)and 11(b) were obtained for elastic perfectly plastic modeling of steel, experimentally measured average geometry and constant,average wall thickness. Design guidance for the cones subjectedto interactive loads, as discussed above, is provided by the ASMEcode [41]. Reference [42], on the other hand, provides the designguidance for the cases of pure axial compression and lateral pres-sure only. Magnitudes of design loads were obtained for all testedcones by using both codes in order to compare them with the test

Fig. 10 View of machined cones: (a ) b526deg, (b ) b514 deg,and (c ) arrangement for combined loading. Adapted from Ref.[103].





data. Estimates of implicitly safe values are shown in Fig. 11(b)for tested models C1,…, C13. The ASME predictions are markedby open circles while the ECCS predictions are marked by aster-isks. It is seen here that for some load configurations the recom-mended values would generate plastic straining under singleincremental loading. This happens, for example, for cones C8, C7,C12, C5/C6, C1/C2 subjected to combined loading as well as toC11 subjected to pure lateral pressure. The ECCS based design

pressure for C11 also falls into the elastic–plastic domain. Thesame applies to cones C3/C4 subjected to pure axial compression.While there is a substantial margin of safety against bucklingunder a single incremental loading, the predictions of failuresinside of the elastic–plastic domain can be of concern under repeating loads. In the latter case, cones can fail through growthof plastic strains if there is no shakedown. As mentioned earlier the second set of experiments was carried out in order to verify

Fig. 11 Combined stability plots for b526 deg cones [106]. Loading paths shown in Fig. 11(a ) while configurations given bythe current Design Codes are superimposed in Fig. 11(b ).

Fig. 12 Combined stability plot for ten, b514 deg, cones [110]





these unexpected results. It is seen in Fig. 12 that some of recom-mended configurations would buckle with plastic strains beingdeveloped. This applies to cones CS1, CS6, CS7 (ASME, Ref.[41]), and CS3/CS4 (ECCS, [42]). Buckling under pure lateralpressure predicted by both codes remains elastic (CS2/CS5).Hence the same observation about safety under repeating loadsremains true here for cones with code-based design loads fallinginside of the elastic–plastic domain. Finally, views of buckledcones C6 and CS1 are seen in Fig. 13.

Initial geometric imperfection in conical shells can lower thebuckling strength and there have been numerous studies address-ing this topic. However, only a handful of tests have been carriedout on cones with deliberately built-in shape deviation from per-fect geometry. It has been generally believed that inward dimple-type axisymmetric shape deviations from perfect geometry are themost dangerous imperfections, i.e., leading to the largest reductionof the buckling load. Reference [112], for example, examined theinfluence of axisymmetric inward-bulge type shape imperfectionon the magnitude of buckling load (axial compression). Theimperfections were deliberately introduced during manufacturing(by electro-deposition of copper). Thirty cones with the top endradius-to-the wall-thickness ratio ranging from 181 to 1115 weretested. The extreme sensitivity of the buckling load to initialimperfections of the order of a fraction of the wall thickness wasconfirmed experimentally. Imperfections found in real structuresare likely to have neither axisymmetric nor have the shape of buckling mode but they rather occur locally. One needs to exert

high degree of skills when assessing the load carrying capacity of complex structures prone to buckling. A three stage approach isadvocated in Ref. [113] for the assessment of imperfection sensi-tive real structure. The effect of localized dimple-type imperfec-tions on the buckling strength of axially compressed cones wasaddressed in Ref. [114]. The FE results showed that buckling loadof the cone with inward axisymmetric imperfection was nearlyequal to the buckling load of local imperfections which extended60 deg or more around the circumference. Twenty high quality ep-oxy conical shells were buckled in Ref. [115] by axial compres-sion. The prime objective here was to develop experimentalknock-down factor against buckling. The paper also contains dataon axially compressed imperfect cylinders (built-in inward dim-ple). Recent numerical results given in Ref. [110] show that out-ward dimple-type shape distortion can be as bad as the

corresponding inward dimple. This directly contradicts the longstanding view that the inward shape imperfections constitute theworst case. A subsequent study [116] considered geometricallyimperfect conical shells subjected to axial compression, externalpressure, or simultaneous action of both loads. Axisymmetricshape imperfection was assumed to be an inward dimple, outwarddimple, or coexisting inward and outward dimples. The profile of inward bulge was described by (see Fig. 14)

d zð Þ ¼0 z zij j b A

2

di cos3 p

b A

z zið Þ

z zij j b A

2

8>><>>:

9>>=>>;

(2)

where di is the amplitude of the inward imperfection, zi is positionof its center along the generator, and b A is the extension of theimperfection along the cone’s slant. The shape of outward axi-symmetric bulge was also considered. Its form was given by Eq.(2) in which zi was substituted by zo, and the slant extension wasassumed to be characterized by, b B (see Ref. [116] for details).From a design point of view, it would be desirable to know inadvance what would be the lower bound response to any possibleshape deviations one could encounter in reality (inner and/or outer). Typical parameters which influence the buckling strengthwere assumed, and the worst scenario was sought using the TabuSearch optimization method. Results were obtained for mild steelcones for which earlier test results on perfect models had beencarried out [103,110]. Interactive diagrams obtained for inwardbulge, outward bulge, and coexisting inward/outward bulges areshown in Fig. 15. It is seen from Fig. 15 that both inward and out-ward imperfections can significantly reduce the load carryingcapacity. However, the largest shrinkage of the interactive dia-gram is obtained for the case of coexisting inward and outwardinitial shape imperfections in cone’s generator. Illustration of imperfect initial geometry together with shape of the generator atthe collapse is plotted in Figs. 15(b) – 15(d ) for selected points. InRef. [118], initial geometric imperfections were taken in the formof the eigenmode, “a single wave” extracted from the eigenmodeand localized smooth dimple modeled analytically. Load carrying

Fig. 13 Photographs of collapsed cones by external hydrostatic pressure(b526 deg for C6 model, b514 deg for CS6 model)

Fig. 14 Geometry of inward, axisymmetric dimple imperfection[116]





capacity of imperfect models was computed along the combinedstability domain using the finite element proprietary code. The FE

results showed that buckling strength of axially compressed andimperfect cone was only 55% of geometrically perfect model.Buckling strength of a cone subjected to lateral pressure; on theother hand, amounted to 43% of the corresponding value of per-fect model. However, it was the shrinkage of stability plot of imperfect cone which was found to be significant. For imperfectcones subjected to combined axial compression and external pres-sure, the collapse envelope shrunk by 48% with the elastic sub-setbeing reduced by 51%. Numerical study into imperfection sensi-tivity of buckling loads for cones with the semiangle, b ¼ 30, 55,60, 65, and 70 deg for the case of axial compression, was only car-ried out in Ref. [119]. Imperfections were taken in the form of eigenshapes and the effect of different boundary conditions onbuckling was examined. Results are comparable to those obtainedin Ref. [118].

