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Review Articles

Strength, Stability, and Optimization of Pressure Vessels: Review of Selected Problems

[+] Author and Article Information
J. Błachut

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

K. Magnucki

Institute of Applied Mechanics, Poznań University of Technology, ul. Piotrowo 3, 60-965 Poznań, Poland

Appl. Mech. Rev 61(6), 060801 (Oct 07, 2008) (33 pages) doi:10.1115/1.2978080 History: Received February 04, 2008; Revised February 14, 2008; Published October 07, 2008

Recent research effort into some aspects of strength, static stability, and structural optimization of horizontal pressure vessels is reviewed in this paper. Stress concentrations at the junction of cylinder-ellipsoidal end closures are covered in detail. This in turn establishes efficient choices for wall thicknesses in the vessel. Detailed account of stresses for flexible supports of a horizontal cylindrical shell is provided. Dimensions of support components, which assure the minimum stress concentrations between a horizontal shell and its support, are calculated. In particular, the wall thickness is found for vessels being loaded by the weight of its content and placed on two supports. Stability issues are also reviewed in this paper. In particular, attention is paid to the stability of cylinder under external pressure and to the stability of end closures. The latter are loaded by internal or external pressure. Apart from buckling and plastic loads, the ultimate load carrying capacity, i.e., burst pressure, for internally pressurized heads is also examined. On a practical side, aboveground and underground cases are discussed. In the latter case of underground vessels the reinforcement by internal rings is assessed. The optimization part of this paper deals with the effective choice of the end closure depth and the shape of its meridian. The overriding aim here is to examine the stress concentrations and the ways in which they can be mitigated. The optimal shape of closures is also searched for, with respect to the maximum buckling pressure for a given mass of the head. In the case of internal pressure the maximum of plastic load is sought within a specified class of meridional profiles. Finally, optimal sizing of whole vessels is discussed for slender and compact geometries. Extensive references are made to relatively recent and ongoing work related to the above topics. This paper has 287 references and 50 figures.

Copyright © 2008 by American Society of Mechanical Engineers
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References

Figures

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Figure 1

Classification of horizontal cylindrical tanks

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Figure 2

Illustration of an aboveground tank

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Figure 3

Rail cistern of transport

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Figure 4

Tank for bimodal form of transport

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Figure 5

(a) Geometry of torispherical shell. (b) Change of meridional, R1, and hoop, R2, radii in the spherical cap and in the knuckle parts, respectively.

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Figure 6

(a) Geometry of a:b ellipsoidal dome. (b) Change of meridional, R1, and hoop, R2, radii of curvature in a:b ellipsoidal head.

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Figure 7

Forces at the junction between ellipsoidal head and cylindrical shell due to internal pressure

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Figure 8

Plots of dimensionless equivalent Huber–Mises stresses in (a) ellipsoidal and in (b) cylindrical shells for the cases of x1=1, x2=100, and β=0.5

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Figure 9

The effect of equalization of maximal Huber–Mises equivalent stresses in (a) ellipsoid and in (b) cylinder for the cases of x1,ef=0.645, x2=100, and β=0.5

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Figure 10

Effective wall thicknesses ratio x1,ef versus relative depth, β, of ellipsoidal vessel domed end

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Figure 11

Stress concentration factor, αSC, versus the relative depth, β, for the effective choice of parameter x1,ef and for 75⩽x2⩽200

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Figure 12

Sketch of three typical supports for horizontal vessels (adapted from PD 5500 (15))

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Figure 13

Details about arrangements in a horizontal pressure vessel with saddle support

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Figure 14

Geometry of the saddle support and the corresponding FEM model

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Figure 15

Influence of the width of the saddle support bed, b0, on the stress level (V0=300m3)

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Figure 16

Influence of the height of the saddle support bed, e, on the stress level (V0=300m3)

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Figure 17

Influence of the vessel slenderness ratio, L∕a, on the stress level (V0=300m3)

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Figure 18

Horizontal cylindrical shell subjected to hydrostatic pressure due to liquid self-weight and simultaneous action of internal pressure

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Figure 19

Differences in wall thickness between the current design code values given by Eq. 16 and values given by Eq. 14 for a storage tank with ellipsoidal heads

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Figure 20

(a) Deformed shape of externally pressurized torispherical end closure just prior to bifurcation buckling (not to scale). (b). Eigenshape of externally pressurized torispherical end closure with n=17 circumferential waves. (c) View of CNC-machined, mild steel torisphere after collapse by uniform external pressure (snap-through). (d) View of copper/steel/copper layered hemispherical dome after collapse by external pressure. Inward dimple attributed to the initial shape and wall thickness imperfections.

