[Kryvoshapko_S.N.]_Research_on_General_and_Axisymm(Bookos.org)[1].pdf

1

wdpmletneoiu

ps

pmnbbsopss2

sC

w

3

Downlo

S. N. KrivoshapkoPeoples’ Friendship University of Russia,

6 Miklukho-Maklaya Street,Moscow 117198, Russia

Research on General andAxisymmetric Ellipsoidal ShellsUsed as Domes, PressureVessels, and TanksThe principal advances in the design and construction, as well as the static, vibrational,and buckling analysis of thin-walled structures and buildings in the shape of general andaxisymmetric ellipsoidal shells are summarized in this review. These shells are particu-larly useful as internally pressurized vessels or as heads and bottoms of cylindrical tanksand vessels. Reinforced concrete and structural steel domes of buildings, air-supportedrubber-fabric shells, and underwater pressure vessels are also made in the form of ellip-soidal, shells. Knowing the geometry of ellipsoids, one can solve various problems inphysics, optics, and so on. Basic results of theoretical and experimental investigations ofthe stress-strain state, buckling, and natural and forced vibrations contained in 209references are presented in the review. The influence of temperature on the stress-strainstate of the shells in question is also discussed. Some parts of the review are also devotedto an analysis of the literature on the stress-strain state of ellipsoidal and torisphericalheads of pressure vessels with openings. �DOI: 10.1115/1.2806278�

IntroductionTriaxial ellipsoids, and especially ellipsoids of revolution, are

ell studied and widely known closed surfaces of the second or-er. These surfaces or their segments used properly can have ex-ressive architectural forms. Spherical shells are used widely inodern designs because of their simple form, which, in turn, al-

ows accurate methods of analysis. Spherical surfaces are degen-rate ellipsoids of revolution and they will not be considered inhis review. In various parts of the world, covering shells of newonspherical form have appeared including general ellipsoids andllipsoids of revolution. Buildings and constructions in the formf ellipsoids of revolution have some advantages in distributingnternal stress resultants. Thin-walled spheroidal shells are oftensed as bottoms or heads of reservoirs and various vessels.

The ellipsoidal form sometimes helps in solving problems inhysics �1–7�, mechanics of fluids �8–10�, acoustics �11,12�, ando on �13,14�.

Krivoshapko in his monograph �15� with 134 references tried toay attention to shells in the form of ellipsoids of revolution. Theain focus was on constructive features of erected buildings. This

ew, present review is not a repetition of the published material,ut is new with special attention paid to static, vibration, anduckling problems of general ellipsoidal shells and ellipsoidalhells of revolution and to an analysis of tendencies and featuresf their design. In preparing this review, the author used materialsublished in scientific and technical journal and proceedings, andtudied monographs, reports of scientific conferences, and othercientific and technical literature mainly for the period 1975–005.

1.1 Equations of Ellipsoidal Surfaces. Ellipsoids are theecond order surfaces. A general equation of the second order inartesian coordinates x, y, z has the form

a11x2 + a22y2 + a33z

2 + 2a12xy + 2a13xz + 2a23yz + 2a14x + 2a24y

+ 2a34z + a44 = 0

here aik=aki; k=1,2 ,3 ,4.

Transmitted by Associate Editor J. Simmonds.

36 / Vol. 60, NOVEMBER 2007 Copyright ©

aded 21 Jan 2008 to 203.88.129.4. Redistribution subject to ASME

1.1.1 Triaxial Ellipsoid. A surface is called an ellipsoid if in aright-handed Cartesian system of coordinates it has the canonicalform

x2

a2 +y2

c2 +z2

b2 = 1

An ellipsoid may also be defined by the parametric equations

x = x�u,v� = a sin u cos v y = y�u,v� = c sin u sin v

z = z�u� = b cos u

Assuming a=6379.351 km, b=6356.863 km, and c=6378.139 km, one can obtain the triaxial ellipsoid of Krasovskiy,which is the most perfect model of the earth surface. The differ-ence between Krasovskiy’s ellipsoid and the earth’s form does notexceed 100 m.

A radius vector of an ellipsoid referred to the geographical sys-tem of curvilinear coordinates may be written as �16�

r = r��,�� = acos �

ch �i + c

sin �

ch �j − b tan �k �1�

All plane sections of an ellipsoid are ellipses. Two families oflines of principal curvature of triaxial ellipsoids are shown in Fig.1.

A paper of Krzyzanowski �17� devoted to research of stereo-graphic projections of ellipsoids is of interest. Narzullaev �18�approximated an ellipsoid by elliptical cones and after that con-structed its approximate development assuming a per-unit error ofareas of an ellipsoidal surface. Knabe and Möller �19� looked forthe lines of equal illumination on an ellipsoid under all possibledirections of lighting. Some additional properties of ellipsoids arestudied in Refs. �20–22�.

1.1.2 Ellipsoids of Revolution. An ellipsoid of revolution maybe formed by rotation of an ellipse

x2

a2 +y2

b2 = 1

around the z axis, yielding a surface that may be represented as

2007 by ASME Transactions of the ASME

license or copyright; see http://www.asme.org/terms/Terms_Use.cfm

Ae

Ibet

sPpr

2t

mts

Fa

A

Downlo

x2 + y2

a2 +z2

b2 = 1

n ellipsoid of revolution may also be defined by the parametricquations

x = x�u,v� = a cos u cos v y = y�u,v� = a sin u cos v

z = z�v� = b sin v �2�

f b�a, one has an oblate ellipsoid of revolution �Fig. 2�a��, if=a, one has a sphere �Fig. 2�b��, but if b�a, we have a prolatellipsoid �Fig. 2�c��. Parallels and meridians of the ellipsoid arehe lines of principal curvature.

Chervyakov and Mutriskov �23� carried out a geometrical con-truction of the lines of cut for developing an ellipsoidal surface.alamutoglu �24� demonstrated a system of the central-and-polarrojections, the fundamental surface of which is an ellipsoid ofevolution.

Examples of Buildings and Constructions Erected inhe Form of Ellipsoids

Smirnov et al. �25� explored the possibilities of using pneu-atic rubber-fabric coverings in the form of ellipsoids of revolu-

ion, which are supported in their final position by internal pres-

ig. 1 Two families of lines of principle curvatures on a tri-xial ellipsoid

ure. Zryukin et al. �26� ascertained that it is necessary to use

pplied Mechanics Reviews


pneumatic ellipsoidal shells with a ratio of semiaxes equal to0.7071 �b=0.7071a� to avoid wrinkling or another forms of theloss of stability. Royles and Llambias �27� studied the stability ofspheroidal shells used in a deep-water apparatus. The methods ofdesign, manufacture, and model testing of the ellipsoidal tanks forstoring the fuel of the firm “Pressure System Inc.” are described inRef. �28�.

2.1 Buildings and Constructions in the Form of Tri-AxialEllipsoids. Shed coverings erected from shells of double curva-ture were first applied in Russia for covering a factory shop in1950. The cover took the form of the triaxial ellipsoid over cellswith spans of 12�21 m �29�. The first full scale test of the con-struction of the shed roof was also carried out in 1950. The roofconstructions were driven to complete destruction. Later, severalindustrial buildings for light industry were erected on a mesh of12�21 m2 columns using shed construction techniques. Theshells had a 6 cm thickness. Figure 3 shows the model of shuttersfor the reinforced concrete roof of industrial building designed by“Tekstilproekt.” A shell analysis using boundary elements wasmade taking into account bending and torsional moments andjoint action. However, in 1956, part of the poured-in-place rein-forced concrete shed covering an area of 5000 sq m �ten shells�collapsed. Shkinev �30� described in detail the reasons for thisfailure. Insufficient rigidity and a large ice load were the maincauses. The damaged part of the roof cover was then rebuilt.

2.2 Materials Used for Erection of Ellipsoidal Shells ofRevolution. At present, it is known that metal, reinforced con-crete and cement, impregnated textile, and fibrous composite ma-

Fig. 2 Ellipsoids of revolution: „a… the oblate ellipsoid of revo-lution, „b… the sphere, „c… the prolate ellipsoid of revolution

terials are applied for construction of thin-walled shells in the

NOVEMBER 2007, Vol. 60 / 337


f

laPosfbiTtsgogrd

Ff

3

Downlo

orm of a spheroid.A dome of the hall “The Century” in Vrotzlaw made of mono-

ithic reinforced concrete was built in 1912. It has a span of 65 mnd became the first building whose span exceeded the span of theantheon in Rome, which is equal to 43.3 m. From this momentn, a new material called reinforced concrete was used for con-truction of new large-span thin-walled shells. After 1945, rein-orced concrete shells in the form of ellipsoids of revolution wereuilt. Shells made of fibrous composite materials are also of greatnterest as these materials can operate in extreme conditions.echnological processes of contact, elastic, rigid forming, con-

inuous winding on immovable or rotating mandrels generatehells quite of any form with arbitrary changing thickness fromlass-, coal-, boro-, organoplastics and having arbitrary anisotropyf materials �31�. A widening application of rubber-fiber, homo-eneous, and reinforced films for ellipsoidal air-supported shellsequir additional experimental investigations and more accurateefinition of special parameters of wind pressure �25�.

ig. 3 A model of forms of reinforced concrete covering theactory shop designed by “Tekstilproekt”

Fig. 4 Geometrical parameters of the main te
in Istra „Russia, Moscow Region…
38 / Vol. 60, NOVEMBER 2007


2.3 Buildings With Metal Dome Covers. As an example ofthe practical application of the form of an ellipsoid of revolutionone may describe a project of the building of the main test benchof All-Russian Electrotechnical Institute in Istra of the MoscowRegion. It was designed in a form very close to an ellipsoid ofrevolution of 234 m diameter, with total height of 112 m �32�.The shell’s equator was 23 m from the floor level �Fig. 4�. Melni-kov and Saveliev �33� described an analogous net ellipsoidal shell.A membrane made of rolled steel with a thickness of 0.15 cm waswelded on an external belt steel. The membrane was intended tocarry the wind and snow loads and also to be a protecting con-struction. However, the shell collapsed immediatly after its erec-tion in 1984.

A sports hall, round in plane, covered by a ribbed dome ofellipsoidal shape of 82.3 m in diameter and 15.24 m high wasbuilt in Atlanta �Fig. 5� �34�. 32 steel half-arcs of ellipsoidal formplaced at the earth level on the upper edge of the foundations arethe main bearing constructions of the cover. Gohar-Harmandaryan�35� said that pure sketchiness of the constructive form of thissport hall gives to it very downcast appearance.

A dome of the covered stadium in San Paulu �Brazil, 1958� hasa form of an ellipsoid of revolution with an 80 m diameter. Thedome cover was made of metal with the application of steel grid-work half-arcs of elliptical form. The reinforced concrete ringresists the thrust of the dome.

An interior of a building of the pavilion of a travelers firminsurance at the international exhibition in New York was a closedellipsoidal volume. The steel spatial bearing construction con-sisted of 24 pre-cast steel ribs of ellipsoidal form. Zetlin �36�described the process of assembling the framework of construc-tion.

2.4 Buildings With Reinforced Concrete Shell Coverings.

ench of All-Russian Electrotechnical Institute

Fig. 5 Sport hall in Atlanta „USA…

st b

Transactions of the ASME


IthTtDs4p

setamotmo5�wfb

tsoTosoF

cc

Vwc

F„

A

Downlo

n Munich �Germany�, a building of the University with a dome inhe form of an ellipsoid of revolution was erected. The buildingas a rectangular plan with the dimensions of 16.75�13.04 m2.he dome is supported by four arcs bounding its plan. The arc

hickness near the dome pole is 55 cm and the arc width is 80 cm.ischinger �37� described the University’s dome and presented

ome photos of it. A compressed central ring has a height of0 cm and is reinforced by 6 bars with a 2 cm diameter, which arelaced along internal and external contours of the ring.

A roof cover of a rubber factory in Brinmore �GB, 1947� con-ists of 9 �3�3� ellipsoidal monolithic reinforced concrete shellsach of which covers an area of 18.6�25.5 m2 �Fig. 6�. The shellhickness is 7.5 cm, the shell rise is 2.4 m. Additional informationbout the roof cover of this factory can be found in Ref. �38�. Theain dome of a nuclear center near Munich �1957� is in the form

f an ellipsoidal shell of revolution �Fig. 7�. It is the most effec-ive form of a dome, which satisfies the design functional require-

ents. The thickness of the monolithic reinforced concrete shellf the dome is 10 cm, designed to take an internal pressure of0 Mpa. A circular foundation with a cross section of 6030 cm2 supports the shell �35�. An analogous type of reactoras built in the State of New Jersey in 1958. The complex differs

rom the Munich center only by the placing of the subsidiaryuildings.

A religious building in Jerusalem �Israel, 1957� was erected inhe form of a domelike monolithic reinforced concrete shell on aquare plan with rounded angles �Fig. 8�. An upper ellipsoidal partf the shell passes smoothly into the lower cylindrical part �35�.orroja �39� used an ellipsoidal dome of revolution in the designf the chapel. The dome is supported by eight ellipsoidal elementsimilar to the form of the main dome. A layered ellipsoidal shellf revolution is used as a coffee-house in Domby �the Caucasus�,ig. 9.Methods of erection of ellipsoidal domes by placing a layer of

oncrete paste and steel bars on a pneumatic membrane are dis-ussed by Roessler and Bini �40�.

2.5 Ellipsoids of Revolution in Construction of Pressureessels and Tanks. Thin-walled component shells of revolutionith ellipsoidal elements are widely used in chemical, metallurgi-

al, transport machinery constructions, building, and other spheres

Fig. 6 A roof cover of rubber factory in Brinmore „GB…

ig. 7 The main dome of nuclear center near Munich
Germany…


of human activity �41�. Pressure vessels are usually made of twoends called heads and a cylindrical shell. These vessels may bedesigned to be operating in vertical or horizontal positions. Allexternal tank heads must have a form of an ellipsoid of revolutionin which the major axis must be equal to the diameter of the shelland the minor axis must be one-half the major axis. Internal com-partment tank heads may be 2:1 ellipsoidal, 3:1 ellipsoidal, orflanged and dished to thickness. The ASME flanged and dishedhead or Code flashed and dished head requires the dish radius tobe no greater than the diameter and a knuckle radius of no lessthan 6% of the diameter or three times the metal thickness, which-ever is greater. Ellipsoids of revolution can be seen in forms ofbottom and head of a cylindrical vertical vacuum camera used inbuilding. Simpson and Antebi �42� present a sketch and describeone of these cameras consisting of a cylinder with 12.8 m diam-eter and 17.8 m high strengthened by rigid circular ribs placedalong the height. There are hundreds of web sites on the Internetwhere reservoirs, pressure vessels, water tanks of ellipsoidalforms are recommended for the application. For example, onemay refer to web sites of Compliance Resource Center, ICEMEngineering Company Ltd., AEA Technology �The innovationbusiness�, Pittsburgh Tank and Tower Co., Inc., and so on. InRussia, the constructions of vessels and tanks recommended forapplications are presented in the special State standards calledGOST, for example, in GOST 26421-90 “Ellipsoidal flanged bot-toms made of aluminum,” GOST 6533-78 “Ellipsoidal bottoms.”In the USA, the National Board of Boiler and Pressure VesselsInspector is an organization primarily responsible for training andcertification of ASME Code pressure vessels inspector. It alsomaintains on file Manufacturers Data Report Forms for all pres-

Fig. 8 A religion building in Jerusalem „Israel…

Fig. 9 The coffee-house in Domby „the Caucasus…

NOVEMBER 2007, Vol. 60 / 339


s

tpo

3l

lsoGtcwsgswtvtftLe

Tctmamlw

ibiuaoelk

rop

3

Downlo

ure vessels registered with it.Drop-shaped tanks �Fig. 10� are not ellipsoids of revolution, but

heir form is very close to this. The drop-shaped surfaces are notresented in this review. Information on the application and meth-ds of analysis of these reservoirs can be found in Ref. �43�.

