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Research Papers: Design and Analysis

Buckling of Externally Pressurized Prolate Ellipsoidal Domes

[+] Author and Article Information
P. Smith, J. Błachut

Mechanical Engineering, The University of Liverpool, Liverpool L69 3GH, UK

J. Pressure Vessel Technol 130(1), 011210 (Jan 24, 2008) (9 pages) doi:10.1115/1.2834457 History: Received August 03, 2006; Revised November 04, 2006; Published January 24, 2008

Details are given of a numerical and experimental study into buckling of steel ellipsoidal domes loaded by static external pressure. A range of geometries and thicknesses of domes is examined, as is the influence of different boundary conditions. Shells are examined on the basis of having the same mass. The main focus of the study is on prolate domes, i.e., those taller than a hemisphere of the same base radius. Numerical predictions are confirmed by pressurizing six laboratory scale prolate domes to destruction. Details are given of the manufacture and test procedure for the domes. The adverse effects of variations in shape and wall thickness are discussed, and finite element predictions are made for geometrically imperfect domes. Correlation between the two sets of results is good. Numerically and experimentally obtained results are related to the current design codes: ASME Boiler and Pressure Vessel Code, Sec. 8, Division 2 (described hereon as ASME VIII), PD5500, and ECCS recommendations (ASME B&PV Code, 2004 ed., Sec. 8, Division 2, New York, NY; BSI 2003 “Published Document PD5500: Specification for Unfired Pressure Vessels  ,” BSI London; European Convention for Constructional Steelwork Recommendations, 1988 “Buckling of Stell Shells-European Recommendations  ,” ECCS-TWG 8.4, 4th ed., Brussels), which at present make no provision for prolate domes. Suggestions are made for the possible inclusion of such domes into the standards.

Copyright © 2008 by American Society of Mechanical Engineers
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Figures

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Figure 1

Geometry and notation of (a) prolate ellipsoidal shell and (b) hemisphere

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Figure 2

Elastic buckling pressures of complete ellipsoidal shells

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Figure 3

Buckling pressures of externally pressurized ellipsoidal domes. Results are normalized by the hemisphere, which was seen to fail by axisymmetric collapse. (R∕tH=50,pH=11.01MPa).

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Figure 4

Buckling mode of (a) oblate and (b) prolate ellipsoidal domes. The oblate buckling mode is axisymmetric (n=0), and the number of circumferential waves for prolate ellipsoid is n=8.

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Figure 5

Elastic-plastic buckling loads for four different thickness ratios of reference hemisphere. All results are normalized by the reference hemisphere R∕tH=50, for which pH=11.01MPa.

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Figure 6

First yield of middle surface of clamped prolate ellipsoidal domes. Note that D=2A.

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Figure 7

Influence of edge boundary conditions on buckling strength of ellipsoidal domes. Results are normalized by the reference hemisphere R∕tH=50, for which pH=11.01MPa.

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Figure 8

Imperfection sensitivity of an A∕B=0.5 ellipsoidal dome to eigenshape imperfections. The dome is more sensitive to mixed mode imperfections. pbifperf=4.17MPa(n=13).

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Figure 9

Imperfection sensitivity of a fully clamped hemisphere to eigenshape imperfections. The dome is more sensitive to mixed mode imperfections, but less sensitive than an equivalent elliptical dome. pcollperf=11.01MPa.

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Figure 10

Domes E1 and E1a (A∕B=0.5) in as machined state

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Figure 11

Domes E3 and E3a (A∕B=0.65) in as machined state

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Figure 12

Domes E2 and E2a (A∕B=0.8) in as machined state

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Figure 13

Thickness contours for Domes E1, E3, and E2 as viewed from above. Darker areas are thinner; note the three areas of thinner material at the base of E1 and E3. (a) E1, A∕B=0.5 (b) E3, A∕B=0.65, and (c) E2, A∕B=0.8.

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Figure 14

Volume change of A∕B=0.8 ellipsoid during pressurization.

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Figure 15

Nominally identical pairs of Domes E1, E1a (A∕B=0.5), E3, E3a (A∕B=0.65), and E2, E2a (A∕B=0.8) after testing

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Figure 16

View of collapsed domes from above with corresponding plots of thickness. Note how failure occurs in an area of decreased shell wall thickness.

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Figure 17

Pressure versus apex deflection for E1 (from ABAQUS S9R5).

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Figure 18

Buckling mode of Dome E1 (A∕B=0.5) with variable thickness. Instability is local and at the base of the shell ((a), side view; (b), top view).

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Figure 19

Factor K0 proposed for determining equivalent radius of prolate ellipsoidal domes. Also shown are the current factors used for oblate domes

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Figure 20

Proposed radius used for calculation of pressure resistance of prolate ellipsoidal domes in PD5500

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Figure 21

Current allowed ellipsoids (right of hemisphere) and suggested inclusion of prolate domes (left of hemisphere). All domes have the same mass. Points a, b, and c are experimental points (two tests per point).

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