It appears that, so far, there has been no experimental verifica-tion of imperfection sensitivity of buckling loads for cones subjected to simultaneously acting axial compression and externalpressure.

Finally, Ref. [117] details development of a test rig for bucklingtests of shell components subjected to pressure and/or axial (cen-tric/eccentric) loading. Test results were obtained for conicalshells made from Mylar. These delivered interactive stability dia-grams. The effect of off-axis axial compression on the bucklingstrength of pressurized cone was also investigated. Shape meas-urements gave information on the quality of tested models.

4 Domed Ends—External Pressure

4.1 Hemispheres, Torispheres, and Toricones. End clo-sures onto cylindrical vessels can take different shape forms rang-ing from flat plates to domed ends. The latter can includespherical caps, hemispherical, ellipsoidal, torispherical or toricon-ical shapes. These shells also find way into other specializedapplications, e.g., large outer space mirrors. When the loading issuch that the internal stress is dominated by membrane stressresultants and their associated, relatively high stretching stiffnessthen the load carrying capacity becomes very efficient. In doublycurved ends, the curvature of a shell’s mid-surface, together withthe high ratio of stretching to bending stiffness, generally leads toa nonlinear interaction of membrane and bending effects. In caseslike that, the load carrying capacity of domed ends stronglydepends on their geometry, boundary conditions, material

behavior, type of applied load and the presence, or absence, of ini-tial geometric imperfections. Static stability of domed ends has

been researched for decades both theoretically and experimen-tally, e.g., Ref. [120]. Review of past efforts in this area can befound, for example, in Refs [7,121]. It is worth noting here thatthe first tests on externally pressurized spherical caps were at theend of the 19th century. Buckling tests on domed ends of other shapes have continued until today and they were driven by varietyof reasons. For example, in view of raised concerns that aluminumcaps tested in Ref. [122] had the base diameter only between20 mm and 50 mm, it was decided to re-examine the sudden dropin buckling strength around the shallowness parameter k, k ¼ 4.0,where k 2 3 1 t2ð Þ½ 1=4ð H =t Þ1=2

(see Fig. 16 for notation). Sixmild steel caps with 200 mm diameter were carefully machinedfrom a billet. Each model had an integral heavy base ring, and joining arrangement of the shell with base ring is shown inFig. 16. Three rings were designed to fail elastically and the

remaining three to fail within the elastic–plastic range. Six testpoints are superimposed in Fig. 16 on the original 1963 test databy Krenzke and Kiernan. It is seen that indeed there is a minimumof the load carrying capacity for k 4.0, irrespective whether it isin the elastic or elastic–plastic regime. Further details are avail-able in Ref. [123]. An interesting piece of experimentation isreported in Ref. [124]. Spherical caps manufactured from brasswith the wall thickness, t ¼ 0.4 mm, were subjected to outward ra-dial load applied at the base of the cap. Experimental bucklingloads compared favorably with the theory provided.

An empirical approach to design of externally pressurizedhemispheres, adopted by British Standards Institution’s BSI 5500(now PD 5500), has been reviewed in Ref. [24]. The paper addresses elastic and elastic–plastic buckling. It points to the mostconsistent safety factor over the experimental data, together with

a definition of allowable shape that includes both overall shapeallowance and a local defect parameter. Extension of PD 5500 toexternally pressurized torispheres triggered concerns about thesafety factor for sharp knuckle torispheres. Experimental resultsof Ref. [125] demonstrate the kind of issues (see Fig. 17). It isseen here that several test points plot below the lower bound curveto all previously known experimental points, i.e., domes 1A, 1C,4A, 4C, 7A, and 7C (see Table 2 for model descriptions). Whiledomes 1A and 1C are outside the stipulated lower limit onr / D ¼ 0.06, it is the remaining heads tested at Brown University,which surprisingly fell below the PD 5500 design curve (after being multiplied by the 1.5 safety factor) which raised the concernabout the universality of the proposed experimental-lower-bound.In view of these findings, it has transpired there was not much

Fig. 15 The worst interactive stability plots for different imperfection profiles (a ). Collapsed shapes at points a, b, and c [116].





information in the literature on torispherical heads with knuckleradii, r , in the region of lower limit of r / D ¼ 0.06. In response tothis, two research programs were carried out in Liverpool. Thefirst program concentrated on 24 torispherical heads, some hot

pressed and some cold spun, to acquire information in this range[126 – 128]. The heads had R / t ratios between 85 and 330 and theywere about 0.75 m in diameter. A few of the test results fell belowthe recommended PD 5500 design curve [23] (after they had been

Fig. 17 Safe and unsafe domains in PD 5500 code. Models 1A and 1C fall outside admissible geometry stipulated by PD 5500(a ). Test data for ten machined and two spun torispheres is plotted in (b ) (p e ¼ 1:21Et 2=R 2s and p yss ¼ 2ryp t =R s ) adapted fromRef. [138].

Fig. 16 Plot of buckling load versus shallowness parameter, k. Also, view of col-lapsed cap and joining arrangements between the cap and integral base flange(adapted from Ref. [123]).

Table 2 Geometry, material properties, collapse pressures, and parameters K and D for externally pressurized torispheres

E r yp pexp tl

Dome r / D Rs / D Rs / t L / D (GPa) (MPa) K D Ref.

1 0.059 1.01 563.64 0.37 207.0 370.0 0.128 0.643 0.105 [120]1A 0.0427 0.749 128.99 0.20 192.4 220.4 0.662 4.387 0.173 [125]1C 0.0427 0.749 126.09 0.20 207.0 435.2 1.172 2.445 0.182 [125]4A 0.0492 1.235 61.47 0.33 192.4 243.6 3.862 8.329 0.522 [125]4C 0.0492 1.235 61.47 0.33 207.0 365.4 5.655 5.974 0.510 [125]