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Figure 21

(a) Photograph of internally pressurized torispherical head after buckling. (b) Side view of the aluminium torispherical head after it was pressurized to the plastic load level. (c) Photograph of a torispherical end closure manufactured from aluminum alloy after burst.

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Figure 22

Test data for ten machined (r∕D≈0.06, Rs∕D≈0.75, and L∕D≈0.05) and two spun (r∕D≈0.06, Rs∕D≈1.0, and L∕D≈0.075) torispheres plotted in the PD 5500 format (pe=1.21Et2∕Rs2 and pyss=2σypt∕Rs), where σyp is the yield point). Also, geometry of a torispherical end closure is defined in the insert.

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Figure 23

View of failed domes: (a) machined model, (b) spun model SD1, and (c) numerically predicted shape of spun dome SD1 at collapse

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Figure 24

(a) Sensitivity of the buckling pressure to the amplitude of increased radius and affine to eigenshape imperfections. Also, view of the torisphere at (b) bifurcation buckling and (c) at collapse.

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Figure 25

(a) Draped woven fabric over a quarter surface of torispherical end closure. (b) View of the fiber distortion in the outer ply of a 36-ply torisphere (left) and the dome before being lowered to the test tank (right).

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Figure 26

(a) Illustration showing how the ultimate plastic strain, εpu, was established for aluminum alloy 6061. (b) Comparison of load versus apex deflection curves for internally pressurized aluminum torisphere (r∕D=0.08, Rs∕D=1.0, and D∕t=26). Location of experimental and computed burst pressures is also shown.

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Figure 27

Buckled shape of the ellipsoidal head under external pressure—linear solution

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Figure 28

Critical pressure, pcr(ext), for ellipsoidal heads with relative depth β subjected to external pressure

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Figure 29

Deformed shape of ellipsoidal head, β=b∕a=0.25, under external pressure—nonlinear solution. Note: Deformation process and the loss of stability were elastic.

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Figure 30

Buckled shape of ellipsoidal head, β=0.25, under internal pressure—linear solution

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Figure 31

Critical pressure, pcr(inl), versus relative depth, β, of internally pressurized ellipsoidal heads

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Figure 32

Deformed shape of the ellipsoidal head, β=0.25, under internal pressure—nonlinear solution

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Figure 33

Geometry of the aboveground and underground horizontal cylindrical vessels

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Figure 34

Relative critical thickness of aboveground vessels with additional external pressure p01=0.01MPa

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Figure 35

Relative critical thickness of aboveground vessels with additional external pressure p01=0.1MPa

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Figure 36

Geometry and loading of underground vessel

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Figure 37

Section of T-ring reinforced cylindrical shell in underground storage vessel. Note that (30×t2)-length of the cylinder is used when calculating the value of the ring’s second moment of area, Jz.

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Figure 38

Critical thicknesses, t2,cr, of ring-stiffened underground horizontal cylindrical vessels (nr≡number of internal reinforcing rings)

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Figure 39

Relative critical thickness of the wall of underground cylindrical tank (h0=1m)

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Figure 40

Plot of dimensionless mass of ellipsoidal head versus its dimensionless depth, β

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Figure 41

Illustration of four types of a joint between elliptical head and cylindrical shell. The magnitudes of the peak stress in the ellipsoidal and cylindrical shells are the same for all four cases.

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Figure 42

Meridional shape of a dished dome

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Figure 43

Variation of principal radii in a dished head

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Figure 44

Variation of equivalent stresses in a dished head

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Figure 45

Contours of optimal solution for elastic analyses. Each contour represents torispherical heads of the same weight.

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Figure 46

(a) Contours of optimal elastic-plastic solutions for D∕t=300. Test points 6 (reference hemisphere), 7 (optimum), and 8 (pesimum) are shown. (b) Photograph of collapsed heads in (a).

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Figure 47

A set of slender and thick aboveground horizontal cylindrical tanks

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Figure 48

Sketch of a typical domain of allowable solutions for a fixed volume of a vessel. Note how stability and strength constraints restrict the design space to the shaded area.

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Figure 49

Optimal proportions of length, L, to diameter, d, of horizontal cylindrical tanks

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Figure 50

Optimal horizontal cylindrical vessel with the capacity of V0=200m3

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