Theoretical Studies of the Stress-Strain State of El-ipsoidal Shells

3.1 Membrane (Momentless) Shell Theory of General El-ipsoidal Shells. Using membrane shell theory, Vlasov �44� gave aolution of a static problem of analysis of a shell having a secondrder middle surface referred to lines of principal curvatures.oldenveizer �16� held that the reduction of a static problem to

he Poisson equation for shells of positive Gaussian curvatureonducted by Vlasov �44�, Sokolovskiy �45�, and Rabotnov �46�as carried out in a complicated way. For an arbitrary membrane

hell formed by the surface of the second order referred to aeographic nonorthogonal system of curvilinear coordinates �1�, aolution may be reduced to integration of the Poisson equationith the help of a change of variables �16�. Pavilaynen �47� used

he variant of a membrane shell theory derived by Pucher. Pa-ilaynen reduced the Pucher equations to the one governing equa-ion of the second order with variable coefficients using a stressunction. This equation may be applied for the determination ofhe main stress state of shells in the form of a three-axial ellipsoid.ogan and Hourani �48� described a membrane theory for layeredllipsoidal shells.

3.2 Membrane Theory of Ellipsoidal Shells of Revolution.hin-walled shells in the form of an ellipsoid of revolution are alass of shells of revolution of nonzero Gaussian curvature and sohe methods of analysis derived for general shells of revolution

ay be used for them. A history of development of methods ofnalysis of general type shells of revolution, their optimal forms,ethods of theoretical analysis of their stress-strain state, buck-

ing problems, and vibration are presented in Krivoshapko �49�ith 41 references.In analyzing a smooth dome of revolution loaded by the grav-

ty, or axisymmetrical uniform loading, with the help of mem-rane theory, one may show that compressive stresses �� appearn the meridian direction. Compressive stresses �� appear at thepper zone in the circular direction, but tensile stresses �� appeart the lower zone of shells with the exception of parabolic domesf revolution. An analysis of thin membrane shells in the form ofllipsoids of revolution subjected to an axisymmetrical uniformoad q is not difficult because normal stresses are determined by

Fig. 10 The d

nown analytical formulas. Having two equilibrium equations,



N�/R1 + N�/R2 = − q �3�

2�rN� sin � + Q = 0 r = R2 sin � �4�

where N� is a normal stress resultant in the direction of a meridianwith the principle radius of curvature denoted as R1, N� is a nor-mal stress resultant in the direction of a parallel with the principalradius of curvature denoted as R2, Q is the resultant of an axisym-metrical load denoted as q and acting above the given cross sec-tion �=const. From Eq. �4� and then Eq. �3� one obtains

N� = −Q

2�R2 sin2 �and N� = − R2�q +

N�

R1� �5�

Equation �3� is called the Poisson equation. The formulas for thedetermination of the principal radii of curvature and sin � are

R1 =��b2 − a2�r2 + a4�3/2

a4b, R2 =

��b2 − a2�r2 + a4�1/2

b

sin � = r/R2 �6�

For example, we can obtain the following values of stresses foran ellipsoidal shell of revolution caused by uniform internal pres-sure q

�� =q

2b�r2�b2 − a2� + a4 �� =

q

2b

2r2�b2 − a2� + a4

�r2�b2 − a2� + a4�7�

where is the shell thickness. Equation �7� show that the normalstress ��0, but the normal stress �� changes sign on the parallelfor which

2r2�b2 − a2� + a4 = 0 �8�

A membrane analysis of a shell having the form of an ellipsoidof revolution is given by Dishinger �37�. A stress state was deter-mined for a shell subjected to internal uniform pressure and asnow load. Dishinger used also a graphic-analytic method for ananalysis of a symmetrical dome subjected to wind pressure. Or-thographic representations of the internal membrane stress result-ants in a shell with ellipsoidal middle surface when a ratio ofhalf-axes is a :b=1.67 and subjected to internal pressure are pre-sented by Timoshenko and Woinowsky-Krieger �50�. Examples ofanalyses are given also in the monograph �51� where four differ-ent types of domes are studied that are spherical or parabolic, andellipsoidal domes consisting of half of an ellipsoid, and also ellip-soidal domes consisting of the lower part of the ellipsoid. In thelast type of domes, tangent straight lines to meridians along the

-shaped tank

circular edge are not vertical. It was found that a parabolic dome



rcbsdimess�iyrsw�lpc

s�atltsiCshfvaa�ai

sLosircTtam

A

Downlo

equires the most area of the thrust ring. An ellipsoidal domeonsisting of half of an ellipsoid does not demand a thrust ring,ut it has considerable circumferential tensile stresses near thehell edge. Circumferential tensile stresses near the edge of aome consisting of the lower part of the ellipsoid will be less thann the previous case. In a monograph �51�, the description of the

ethod of affine transformation for the determination of the gen-ral membrane stress-strain state in an ellipsoidal dome with aupporting ring is presented. An affine transformation reduces thehell in question to a spherical auxiliary shell. Novozhilov et al.51� showed that the equations of membrane shell theory can bentegrated in quadratures if the shell suffers wind loads. Stol-archuk �52� found a form and a wall thickness for a shell ofevolution of minimal weight subjected to uniform pressure. Thishell has a weight advantage some about 4–9% in comparisonith ellipsoidal shells of revolution. Petuhov and Shevchenko

53� obtained an analytical solution for a membrane shallow el-ipsoidal shell under gravity, snow, and wind loads. They appliedarametrical equations �2� for the determination of geometricalharacteristics of the middle surface.

Additional information on results of the analysis of ellipsoidalhells using a membrane shell theory may be found in Pavilaynen54�, Chausov �55,56�, Clark and Reissner �57�, Schmidt �58�, andlso in papers �59,60�. Pavilaynen �54� used the governing equa-ions of Novozhilov �61�. He analyzed the influence of the shal-owness of the dome on the value and character of distribution ofhe membrane stress resultants and showed that a membrane stresstate under small rigidity of the supporting ring could be realizedn a dome of revolution in the form of an ellipsoidal segment.hausov �55,56� applied a method of Vlasov �62,63� for an analy-

is of the part of an ellipsoidal shell of revolution limited by aorizontal cut and a vertical section and loaded by a vertical pointorce. He �64� analyzed a membrane shell theory from the point ofiew of its application and discussed the opinions of some authorsbout this theory. An analytical solution for an ellipsoidal shellcted upon by internal pressure is presented in a paper of Kuotong65� who used the membrane state and the equations of the bound-ry effect. An ellipsoidal thin shell supported by a pencil of tubess studied in this paper.

Tibor �66� derived the boundary conditions when a membranetress state appears under an arbitrary uniform transversal load.ogan and Hourani �48� presented equations for the determinationf a linear stress-strain state in layered anisotropic ellipsoidalhells of revolution. A shell resting on one edge and subjected tonternal pressure was analyzed as a numerical example. Equilib-ium of an orthotropic membrane ellipsoidal shell of revolutionaused by internal pressure was determined by Mamedov �67�.he tridimensional stress problem was solved by methods of the

heory of elasticity. Formulas for the determination of a shell formnd its change of thickness after deformation are presented. The

Fig. 11 Cylindrical tanks with an

aterial of the shell is assumed to be linearly elastic and it under-



goes large deformations before destruction. An analogous problemwas also studied by Ganeeva and Skvortsova �68�.

3.3 Membrane Stress State of Ellipsoidal Heads and Bot-toms of Cylindrical Tanks and Vessels. Let us use the materialsin a book by Avdonin �69� and rewrite the formulas for the deter-mination of normal internal forces in membrane elliptical bottomsand heads of cylindrical tanks with a radius denoted by a. Assumethat the elliptical bottoms and heads are attached to the cylindricaltank and the tank itself is joined to the foundation. The tank isfilled by a liquid with density of �Fig. 11�. If the lower bottomin the form of an ellipsoid of revolution is subjected to internalhydrostatic pressure �Fig. 11�a�� then formulas �5� give

N� =

2br2�2

3a2b + Hr2 −

2

3b�a2 − r2��1 −

r2

a2��b2 − a2�r2 + a4

N� =H�2�b2 − a2�r2 + a4�

2b��b2 − a2�r2 + a4

+ r2��b2 − a2�r2 + a4� + a4�a2 − r2�/3��1 − r2/a2 − a6/3

r2��b2 − a2�r2 + a4

�9�

At the apex of an ellipsoidal bottom, one obtains

N� = N� = a2�1 + H/b�/2

assuming r=0. Assuming r=a, it is possible to obtain normalresultants at the zone of an equator:

N� =a

2�2

3b + H� N� =

a

2b�2b2 − a2

bH −

2a2

3� �10�

Considering a head in the form of an ellipsoid of revolutionsubjected to internal hydrostatic pressure �Fig. 11�b��, one can useformulas �5�

N� =a2

b3r2�H3

6− b2�H

2−

b

3�1 −

r2

a2��1 −r2

a2��b2 − a2�r2 + a4

N� =

6�H − b�1 −

r2

a2��b2 − a2�r2 + a4

+a6

b3r2��b2 − a2�r2 + a4�b2�H

2−

b

3�1 −

r2

a2��1 −r2

a2�−

H3

6 �11�

psoidal bottom „a… and a head „b…
elli
NOVEMBER 2007, Vol. 60 / 341


te

m

tsqctp

a=

TodsA

drbw

spAsi�crsntaGestaasGsow

3

Downlo

Assuming r=a in formulas �11�, one can obtain expressions forhe determination of normal stress resultants at the equator of anllipsoid of revolution:

N� =aH3

6b2 N� = aH�1 −a2H2

6b4 �If H=�2b2 /a, then a circular normal resultant has the maxi-um value

N�,max = 2�2b2/3

In the same book by Avdonin �69�, formulas for the determina-ion of meridional u and normal w displacements of the middleurface of an ellipsoidal shell caused by internal uniform pressure

are presented. A constant of integration was found due to theondition that a meridional displacement u on the equator is equalo zero. So assuming r=0, we can determine displacements at theole

u�r = 0� = 02Eb2w�r = 0�

qa2 = �1 − ��a2 − �b2 − a2��1

−1 − 2�

2

b2

a2�ln��1 −b2

a2 + 1� − lnb

a −

1 − 2�

2

nd on the equator if to take r=a: u�r=a�=0; w�r=a�qa2 /2E�2−�− �a2 /b2��.For example, assuming a /b=2, �=0.3 one calculates

w�r = 0� = 3qa2

Ew�r = a� = − 1.15

qa2

E

his means that the ellipsoid stretched in the direction of an axisf rotation and shrank in the radial direction. Formulas for theetermination of hoop and axial stresses for a cylinder and ellip-oidal ends under internal pressure is presented in the web site ofEA Technology.Novozhilov et al. �51� considered a membrane theory of cylin-

rical pressure vessels with two bottoms in the form of surfaces ofevolution. It was found out that an ellipsoidal construction of theottom better complies with the membrane theory in comparisonith a spherical construction.

3.4 Shell Bending Theory. The bearing capacity of stiffenedpherical, ellipsoidal, and parabolic domes on a square plan sup-orted at separate points is studied in a dissertation of Berg �70�.xisymmetrical and nonaxisymmetrical loads such as gravity,

now, and wind loads are taken into account. An estimation of thenfluence of geometrical parameters is given. Ermakovskaya et al.71� were involved in the investigation of nonlinear axisymmetri-al deforming concentrically loaded elastic ellipsoidal shells ofevolution with a nonclosed meridian. They applied a method ofuccessive approximation and finite differences. The influence ofonlinearities, boundary conditions, shell forms, type, and charac-er of the loading on the distribution of displacements, stresses,nd deformations in the area of their concentration was studied.orlach and Mokeev �72� demonstrated the advantages of an it-

ration process for axisymmetrically deformed nonshallow ellip-oidal shells of revolution of constant thickness stretched alonghe larger half-axis. The circular edge is simply supported andllows the radial displacements, but the top is simply supportednd immovable. An external load on the shell is distributed sinu-oidally. Displacement errors within 0.01% require 25 iterations.aneeva et al. �73,74� analyzed an isotropic oblate ellipsoidal

hell closed in the circular direction and with a ratio of half-axesf 0.5 �b /a=0.5�. The wind load on a shell of constant thickness
as described with the help of the following expressions:


Z = P if � � �0,�0� and Z = P�1 + cos � sin�� − �0��

if � � ��0,�/2� �12�

where � is an angle of the axis of rotation with a normal to thesurface. General expressions for this problem were obtained onthe basis of the theory of Kirchhoff–Love taking into consider-ation geometrical nonlinearity under moderate rotation and physi-cal nonlinearity under a theory of small elastic-plastic deforma-tions for compressed material. The unknown functions wereexpanded into trigonometric series along a circular coordinate. Astress state of plastic space relaxed by an ellipsoidal cavity wasstudied by Efremov �75�. He decomposed the equation of an el-lipsoidal surface, referred to spherical coordinates, in terms of thepowers of a small parameter up to the second order. After that, hewrote a series in Legendre polynomials equilibrium equations andthe full plasticity conditions were taken in the form of Ivlev.

An interesting analysis of an orthotropic ellipsoidal shell on anelastic foundation was realized by Paliwal et al. �76�. Murakami etal. �77� showed the advantages of finite element method �FEM�for an analysis of an elliptical shell of revolution subjected tovertical and horizontal loads and resting on a system of bar sup-ports. Wu and Wang �78� presented indices of technical and eco-nomical effectiveness of their method of forming of ellipticalpressure reservoirs and compare the calculated results with theexperimental values of stresses and elastic deformations appearingunder internal pressure.

