The paper is a review work devoted to dished heads of various meridian shapes. Geometry of the shells of revolution, the membrane state, and the edge effect occurring in the shells are described. Exemplary analytical and numerical finite element method (FEM) studies of torispherical, ellipsoidal, Cassini-ovaloidal, and untypical special dished heads are presented. The results of the above-mentioned two methods are compared. Moreover, numerical research of elastic buckling of the above-mentioned selected heads under external pressure is carried out. Literature related to each of the considered head types is quoted and discussed, with special attention paid to the works developed in the 21st century. In concluding remarks, the stress concentration and buckling of these structures are commented, with consideration of the head meridian shapes.

References

1.
Ross
,
C. T. F.
,
1990
,
Pressure Vessels Under External Pressure: Statics and Dynamics
,
Elsevier Applied Science
,
London
.
2.
Spence
,
J.
, and
Tooth
,
A. S.
, eds.,
1994
,
Pressure Vessel Design, Concepts and Principles
,
E & FN Spon
,
London
.
3.
Magnucki
,
K.
,
1998
,
Strength and Optimization of Thin-Walled Vessels
,
PWN-Polish Scientific Publishers
,
Warsaw, Poland
(in Polish).
4.
Teng
,
J. G.
, and
Rotter
,
J. M.
, eds.,
2004
,
Buckling of Thin Metal Shell
,
Spon Press, Taylor & Francis Group
,
London
.
5.
Lewiński
,
J.
,
2012
, “
Optimal Shaping of Convex Heads of Pressure Vessels
,” Poznan University of Technology, Poznan, Poland (in Polish).
6.
Vullo
,
V.
,
2014
,
Circular Cylinders and Pressure Vessels: STRESS Analysis and Design
,
Springer International Publishing
, Cham,
Switzerland
.
7.
Zingoni
,
A.
,
2017
,
Shell Structures in Civil and Mechanical Engineering
, 2nd ed.,
ICE Publishing
, London.
8.
Krivoshapko
,
S.
,
2007
, “
Research on General and Axisymmetric Ellipsoidal Shells Used as Domes, Pressure Vessels, and Tanks
,”
ASME Appl. Mech. Rev.
,
60
(
6
), pp.
336
355
.
9.
Blachut
,
J.
, and
Magnucki
,
K.
,
2008
, “
Strength, Stability, and Optimization of Pressure Vessels: Review of Selected Problems
,”
ASME Appl. Mech. Rev.
,
61
(11), p.
0608011
.
10.
Pietraszkiewicz
,
W.
, and
Konopińska
,
V.
,
2015
, “
Junctions in Shell Structures: A Review
,”
Thin-Walled Struct.
,
95
, pp.
310
334
.
11.
Zingoni
,
A.
,
2002
, “
Discontinuity Effects at Cone-Cone Axisymmetric Shell Junctions
,”
Thin-Walled Struct.
,
40
(
10
), pp.
877
891
.
12.
Mazurkiewicz
,
Z. E.
, and
Nagórski
,
R. T.
,
1987
,
Shells of Revolution
,
PWN-Polish Scientific Publishers
,
Warsaw, Poland
(in Polish).
13.
Ventsel
,
E.
, and
Krauthammer
,
T.
,
2001
,
Thin Plates and Shells: Theory, Analysis, and Applications
,
Marcel Dekker
,
New York
.
14.
Camilleri
,
D.
,
Mackenzie
,
D.
, and
Hamilton
,
R.
,
2008
, “
Evaluating Plastic Loads in Thorispherical Heads Using a New Criterion of Collapse
,”
ASME J. Pressure Vessel Technol.
,
130
(
1
), p.
011202
.
15.
Magnucki
,
K.
,
Szyc
,
W.
, and
Lewiński
,
J.
,
2002
, “
Minimization of Stress Concentration Factor in Cylindrical Pressure Vessels With Ellipsoidal Heads
,”
Int. J. Pressure Vessels Piping
,
79
(
12
), pp.
841
846
.
16.
Zingoni
,
A.