7A 0.0501 0.989 45.189 0.37 192.4 243.6 5.793 11.33 0.576 [125]7C 0.0501 0.989 45.189 0.37 207.0 365.4 7.586 8.126 0.503 [125]P2/1 0.060 1.05 143.99 0.05 208.0 430.0 1.71 2.178 0.307 [128]P2/2 0.058 1.05 148.60 0.05 208.0 430.0 1.78 2.110 0.330 [128]P2/3 0.058 1.03 146.00 0.05 208.0 430.0 1.78 2.148 0.324 [128]P2/4 0.059 1.04 148.60 0.05 208.0 430.0 1.67 2.110 0.309 [128]P4A/1 0.060 0.775 83.0 0.09 212.0 426.0 4.65 3.887 0.485 [127]P4A/2 0.062 0.764 83.0 0.09 212.0 426.0 4.62 3.887 0.482 [127]





multiplied by the 1.5 safety factor, see domes P2 and P4 inFig. 17). This led to the second set of tests on 16 torispherical and2 hemispherical end-closures. Nominal dimensions of thesedomes were as follows: diameter of approximately D 800mm,and wall thickness, t 6 mm. The ratio of the radius of sphericalportion, Rs, to diameter D, varied from Rs / D ¼ 0.75 to Rs / D ¼ 1.0.The ratio of the knuckle radius, r , to diameter, D, varied from r / D ¼ 0.064 to r / D ¼ 0.18. Each dome had L ¼ 50 mm cylindricalflange. Some of tested heads were petal-welded simulating a pro-cedure commonly used for fabrication of large end-closures. Thisis illustrated in Fig. 18. Eight pressed petals and one spherical cap

are shown in Fig. 18(a). The wooden rig seen in Fig. 18(b) wasused to assemble the segments and then weld them. The shellsinvestigated were expected to be sensitive to deviations from theperfect shape (see, for example, Refs [129 – 135]). It was, there-fore, decided to carefully monitor shell’s shape prior to testing,i.e., after pressing, cutting, and welding. The shape of each headwas measured along 72 meridians at 2.5 deg intervals within thespherical cap and 5 deg intervals within the knuckle. The cylindri-cal portion was scanned at 5 mm intervals. The shape measure-ments were made on the shell’s inside surface using an LVDTtransducer and a computer based data acquisition system. On av-erage, there were about 2000 measured points per dome. Thesame grid was used to measure the dome’s thickness using an ul-trasonic probe. Material properties were obtained from test plateswhich had accompanied the heads through their heating and heat-

treatment cycles. Results were obtained from two specimens per test plate and, in modeling the stress–strain curve for the use insubsequent numerical calculations, a multisegment technique wasused. Experimental value of the Young’s modulus was 189.0 GPawith 0.2% proof stress being, r yp¼ 665 MPa. The above data wasincorporated in several different ways into the numerical analyses.Detailed description of manufacturing, pretest measurements, andtesting is given in Ref. [136]. Additional information is given inRef. [137]. It has been found that there was no definitive patternto the relative collapse pressures of the welded and nonweldedheads. Furthermore, the collapse pressure of all tested domes,were higher than the predictions of PD 5500 multiplied by thesafety factor of 1.5 (although some of the heads, as delivered, didnot pass the PD 5500 allowable tolerances on shape). Calculationsshowed that the loss of strength, in almost all cases, was due toformation of a single, localized, dimple which gradually deepened

on the post-collapse path. This resembled closely the experimentalcollapse mode (see Figs. 18 and 19). The ratios of the experimen-tal to the predicted collapse pressures by the FE were in the rangefrom 0.85 to 1.14 (for torispheres) and (1.02, 0.91) for two weldedhemispheres. Hemispheres described in Ref. [136] were relativelythick, i.e., with R / t 62 and manufacturing the whole head wouldbe difficult due to the possibility of wrinkling. Thinner hemi-spheres on the other hand could be spun. Here, a series of 0.58 mdiameter with R / t varying between 190 and 780 were examined inRef. [131] both experimentally and numerically. One of the mainconclusions made was that one should use the minimum shell

thickness for design purposes and not rely on the average wallthickness since three test results plotted below the design curvewhen the average wall thickness was used. Additional computa-tions have indicated that torispheres with sharp knuckle, r / D 0.06, might collapse well below the recommended design curve[138]. A series of 12 collapse tests were carried out on torisphereswith r / D ¼ 0.6 in order to verify these findings. There were twomodels having nominally the same geometry. Ten heads weremachined laboratory models and further two heads were industri-ally spun. Experimental results are shown in Fig. 17. It is seen inFig. 17(b) that all experimental data falls below the recommendeddesign curve. The implication of this is that the radius of theknuckle in the torisphere needs to be considered in the calcula-tions/design. Four titanium alloy spheres made by welding twohalves were collapsed by external pressure as a part of

research into deep sea vehicle [139,140]. Spheres had theinternal diameter of 500 mm, the wall thickness of about 9.0 mm,and they disintegrated under “implosion-type” failure, at pressures55.0–58.0 MPa. Experimental results compared satisfactorily withthe design equations which took into account shape and wallthickness imperfections.

It is worth noting here the use of multilayer materials for theconstruction of domed ends. These could be entirely layered metalconstructions, multiply constructions assembled from fiber rein-forced plastics or hybrids. The first case can be illustrated byrecent buckling tests on spun hemispheres and torispheres fromcopper-steel-copper flat sheets, as described in Refs. [78,141,142].Flat sheets of three layer copper-steel-copper hybrid materialwere manufactured by rolling diffusion. The resulting bonding of layers was strong enough to withstand manufacturing of torispher-ical and hemispherical domes by spinning. No debonding was

Fig. 18 Various stages of manufacturing of a hemisphere to be externally pressurised ((a ) and(b )). View of collapsed hemisphere (c ).

Fig. 19 Photographs of collapsed petal-welded and plain, nominally identical, torispheres





found in post-collapsed heads. Filament winding has been widelyused for manufacturing pressure vessels to be loaded by internalpressure. Wrapping of extruded cylindrical barrels by woven FRis also known. The use of FRP as a material for manufacturingdomed ends to be subjected to external pressure has a muchshorter history. The use of patches of woven CFRP for reinforce-ment of deliberately made imperfect, steel torispheres has beenstudied in Ref. [143]. Mild steel heads were machined with the

increased-radius flat patch at the apex. They were then reinforcedon inside with the aim of restoring the initial buckling strengthwhen subjected to external pressure. Details about damaged bybuckling, laboratory size hand laid-up domes, which wererepaired and then retested are available in Ref. [144]. Variousaspects of manufacturing and buckling/collapse of externally pres-surised domed ends can be found in Refs. [145 – 154]. These refer-ences consider shells of about 200 mm and 800mm diameter,their manufacturing, testing and theoretical analyses. The larger heads were either filament wound or draped from woven fabric. Inboth cases, the prepreg material was used (predominantly CFRP),and there were no internal liners when loaded by external pres-sure. Some heads, both torispherical and hemispherical, werelaminated using petalled segments as illustrated in Fig. 20. Thesesegments of woven cloth were butt-jointed and it is seen in Fig.

20 that failure is due to large crack running in the hoop directionwhile the butt-joining was not affected. Although the heads wereaxisymmetric, their material properties were not. Fiber distortionand the wall thickness were found from the draping algorithm.Fiber orientation was measured along a number of meridiansusing specially constructed jig. Measured distribution of the fiber-angle compared well with predictions given by the draping algo-rithm [151]. Figure 21 shows how the woven fabric distorts whendraped over doubly curved surfaces. Draping process could beenhanced by moving the focal point away from the apex in order to mitigate the wall thickness build-up around the edges[151,152,155].