3.4.1 Ellipsoidal Shells of Revolution With ChangingThickness. By choosing linearly changing thickness of thin ortho-tropic ellipsoidal shell of revolution with a circular opening, orwith changing rigidity of a stiffened ring, one can design shellsequal in strength with the same maximum stresses on the contourof the opening and in a zone of the equator �79�. The influence ofchangeability of thickness, the geometrical parameters of a shell,the rigidity of stiffening circular elements, and a value of theacting load on the distribution of stresses in spheroidal shellsmade of orthotropic composite material was studied in Ref. �80�.Golushko and Nemirovskiy �31� used a criterion of constancy ofspecific potential energy for the investigation of thin-walled elas-tic reinforced shells with changing thickness. They derived thenecessary minima of the distribution of thickness, intensities, andangles of reinforcement of shells of revolution caused by constantinternal pressure. The material was assumed to be orthotropic,quasihomogeneous along the thickness, and heterogeneous alongthe shell meridian. The problem was solved using the hypothesisof Kirchhoff–Love. The authors �31� show that a design of thin-walled constructions of minimal self-weight, based on the criteriaof constancy of specific potential energy for elastic constructionsand constancy of specific power of energy of dissipation for plas-tic constructions working in the condition of formed creep, is themost efficient.

3.4.2 Layered Ellipsoidal Shells of Revolution. The stress-strain state of a thin anisotropic layered shell of variable thicknessin the form of a spheroid was studied by Grigorenko andVasilenko �81�. A method of analysis of an elastic layered ellip-soidal shell with the help of a method of linearization and a stablenumerical method of discrete orthogonalization was presented byAbramidse �82�, who studied the influence of boundary conditionson the stress-strain state of a shell. Kamalov and Teregulov �83�considered the stress-strain state of a thin ellipsoidal shell of revo-lution consisting of isotropic and anisotropic layers with variablethickness and rigidity. The layers were joined by a thin glue layer.The stress state was sought under internal pressure. Vohmyanin�84� solved a problem of rational design of an ellipsoidal two-layered shells of revolution with variable thickness assuming thecondition of minimum of weight and proceeding from the require-ment that values of the Mises–Hill function on external surfacesshould not exceed the limit of proportionality. A problem of elas-
tic, nonlinear deforming in a locally loaded thin-walled trilayered


e�lspsgcswsfilrtvmtmcciacpatsa=

ar

wroc

Hosdis�tsimttc

wf

vs

A

Downlo

llipsoidal shell is discussed in a book of Grigolyuk and Mamay85�. They tried to determine the limits of applicability of theinear shell theory equations for the determination of a stress-train state of a locally deformed ellipsoidal shell and studied theossibility of using a geometrically nonlinear theory for such ahell. Grigolyuk and Mamay �85� claimed that it is not possible toive a correct and simple estimation to the influence of geometri-al nonlinearity on a stress-strain state of the shell in question inpite of the seeming simplicity of the problem. A prolate spheroidith a=158 cm and b=300 cm was taken for a numerical analy-

is. External layers of the ellipsoid are assumed to be made ofber-glass plastics and a filler made of sphere plastic. The shell is

oaded at the top by external pressure uniformly distributed on aound area. The total value of the axial force is constant and equalo P=q�r2=qS=20 kN. The results of an analysis showed thatalue of the loading area denoted as S influenced considerably theaximum value of the components of the displacement vector. So

ransition from S=900 sq cm to S=100 sq cm increases the nor-al deflection w up to 2.38 times. However, this effect has a local

haracter. The analysis showed that the linear solution gives ex-essive values for the components of the displacement vector. Thenfluence of nonlinearity is rather moderate. A case of loading ofn ellipsoidal shell of revolution by distributed load along its cir-ular zones was also considered �85,86�. It is seen from the graphresented in Ref. �85� that the maximum deflections of shellsnalyzed with the help of nonlinear shell theory are 3–3.5% lesshan deflections determined by linear shell theory. The shell wasubjected to loading at the top �� 3.87 deg; q=2 MPa� andlong the circular zones ��=61.56–66.67 deg; �17.38–31.14 deg; q=0.4 MPa�.

3.4.3 Thick-Walled Ellipsoidal Shells of Revolution. Koupriy-nov �87� shows that or closed thick-walled ellipsoidal shells ofevolution given by parametrical equations

x = f sin � cos �ch y = f sin � cos �ch z = f cos �sh

�0 � � 0 � � 0 ��

here f is a focus distance and limited by two oblate ellipsoids ofevolution with =1 and =2, may be analyzed with the helpf a method of initial functions. He examined a shell under twoompressive axisymmetrical distributive loads.

3.5 Stress-Strain State of Reservoirs With Ellipsoidaleads and Bottoms. Ellipsoidal, torispherical, and ideal bottoms

f Bitseno of cylindrical reservoirs under internal pressure aretudied in Ref. �88� with the help of FEM. A stress state wasetermined by taking into consideration a boundary effect appear-ng at the junction of a bottom with a cylindrical body. Equivalenttresses were calculated by an energy strength theory. Eltyshev89� derived simple approximate formulas for the determination ofhe boundary effect zone in cylindrical orthotropic pressure ves-els with elliptical bottoms caused by internal pressure p. Assum-ng only membrane stresses, we find a discontinuity of displace-

ents at the junction of a bottom with a cylindrical body. It meanshat distributive transversal forces and bending moments M1 haveo act at this zone to equilibrate the discontinuity. Following thisonditions, Eltyshev �89� obtained

W = −e−�z

16�4D11

b2

a2 p cos �z M1 = − D11d2W

dz2 =p

8�2

b2

a2e−�z sin �z

here W is the radial displacement of the cylindrical middle sur-ace,

� =�4 3

h2b2��2

�1− �2

2�i are Poisson coefficients of the orthotropic material of the ves-
el, b is the radius of the cylindrical body, and h is the thickness of


shell wall. The bending moment M1 obtains its maximum value atthe point with z=� /4� because the first derivative of M1 at thatpoint is equal to zero, but the second one is negative. The formu-las so obtained may be used as a particular case for the descriptionof the boundary effect in isotropic cylindrical vessels with spheri-cal bottoms.

A stress-strain state of vessels consisting of a cylindrical shelland an elliptical bottom or a head is studied in �90,91�, in a paperof Simpson and Antebi �42� with the help of FEM, and in Gu-odong �92� by using a complex representation of the equations ofa thin shell theory containing a small parameter. However, Gu-odong �92� does not present numerical results. Cloclov �93� alsoanalyzes the junction of a cylinder and an ellipsoid of revolution.Yakovlev �94� carried out a study of the deformation of an elasticsystem consisting of an ellipsoidal bottom, a ring stiffened a jointat the bottom and a conical shell, and a conical shell by itself. Byconsidering the differential relations for a plane circular ring, heobtained expressions for internal forces, moments, and angles ofrotation of the cross section. Later, on approximate solutions forthe ellipsoidal bottom and the conical shell were determined by anasymptotic method on the basis of two homogeneous differentialequations, which describe nonaxisymmetrical deformation of ashell of revolution caused by an arbitrary boundary load. Formu-las for the determination of displacements of the middle surfacewere also presented �95�.

Kantor and Belov �96� offered a method for determining thebearing capacity of ellipsoidal bottoms near the area of supportingpillars. The behavior of an ellipsoidal head of a boiler is studied inHu �97� from a point of view of the theory of elasticity. Chao andSutton �98� described a method of numerical solution of theLove–Meissner differential equations of equilibrium of thin shellsin the form of an ellipsoidal head on a cylindrical vessel subjectedto internal pressure. The head has a radial branch pipe. An ellip-soidal shell of revolution as a part of a closed pressure vessel isdescribed in Ref. �99�. The design of ellipsoidal cups of cylindri-cal pressure vessels is considered also by Smith �100�. A methodof strengthening the openings at the top of ellipsoidal bottoms,which considerably decreases a coefficient of stress concentrationis demonstrated in Ref. �101�. The strengthening was selected inthe form of a ring, the upper half of which had a smaller thicknessbecause of the stiffening influence of the cap. Alekseeva andGaneeva �102� considered an axisymmetrical problem of largedeflections and stability of ellipsoidal bottoms weakened by a cen-tral opening, assuming that an external vertical force acted on thestiffened ring. They considered a geometrically nonlinear prob-lem. A numerical study of the dependence of the critical force onthe geometrical parameters of a bottom and the dimensions andrigidity of the opening was carried out.

3.5.1 Elastic-Plastic Behavior of Ellipsoidal Shells. The de-formation produced by increasing internal pressure of a constantthickness consisting of a cylindrical reservoir with an ellipsoidalbottom shell was studied by Bandurin and Nikolaev �103� with thehelp of a variant of a method of step by step loading. The numeri-cal analysis was realized by the application of a method of super-elements of special type in conformity with axisymmetrical de-forming elastic-plastic thin shells of revolution, taking intoconsideration variable time steps. The result obtained in this papercorroborates the results of numerous experiments according towhich a dangerous zone propagates from an area of the ellipsoidalbottom to an area of the cylindrical shell if the pressure increases.A kinematic method of a theory of ultimate equilibrium is used byGerasimov �104� for study of bearing capacity of ellipsoidal shellsweakened by openings with flanges. The collapsed part of theshell is treated by a kinematically changing mechanism. A numeri-cal analysis of elastic-plastic behavior of pressure vessels withelliptical and torispherical heads with different geometrical pa-rameters is presented by Yeom and Robinson �105� with 12 refer-ences. They used FEM. Sorkin �106� demonstrates an example of
calculation of residual stresses in an ellipsoidal bottom. The bot-
NOVEMBER 2007, Vol. 60 / 343


tuo

ot�earads

wl2safmobroistci�Twtht

sa=aroi

Fo

3

Downlo

om was considered as a shell with rapidly changing deformationnder compression of its layers. It was assumed also that a changef radii of curvatures was insignificant.

3.6 Ellipsoidal Shells of Revolution With Openings. Papersf Galustchak and Koshevoy �79�, Simpson and Antebi �42�, An-ipov et al. �101�, Alekseeva and Ganeeva �102� and Gerasimov104� devoted to analysis of ellipsoidal shells of revolution weak-ned by circular openings were mentioned before. Mileykovskiynd Selskiy �107� studied the behavior of an ellipsoidal shell ofevolution formed by rotation of the ellipse x2 /b2+z2 /a2=1round the x axis �Fig. 12�. Using a geographical system of coor-inates �, � they obtained the following parametrical equations ofurface of an ellipsoid of revolution

x = � sin � cos � y = � sin � sin � z = � cos � �13�

here �=a /�1+� sin2 � cos2 �, �=a2 /b2−1, and the coordinateines did not coincide with the principal lines of curvature �Fig.�c��. They assumed that a shallow ellipsoid of revolution wasubjected to internal pressure. An elliptical opening at the top wasssumed to be closed by a rigid head passing only a transverseorce to its contour �=�o. Stress resultants were obtained by aembrane shell theory. In the general case a stress state near the

pening would be formed by a simple boundary effect, plus mem-rane and bending stress states. The investigations showed that aatio M� /N� at the contour increased with the increase size of thepening. M� and N� are bending moment and normal force actingn the direction of parallels �. The determination of the momenttate was reduced to the determination of a harmonic function onhe contour �=�o under the given boundary condition along theontour. A prolate ellipsoidal shell �Fig. 2�c�� with a central open-ng �=�1 under wind load �12� was considered by Ganeeva et al.73� taking into account geometrical and physical nonlinearities.hey assumed a low linear hardening for an isotropic material. Itas assumed that an edge of the opening �1=0.276 was free, but

hat the equator �N+1=� /2 was rigidly fixed. The shell consideredad a constant thickness h=h0 if �� 0.276;1.54� and variablehickness

h = h0�1 + �� − 1.54�/0.03� if � � �1.54;�/2� �14�

The calculation showed that the largest intensity of stresses in ahell of constant thickness, namely, �i=��11

2 +�222 −�11�22+3�12

2 ,ppeared at the fixed edge �N+1=� /2 on the meridian �=0, zh /2. A prolate ellipsoid behaved nearly like a membrane shelllong a considerable part of the meridian �� 0.793;1.54��. Theesults show that the increase of thickness �14� near a small areaf the fixed edge decreases the displacements along all the merid-

ig. 12 An oblate ellipsoid of revolution with an ellipticalpening given by parametrical equations „13…

an. The point of maximum �i moved to the beginning of thick-



ening �=1.54, �=� /2, z=h /2 and the level of intensity ofstresses was reduced by 45%. At the same time, the volume of theconstruction rose up by 6%. The concentration of stresses at thearea of a central opening was also studied in Ref. �108�.

Considering a problem of ultimate equilibrium of a prolate el-lipsoidal shell with a flanged opening at the shell pole, Gerasimov�109� assumed a form of failure as concentrated lengthening alongan axis of rotation near the area of the pole and determined theultimate internal pressure using an equation of energy balance. Herecommended this approach for ellipsoidal shells weakened byopenings of large diameters. Another scheme of failure with theappearance of a ring bulge near the area of a flanged opening canbe used for shells with openings of small diameter. Gerasimovalso derived a formula for the calculation of ultimate internal pres-sure for this case.

Gramoll �110� analyzed the stress-strain state of open-endedcomposite shells and the results obtained were compared to finiteelement results for three dome types.