,
Enoma
,
N.
, and
Govender
,
N.
,
2015
, “
Equatorial Bending of an Elliptic Toroidal Shell
,”
Thin-Walled Struct.
,
96
, pp.
286
294
.
17.
Jasion
,
P.
, and
Magnucki
,
K.
,
2015
, “
Elastic Buckling of Cassini Ovaloidal Shells Under External Pressure—Theoretical Study
,”
Arch. Mech.
,
67
(
2
), pp.
179
192
.http://am.ippt.pan.pl/index.php/am/article/view/v67p179
18.
Magnucki
,
K.
, and
Lewiński
,
J.
,
2000
, “
Fully Stressed Head of a Pressure Vessel
,”
Thin-Walled Struct.
,
38
(
2
), pp.
167
178
.
19.
Wittenbeck
,
L.
, and
Magnucki
,
K.
,
2008
, “
Strength Shaping of Dished Heads of Pressure Cylindrical Vessels
,”
Ninth International Conference on Computational Structures Technology
, Athens, Greece, Sept. 2–5, Paper No. 138.
20.
Lewiński
,
J.
, and
Magnucki
,
K.
,
2010
, “
Shaping of a Middle Surface of a Dished Head of a Circular Cylindrical Pressure Vessel
,”
J. Theor. Appl. Mech.
,
48
(
2
), pp.
297
307
.http://www.ptmts.org.pl/jtam/index.php/jtam/article/view/v48n2p297
21.
Magnucki
,
K.
,
Rodak
,
M.
, and
Jasion
,
P.
,
2017
, “
Strength and Buckling of an Untypical Dished Head of a Cylindrical Pressure Vessels
,”
11th International Conference on Shell Structures: Theory and Applications
, Gdansk, Poland, Oct. 11–13, pp. 379–382.
22.
Ortega
,
N.
, and
Robles
,
S.
,
2005
, “
Form Finding of Revolution Shells by Analyzing Mechanical Behavior by Means of the Finite Element Method
,”
J. Strain Anal. Eng. Des.
,
40
(
8
), pp.
775
784
.
23.
Banichuk
,
N.
,
2007
, “
Optimization of Axisymmetric Membrane Shells
,”
J. Appl. Math. Mech.
,
71
(
4
), pp.
527
535
.
24.
Kisioglu
,
Y.
,
Brevick
,
J. R.
, and
Kinzel
,
G. L.
,
2008
, “
Minimum Material Design for Propane Cylinder End Closures
,”
ASME J. Pressure Vessel Technol.
,
130
(
1
), p.
015001
.
25.
Zingoni
,
A.
,
2009
, “
Simplification of the Derivation of Influence Coefficients for Symmetric Frusta of Shells of Revolution
,”
Thin-Walled Struct.
,
47
(
8–9
), pp.
912
918
.
26.
Zingoni
,
A.
,
Mokhothu
,
B.
, and
Enoma
,
N.
,
2015
, “
A Theoretical Formulation for the Stress Analysis of Multi-Segmented Spherical Shells for High-Volume Liquid Containment
,”
Eng. Struct.
,
87
, pp.
21
31
.
27.
Zingoni
,
A.
,
2015
, “
Liquid-Containment Shells of Revolution: A Review of Recent Studies on Strength, Stability and Dynamics
,”
Thin-Walled Struct.
,
87
, pp.
102
114
.
28.
Wilczynski
,
B.
,
2005
,
Strength Optimization of Machine and Structural Components
,
Koszalin Publishing House
,
Koszalin, Poland
.
29.
Trendafilova
,
I.
, and
Ivanova
,
J.
,
1995
, “
Loss of Stability of Thin, Elastic, Strongly Convex Shells of Revolution With Initial Imperfections, Subjected to Uniform Pressure. A Probabilistic Approach
,”
Thin-Walled Struct.
,
23
(
1–4
), pp.
201
214
.
30.
Singer
,
J.
,
Arbocz
,
J.
, and
Weller
,
T.