The inclusion of toroidal (knuckle) segment between cylinder and conical vessel end closure is a natural way leading to

diffusion of the stress jump at the junction (see, for example,Refs. [38,156]). Results based on parametric studies carried out inRef. [156] show that the inclusion of the knuckle can significantlyincrease elastic buckling strength. Larger the (r / D)-ratio larger isthe buckling strength. For a wide range of the apex semiangle,30deg b 75 deg, toricones are stronger than the correspondingcones alone. However, there is very little experimental data insupport of the knuckle’s role and its influence. With this in mind,

eight steel toriconical shells have been buckled by quasi-staticexternal pressure in order to measure this problem [157]. The di-ameter of all models was 200 mm at the base and their wall thick-ness was 2 mm. The apex semiangle was b ¼ 45 deg for all shells.A summary of experimental results, together with numerical esti-mates of bifurcation buckling pressures, are provided in Table 3.It is seen here that experimental buckling pressures vary onlybetween 3.9 and 4.4 MPa despite large variation in shape. The ra-tio of experimental to numerically predicted values of bucklingloads varied between 1.03 and 1.18. Numerical estimates of buck-ling loads given in Table 3 are based on overall average geometry,axisymmetric model and the use of elastic–plastic modeling of

Fig. 20 Female molding tool after the first ply being draped. Also, view of hemisphere after thecollapse test.

Fig. 21 Pattern of distorted fibers after draping (a ). One quarter of draped fabric superimposedon the FE grid (b ) [151].

Table 3 Comparison of test data with computed results forexternally pressurised steel cones/toricones

pfailureexp tl pfailure

numerical

pfailure

design

pfailure

exp tl

pfailurenumericalModel r/D (MPa)

T1 0.2 4.38 3.95 6.39 1.11T1a 0.2 4.35 3.97 6.39 1.10T2 0.1 4.14 3.52 5.20 1.18T2a 0.1 4.10 3.50 5.20 1.17T3 0.05 3.86 3.68 4.55 1.05T3a 0.05 3.86 3.42 4.55 1.13T4 0.0 4.14 4.03 5.16 1.03T4a 0.0 4.14 3.99 5.16 1.04





material. Views of buckled toricones are provided in Fig. 22. Ref-erence [156] provides design equations for elastic buckling pres-

sures derived from parametric studies, and these are extended toplastic region as well. According to Reference [156], plastic buck-ling pressure of a toricone, p

pltoricone, can be approximated by the

following expression:

pfailuredesign ¼ 0:5 po þ 0:4 pel

toricone (3)

where elastic buckling pressure, peltoricone, is approximated by

peltoricone ¼ 13:68 1 t2

0:75 E sin bð Þ cos bð Þð Þ1:5 t

D

2:5

(4)

and D is given by

D ¼ D þ 2r cos bð Þ 1ð Þ þ 2 ffiffiffiffi

rt

p sin bð Þ (5)

The quantity, po is referred to as “the yield pressure,” and it corre-sponds to the pressure which causes the spread of plastic strainsanywhere in the shell reaching half of the wall thickness. The val-ues of, po, for different values of (r / D), the yield point of material,r yp, Young’s modulus, E, and the ( D / t )-ratio can be read fromdesign diagram. When read from Fig. 6 in Ref. [156] these valuesare: 4.95 MPa, 2.81 MPa, 2.25 MPa, 2.03 MPa for models T1/T1a,T2/T2a, T3/T3a, T4/T4a, respectively. The resulting failure pres-sures are given in column 5 of Table 3. All estimated values aresignificantly higher, ranging from 18 to 47%, than the test data.However, it needs to be said that design the equations, Eqs.(3) – (4), have been obtained for geometrically perfect shells, elas-tic perfectly plastic modeling of material, and for cone/toriconesbeing supported by cylindrical shell unlike here [157] where theheavy base ring simulated clamped boundary conditions. Further-more, reading values of, po, from a nomogram was not accurate.However, the tendency of buckling pressure variation with theknuckle size was similar between the Eqs. (3) – (4) and the currenttest data.

4.2 Ellipsoids. Elliptical shells of revolution can be used inspecialized applications, e.g., in pressure hulls for rescue-typesubmersibles. Early test data is available in Refs. [158,159] wheretwo prolate ellipsoids, machined from 7075-T6 aluminum, werecollapsed under incremental external pressure (see Fig. 23) for their view after collapse. The semiaxes were B¼ 75mm andA ¼ 25mm, while the wall thickness was 0.76mm. Despite

different arrangements for boundary conditions around the equa-tor, the collapse pressures differ only by about 10%. Results of the

buckling tests on an additional 33 machined epoxy resin modelstogether with the underlining theory are available in Ref. [160].Elliptical shells of revolution can also be used to close the ends of externally pressurized vessels, with typical applications in thesubmersibles and space vehicle industry. Details about a recentnumerical and experimental study into buckling of steel ellipsoi-dal domes loaded by static external pressure can be found in Refs.[77,161,162]. A range of geometries and thicknesses of domeswas examined, as was the influence of different boundary condi-tions. Shells were examined on the basis of having the same mass.This meant that all shells were analyzed on a like for like basis,and as such, each dome’s performance was easily quantified. Nu-merical analyses of both perfect and imperfect shells were carriedout. Two kinds of imperfections were considered: deterministicimperfections derived from measured dimensions and eigenmode

imperfections. The main focus was on prolate domes, i.e., thosetaller than a hemisphere of the same radius (Fig. 24). Numericalpredictions were confirmed by pressurizing six laboratory scaleprolate domes to destruction. Three current design codes summar-ized in these studies included: ASME VIII, PD5500, and theECCS. At present, prolate domes are not included in the threecodes. The method of calculating design pressures was outlinedand recommendations were made for the possible inclusion of prolate ellipsoids into the codes. Currently allowed ellipsoids(right of hemisphere) and suggested inclusion of prolate domes(left of hemisphere) are depicted in Fig. 24. All domes in this fig-ure have the same mass. Points (t 1, t 2, t 3) are experimental points(two tests per point). View of three nominally identical pairs of prolate ellipsoids after testing is shown in Fig. 24. The ratio A/Bwas 0.8, 0.65 and 0.5 for ellipsoids t 1, t 2, and t 3, respectively. It is

also worth mentioning that the new European design rules for externally pressurized vessels, EN 1993-1-6/Eurocode3, Part 1.6,Refs. [163,164], do not contain prolate ellipsoidal shells. Oblateellipsoids, especially with (B/A)-ratio of two, have been fre-quently used as closures on cylindrical vessels. Oblate ellipsoidswith other (A/B)-ratios have not been widely used. The sameapplies to domed ends whose generators were subject of structuraloptimization. References [165,166] are two examples where theshape of the meridian was searched for the best performance andthe optimum was subsequently subject of experimental verifica-tion. The latter reference examined the performance of externallypressurized ellipsoids through the formal optimization process. Inthis study, the highest buckling load was to be found within arange of geometries of generalized ellipsoidal domes loaded by

Fig. 22 View of cone T4 as machined (a ), and after collapse (b ). Buckled toricones T2 and T2aare seen in (c ) and (d ). External pressure in all cases.