4 Stability of Ellipsoidal ShellsThe loss of stability of an ellipsoid of revolution caused by

uniform normal pressure was first analyzed by Gekkeler who sug-gested

qcr =1.21Eh2

a2�a2/b2 − 2�

However, Mushtari �111� showed that Gekkeler’s result was erro-neous. Mushtari’s equation for the determination of the bucklingpressure,

qcr =2Eh2

�3�1 − �2�

1

R22 − 2R1R2

�15�

where R1 and R2 are the functions of the meridian length, wascorroborated by the results of Shirshov �112� and Tovstik �113�.Tovstik �113� studied the buckling of thin, strictly convex shells ofrevolution with a membrane stress state. The problem was solvedwith the help of linear shell theory, but his method is good onlyfor buckling modes with a great number of waves along the par-allels. The equations describing the loss of stability of a mem-brane stress state were taken in the form

�i=1

3 �Lij +h2

12Nij�uj +

1 − �2

EhXj = 0 j = 1,2,3

where u1=u�s ,��, u2=v; u3=w are the components of the elasticdisplacement vector in the direction of meridian, parallel, and nor-mal directed inside the ellipsoid; and Lij, Nij are the linear differ-ential operators of shell theory. The load terms Xj were taken inthe form proposed by Vlasov. The unknown displacements werewritten by Tovstik �113� as

u�s,�� = u�s�eim� v�s,�� = − iv�s�eim� w�s,�� = w�s�eim�

Later on, Mushtari and Korolyov proposed a new formula forinternal critical pressure,

qcr =1.21Eh2

b2�a2/b2 − 2�a � b �16�

and for external critical pressure: qcr=1.21Eh2b2 /a4.Avdonin �69� presented his version for a shell in the form of

ellipsoid of revolution subjected to internal pressure:

qcr =0.358Eh2

b2�a2/b2 − 2�a � b �17�

Comparing the formulas �16� and �17�, one can see that they areof the same form, but have different coefficients. Avdonin �69�supposed that his boundary conditions on the contour of dimples
and bulges are coordinated satisfactorily with the experiment and


tmcw

wtaou

w

il

wasa

i=

we2pstp

r�

A

Downlo

hat is why the formula �17� should yield satisfactory results. For-ula �16� was obtained for more rigid boundary conditions on the

ontour of the dimples and that is why the results of calculationsould be overstated.It is known that an ellipsoid of revolution has

R1 =a2

�sin2 � + 2 cos2 ��3/2 R2 =a

�sin2 � + 2 cos2 ��1/2

B = R2 sin � = b/a �18�

here � is the angle between the axis of rotation and the normalo the surface, a and b are the semiaxes, and B is the distance frompoint of the surface to the axis of rotation. For the determinationf principle curvatures of an ellipsoid of revolution, Guodong �92�ses the following formulas:

R1 = R�1 − K sin2 ��−3/2 and R2 = R�1 − K sin2 ��−1/2

here R=a2 /b, K=1−a2 /b2.For external pressure �q�0� if R2�R1 and for internal pressure

f R2�2R1, Tovstik �113� proposed the following value of criticaload:

qcr =Eh2�R2/2 − R1�−1

�3�1 − �2�R2�1 + � h

2R2�1/2

�R1

�12�1 − �2��1/4� R1 − R2

R1 − R2/2�1/2�R1N2

d2

ds2� 1

R1N2�

+ �R1

B2

d

ds

B2

R1�21/2

+ 0��3/2� �4 =h2

12�19�

here � is a small parameter, N1=R2q /2, N2= �R2−0,5R22 /R1�q

re normal internal forces determined with the help of membranehell theory. For external pressure if R2�R1, it is necessary topply another formula �113�

qcr = −2Eh2

R22�3�1 − �2�

�1 +�R2

�1 − �2�1/4�N1d2

ds2

1

R2N1�1/2

+ 0��3/2� �20�

In his paper �114�, Tovstik presented the formulas �19� and �20�n another form. For example, analyzing a prolate spheroid �b /a�1� under external pressure, he obtained

qcr =2Eh2

a2�3�1 − �2��22 − 1��1 + �h

a�1/2 �2 − 1��42 − 1�1/2

�4 3�1 − �2��22 − 1�

+ O�h*�n ��4 12�1 − �2�

�a

h�1/2

=b

ab � a

�21�

here n is a number of waves in the circumferential direction. Thellipsoid, subjected to internal pressure, can lose its stability if2�1. In the case 1�2��2, the equator will be the weakestarallel, qcr and the value n are determined with the help of theame formulas �21� as in the case of external pressure. If 2�1hen Tovstik �114� proposes to determine critical values of internalressure qcr and n using formulas

qcr = −16Eh22

a2�3�1 − �2��1 + �h

a�1/2� 193�1 − 42�

16�12�1 − �2�+ O�h*�

n �� 6

1 − 2�4 12�1 − �2��a

h�1/2

�22�

If 2=1 then formulas �21� and �22� can be used only for aough calculation of qcr and n. The equator is the weakest parallel
�=� /2� for the prolate spheroid, subjected to external pressure,


as was mentioned in �113� and also in Surkin’s paper �115�. For anoblate spheroid, the main term of formula �20� is consistent withformula

qhp = −2E

�3�1 − �2�� h

R�2

R =a2

b�23�

presented in monograph of Mushtari and Galimov �116�, where itis shown that one dimple is formed at the top of the shell as aresult of the loss of stability. Tovstik �113� showed that dimplesare formed not on the equator, but on the parallel �1

=arcsin �32 / �1−2� if the ratio of semiaxes of oblate spheroid isb /a�1 /2. In �113� he has also studied the stability of a shellunder the action of axial force and under the action of a torqueapplied on the shell ends.

Pogorelov �117� obtained the formula for the critical pressure,

qcr =2Eh2

�3�1 − �2�R2

B2�R2 − 2R1��24�

assuming that a loss of stability took place always near the equa-tor. The same assertion can be found in a monograph of Volmir�118�, but Mushtari and Galimov �116� and Tovstik �113� showedthat dimples can appear elsewhere than on the equator. If a loss ofthe stability takes place on the equator then B=R2=a and formula�24� gives a result differing from Eq. �15� only by the multiplier�1−�2. The difference between formulas �15� and �24� becomesvery substantial if �1 /2. A formula for the determination ofcritical load presented by Fidrovskaya �119� corresponds to theexperimental data of American authors better then their solution.

The stability of ellipsoids of revolution subjected to uniformpressure was studied also by Pogorelov �120� and Danielson�121�.

Having examined the stability of ellipsoidal shells of revolutionunder external uniform pressure, Surkin �115,122� first consideredthe possibility of a local loss of stability of a prolate ellipsoid ofrevolution. Buckling pressures pm and pk were determined withthe help of an energy method with application of nonlinear shelltheory. The pressure pm corresponds to equal levels of total shellenergy in the zero and nonlinear states. The pressure pk corre-sponds to the case when stable and unstable states are combinedin one state in which the functional of energy F has a parabolicpoint; i.e., the first and the second variations of F are equal tozero. The problem was solved in general completely. The solutionof the problem was carried out by the Ritz–Timoshenko methodwith the assumption that the center of the bulge was on the shellequator and the area of the bulge has an elliptical form extendedalong the meridian. The comparison showed that Mushtari’s for-mula �15� gave a lesser value than nonlinear shell theory if =b /a=R1 /R2�3. Sachenkov �123� noted that the influence of thenonlinear factor decreased with the increase of lengthening of theshell. Hyman �124�, using the energy functional F derived bySurkin �115�, made a buckling analysis of a prolate spheroid withfinite displacements. The displacements for a bulge in the equato-rial zone were taken in the form of trigonometric series, whichsatisfied geometrical boundary conditions of continuity of the con-tour. Hyman �124� confirmed that good results could be obtainedwhen using ten terms of a trigonometric series. Alumyae �125�examined the stability of a part of an ellipsoidal shell of revolu-tion under hydrostatic pressure. The ends of the shell were closedby rigid diaphragms. The buckling pressures were determined bythe stability equations presented by Mushtari and Vlasov.

Having used the stability equations presented in Mushtari andGalimov �116�, Bakirova �126� determined the upper critical loadfor a prolate ellipsoidal shell with the ratio of semiaxes 1 b /a
2.41. She obtained
NOVEMBER 2007, Vol. 60 / 345


otlFdld

Kepfaptl�pasitpoc

s

Smsescacdcv

Ft„

3

Downlo

qloc =2E

�3�1 − �2�

h2

b2 �25�

n the basis of a shell theory for local loss of stability. Her inves-igations resulted in the conclusion that the shell theory for localoss of stability gave understated values of the upper critical load.or example, if h /a=1 /50, b /a=11.4, then a critical pressure,etermined according to formula �25�, is less then one-third asarge as the results obtained by Bakirova and Surkin �127�. Thisifference increases if the elongation of the ellipsoid is increased.

Using the method of Bubnov–Galerkin and linear shell theory,rivosheev �128� derived a system of homogeneous algebraic

quations for the determination of critical external pressure. Theaper does not have numerical results. The upper critical pressureor ellipsoidal shells of revolution was determined by Krivosheevnd Murtazin �129� on the basis of shallow shell theory with ap-lication of the method of collocation. In this paper, a formula forhe determination of critical pressure was derived for a very pro-ate spheroid. The analogous problem is considered in Krivosheev130� where the equations of neutral equilibrium of nonshallowrolate spheroids are used. Kabritz and Terentiev �131� consideredgeometrically nonlinear problem of compression of a simply

upported elastic semiellipse of revolution. The analysis algorithms based upon a combination of methods of iterational continua-ion by parameter, Newton–Kantorovich, and orthogonal run. Theath tracing of ultimate points was realized with the help of anriginal realization of the idea of a change of the parameter ofontinuation.

Gulyayev et al. �132� studied critical states of thin ellipsoidalhells in simple and compound rotations.

4.1 Connection Between Stability of Ellipsoidal andpherical Shells. The stability problem of ellipsoidal shells hasuch in common with spherical shell stability. Rachkov �133�

howed that a dimensionless parameter p of critical pressure of anlliptical shell can be calculated by p= po /k2, where po is a dimen-ionless parameter of critical pressure for a spherical shell; k is aoefficient, depending on the ratio of the semiaxes of the ellipsend of the shell thickness �Fig. 13�. Rachkov �133� used the cal-ulation method and the form of a dimple as Volmir �118� hadone, and that is why the value of the dimensionless parameter ofritical pressure po=0.31 for a spherical shell agreed with the

ig. 13 The determination of the k coefficient depending onhe ratio of semi-axes of the ellipse and of the shell thicknesss…

alue obtained by Volmir. The results so obtained are correct pro-



vided that critical loading does not exceed the limit of proportion-ality for the shell material. However, most of the bottoms used inchemical and oil machine building lose their stability beyond thelimit of proportionality

4.2 Composite Ellipsoidal Shells. The matrix form of repre-sentation of the governing system of local stability equations tak-ing into account shearing for prolate and oblate composite ellip-soids of revolutions is presented in Abdulhakov and Ganiev �134�.The shell has the pole openings closed by rigid covers. The solu-tion was carried out by the method of finite differences. It wasproved that a decrease of the length of semiaxis of the ellipsoid ofrevolution results in an increase of buckling pressure, but an in-crease of rigidity of the binder results in an increase of the valueof external pressure by an entire order. A composite shell, shownin Refs. �134,135�, was produced by winding. The law of changeof the angle of winding is given as a function of the current radius�136�.

Local stability of glass-fiber plastic shells of revolution pro-duced by winding with due account of transverse shearing wasalso considered by Ganiev �137�. The analysis is based upon thefive equations of neutral equilibrium of the shell of revolutionassuming a local loss of stability. A system of the five governingequations with three unknown components of the vector of elasticdisplacements u, v, w and two unknown transverse shearing �1,�2 was obtained by substitution of values of the internal forcesand moments into the equilibrium equations. The tangential nor-mal forces were determined on the assumption of a membraneinitial state of the shell. The components of the displacement andshearing were taken in the form

u = A1 cos m�� − �0�cos n� v = A2 sin m�� − �0�sin n�� − �0�

w = A3 sin m�� − �0�cos n� �1 = A4 cos m�� − �0�cos n��

− �0�

�2 = A5 sin m�� − �0�sin n�� − �0� m = m1�� − 2�0�−1

This form of representation allowed satisfaction of the bound-ary conditions on the circular ends �=�0 and �=�−�0 sup-ported by sliding hinges. The critical value of external uniformpressure was determined by means of selection of the numbers ofhalf-waves m and n. The values of external buckling pressure withand without taking into account the deformation of transverseshearing were presented for an ellipsoid of revolution with chang-ing thickness along the meridian direction, with a pole opening of0.3 m in radius and with a radius of equatorial cross section equalto 1 m. Ganiev �137� examined all three possible forms of anellipsoid of revolution �Fig. 2� produced from materials with Ef=7.5�104 MPa and Eb=3�102 MPa; � f =�b=0.25. As a result,it was ascertained that taking into account the transverse shearingdoes not give any considerable corrections to the results obtainedwithout considering it. It should be noted that considering thetransverse shearing results in a decrease of the value of externalbuckling pressure.

Additional information about the behavior of ellipsoidal shellsmade of composite materials can be found in Vinson �138�.

4.3 Orthotropic Ellipsoidal Shells. Vasiliev and Ivanov�139� examined the stability of ellipsoidal shells of revolution oforthotropic structure connected along the inner surface with athick-walled elastic isotropic filler and subjected to separate ex-ternal loading. Supposing that the shell is a shallow one, the au-thors determined the buckling pressure and optimal angles of re-inforcing depending on the geometrical and physicalcharacteristics of construction. The stability of the orthotropic el-lipsoidal shell of revolution under uniform internal pressure andhaving two axisymmetrical pole openings was described byGaniev et al. �140,141� under the condition that the initial stress-
strain state of the shell is momentless one. The deformation of


tltdendceuttgcan

gtspsccHpnt�aomsptt�nsn

lbtepcbashobinmtttpdthr

Lrv

A

Downlo

ransverse shearing was taken into consideration. Determining theocal buckling reduces to the integration of five differential equa-ions of neutral equilibrium in terms of the components of theisplacement vector and the shearing. For the solution of thesequations, the method of finite differences was used. Based on theumerical results so obtained, the following conclusions wererawn: �1� the loss of stability with the formation of many wavesan take place in the area of compression. That is in the area of thequator for oblate spheroids of constant thickness with some val-es of radii of curvatures under the action of internal pressure; �2�he values of qcr for oblate shells change in proportion to square ofhe shell thickness; �3� a buckling problem for a specific shell withiven geometrical characteristics must be solved separately be-ause an empirical analytical relationship of qcr to all geometricalnd physical characteristics of an orthotropic ellipsoidal shell wasot determined.

Ganeeva and Kosolapova �142� studied the influence of theeometrical parameters of the shell, the characteristics of the ma-erials, and the deformation of transversel shearing on the stress-train state and on nonaxisymmetrical buckling of thin orthotropicrolate spheroids subjected to uniform external pressure. It washown that the influence of a value of r0 /a on nonaxisymmetricalritical loads for prolate ellipsoidal shell is unimportant, for allonsidered values of E=E2 /E1 in contrast to oblate spheroids.ere r0 is the radius of the pole opening. This effect can be ex-lained because the formation of the waves at the moment ofonaxisymmetrical loss of stability of a prolate ellipsoidal shellakes place mainly in the equator area. Ganeeva and Kosolapova142� presented also the orthographic representations of flexurend normal stresses along the meridian appearing at the momentf nonaxisymmetrical buckling. The greatest normal displace-ents can be observed on the equator. The greatest normal

tresses are on the equator if E=1; but if E=2, they are near theole opening. The greatest dimensionless stresses �11 /E1 are nearhe pole opening if E=0.5, but the greatest �22 /E1 is on the equa-or. The values of �13 /E1 are considerably less than �11 /E1 and22 /E1. The calculation results showed that at the moment ofonaxisymmetrical loss of the stability the shell is in a membranetate along the whole meridian, with the exception of the areasear the pole openings.