,
2002
,
Buckling Experiments, Experimental Methods in Buckling of Thin-Walled Structures
, Vol.
2
,
Wiley
,
New York
.
31.
Teng
,
J. G.
, and
Hong
,
T.
,
2006
, “
Postbuckling Analysis of Elastic Shells of Revolution Considering Mode Switching and Interaction
,”
Int. J. Solids Struct.
,
43
(
3–4
), pp.
551
568
.
32.
Ma
,
Y. Q.
,
Wang
,
C. M.
, and
Ang
,
K. K.
,
2008
, “
Buckling of Super Ellipsoidal Shells Under Uniform Pressure
,”
Thin-Walled Struct.
,
46
(
6
), pp.
584
591
.
33.
Amiri
,
S. N.
, and
Rasheed
,
H. A.
,
2012
, “
Plastic Buckling of Moderately Thick Hemispherical Shells Subjected to Concentrated Load on Top
,”
Int. J. Eng. Sci.
,
50
(
1
), pp.
151
165
.
34.
Smith
,
P.
, and
Blachut
,
J.
,
2008
, “
Buckling of Externally Pressurised Prolate Ellipsoidal Domes
,”
ASME J. Pressure Vessel Technol.
,
130
(
1
), p.
011210
.
35.
Blachut
,
J.
,
2009
, “
Buckling of Multilayered Metal Domes
,”
Thin-Walled Struct.
,
47
(12), pp.
1429
1438
.
36.
Blachut
,
J.
,
2011
, “
Buckling of Externally Pressurized Shallow Spherical Caps From Composites
,”
Mech. Adv. Mater. Struct.
,
18
, pp.
96
105
.
37.
Blachut
,
J.
,
2013
, “
Experimental Perspective on the Buckling of Pressure Vessel Components
,”
ASME Appl. Mech. Rev.
,
66
(
1
), p.
010803
.
38.
Blachut
,
J.
,
2015
, “
Locally Flattened or Dented Domes Under External Pressure
,”
Thin-Walled Struct.
,
97
, pp.
44
52
.
39.
Blachut
,
J.
,
2016
, “
Buckling of Externally Pressurized Steel Toriconical Shells
,”
Int. J. Pressure Vessels Piping
,
144
, pp.
25
34
.
40.
Blachut
,
J.
,
2016
, “
Buckling of Composite Domes With Localised Imperfections and Subjected to External Pressure
,”
Compos. Struct.
,
153
, pp.
746
754
.
41.
Blachut
,
J.
,
2017
, “
Composite Spheroidal Shells Under External Pressure
,”
Int. J. Comp. Methods Eng. Sci. Mech.
,
18
(
1
), pp.
2
12
.
42.
Blachut
,
J.
,
2017
, “
Bifurcation, Plastic and Plastic Instability Loads for Cones Subjected to Combined Pressure and Axial Force
,”
Int. J. Pressure Vessels Piping
,
158
, pp.
20
28
.
43.
Jasion
,
P.
, and
Magnucki
,
K.
,
2015
, “
Elastic Buckling of Clothoidal-Spherical Shells Under External Pressure—Theoretical Study
,”
Thin-Walled Struct.
,
86
, pp.
18
23
.
44.
Hutchinson
,
J. W.
,
2016
, “
Buckling of Spherical Shells Revisited
,”
Proc. R. Soc. A
,
472(
2195).
45.
Zhang
,
J.
,
Zhang
,
M.
,
Tang
,
W.
,
Wang
,
W.
, and
Wang
,
M.
,
2017
, “
Buckling of Spherical Shell Subjected to External Pressure: A Comparison of Experimental and Theoretical Data
,”
Thin-Walled Struct.
,
111
, pp.
58
64
.
46.
Zhang
,
J.
,
Wang
,
M.
,
Wang
,
W.
, and
Tang
,
W.
,
2017
, “
Buckling of Egg-Shaped Shell Subjected to External Pressure
,”
Thin-Walled Struct.
,
113
, pp.
122
128
.
47.
Kruzelecki
,
J.