Fig. 23 View of collapsed ellipsoidal shells (adapted from Ref. [159]). External pressure.





external pressure. Generalized ellipsoids are a variation of stand-ard ellipsoids in that the exponents used to size the dome are vari-ables. A generalized ellipse is of the form

x

A

v1þ y

B

v2¼ 1 (6)

For a given dome geometry A and B, one has a two-dimensionaldesign space of variables v1 and v2. In terms of geometry of dome,when v1¼ v2¼ 1 the dome is conical. When v1¼ v2¼ 2 the domeis a standard 1:2 ellipsoid. Additionally if A¼ B the domebecomes hemispherical. As v1 and v2 are further increased, thedome tends towards a closed cylinder. Details about the materialproperties can be found in Ref. [166], and they were modeled asbeing elastic–perfectly plastic. The boundary conditions at thebase of the dome were set to fully clamped.

A hemisphere of the same material properties, boundary condi-tions, and thickness ratio D / t H ¼ 100 had a failure pressure of

p H ¼ 11.01 MPa. Mode of failure for this shell was axisymmetriccollapse occurring at the base of the shell. This hemisphere(hereon termed the reference hemisphere) was the benchmark for assessing the pressure resistance of generalized ellipsoids.

The condition of constant mass was imposed on all shells,meaning that thickness of the ellipsoids is determined from itsshape. Where mass is calculated from:

mshell ¼ mref ; with m ¼ St q (7)

where S is surface area and was calculated by integrating numeri-cally. This condition means that taller, prolate shells will have athinner wall due to their larger surface area. Conversely, shallow,oblate domes will be thicker than the reference hemisphere. In allcases, the shell wall thickness was kept constant as one movedalong the meridian.

The modes of failure considered here were bifurcation buck-ling, pbif , and axisymmetric collapse, pcoll. The lower of these twofailure loads was taken to be the critical failure load, pcr .

The optimization problem can be formally expressed as

poptðvÞ ¼ max pcr ðv1; v2; v3Þ (8)

subject to the constraints

1:5 v1 2:5 (9)

1:5 v2 2:5 (10)

0:3 v3 4:0 (11)

where

pcr

¼ pcr = p H (12)

v3 ¼ A= B (13)

v ¼ ðv1; v

2; v3Þ (14)

The adaptive Tabu search method was employed to determine val-ues of v1 and v2, which give the maximum pressure resistance of ellipsoidal domed ends. The results of the Tabu search are shownin Fig. 25. Also shown in Fig. 25 are the failure pressures for standard ellipsoids (i.e., v1¼ v2¼ 2) of the same material, bound-ary conditions and mass. The optimum shell corresponds to point“a” in Fig. 25 and has geometry of v* ¼ (1.92857, 1.57937, 1.4).The optimal dome is 20% stronger than the reference hemisphere,and has a failure pressure of pmax ¼ 12.84 MPa, mode of failure isaxisymmetric collapse.

To confirm numerical results obtained, four geometries of the

dome were machined from a mild steel billet (global optimumplus further three geometries). The geometries of dome machinedcorrespond to points “a,” “b,” “c,” and “d” in Fig. 25, and their nominal dimensions are given in Ref. [166]. All ellipsoids had thesame mass. Domes were machined in pairs, to demonstrate repeat-ability of the experiment and also act as a safeguard should one of the pair be damaged, e.g., during manufacture. The domes weremachined with integral flanges in order to attach them to a baseplate for testing, and also to make sure that no radial movement of the base was allowed during testing. Before testing, domes werecarefully measured for any variations in shape and thickness— details are in Refs. [166,167].

The experimental failure pressures of all the domes tested arelisted in Table 4. Also given are numerical predictions made usingBOSOR5 based on average thickness. The numerical results are

normalized by the experimental pressures and shown in parenthe-ses. The numerical predictions are all within eight percentage of the experimental values.

It is worth noting here that structural optimization of pressurevessel components subject to buckling constraints poses multifac-eted challenge. At the structural analysis level, significant insightis needed into the effects of, for example, the effects of initial geo-metric imperfections, nonlinear preloading, boundary conditions,or follower-type loading on the type and magnitude of buckling.At the optimization level, there is usually very little knowledgeabout the nature of design space. For the two cases discussed inthe case of ellipsoids the design space proved to be nonconvex. Insituations like these the use of zero-order approach appears to be avery efficient approach (Tabu search in this case). A reliable

Fig. 24 Currently allowed and proposed ellipsoids (a ), prolate and oblate geometries sketched in (b ), (c ), and view of threepairs of elliptical domes after tests (d )





reanalysis could be adopted (usually with a substantial number of man-years of development and quality assurances). The aboveillustrates how this approach provided results for componentswith complicated buckling behavior. Ellipsoidal shells have

recently been considered for tanks in rocket launchers. For exam-ple, an overall reduction in space vehicle mass has motivatedparametric studies of buckling performance of oblate bulkheadsfor a propellant tank [168]. The paper considers both the oblate el-lipsoidal shell and the combination of cylinder and bulkhead withbuckling being one of prime design considerations. The bucklingof the three ring-reinforced prolate ellipsoids subjected to externalpressure is reported in Ref. [169]. Three shells, of 200mm basediameter, were cast using thermoset polyurethane plastic. Theywere then reinforced by equally spaced and externally attachedrings (ten in two domes and 13 in the third model). All threedomes failed through elastic bifurcation buckling. The reinforcedshells were at least four times stronger than their plain counter-parts (based on the FE estimates).