4.4 Multilayer Shells. The local stability of two-layered el-ipsoidal shells of revolution was studied by Haliullin �143� on theasis of the equilibrium equations of multilayer, orthotropic ma-erials. He derived a formula for the determination of the criticalxternal pressure and analyzed the influence of geometrical andhysical characteristics on the value of this pressure. The numeri-al analysis of a three-layered prolate spheroidal shell carried outy Grigolyuk and Mamay �85� with the help of a package ofpplied programs of the integrated system KIPR-IBM-PC/AT 2.0

howed that a shell, under uniform external pressure and having ainge support along the equator, lost stability with the formationf n half-waves in the circular direction. The problem was solvedy the application of geometrically nonlinear shell theory. Thenvestigations were also made for a shell subjected to aerody-amic loading. This load appears when the shell is a fairing andoves with small velocities and shallow angles of attack. The

hird numerical analysis was made under the condition of simul-aneous loading of a shell by hydrostatic pressure, a point load athe shell top, and two axisymmetrical point forces lying in normallanes. In this case, two important results were obtained: �1� theeformation of the shell by point forces decays quickly with dis-ance from the point of contact with the force; �2� the influence ofydrostatic pressure provoking the bending of the whole shell isather great.

4.5 Joint Action of External Pressure and Additionaloading. If one wants to examine buckling of a nonshallow sphe-

oidal shell subjected to uniform external pressure and to circular
ertical loading along the ring stiffener of the central opening,


then one can use the paper �144� where an algorithm is based onthe method of linearization and orthogonal run. Mikhasev �145�studied a problem of local loss of stability of a thin shallow pro-late truncated spheroid loaded by axisymmetrical normal pressureand by a horizontal force at the top shell edge. The shell was in amembrane stress state. The load on the top shell edge was createdby a body rigidly fixed to the top edge of the shell where a hori-zontal point force and moment, acting in the plane of this forceand the axis of the ellipsoid, were applied at this body. The lowershell edge was rigidly fixed. In this paper �145�, the results ofTovstik �146� are used. Mikhasev affirms that the loss of stabilitycan take place simultaneously both in the neighborhood of onepoint of the equator and near two symmetrical points, dependingon the values of the loading parameters. If by chance this point isfar from the edges, a form of the loss of stability is plotted withthe help of asymptotic methods. As it seen from a table presentedin Ref. �145�, the buckling form is changed as the external hori-zontal force or moment is increased, holding the internal pressureconstant. The smallest value is insignificantly lowered. The el-lipse, determining the area of the lost of stability, is widened in thedirection of the meridian and narrowed in the direction of theparallel. The number of dimples filling the ellipse is decreased.Considering an ellipsoidal shell of revolution �18� with variablethickness

h�s� = h�1 + �� − �/2�2� � � 1 �26�

one can prove that a parallel with minimal thickness along thediameter will be the most sexsitive parallel and the critical valueof an axial tensile force P�0, and the number of waves �n� in thecircular direction can be calculated by a formula presented byTovstik �144�:

Pcr =2�Eh2

�3�1 − �2��1 +�h

a

a2

b2

�� − 1

b2�4 3�1 − �2�+ O�h*�

n � �4 12�1 − �2��a/h/ �27�

Let the same ellipsoid of revolution �18� of variable thickness�26� be under an axial compressive force P�0. Then the criticalvalue of this force is proportional to the modulus, in agreementwith the critical tensile force �27�, though the form of the loss ofstability would be quite different. The torsion of an oblate sphe-roidal shell with rigidly fixed ends was studied in Ref. �113�, butthe buckling modes of one with constant and variable thicknessloaded simultaneously by an axial force P and uniform pressure qcan be found in Ref. �114�. In this monograph, a formula for thedetermination of the critical values Pcr and qcr is presented.

4.6 More Accurate Methods of Buckling Analysis of Ellip-soidal Shells. Ganiev and Cherevatskiy �147� showed that in themethod used to determine critical loads in local stability theory, itwas necessary to fulfill an additional inequality involving thewave number of a buckling mode along the meridian. They de-rived an accurate expressions for the critical values of internalpressure on an oblate ellipsoid. Wunderlich et al. �148� consideredthe bifurcation pressure of a spheroid and showed that the semi-axial ratio 2:1 is the most optimal form. Alekseeva and Ganeeva�149� obtained numerical results for ellipsoidal segments withhinge fastening and also rigid fixation of boundary contours. Theyinvestigated also the influence of geometrical parameters on thevalues of the critical loads. The equations of neutral equilibriumwere integrated �150� by first decomposing the unknown functionsinto trigonometric series along the arc coordinate and then usingthe method of orthogonal run. They took into consideration onlyuniform external pressure. A stability investigation of spheroidsby FEM was carried out by Royles and Llambias �27�. Ross et al.�151� studied the buckling of plastic hemiellipsoidal dome shellsunder external hydrostatic pressure.

Component shells of revolution subjected to the action of inter-

NOVEMBER 2007, Vol. 60 / 347


nmsecmebi�dapl2ibsitmtrse

wbmc

5B

hosajcSc�lupoptalb

6

lsm

a

3

Downlo

al pressure are studied in Ref. �152� with 11 references, using theethod of curvilinear nets. This method gave an opportunity to

tudy a component shell without its dismemberment into separatelements, and eliminated the necessity of introducing additionalonditions of contact. Besides, the rate of convergence of the nu-erical method increases considerably because the digitized gov-

rning equations satisfy conditions of rigid displacement. The sta-ility of a whole closed shell in the form of a spheroid undernternal pressure was solved as a test example. Grigorenko et al.152� also presented the results calculated by the formula �16�erived by Mushtari �153� on the basis of a linear approach, andlso data of Brown and Kraus �154�, and Galletly �155�, and com-ared these results with theirs. The computed values of criticaloads for the closed ellipsoidal shells with the different ratiosa /h=700;750;1000;1250;1500 showed considerable differencen the results obtained by Grigorenko et al. from these presentedy Galletly �155�. This difference is especially large in the disper-ion of the lesser values of the ratio 2a /h, i.e., for thick shells, andt is conditioned by large displacements. However, the results ob-ained due to the method of curvilinear nets coincide approxi-

ately with data in the paper of Brown and Kraus �154� based onhe linear relationships of shell theory. Having used the Rayleighelations and a static criteria of stability, Magnucki et al. �156�olved a stability problem for an ellipsoidal cup with stiffeneddges.

It should be noted that a review of Bakirova and Surkin �157�ith 28 references is devoted to theoretical investigations on sta-ility of ellipsoidal shells. This paper contains an additional infor-ation on the investigations of Kan, Cohen, Danielson �121�, Sa-

henkov, and Bakirova.

Stability Problems of Ellipsoidal Shell Heads andottoms of Cylindrical Reservoirs and Pressure VesselsHeads may be of many shapes such as flat, conical, toriconical,

emispherical, torispherical, reverse dished, or ellipsoidal. Grig-renko et al. �152� described a problem of stability of a cylindricalhell with ellipsoidal heads. They ascertained that initial dimplesppeared because of the loss of stability in a zone around theuncture of the ellipsoid with the cylinder, because in this zone theompressive stresses are high due to internal pressure in the shell.imple formulas for buckling pressure and comparison of analyti-al and experimental data are given in Roshe, Alix, and Autrusson158�. Magnucki and Szyc �159� investigated the stability of el-ipsoidal heads of cylindrical tanks under external and internalniform pressure. New formulas determining values of criticalressure were proposed. The calculation was executed by meansf FEM using the COSMOS-M system. They took account oferturbation of the stress-strain state appearing at the junction ofhe cylindrical and ellipsoidal shells �160�. The results of linearnd nonlinear stability analysis were compared with existing ana-ytical solutions. Additional information on buckling analysis cane found in Ref. �154�.

Torispherical ShellsPartial torispherical shells are often used instead of oblate el-

ipsoids because torispherical shells are easy to fabricate and con-ist of closed fragment of a circular torus and two identical seg-ents of the sphere �Fig. 14�.The parametrical equations of the middle surfaces of a sphere

nd torus can be written in the following form:

x = R cos � cos � y = R cos � sin � z = R sin � − �R − r�sin �0

�0 � � − �0 �a sphere�

x = �r0 + r cos ��cos � y = �r0 + r cos ��sin �; z = r sin �

− �0 � �0 �a torus�



A shell height may be defined by �Fig. 14�

b = R − �R − r�sin �0

where �0=arccos r0 /R−r and the equator radius is a=r0+r. Atorispherical surface is characterized by a comparative coefficient

com = b/a

The connection of a toroidal zone with a spherical segment isfulfilled without a bend of the middle surface.

Bentebba et al. �161� presented the results of a buckling analy-sis of component shells based on geometrical nonlinearity. Theproblems are solved by a modified method of finite differenceseliminating rigid displacement errors. Using a method of curvilin-ear nets, Gulyayev et al. �162� studied the stability of torisphericalshells of constant thickness using a geometrically nonlinear analy-sis. The elimination of an error of approximation of covariantderivatives of the functions of rigid displacements results in con-siderable improvement of the numerical results of the method ofcurvilinear nets in comparison with the solutions of a traditionalmethod of finite differences. A buckling analysis was made fortwo types of shells. In the first case, a radius R of the sphericalsurfaces and a shell height 2b along an axis of rotation remainsinvariable but the geometrical parameters r, r0, and an angle �0limiting the torus belt of the shell change. In the second case, theparameters r, r0 were constant, but R and 2b varied. Geometricalparameters varied so that the line of the axial cross section of atorisphere differed little from an ellipse with the analogous overalldimensions. The results showed that a critical external pressureand a number of circular waves increased if the parameter R /hdiminished and the ratio of b /h increased, but values of r0 /h=48.98 and r /h=20.0 remained constant. The dimples becomelocalized in the spherical zone of the shell. Critical loads for tori-spherical shells with r0 /h=489.8 and r /h=200 were obtained.The shells were loaded by internal pressure. It was ascertainedthat the critical load and the number of circular waves increasedwhen R decreased or b increased. The buckling zone occurredaway from the joint in the direction of the torispherical part of theshell. The critical loads turned out greater than in the case ofexternal pressure. The values of critical loads so obtained divergewith critical loads for ellipsoids of revolution the principal curva-tures of which coincide at the equator with corresponding princi-pal curvatures of the torispheres in question if the parameter R isdecreased. Analyzing the nonlinear stability of component shellsof revolution with the help of the method of curvilinear nets,Zhadrasinov �163� came to the conclusion that an empirical ex-pression �16� for the determination of a critical pressure in ellip-tical shells subjected to internal pressure presented by Mushtaripredicts a buckling pressure with satisfactory accuracy for oblateclosed ellipsoidal �a :b=2:1� and torispherical shells.

Let us suppose that a railway cistern consisting of a cylindricalbody and two bottoms is subjected to internal pressure p. In thatcase, the largest normal stresses appearing in the junction of thecylindrical part with the torispherical bottom may be calculated

Fig. 14 A torispherical shell

with the help of empirical formulas presented by Vershinskiy,



T

aatbwbpnapft

7

sanadewsvhbtte

8

sIltAmd�sppwitfzivbmtdwssl

A

Downlo

� =pR

2h

1

�0.26 + 6.2r/R − 0.23r2/R2�if r/R = 0.01 − 0.13

� =pR

2h�1.21 − 2

r

R� if r/R = 0.13 − 0.25

hese stresses decrease if R is decreased or r is increased.The monograph of Gulyayev et al. �162�, the papers of Aylward

nd Galletly �164�, Galletly �165�, and Shield and Drucker �166�re also devoted to the investigation of cylindrical shells joined toorispherical bottoms. In Ref. �164�, the stability of torisphericalottoms under external and internal pressures was investigatedith the help of the computational programs BOSOR 4 and BOSOR 5

ased on a finite difference energy method and the results of ex-erimental tests of stability of several torispherical heads. Lami-ated composite torispherical shells attracted the attention of Soricnd Smojver �167�. Information on the sensitivity of internallyressurized torispherical shells to initial imperfections can beound in Ref. �168�. The papers �88,105� contain materials onorispherical shells that were already mentioned.

Thermal LoadingVlaykov �169� studied the stress-strain state of thick-walled

hells of revolution under asymmetric thermal loading. He appliedn approach based on reducing a three-dimensional problem to aumber of one-dimensional problems by means of expansion ofll factors in a Fourier series along a circular coordinate and aifference approximation along the shell thickness. A numericalxample was given of approved to a thick-walled ellipsoidal shellith rigidly fixed ends. Horoshun et al. �170� analyzed the stress-

train state of an ellipsoidal heat-sensitive shell of revolution ofariable thickness. They presented an analysis of the influence ofeat sensitivity, a change of thickness, conditions of heating, andoundary conditions on the design. It was shown that neglect ofhe dependence of physical-and-mechanical characteristics of ma-erials on the temperature can sometimes result in considerablerrors.

Free and Forced VibrationsFree �natural� nonaxisymmetrical vibrations of a thin elastic

hell in the form of an ellipsoid of revolution were studied byvanov �171�. The shell was closed in the circular direction andimited by two parallels. Ivanov �171� described the influence ofhe shell thickness and the shell form on the minimal frequencies.n analytical method of solution of the problem of free axisym-etrical vibrations of thin ellipsoidal shells of revolution for three

ifferent of boundary conditions is presented by Kosawada et al.172�. An accurate method using the expansion of the functions oftresses and displacements in power and trigonometric series isroposed in a paper of Suzuki et al. �173� for the solution of aroblem of axisymmetric free vibrations of shells of revolutionith variable curvature and thickness along the meridian, includ-

ng an elliptical meridian. The main differential equations of mo-ion and boundary conditions are derived from the variation of theunctional of Lagrange. An analogous problem was solved in Su-uki et al. �174,175�, but for thick shells. In these papers, thenfluence of inertia of rotation and shear deformation on the freeibration frequencies was studied. Both edges were supposed toe rigidly fixed. Hayek and Boisvert �176� presented the axisym-etrical dynamic response for damped shells of various eccen-

ricities and thickness under point and ring surface forces. Theyerived the resulting five coupled partial differential equations,hich were self-adjoint and positive definite. For the case of axi-

ymmetrical motion, these were solved numerically for variouspheroidal shell eccentricities and thickness-to-length ratios for a
arge number of modes.


Some additional information on free vibration of axisymmetri-cal shells may be found in Filippov �177�, Leissa �178�, DiMaggioand Rand �179�, Ross and Johns �180�.