, and
Proszowski
,
R.
,
2012
, “
Shape Optimization of Thin-Walled Pressure Vessel End Closures
,”
Struct. Multidiscip. Optim.
,
46
(5), pp.
739
754
.
48.
Hsieh
,
M. F.
,
Moffat
,
D. G.
, and
Mistry
,
J.
,
2000
, “
Nozzles in the Knuckle Region of a Torispherical Head: Limit Load Interaction Under Combined Pressure and Piping Loads
,”
Int. J. Pressure Vessels Piping
,
77
(
13
), pp.
807
815
.
49.
Skopinsky
,
V. N.
, and
Smetankin
,
A. B.
,
2003
, “
Parametric Study of Reinforcement of Pressure Vessel Head With Offset Nozzle
,”
Int. J. Pressure Vessels Piping
,
80
(
5
), pp.
333
343
.
50.
Schindler
,
S.
, and
Zeman
,
J. L.
,
2003
, “
Stress Concentration Factors of Nozzle–Sphere Connections
,”
Int. J. Pressure Vessels Piping
,
80
(
2
), pp.
87
95
.
51.
Bae
,
H.-Y.
,
Kim
,
Y.-J.
,
Kim
,
J.-H.
,
Lee
,
S.-H.
,
Lee
,
K.-S.
, and
Park
,
C.-Y.
,
2014
, “
Three-Dimensional Finite Element Welding Residual Stress Analysis of Penetration Nozzles—Part I: Sensitivity of Analysis Variables
,”
Int. J. Pressure Vessels Piping
,
114–115
, pp.
1
15
.
52.
Simon
,
J.-W.
,
Chen
,
G.
, and
Weichert
,
D.
,
2014
, “
Shakedown Analysis of Nozzles in the Knuckle Region of Torispherical Heads Under Multiple Thermo-Mechanical Loadings
,”
Int. J. Pressure Vessels Piping
,
116
, pp.
47
55
.
53.
Lu
,
M.-H.
,
Yu
,
J.-S.
, and
Chen
,
J.-J.
,
2014
, “
The Effect of Analysis Model on the Stress Intensity Calculation for the Nozzle Attached to Pressure Vessel Under Internal Pressure Loading
,”
Int. J. Pressure Vessels Piping
,
117–118
, pp.
9
16
.
54.
Lewiński
,
J.
,
2014
, “
Equivalent Stress in a Pressure Vessel Head With a Nozzle
,”
J. Theor. Appl. Mech.
,
52
(
4
), pp.
1007
1018
.
55.
Lewiński
,
J.
,
2015
, “
The Effect of Manhole Shape and Wall Thickness on Stress State in a Cylindrical Pressure Vessel
,”
J. Theor. Appl. Mech.
,
53
(
1
), pp.
59
68
56.
Jasion
,
P.
, and
Magnucki
,
K.
,
2015
, “
Stability of an Ellipsoidal Head With a Central Nozzle Under Axial Load
,”
Arch. Civ. Eng.
,
LXI
(
2
), pp.
89
98
.
57.
Chapuliot
,
S.
,
2016
, “
Stress Intensity Factor Calculation in Sharp and Beveled Edge Nozzle Corners
,”
Int. J. Pressure Vessels Piping
,
141
, pp.
11
18
.
58.
Blazejewski
,
W.
,
2013
,
Composite Pressure Vessels Reinforced With Fibers According to Mosaic Patterns
,
Wroclaw Publishing House
,
Wroclaw, Poland
(in Polish).
59.
Cichy
,
R.
,
Krolikowski
,
J.
, and
Medwid
,
M.
,
2012
, “
Requirements for Transport of Liquid Fuels Using Bimodal Technology in Poland, With Consideration of ADR and RID Regulations
,”
Rail Veh.
,
4
, pp.
29
34
(in Polish).
60.
Medwid
,
M.
, and
Cichy
,
R.
,
2017
,
Technical Means of the Rail-Road Combined Transport
,
IPS
,
Poznan, Poland
(in Polish).
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