4.3 Toroids. Toroidal shells of closed cross-section havebeen used commonly for liquid or gas storage in vehicles, aero-space structures and in specialized underwater applications. Dueto its compacted shape, a toroid is also suitable for breathing ap-paratus. It has proven to be a very desirable storage vessel inspace-limited applications since it permits various systems struc-tures to be routed through its central opening [170]. Thin-walledtoroids, when subjected to external pressure, or vacuum, canbuckle. Early buckling tests are described in Ref. [171], and theknown experiments are listed in Ref. [172]. Results of extensiveparametric studies into stability of externally pressurized toroids(with circular and noncircular cross-sections; with perfect andimperfect geometries; metallic as well as from composites), can

be found in Refs. [173 – 178]. Consider a closed toroidal shell, of uniform wall thickness, t , and being subjected to uniform externalpressure, p. Assume that the mid-surface radius of the tube is, r ,and the distance from the shell’s axis of rotation to the center of the cross section is, R, as sketched in Fig. 26. It is not immediatelyobvious what boundary conditions should be applied to toroidalshell shown in Fig. 26. This problem has been addressed in a num-ber of previous studies. One particular approach, adopted in Ref.[176], was based on a series of numerical runs in which variouscombinations of restraints were imposed. The boundary condi-tions (BCs), which gave the lowest buckling pressure, were there-fore identified numerically. Results shown in Fig. 26 have beenobtained for BCs applied at the inner and outer perimeters of theequatorial plane. The prebuckling and buckling boundary condi-

tions were different. At the prebuckling phase of the analysis:u ¼ 0.0, v w b free. They were applied at the inner equator only. The outer equator was left unconstrained. The BCs at thebuckling phase of computing were: u v w b free at boththe inner and outer equators. Typical results can be seen in Fig. 26where in the insert, Fig. 26(a), prebuckling shape is shown, whilein 26(b) the buckling mode is plotted. Further details and other results are given in Ref. [176]. Results seen in Fig. 26 show thatbifurcation buckling takes place for toroids with “bigger” ( R / r )-ratios. More compact toroids fail by collapse. The transition frombifurcation to collapse depends not only on the (r / t )-ratio but alsoon the yield point of material, r yp. The transition value, ( R / r )tr ,between bifurcation and collapse can be calculated from the fol-lowing equation:

Fig. 25 Failure pressures of optimized generalized ellipsoids. Also shown is the failure of standard ellipsoids (m 15m 252.0).Points a, b, c, and d denote experiments (two tests per point) [ 166].

Table 4 Experimental and numerical values of buckling loadsof CNC machined mild steel domes (numerical values are nor-malised by experimental values and given in parentheses)

Model pexptl (MPa) pnumerical (MPa)

a a1 13.24 12.88 (0.97)a2 13.46 12.90 (0.96)

c c1 7.98 8.18 (1.03)c2 8.14 7.98 (0.98)

b b1 10.21 9.77 (0.96)b2 10.14 9.69 (0.96)

d d1 5.07 5.47 (1.08)d2 5.07 5.39 (1.06)

Fig. 26 Bifurcation and collapse pressures for toroidal shellwith circular cross-section and r / t 518.74 [177]





R

r tr ¼

1:13

þ0:0486 r yp= E

1:71

ðr =t

Þ

2:34(15)

Equation (15) has been derived in Ref. [176] by curve-fitting of numerical data generated by wide parametric studies.

The available literature suggests that experimental results onexternally pressurized closed toroidal shells are rare (see for example Refs. [172,179]). Details about manufacturing, pre-experiment measurements, and tests on three steel toroids subjected to external pressure are reported in Refs. [177,179]. Twomodels, TS1 and TS2, were manufactured from mild steel by spin-ning two halves and then welding them around the inner and outer equatorial perimeters (see Fig. 27). The third model, TE1, hasbeen assembled by welding four 90-deg stainless 316/316L steelelbows (see Fig. 28). Prior to tests, shells were filled with oil and

the inside of shells was open to the atmosphere during the applica-tion of external pressure. The aim here was to avoid “animplosion” type collapse of shells. The external pressure wasapplied at D p ¼ 0.04 MPa. The end of the load bearing capacitywas associated with a sudden loud bang, large outflow of oilthrough the vent, and big drop in pressure. Photographs of shellsafter removal from the pressure tank are depicted in Fig. 27 andFig. 28. Experimental collapse pressures are given in Table 5 to-gether with numerical estimates based various modeling assump-

tions. It is seen that a reasonable agreement has been obtainedbetween experimental failure pressures and numerically predictedvalues. The ratio of ( pexp tl / pnum) varies between 0.82 and 1.0while results, which have so far been published in the literature,oscillate between 0.40 and 1.40.

5 Domed Ends—Internal Pressure

According to a recent report by the US Department of Energy[180], there are eight failure modes to be found in internally pres-surized vessels, and they are: (i) excessive elastic deformation,including elastic instability, (ii) excessive plastic deformation,(iii) brittle fracture, (iv) stress rupture/creep deformation, (v) plas-tic instability–incremental collapse, (vi) high strain–low cycle fa-tigue, (vii) stress corrosion, and (viii) corrosion fatigue.

The current section will review recent research effort related toelastic/elastic–plastic buckling, excessive plastic deformation (plasticloads), and plastic instability–incremental collapse (burst pressure).

5.1 Buckling. Domed ends onto internally pressurized cylin-ders appear in much wider engineering applications than exter-nally pressurized counterparts. Their safe and efficient design hasattracted a significantly larger amount of research effort since, onone hand, the thinner heads are prone to buckling, and the transi-tion region between bifurcation buckling and axisymmetric yield-ing is not well defined; however, it is of practical relevance, onthe other hand. In addition, studies to develop inelastic and limitanalyses have also been carried out. Previous work in this areahas been well documented and regularly reviewed. A goodsource of relevant information can be found in Refs.

[2,7,20,22,27,181 – 188].An approximate expression for the limit pressure correspondingto appreciable plastic deformations in internally pressurized tori-spheres can be found in Ref. [189]. Experimental verification of plastic limit analysis for ten torispherical and for three toriconicalshells was discussed in Ref. [190]. Experimental work into thefailure of 12 torispherical ends subjected to internal pressure isreported in Ref. [191]. All models were manufactured from alumi-num alloy and some of them failed through asymmetric bifurca-tion buckling while the thicker models failed by plasticdeformation. Results of a parametric study into elastic buckling,and first yielding, for torispherical shells were studied in Ref.[181]. Comparison of test and theory for asymmetric elastic–plas-tic bifurcation buckling of torispheres is available in Ref. [185].

Fig. 27 Two spun halves prior to welding into TS1. Toroids TS1 and TS2 after collapse [177].