Some conclusion, from the application of the differential equa-tions for free nonaxial vibration of shells in the form of ellipsoidsof revolution are described also in Ref. �181�, where the displace-ments of middle surface points are presented in the form of prod-ucts of exponential and trigonometric functions. The applicationof this method is connected with two constraints: �1� the depen-dencies are correct only for convex shells and �2� the applied loadmust depend only on one parameter. Investigations of the forcedvibrations of ellipsoidal shells fixed on a rigid shaking base are ofinterest �182�. The peculiarities of the numerical determination ofthe lowest frequency of natural vibrations of anisotropic layeredellipsoidal shell of revolution subjected to axisymmetrical loadingare discussed in a paper of Bespalova and Grigorenko �183� onthe basis of the theory of small vibrations. The solution of non-linear dynamic problems is achieved by decomposition into a non-linear axisymmetrical problem of statics and a linearized problemof free vibration parametrically connected with it. The stability ofa forced axisymmetrical vibrations of thin elastic shell in the formof an oblate spheroid subjected to load normal to its surface andharmonic in time is studied in Ref. �184�.

An analytic investigation of the vibration parameters of shallowellipsoidal domes of variable thickness with fixation at the upperpoint or the central concentric area was accomplished by Lin et al.�185� who determined a numerical solution of the differentialequations of motion using the energy approach and the Ritzmethod under corresponding boundary conditions. Using also theRitz method, Kairov �186� considered a vibration problem forthin-walled developable shells and for shells with middle surfacesformed by the rotation of the second order curves. He carried outnumerical investigations of the influence of the form of a meridianand attached bodies on natural frequencies and modes of vibra-tions of the shells. Natural frequencies and free vibration modesof vessels consisting of ellipsoidal shells of revolution and circu-lar plane heads are presented in a paper of Suzuki et al. �187�,where a vibration equation is derived by minimization of the func-tion of Lagrange expressed in terms of unknown boundary quan-tities.

Penzes �188� solved a problem of free vibration of a thin ortho-tropic, oblate spheroidal shell by assuming orthotropic membranetheory and harmonic axisymmetrical motion. The differentialequations of motion were reduced to a single ordinary second-order differential equation with variable coefficients, which wassolved by Galerkin’s method. Gerasimov and Lyukshin �189� ana-lyzed the action of pulsed, high-rate, axisymmetrical loads ontruncated ellipsoidal shells of revolution with filler. One of theshell edges is joined to a rigid diaphragm and another one isjoined to a rigid bottom on which an axial pulsed load acts. Theequations of motion in a mixed form including shearing and iner-tia of rotation are used, and the stress—deformation ratios areapplied according to a theory of small elastic—plastic deforma-tions. They took into consideration only the normal resultant ofthe filler reaction. A solution of the equations of axisymmetricalmotion of the shell is found with the help of a method of straightlines and the Runge–Kutta method. Shulga and Meish �190� de-veloped a variant of the theory of vibration of three-layered shellsunder axisymmetrical loading using independent kinematical andstatical hypotheses for every layer, taking into account transversenormal and shear deformations in the filler. Using the variationalprincipal of Reissner for dynamic processes, they obtained thenonlinear vibration equations. Dzama and Egarmin �191� demon-strate a method of determining the velocity of procession of stand-ing waves in shells having the form of the second order surfacesof positive curvature, including ellipsoids of revolutions. Vibra-tions of an oblate ellipsoidal shell with a central rigid insert rotat-ing with constant angular velocity relatively the vertical axis of
symmetry were studied by Gulyayev et al. �192�. Hayek and Bois-
NOVEMBER 2007, Vol. 60 / 349


vntpcoo2vltd

9l

aenrccpao�

wwnptcecwseuptftrnTstrSphom

wmrsc�rs ortw

3

Downlo

ert �193� set up equations of motion of shells of constant thick-ess under uniform surface forces and moments. The derived sys-em of five partial differential equations is self-adjoint andositively definite. Michelitsch et al. �194� determined a dynami-al potential for the inner points of an ellipsoidal shell. A reviewf Kubenko and Kovalchuk �195� devoted to nonlinear problemsf vibrations of thin shells is of great interest. The review contains23 references. The authors first describe free vibrations, and thenibrations under the action of external periodical and parametricaloads. The determination of nonlinear dynamic shell characteris-ics are discussed based on the analysis of experimental data ofynamic tests of the shells.

Experimental Studies of Ellipsoidal Shells of Revo-ution

E Torroja �Spain� is an active follower of experimental-and-nalytical methods of design of new shell forms. He thought thatxperimental model investigations gave an opportunity to find aecessary shape of a shell and to determine the starting data for itsealization in nature. He supposed precise experimental resultsould be superior to analytical results, and that an experimentould reveal hidden construction flaws. However, Kanderla sup-oses, contrary to Torroja, that it is necessary to apply formsmenable to simple and accurate methods of analysis, and thatnly this may be a basis for the application of a selected form35�.

The first information a wind tunnel test for the determination ofind loads on rigid models in the form of ellipsoids of revolutionas presented in Smirnov et al. �25�. It was motivated by theecessity of determining accurate values of the wind load onneumatic rubber-and-fabric shells. It is very important becausehe compression caused by a wind load can exceed the tensionaused by inner surplus pressure. A rigid model of the truncatedllipsoidal shell of revolution was used for testing. Models wereonstructed with heights of �0=h0 /b=1.72, �0=1, and �0=0.71,here h0 is a height of the elliptical part of the model, and b is a

mall vertical semiaxis of the ellipsoid. A ratio of semiaxes of anllipse was taken as a /b=200 /140=1.43. This ratio is close to theltimate value �a /b�1.41� for soft shells and determines the ap-earance of the one-axis stress areas and a zone of wrinkles. Theruncated ellipsoidal models had base diameters equal to 0.282 mor h0=0.1 m, 0.14 m, 0.24 m, 0.384 m, and 0.4 m. The distribu-ion of pressure on the model surfaces was determined over aange of flow velocities from 10 m /s to 45 m /s and Reynoldsumber Re��1.6–8.2��105 for five values of the velocity head.he flow about the model with comparative height of �0=1 has apecial character. A flow transition ensues if Re�4.1�105. Inhat case, values of lift coefficient and a coefficient of frontalesistance decrease considerably. Analyzing data obtained bymirnov et al. �25� established that �1� a pattern of distribution ofressure on the model surface depends considerably on its per-uniteight, and �2� distribution of pressure does not depend practicallyn Reynolds number, and that is why the data of model testingay be transferred to the real world.Experimental control of results determinined by formula �15�

as carried out by Healey �196�. He tested to failure an aluminumodel of a shell in the form of an ellipsoid of revolution with the

atio of semiaxes b /a=3, but with h /2a=0.015; 2a=5.1 cm. Thehell models collapsed under a pressure of only 20–38% of theritical pressure determined by formula �15�. Hyman and Healey197� presented results of the tests of 33 models of ellipsoids ofevolution made of epoxy resin and subjected to external hydro-tatic pressure. The models had 1 b /a 4 and 0.012 h /a

0.103. The 22 models collapsed under pressures that were 85%f the theoretical values. Bakirova and Surkin �198� described theesults of testing of series of prolate ellipsoidal shells of revolu-ion which had been subjected to external pressure. The shells
ere made of an epoxy compound and reinforced by a kapron net.


The technology of manufacture of such shells was described. Us-ing the materials of testing Bakirova and Surkin proposed a buck-ling pressure-geometrical parameter ratio.

Two experimental devices described by Rachkov �199� weremade for carrying out of series of tests of oblate ellipsoidal shellsof revolutions of 0.15 m and 0.3 m in diameter. A sketch of thedevice is given in Fig. 15. Two lots of elliptical bottoms made ofsteel were tested for the determination of buckling pressure. Thebottoms were 0.3 m in diameter and had thickness varying from1 mm to 6 mm, but the bottoms of 0.15 m in diameter had thick-ness varying from 1 mm to 4 mm. The ratio of the convex part ofthe bottom to the diameter was equal to 0.25. The loss of stabilityof the bottoms was accompanied by the formation of axisym-metrical dimples in the center of a bottom. The pressure-normaldisplacement ratio for thin-walled models was linear until the lossof stability. The relation for thick-walled bottoms remained linearonly for the initial part. Rachkov believes that experimental datacan give an answer for the correct choice of the form of a dimplefor additional study of the loss of stability.

The loading of ellipsoidal shell of revolution by a uniform loadalong ring strips may be realized on a special test bench �200�.The loading of the shell is produced with the help of hydraulicjacks which settle by rows so that they form six strips of the loads.Glushko �201� presented the results of experimental testing thestress-strain state of a large-overall, metal, four-layered, ellipsoi-dal bottom manufactured layer-by-layer hot punching. The diam-eter of the bottom is 1.2 m and its total thickness is 8 cm. Thebottom was subjected to momentary static loading by internalpressure. It was noted that the most stressed zone of the multilayerbottom was on the internal surface at a distance of 0.25–0.55 mfrom the pole.

Royles and Llambias �27� compared theoretical results with ex-perimental results for glass-plastic shells. The shells had an inter-nal pressure and were subjected to hydrostatic external pressure.Murakami et al. �77� described a testing device which permittedelliptical cups to be loaded by forces from above and by horizon-tal forces from below imitating seismic actions. They presentedphotos of the cups loss of stability. Test results were comparedwith theoretical results obtained with the help of FEM. Numerical

Fig. 15 A sketch of the experimental plant consisting of abody made of thick-walled cylinder „1…; supporting ring „2…; anelliptical bottom „3…; thick-walled plane head „4…; rubber camera„6…; a manometer „7…

and experimental methods of determining residual stresses in el-



l�atAbaisTeSoWamwcgsswa

1

pne1Ztroetomat

1

otevssrsp

pttttfomcn

ammp

A

Downlo

ipsoidal glass shells were described in a paper of Bazilevich202�. Solodilov �203� presented data about resonant frequenciesnd modes of vibrations of an ellipsoidal closed shell of revolu-ion filled with water, and also without water, but placed in water.n analogous problem was solved by Ross and Johns �180,204�ut by an analytical method. Infimovskaya and Ponomaryov �205�nalyzed the vibration of an ellipsoidal shell of revolution contact-ng a fluid. It had an equatorial radius equal to 6 cm, the otheremiaxis equal to 15 cm, and a wall thickness equal to 0.05 cm.he shell was fixed in the area of poles and was excited by anlectromagnet vibrator in the sound range. The conclusions ofolodilov �203� about the optimal dimensions of a steel cubic tankf 1 m3 in volume filled by water were taken into consideration.hen only the external surface of the ellipsoid contacts a fluid, an

dditional decrease of resonance frequencies and a shift of theinimum of the lower dispersion curve toward larger number ofaves along the parallel was observed. Amplitude-frequency

urves of a shell vibrating in air and a shell vibrating in liquidive quite different pictures. The excited vibrations of the testedhell for low-frequency resonance forms are characterized by con-iderable reduction of the frequencies if air is replaced by theater on the external or internal surface of the ellipsoid. However,

ppreciable changes in the flexure forms were not observed.

0 Review PapersBesides the works presented in the review and devoted to ex-

erimental investigations of spheroidal shells, it is necessary toote a voluminous review paper of Singer �206�, who analyzed 73xperimental work on shell stability published in the 1970s and980s. His review also contains materials on ellipsoidal shells.arutskiy and Sivak of the Timoshenko Institute of Mechanics of

he National Academy of Science of the Ukraine �207� showed theole of experimental investigations in the development of methodsf shell analysis. They presented 47 published papers devoted toxperimental investigations of the dynamics of shells of revolu-ion. As a rule these experiments were carried out for the purposef assessing the trustworthiness of analytical results. The experi-ental results are shown to motivate the development of more

ccurate methods of analysis and further experimental investiga-ions.

1 ConclusionsThe thin-walled and structural shells considered in this review

ccupy an important place in the architecture of public and indus-rial buildings. They also serve as the heads, cups, bottoms, andnds of reservoirs and vessels. The author has tried to assemblearious investigations on this widely applied class of ellipsoidalhells. The review presents a brief survey of analysis of ellipsoidalhells with references. The author hopes that the presented mate-ials can reduce literature searches and can point to future re-earch. It will be easier now for design and analysis engineers torepare broadened analyses for each shell of interest.

Most structural engineers prefer popular finite element analysisrograms demanded by modern practice. But it would be not ra-ional to rely only upon this method. Kiselyov’s research �208� onhe analysis of truncated axisymmetric ellipsoids subjected to in-ernal pressure reaffirms the well known fact that the stresses ob-ained with the application of traditional approximation differrom real stresses if displacements of the shell as a rigid body canccur. He used a vector approximation and a volume finite ele-ent with eight angles having a matrix of rigidity 96�96. As a

onsequence of his results, continue to researchers for new, alter-ative methods of analysis.

This review may be useful not only for specialists in strengthnalysis and the design of shells, but also for teachers because theodern training of engineers must include an acquaintance withodern design, strength analysis, and erection of shells of com-

licated form �178,209�.



References�1� Beattie, W. H., and Tisinger, R. M., 1969, “Light-Scattering Functions and

Particle-Scattering Factors for Ellipsoid of Revolution,” J. Opt. Soc. Am., 59,pp. 818–821.

�2� Latimer, P., 1975, “Light Scattering by Ellipsoid,” J. Colloid Interface Sci.,53, pp. 102–109.

�3� Podosenov, S. A., and Sokolov, A. A., 1994, “Calculation of Dipole Transfor-mation Factor in the Form of Ellipsoid of Revolution and Dispherical Dipole,”Metrologia �2�, pp. 30–38, in Russian.

�4� Baranov, A. S., 1993, “Local Waves on a Pulsating Ellipsoid of Revolution,”Astron. Rep., 37�6�, pp. 658–662.

�5� Borisov, A. V., Mamaev, I. S., and Kilin, A. A., 2002, “A New Integral in theProblem of Rolling a Ball on an Arbitrary Ellipsoid,” DAN, 385�3�, pp. 338–341.

�6� Belinskii, A. V., and Plokhov, A. V., 1994, “High-Speed Framing CamerasWith a Spherical Rotating Mirror,” Quantum Electron., 24�3�, pp. 250–252.

�7� Rakhmatulin, M. A., Kotova, S. P., and Filkin, V. V., 2002, “Interaction ofLightly Focused Laser Beam With Transparent Dielectric Particles in theShape of Ellipsoid of Revolution,” Proceeding of SPIE, Saratov Fall Meeting,Laser Physics and Photons, Spectroscopy and Molecular Modeling, pp. 47–51.

�8� Lorenzani, S., and Tilgner, A., 2003, “Inertial Instalibities of Fluid Flow inPrecessing Spheroidal Shell,” J. Fluid Mech., 492, pp. 363–379.

�9� Stergiopoulos, S., and Aldridge, K. D., 1984, “Ringdown of Inertia Waves in aSpheroidal Shell of Rotating Fluid,” Phys. Earth Planet. Inter., 36, pp. 17–26.

�10� Schmitt, D., and Jault, D., 2004, “Numerical Study of a Rotating Fluid in aSpheroidal Container,” J. Comput. Phys., 197�2�, pp. 671–685.

�11� Werby, M. F., and Sidorovskaia, N. A., 1994, “Evidence for the Existence ofStrong Bending Modes for Signals Scattered at Oblique Incidence from Sphe-roidal Shells,” ASA 128th Meeting, Austin, TX.

�12� Michelitsch, T. M., Gao, H., and Levin, V. M., 2003, “On the Dynamic Poten-tials of Ellipsoidal Shells,” Q. J. Mech. Appl. Math., 56�4�, pp. 629–648.

�13� Milch, T., 1973, “Forces and Moments on a Three-Axial Ellipsoid in PotentialFlow,” Isr. J. Technol., 11�1–2�, pp. 63–74.

�14� Shelton, G. P., 2003, “Aerial Pyrotechnic Device Having High CapacityShell,” United States Patent Application 20030230211, Kind Code A1, Dec.18.

�15� Krivoshapko, S. N., 1999, “Ellipsoids of Revolution in Constructions of Build-ings and Structures,” Building Materials and Constructions, Vol. 4, p. 40, inRussian.

�16� Goldenveiser, A. L., 1947, “Membrane Theory of an Analysis of Shells Coin-ciding With the 2nd Order Surfaces,” PMM, Vol. XI�2�, pp. 285–290, inRussian.

�17� Krzyzanowski, W., 1976, “Rzut Stereograficzny Powierzchni AlgebraicznychRzedu Drugiego, Krzywoliniowych �Elipsoida, Paraboloida Eliptyczna, Hy-perboloida Dwupoułokowa�,” Zesz. Nauk. AGH �532�, pp. 53–61, in Polish.

�18� Narzullaev, S. A., 1974, “On One Method of Developing a Three-Axial Ellip-soid,” Prikl. geom i inzhen. grafika, Kiev, 18, pp. 43–46, in Russian.

�19� Peter, K., and Reinhard, M., 1971, “Über Isophoten des Ellipsoids und desElliptischen Paraboloids,” Wiss. Z. Tech. Univ. Dresden, 20�4�, pp. 967–974.

�20� Vandev, D., 1992, “A Minimal Volume Ellipsoid Around a Simplex,” Dokl.Bulg. Akad. Nauk, 45�6�, pp. 37–40, in Bulgarian.

�21� Shaidenko, A. V., 1980, “Some Characteristic Properties of an Ellipsoid,” SibMat. Zh., 21�3�, pp. 232–234, in Russian.

�22� Kantur, G. E., Istomina, V. A., and Kantur, L. G., 1995, “An Application ofCircular Cross-Sections of the 2nd Order Surfaces for Solution of PositionalProblems With a Three-Axial Ellipsoid, Krasnodar: Kuban. gos. tehn. un-t,”Ruk. dep. v VINITI 11.07.95, No. 2095–B95, p. 16, in Russian.

�23� Chervyakov, A. V., and Mutriskov, A. Ya, 1994, “A Development of a Surfaceof an Ellipsoid of Revolution, Kazan. gos. tehnol. univ.,” Ruk. dep. v VINITI11.02.94, No. 380-B94, p. 4, in Russian.

�24� Mehmet, P., 1976, “Le Systéme de Projection Centrale-Polaire dont la SurfaceFondamentale est un Ellipsoide de Révolution et ses Surfaces D’incidence,Istanbul. tekn. univ. bul.,” Bull. Techn. Univ. Istanbul, 29�1�, pp. 88–101.

�25� Smirnov, A. M., Milovidov, A. S., Kartashov, V. E., and Petrov, E. G., 1980,“Aerodynamical Investigations of Truncated Ellipsoidal Shells,” Stroit. Me-hanica i Raschet Sooruzheniy, 2, pp. 74–76, in Russian.

�26� Zryukin, V. V., Isgorodin, A. K., Pischik, G. F., and Malbiev, S. A., 2004,“Analysis of Compressed-Air Ellipsoidal Shells Made of Woven Materials,”Izv. Ivanovskogo otd. Petrovskoy akad. nauk i isskustv, Sek. Tehn. Nauk, pp.3–5, in Russian.

�27� Royles, R., and Llambias, J. M., 1986, “Buckling Aspects of the Behaviour ofan Underwater Pressure Vessel,” Appl. Solid Mech., Vol. 1, First Meetings,Glasgow, Mar. 26–27, London, pp. 287–303.

�28� Ballinger, I. A., Lay, W. D., and Tam, W. H., 1995, “Review and History ofPSI Elastomeric Diaphragm Tanks,” AIAA Paper No. 2534.

�29� Menshikov, N. G., 1954, “On a Question of Finding the Optimal Form ofLarge Span Reinforced Concrete Coverings,” Tr. MISI, Moscow, pp. 105–114.

�30� Shkinev, A. N., 1984, Crashes in Building, Stroyizdat, p. 320, in Russia.�31� Golushko, S. K., and Nemirovskiy, Yu. V., 1989, “A Fulfilling the Projects of

Reinforced Shell Constuctions of Minimal Weight,” Vichislit. Problemi Me-haniki, pp. 117–130, in Russian.

�32� 1980, Metal Constructions, N. P. Melnikov, ed., Stroyizdat, p. 776, in Russian.�33� Melnikov, N. P., and Saveliev, V. A., 1980, “A Construction Decision of the

Steel Net dome of a Diameter of 236.5 m,” Sostoyanie i perspektivi prime-neniya v stroit. prostranstv. konstrukziy: Tez. dokl., Sverdlovsk, pp. 37–38, in
Russian.
NOVEMBER 2007, Vol. 60 / 351


3

Downlo

�34� Lipnitskiy, M. E., 1981, Dome Coverings for Building in a Condition of SevereClimate, Stroyizdat, p. 136, in Russian.

�35� Gohar-Harmandaryan, I. G., 1972, Large Span Dome Buildings, Stroyizdat,Moscow, p. 150, in Russian.

�36� Zetlin, L., 1966, “New Suspension Systems in the U. S. A. Applied to SpatialConstructions,” Bolsheproletn. obolochki: Mezhd. kongres v Leningrade,Stroyizdat, pp. 135–147, in Russian.

�37� Dishinger, F., 1932, Thin-Walled Reinforced Concrete Domes and Vaults, Gos-stroyizdat, p. 270, in Russian.

�38� Sanchez, A., 1964, Shells, Stroyizdat, Moscow, p. 172, in Russian.�39� 1963, Thin-Walled Construction, G. Colonnetti, ed., p. 96, in Russian.�40� Roessler, S. R., and Bini, D., 1986, “Thin Shell Concrete Domes,” Ann. Appl.

Probab., 8�1�, pp. 49–53.�41� Carvalho, I. A., Jr., and Bastos-Netto, D., 1990, “Weight Analysis of Thin

Ellipsoidal Pressure Vessels,” ASME J. Pressure Vessel Technol., 112�2�, pp.187–189.

�42� Simpson, H., and Antebi, J., 1966, “An Investigation of Complex Shells by aFinite Element Method,” Bolsheprolet. obolochki: Mezd. kongress v Lenin-grade, Stroyizdat, Vol. 1, pp. 151–164, in Russian.

�43� Krivoshapko, S. N., 1998, “Drop-Shaped, Cathenoidal, and Pseudo-SphericalShells,” Montazhn. i spetzial. raboti v stroitelstve, pp. 11–12 and 28–32, inRussian.

�44� Vlasov, V. Z., 1939, “An Analysis of Shells in the Form of Central Surfaces ofthe Second Order,” Plastini i obolochki, GSI, pp. 27–40, in Russian.

�45� Sokolovskiy, V. V., 1943, “Equilibrium Euations of Membrane Shells,” PMM,Vol. VII, No. 4, in Russian.

�46� Rabotnov, Yu. N., 1946, “Some Solutions of Membrane Shell Theory,” PMM,Vol. X, No. 5–6, in Russian.

�47� Pavilaynen, V. Ya, 1971, “Analysis of Nonshallow Spherical and EllipsoidalShells on Rectangular Contour by a Membrane Theory,” Tr. LGU, No. 8, pp.99–108, in Russian.

�48� Logan, D. L., and Hourani, M., 1983, “Membrane Theory for Layered Ellip-soidal Shells,” ASME J. Pressure Vessel Technol., 105�4�, pp. 356–362.

�49� Krivoshapko, S. N., 1998, “Shells of Revolution of Nonzero Gaussian Curva-ture,” Montazhn. i spetzial. raboti v stroitelstve, Vol. 10, pp. 28–31, in Russian.

�50� Timoshenko, S. P., and Woinowsky-Krieger, S., 1963, Plates and Shells, Fiz-matgiz, p. 636, in Russian.

�51� Novozhilov, V. V., Chernih, K. F., and Mihaylovskiy, E. I., 1991, A LinearTheory of Thin Shells, p. 656, in Russians.

�52� Stolyarchuk, V. A., 1977, “The Determination of One Class of Shells of Revo-lution Subjected to Internal Uniform Pressure,” Prikl. problemi prochnosti iplastichnosti, 7, pp. 104–108, in Russian.

�53� Petuhov, V. N., and Shevchenko, N. V., 1980, “On a Stress State of a ShallowEllipsoid of Revolution,” Teplovaya zaschita inzhen. soor. i kommunikatziyKraynego severa, pp. 99–103, in Russian.

�54� Pavilaynen, V. Ya., 1961, “On an Analysis of an Ellipsoidal Dome Under WindLoad by a Membrane Theory,” Stroit. meh. i raschet soor. �3�, pp. 38–42, inRussian.

�55� Chausov, N. S., 1950, “The Application of Vlasov’s Theory to an Analysis ofMembrane Ellipsoidal Domes,” Sb. TsNIIPS, Issledovaniya po voprosam teoriii proektir. tonkosten. konstukziy, pp. 78–104, in Russian.

�56� Chausov, N. S., 1954, “An Analysis of Membrane Shells in the Form of anEllipsoid,” Sb. TsNIIPS, Issledovaniya po stroit. mehanike, in Russian.

�57� Clark, R. A., and Reissner, E., 1957, “On stresses and Deformations of Ellip-soidal Shells Subjected to Internal Pressure,” J. Mech. Phys. Solids, 6�1�.

�58� Schmidt, R., 1959, “A Series Solution for Ellipsoidal Shells,” Trans. ASME,E26�3�.

�59� Ganeeva, M. S., 1979, “On an Analysis of Shells of Revolution,” Statika idinamika obolochek, Tr., seminara, Kazan, 12, pp. 143–152, in Russian.

�60� Shuntian, C., 1982, “Calculation of Membrane Stress of Elliptical Shells,”Water Power �12�, pp. 34–39, in Chinese.

�61� Novozhilov, V. V., 1946, “An Analysis of Shells of Bodies of Revolution,” Izv.AN SSSR, OTN, 7, pp. 949–962, in Russian.

�62� Vlasov, V. Z., 1949, A General Shell Theory and its Application in Technics, p.784, in Russian.

�63� Vlasov, V. Z., 1937, “On Analysis of Shells of Revolution Under ArbitraryNonaxisymmetrical Load,” Proekt i standart �3�, pp. , in Russian.

�64� Chausov, N. S., 1949, “An Application of VZ Vlasov’s Theory to an Analysisof Ellipsoidal and Spherical Domes,” Uchenie tr. TsNIIPS, pp. 135–139, inRussian.

�65� Kuotong, C., 1982, “The Thin Ellipsoidal Shell Supported by Pencil of Tubes,”Chin. J. Mech. Eng., 18�1�, pp. 1–14, in Chinese.

�66� Tarnai, T., 1980, “A Hejak Membranallapotanak Letezesi es EqyertelmusegiFelteteleirol. III. Elliptikus Hejak,” Musz. tud., 56�3–4�, pp. 379–410, in Hun-garian.

�67� Mamedov, I. S., 1966, “A Stress State of an Ellipsoidal Shell of RevolutionWith Large Deformations,” Tr. VI Vses. konf. po teorii obolochek i plastin,Baku, Nauka, pp. 561–564, in Russian.

�68� Ganeeva, M. S., and Skvortsova, Z. V., 2002, “A Membrane Stress-Strain Stateof Ellipsoidal Shells of Revolution,” Tez. dokl., p. 312.

�69� Avdonin, A. S., 1969, Applied Methods of Analysis of Shells and Thin-WalledConstructions, p. 404, in Russian.

�70� Berg, H.-G., 1979, “Tragverhalten und Formfindung Versteifter Kuppelschalenüber Quadratischen Grundriss auf Einzelstutzen,” Diss. Dokt., Ing. Univ. Stut-tgart.

�71� Ermakovskaya, I. P., Maksimyuk, V. A., and Chernishenko, I. S., 1987, “In-
vestigation of Nonlinear Axisymmetrical Deforming of Concentrically Loaded


Ellipsoidal Shells, Prikl. Meh., Kiev,” Ruk. dep. v VINITI 27.05.87, No. 3811-B87, in Russian.

�72� Gorlach, B. A., and Mokeev, B. V., 1977, “Axisymmetrical Deformation ofShells of Revolution With Geometrical Nonlinearity,” Kuybishev, 3, pp. 63–68, in Russian.

�73� Ganeeva, M. S., Kosolapova, L. A., and Moiseeva, V. E., 2000, “A. NumericalInvestigation of Deforming Elastic-Plastic Shells of Revolution With the PoleUnder Nonaxisymmetrical Heat-and-Force Loading,” Aktual. probl. mehanikiobolochek, Tr. Mezhd. konf., pp. 151–157, in Russian.

�74� Ganeeva, M. S., Kosolapova, L. A., and Moiseeva, V. E., 1998, “A. StrengthAnalysis of Elastic-Plastic Shells of Revolution Under NonaxisymmetricalHeat-and-Force Loading,” Aktual. probl. mehaniki obolochek, Tr. Mezhd.konf., Unipress, pp. 35–42, in Russian.

�75� Efremov, V. G., 1996, “A. Stress State of Ideally Plastic Space Weakened byElliptical Cavity,” Izv. ITA Chuvashskoy respubliki, 1�2�, pp. 61–67, in Rus-sian.

�76� Paliwal, D. N., Gupta, R., and Anuj, J., 1992, “The Analysis of an OrthotropicEllipsoidal Shell on an Elastic Foundation,” Int. J. Pressure Vessels Piping, 51,pp. 133–141.