Fig. 28 View of stainless steel toroidal shell, TE1 being low-ered to pressure tank for testing (a ), and the model after col-lapse (b ) [177]

Table 5 Externally pressurised steel toroids: comparisonof experimental and numerical pressures for different

numerical models (15nominal, 25average, 35min, 45max,55variable)

pnum (MPa)

1 2 3 4 5 pexpt(MPa)

TS1 9.44(0) 8.10(0) 5.68(0) 9.40(0) 8.40(c) 8.40TS2 7.10(0) 5.66(0) 3.68(0) 6.64(0) 5.20(c) 4.28TE1 8.68(0) 8.68(0) 7.18(0) 9.64(0) 7.48(0) 7.24





Experimental bifurcation buckling loads for spun torisphericaland ellipsoidal dished ends are compared with the predictions of numerically computed values in Ref. [192]. Buckling tests on twosteel torispherical heads are reported in Ref. [186]. These modelswere large diameter, approximately 4.8 m shells, and they wereassembled from a number of welded petals. A set of 16 bucklingtests on spun steel torispheres was reported in Ref. [193]. Diame-ter of these heads was 0.5 m and the ratio of the knuckle radius todiameter varied from 4 to 10%. The effect of initial shape devia-tions from perfect geometry on the bifurcation buckling was also

discussed. The results of 190 experimental tests on internally pres-surized torispherical heads were collated and analyzed in Refs.[187,188]. Most of the analyzed torispheres failed by asymmetricelastic or inelastic bifurcation buckling. In the paper, attentionwas also paid to the plastic collapse and burst pressures. Approxi-mate equations for elastic and elastic–plastic buckling pressures(torispherical and ellipsoidal heads) can be found in Ref. [20] to-gether with the details about sources of past studies. They containinformation on more than 200 buckling experiments on internallypressurized heads—including buckling tests on large industriallymanufactured domes, e.g., Ref. [186]. The equations would allowa quick estimate of failure pressure. It is worth noting here thefact that computed bifurcation buckling can depend on the choiceof plasticity model, i.e., deformation or J2 flow theory. Discussionof this topic with experimentation on small machined steel headsis available in Ref. [183]. Buckling tests have confirmed the exis-

tence of asymmetric bifurcation as predicted by the deformationtheory while computations based on J2 flow theory concluded thatthere should not be bifurcation buckling. The existing paradoxbetween J2 flow theory and deformation theory of plasticity mani-fests itself in a wider class of structural problems where bucklingis a possibility (see Refs [7,194,195]). The suggestion to use bothapproaches, in order to establish the sensitivity of the predictionsto both models of plasticity, appears appropriate [196]. It is worthnoting here that the unique definition of buckling load can some-times be very subjective; see for example Ref. [183]. One wayforward here was the adoption of the Southwell plot despite sub-stantial plastic straining. In this way, the subjectivity could beremoved from experimentation (see Refs. [183,197,198]). Elasticbuckling can sometimes go unnoticed and, as a result, it can leadto serious consequences. Reference [199] illustrates this for the

case of a rear pressure bulkhead in a wide body civilian aircraft.Due to pressure differences between inside of the cabin and out-side of the fuselage, the rear pressure bulkhead can undergo buck-ling when in the air (i.e., skin within a pocket of reinforcementsby stringers and rings). Upon landing affected skin pops back tothe initial shape. Investigations reported in Ref. [199] concludedthat multiple buckling of the skin within a single skin-pocket wasa cause of in-flight rupture of the rear pressure bulkhead. This ledto the cabin pressure loss and an emergency landing. An interest-ing picture of multiple pocket-type buckling of a skin in a wing of flying glider can be seen on the opening page of Ref. [48] (n.b., avery valuable and multifaceted internet resource on buckling of shells). Spars and rings provide a load bearing structure for theglider’s wing but under flight conditions the skin can bucklewithin a “reinforced pocket,” and in a number of pockets. While

there is no immediate loss of structural integrity, multiple in-outbuckles can potentially lead to a skin’s rupture/tearing.Asymmetric bifurcation is not the only form of design limita-

tion for internally pressurized heads. Under single incrementalloading, plastic load has been introduced as a measure of struc-tural integrity [200] and this concept has been applied to a number of pressure vessel geometries and piping components. The nextsection discusses this concept for internally pressurized torispheri-cal ends and for closed toroidal shells.

5.2 Plastic Loads. There is a continuing discussion aimed atidentification of the best failure criterion for torisphericalheads which do not buckle. The use of plastic collapse loads, as

recommended in Ref. [200] is a frequently adopted approach (seealso Refs. [201 – 203]). The above papers use a common approachfor evaluation of a plastic load, using a graphical relationshipbetween internal pressure and apex deflection. In Ref. [201], fivepairs of mild steel torispheres were used for experimental extrac-tion of plastic loads. The load-deflection curves were used toobtain plastic load, pC1 (twice-the-yield-point-deflection) and pC2

(twice-elastic-slope). All models had the same wall thickness andthe same mass, W T . The latter was related to the mass of a hemi-sphere, W H , having the same diameter. The ratio W T / W H ¼ 0.693was assumed for all tested torispheres and this resulted in a rangeof heads having different geometry. Figure 29 shows the compari-son of experimental and computed values of pC1 (with pC2 beingonly marginally higher and not plotted). Calculations have shownthat calculated plastic collapse pressures differ by no more than

12% from those obtained in experiments. In addition, single meas-urements of the apex deflection proved to be an effective way of establishing the twice-the-end-of-proportionality plastic loads.Incremental loading continued until burst, and most of testeddomes ruptured at the apex with a single long crack (as depictedin Fig. 29). It is seen here that a substantial reserve of strengthbeyond the collapse loads existed. The ratio of ( pburst / pC1) varieshere from 1.93 to 2.81 (or from 1.67 to 2.81 if a premature leak istaken into account). Reference [200] also points out that the evalu-ation of the plastic load should employ a more objective and justi-fied by physics criterion, i.e., for pressure tests, the relevantquantities would be pressure and change of internal volume. Thisapproach was adopted in Refs. [69,204 – 208] in order to obtainplastic loads for torispherical, ellipsoidal and toroidal models(mild steel, stainless steel and AA6061-T1 aluminum alloy). Ref-

erence [204] studied plastic loads based on (i) load versus deflec-tion and on (ii) a work criterion associated with volumetricchanges in internally pressurized steel torispheres. In both cases,end of linearity was adopted as a reference point for calculation of the plastic loads. Plastic loads based on the apex deflection werefound to be comparable to those calculated by using volumetricchanges. This was not true for the knuckle deformations. The cor-responding plastic loads were consistently smaller than the apexor volumetric ones. In addition, it was shown that apex deflectionscannot be used as indicators of the plastic loads for some shapes,e.g., for prolate ellipsoidal heads. As part of the experimentation,three machined torispherical models were tested under monotoni-cally increasing pressure up to burst. The ratio of experimental tonumerical values of plastic loads was between 0.92 and 0.99. As

Fig. 29 Comparison of computed plastic load p C 1 with experi-mental values. Also, plot of experimental burst pressures. Viewof burst models K1, K2, and K6, adapted from Ref. [ 201].





in the previous case, a substantial reserve of strength existedbeyond plastic loads. The ratio of the burst pressure to experimen-tal plastic load varied from 1.9 to 2.3. Generally, a reasonablecomparison of the experimental and numerical results wasobtained. But if these domes were to operate in the post-yieldrange where cyclic load could be present, the role of the plasticload was less clear than the shakedown load. This aspect of pres-sure loading was explored in Ref. [206] for mild steel heads wherethe concept of a first-cycle shakedown pressure was adopted tocompare the relative values of plastic and shakedown loads.Experimentally studied geometries included torispheres (four

heads) and ellipsoidal shells (two prolate and two oblate models).The sequence of loading consisted of the following steps: (i)incremental loading to volumetric plastic load then unload, (ii) cy-cling loading until shakedown achieved then unload, and (iii)incremental loading until burst. Torispherical and elliptical endclosures made from mild steel have not lost their serviceabilitywhen pressurized to the level of plastic loads. Experimental meas-urements of strains at the most stressed point has always led toshake down to purely elastic state at the plastic load, pV 1. Experi-mental burst pressures were from 1.8 to 3.4 times the plastic load, pV 1. Figure 30 illustrates the accumulation of plastic strains for torisphere T8 ( Rs / D ¼ 1.0, r / D ¼ 0.10, D / t ¼ 25) for which experi-mental loads were as follows: plastic load, pV 1¼ 18.0 MPa, shake-down load, pshk ¼ 21.92, and burst pressure, pburst¼ 60.69 MPa.This particular head lost its integrity by a longitudinal crack pass-

ing through the apex (similar to seen in Fig. 29). View of oblateellipsoidal head, E1, after burst is also depicted in Fig. 30. Thisstudy was followed by a search for optimal domes with piece wisedistribution of wall thickness subject to shakedown constraints[209]. In a separate study, the problem of finding the minimummass of domed closures on internally pressurised cylinder wassought [210]. This particular motivation came from light weighton-board tanks to be used in natural gas vehicles. The plastic loadbased on global/volumetric criterion was used and selective exper-imental benchmarking followed using eight mild steel heads. Plas-tic and shakedown loads for domes made from strain-hardeningmaterial were studied in Ref. [207]. Results of numerical calcula-tions were benchmarked experimentally using four torisphericaland one oblate ellipsoidal heads machined from AA6061-T1

aluminum alloy. Shakedown was based here on kinematic strain-hardening, and ellipsoidal model, EAL1, after the burst is depictedin Fig. 30. It appears appropriate here to mention the availabilityof 1904 seminal paper by M. T. Huber, which has been translatedinto English in 2004 as Ref. [211]. This work has formed the basisfor the Huber–Mises yield criterion.

While the role of plastic loads for internally pressurized headsis still being researched, it is the burst pressure which is of greater value from a practical point of view as it gives an indication of themargin of safety for a single incremental loading. This is an im-portant quantity, especially, at a design stage or at an emergency

situation.

5.3 Burst Pressure. A number of studies which are relevantto the issue of burst pressure are available in the literature, and abrief summary can be found in Ref. [212]. A procedure for numer-ical calculation of pressure at tensile plastic instability for inter-nally pressurized axisymmetric pressure vessels has beendeveloped in Refs. [213 – 215]. The proposed pressure is to be anupper bound to the burst pressure that could be achieved in realvessels. While the failure mode caused by plastic instability hasbeen studied analytically and experimentally, it is excessive plas-tic deformation which is a more probable mode of failure thanbursting due to plastic instability (see Ref. [216]). A recent theo-retical and experimental study into burst of internally pressurized

domes can be found in Ref. [212]. The burst pressure is based onexcessive plastic deformation rather than on plastic instability. Itis postulated to use the true plastic strain, eu

p, corresponding to theultimate tensile strength, UTS, for computing the magnitude of burst pressure. One needs not only the magnitude of plastic strainbut also a place where this strain is to be attained. Based on theabove criterion for admissible magnitude of plastic strain, it is fur-ther postulated to define the burst pressure as follows: pburst is thepressure at which the equivalent plastic strain, PEQQ, reaches theultimate plastic strain, eu

p, anywhere at the mid-surface of a struc-ture. A series of calculations have been performed for mild steelshallow spherical caps, tested previously for buckling, Ref. [123],and for torispherical heads tested previously for plastic loads, Ref.[201]. The latter were from mild steel. In addition, four

Fig. 30 History of plastic strains growth versus number of pressure cycles in steeltorisphere T8 (a ). Also views of burst oblate ellipsoidal models: (b ) mild steel, and

(c ) aluminum (adapted from Ref. [206,207]).





torispheres were freshly machined from aluminum alloy AA6061-T1 to complement the series of tests. Hence, burst tests were car-ried out on: six steel and four aluminum torispheres and six spher-ical caps. Computed burst pressures for steel spherical caps werebetween 21.1 and 4.9% below experimental values. At thesame time, the errors between computed plastic instability pres-sures and the experiments were between þ35.1 and 51.8% (over-estimated experimental results). For steel torispheres therespective ranges of errors were: [þ12.1%, þ32.4%] and[þ17.6%, 42.8%]. In aluminum torispheres the ranges were:

[0.4%, 14.0] and [12.7%, þ20.8%]. It is seen here that themagnitudes of computed plastic instability pressures were aboveexperimental values for all tested models. Derivation of plasticinstability load for internally pressurized stainless steel toroid,followed by burst tests of two models, can be found inRefs. [217,218]. Seventeen 45 liter carbon steel toroidal tankswere burst-tested during qualifying procedure for the anticipatedLPG usage [219,220]. Water was the pressurization media. Tanksburst when volume increased by about 11%, and the standarddeviation on average burst pressure of 8.56MPa was about0.220 MPa. The FE assessment of structural integrity was basedon monitoring the maximum deflections at the apex and at outer equatorial plane. Other relevant work in this area can be found inRefs. [221,222].

6 Closure/Conclusions

The paper shows that experiments related to static buckling of pressure vessel components have remained relevant to a widerange of industries over the last decade or so. Skepticism aboutthe future role of experimentation into buckling, and raised two or three decades ago, appears to move towards a two way fruitfulcoexistence with computational mechanics. Nearly all tests quotedin this paper had one sort or another theoretical/numerical part— mostly based on the FE approach. However, not all theoreticalpredictions manifested themselves in close experimental out-comes, and part of the reason can be attributed to experimentation.The known issues like material properties, boundary conditions,loading mechanisms, and test repeatability have gained promi-nence in labs. Pretest measurements and online monitoring of tests

have seen a step change. The empirical knock-down factors arebeing gradually reassessed. Buckling performance of epoxy-basedstructural components, and corrosion affected members, is alsoactively researched.

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exp. perspective of press vessels

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