�77� Murakami, T., Yoshizawa, H., Hiravawa, H., Sakurai, A., and Nakamura, H.,1987, “Buckling Strength of Elliptical Heads With Core Support StructureUnder Vertical and Horizontal Loads,” Struct. Mech. React. Technol.: Trans.Ninth International Conference, Lausanne, 17–21 Aug., Rotterdam, Boston,Vol. E, pp. 179–184.

�78� Wu, H.-F., and Wang, Z.-R., 2000, “Elastic Deformation of Ellipsoidal Shell ofDifferent Axis Ratio Under Pressure,” J. Harbin Inst. Technol., 7�2�, pp. 48–51.

�79� Galustchak, O. V., and Koshevoy, I. K., 1975, “A. Stress State of EllipsoidalShells With Linearly Changing Thickness Weakend by a Ring Opening,” Prikl.Mechanika, 11�5�, pp. 122–125, in Russian.

�80� Georgievskiy, V. P., Guz, A. N., Maksimyuk, V. A., and Chernyshenko, I. S.,1989, “Numerical Analysis of Nonlinearly Elastic State Near the Openings inOrthotropic Ellipsoidal Shells,” Prikl. Mehanika �Kiev�, 25�12�, pp. 47–52, inRussian.

�81� Grigorenko, Ya. M., and Vasilenko, A. T., 1971, “On Solution of the Problemsof Axisymmetrical Deformation of Layer Anisotropic Shells of Revolution,”Prikl. Mech., 7�8�, pp. 3–8, in Russian.

�82� Abramidse, E. A., 1989, “A. Numerical Solution of a Problem on Deformationof an Elastic Layered Ellipsoidal Shell With Taking Into Consideration Trans-versal Shearing,” Prikladnaya mech. �Kiev�, Dep. v VINITI No. 1332-B89.

�83� Kamalov, A. Z., and Teregulov, I. G., 1984, “Analysis of Stresses in BondedShells of Revolution, Kazan: KazISI,” Ruk. dep. v BINITI 19.08.1984, No.5220-84Dep, in Russian.

�84� Vohmyanin, I. T., 2002, “On Rational Design of Two-Layer Shell Under aCondition of Equal Strength,” Problemi optimal. proektirovaniya soor. Dokl. 4Vseros. seminara, Novosibirsk, Apr. 3–5, pp. 79–89, in Russian.

�85� Grigolyuk, E. I., and Mamay, V. I., 1997, Nonlinear Deforming of Thin-WalledConstructions, Nauka, Moscow, p. 272, in Russian.

�86� Mamay, V. I., 1994, “Nonlinear Deforming of Ellipsoidal Shells Under LocalLoading,” Tr. Mezhd. konf. po sudostroeniyu, Oct. 8–12, Vol. C, pp. 242–249,in Russian.

�87� Koupriyanov, V. V., 2000, “General Equations of a Theory of Elasticity Re-ferred to Coordinates of Oblate Ellipsoid of Revolution,” Stroit. mehanikainzhen. konstr. i soor., 9, pp. 35–37, in Russian.

�88� Ahmerov, E. F., Rizvanov, R. G., and Sharafiev, R. G., 1994, “An Investiga-tion of a Stress State of Bottoms of Diverse Form,” Materiali 45 nauch.-tehn.konf. studentov, aspir. i molod. uchenih Ufim. gos. neft. univ., Ufa, p. 52, inRussian.

�89� Eltyshev, V. A., 1987, “A. Boundary Effect in Anisotropic Cylindrical PressureVessels With Elliptical Bottoms,” Chislennie metodi v issledov. napryazheniy ideformaziy v konstrukziyah, Sverdlovsk, pp. 65–69, in Russian.

�90� Lahtin, A. A., 1985, “The Stress-Strain State of a Vessel Under Vertical Over-loading,” Issledov. prostranstvennih konstrukziy, Sverdlovsk, pp. 48–51, inRussian.

�91� Gordon, C. J., and Zhi, Y., 1996, “Stress Analysis of Two-Arc ApproximateEllipsoidal Pressure Vessel Heads and Parameter Optimization,” Int. J. Pres-sure Vessels Piping, 67�2�, pp. 199–202.

�92� Guodong, C., 1985, “The Uniformly Valid Asymptotic Solution of EllipsoidalShell Heads in Pressure Vessels,” ASME J. Pressure Vessel Technol., 107�1�,pp. 92–95.

�93� Cloclov, D., 1983, Recipiente sub presiure: Analisa stării de tensiune si defor-matie, p. 354, in Romanian.

�94� Yakovlev, B. N., 1977, “Asymptotic Investigation of Deformations andStresses of a Ring, Strengthening a Joint of an Elliptical Bottom and a ConicalShell,” Sb. nauch. tr. Dnepropetrovsk. univ, Meh.-mat. fak-t, 8, pp. 153–157,in Russian.

�95� Yakovlev, B. N., 1977, “Asymptotic Investigation of Deformations of Ellipti-cal Bottom and Conical Shell,” Sb. nauch. tr. Dnepropetrovsk. univ, Meh.-mat.fak-t, 8, pp. 157–161, in Russian.

�96� Kantor, B. Y., and Belov, S. A., 1990, “Bearing Capacity of Ellipsoidal Bot-toms in the Area of Support Pillars,” Issled. v oblasti prochnosti him. oborud.,pp. 48–54, in Russian.

�97� Hu, C., 1992, “Theoretical Analysis for the Optimum Structure of RotativeShell of Boilers, Haerbin diangong xueuan xuebao,” J. Harbin Inst. Technol.,15�3�, pp. 242–248.

�98� Chao, Y. J., and Sutton, M. A., 1985, “Stress Analysis of Ellipsoidal Shell With
Radial Nozzle,” Int. J. Pressure Vessels Piping, 21�2�, pp. 89–108.


�

�

�

�

�

�

�

�

�

�

�

�

�

�

�

�

�

�

�

�

�

�

�

�

�

�

�

�

�

�

�

A

Downlo

�99� Golushko, S. K., and Nemirovskiy, Yu. V, 1990, “Rational Designing of Com-ponent Reinforced Shells of Revolution,” Prostranstv. konstruksii v Krasno-yarsk. krae, pp. 101–108, in Russian.

100� Smith, C. O., 1980, “Design of Ellipsoidal and Toroidal Pressure Vessels toProbabilistic Criteria,” ASME J. Mech. Des., 102�4�, pp. 787–792.

101� Antipov, V. A., Lahtin, A. A., Pavlenko, S. T., and Pletnyov, V. P., 1978,“Optimal Strengthening a Round Opening at the Top of a Elliptical Bottom,”Issledovanie prostrans, konstruktsiy, Sverdlovsk, 2, pp. 71–79, in Russian.

102� Alekseeva, O. V., and Ganeeva, M. S., 1977, “Large Displacements andStability of an Ellipsoidal Bottom Under Circular Vertical Loading,”Prochnost i ustoych. obolochek, Tr. semim. Kaz. him.-tehn. in-ta i Kaz. fil. ANSSSR, Vol. 9, pp. 70–75, in Russian.

103� Bandurin, N. G., and Nikolaev, A. P., 1990, “Analysis of AxisymmetricallyLoaded Shells of Revolution With Taking Into Account Material and Geo-metrical Nonlinearities With the Help of F. E. M.,” Rascheti na prochnost,Moscow, Vol. 31, pp. 135–144, in Russian.

104� Gerasimov, V. P., 1982, “Bearing Capacity of Spherical and EllipsoidalShells Weakened by the Flange Openings,” Nadezhnost i dolgovechnostmashin i sooruzheniy, Kiev, No. 2, pp. 23–25, in Russian.

105� Yeom, D. J., and Robinson, M., 1996, “Numerical Analysis of Elastic-PlasticBehaviour of Pressure Vessels With Ellipsoidal and Torispherical Heads,” Int.J. Pressure Vessels Piping, 65�2�, pp. 147–156.

106� Sorkin, L. S., 1985, “The Determination of Residual Stresses,” Problemiprochnosti, Kiev, Ruk. dep. v VINITI 31.5.1985, No 3824-85Dep, pp. 1, inRussian.

107� Mileykovskiy, I. E., and Selskiy, Y. S., 1970, “Stresses Near the Opening ona Surface of a Ellipsoidal Shell,” Tr. VII. Vses. konf. po teorii obolochek iplastinok, Nauka, Moscow, pp. 428–431, in Russian.

108� Ganeeva, M. S., and Moiseeva, V. E., 2002, “Deforming of Shells of Revo-lution of Negative and Positive Gaussian Curvature Under Nonaxisymmetri-cal Loading,” Probl. prochn. i plastichnosti, 64, pp. 46–50, in Russian.

109� Gerasimov, V. P., 1983, “The Ultimate Equilibrium of Ellipsoidal ShellsWeakened by Flanged Opening Under Internal Pressure,” Prochnost idolgovechnost elementov konstruktsiy: Sb. nauchn. tr., Kiev: “Naukovadumka,” pp. 76–78, in Russian.

110� Gramoll, Kurt, �1993�, “Stress Analysis of Filament Wound Open-EndedComposite Shells,” Proceedings of the 34th Struc. Dynamics, La Jolla, CA.

111� Mushtari, H. M., 1951, “On Elastic Equilibrium of a Thin Shell With InitialIrregularities of the Form of a Middle Surface,” PMM, 15�6�, pp. 743–750,in Russian.

112� Shirshov, V. P., 1962, “The Local Stability of Shells,” Teoriya plastin i obo-lochek: Tr. II Vses. konf., Kiev, pp. 314–317, in Russian.

113� Tovstik, P. E., 1970, “A Stability of Shells of Revolution Under Linear Ap-proximation,” Raschet prostranstv. konstruktsiy, Stroyizdat, Vol. XIII, pp.118–138, in Russian.

114� Tovstik, P. E., 1995, Stability of Thin Shells: Asymptotic Methods, Nauka,Moscow, pp. 320, in Russian.

115� Surkin, R. G., 1955, “To a Theory of a Prolate Ellipsoidal Shell of RevolutionUnder External Uniform Pressure,” Izv. Kazan. fil. AN SSSR, Fiz.-mat. ser.,Kazan, Vol. 7, pp. 3–15, in Russian.

116� Mushtari, H. M., and Galimov, K. Z., 1957, “Nonlinear Theory of ElasticShells,” Kazan, pp. 431, in Russian.

117� Pogorelov, A. V., 1967, Geometrical Methods in a Nonlinear Theory of Elas-tic Shells, Nauka, Moscow, pp. 280, in Russian.

118� Volmir, A. S., 1967, The Stability of Deforming System, Nauka, Moscow, pp.984, in Russian.

119� Fidrovskaya, N. N., 2000, “Stability of an Ellipsoidal Shell,” Ukr. ing.-ped.akad., Harkov, Dep. v GNTB Ukraini 20.09.2000, No. 172-Uk2000, pp. 6.

120� Pogorelov, A. V., 1986, Bending the Surfaces and Shell Stability, Nauka,Moscow, pp. 96, in Russian.

121� Danielson, D. A., 1969, “Buckling and Initial Postbuckling Behaviour ofSpheroidal Shells Under Pressure,” AIAA J., 7�5�, pp. 936–944.

122� Surkin, R. G., 1952, “To a Theory of Stability and Strength of Spherical andEllipsoidal Shells, Bottoms, and Membranes,” Ph.D. thesis, in Russian.

123� Sachenkov, A. V., 1962, “On One Approach to the Solution of NonlinearProblems of Stability of Thin Shells,” Nelineynaya teoriya plastin i obo-lochek, Kazan, KGU, pp. 3–41, in Russian.

124� Hyman, B. J., 1965, “Elastic Instability of Prolate Spheroidal Shells UnderUniform External Pressure,” David Taylor Model Basin, Report No. 2105.

125� Alumyae, N. A., 1956, “Critical Pressure of Elastic Shell of Revolution in theForm of Ellipsoidal Surface,” Izv. AN Estonskoy SSR, ser. tehn. i fiz.-mat.nauk, 5�16�, pp. 175–190, in Russian.

126� Bakirova, A. Z., 1973, “On Stability of Closed Prolate Shells of RevolutionUnder Uniform External Pressure,” Tr. semin. po teorii obolochek, Kazan,Kaz. fiz.-tehnol. in-t, 3, pp. 266–277, in Russian.

127� Bakirova, A. Z., and Surkin, R. G., 1973, “On a problem of Stability ofProlate Shells of Revolution Under External Uniform Pressure,” Teoriya obo-lochek i plastin, Tr. VIII Vses. konf. po teorii obolochek i plastin, Nauka,Moscow, pp. 226–230.

128� Krivosheev, N. I., 1975, “On Stability of Ellipsoidal Shells of Revolution,”Prochnost i zhestkost tonkosten. konstrukziy, pp. 74–79, in Russian.

129� Krivosheev, N. I., and Murtazin, R. Z., 1975, “On Stability of EllipsoidalShells,” Tr. seminar. po teorii obolochek, Kazan: Kaz. fiz.-tehn. in-t, pp.125–129, in Russian.

130� Krivosheev, N. I., 1977, “On Stability of Prolate Ellipsoidal Shells,”Prochnost i zhostkost tonkost. konstruktsiy, Kazan: Kaz. fiz.-tehn. in-t, Vol. 2,
Ruk. dep.v VINITI 25.10. 1977, No. 4101-77Dep, pp. 114–120, in Russian.


�131� Kabritz, S. A., and Terentiev, V. F., 1988, “On Some Features of the Appli-cation of Methods of Continuation on a Parameter for a Problem on a Ellip-soidal Shell of Revolution,” Matem. metodi, metodi resheniya i optim. proek-tir. gibkih plastin i obolochek, Saratov, pp. 9–11, in Russian.

�132� Gulyayev, V. I., Soloviov, L. L., and Belova, M. A., 2004, “Critical States ofThin Ellipsoidal Shells in Simple and Compound Rotations,” J. Sound Vib.,270�1�, pp. 32–340.

�133� Rachkov, V. I., 1966, “Stability of Ellipsoidal Bottoms Under External Pres-sure,” Voprosi prochn. v himich. mashinostr., Tr. NIIHIMMASh, No. 50, inRussian.

�134� Abdulhakov, M. K., and Ganiev, N. S., 1990, “Stability of Composite Shellsof Revolution,” Raschet plastin i obolochek v him. mashinostr., Kazan: Kaz.him.-tehn. in-t, pp. 3–8, in Russian.

�135� Abdulhakov, M. K., 1987, “Stability of a Composite Ellipsoidal Shell WithTaking Into Account Deformation of Transverse Shearing,” Kazan: Kaz.him.-tehn. in-t, Ruk. dep. v VINITI No. 6815-B87, 23.09., 87, pp. 13, inRussian.

�136� Segal, B. L., and Cherevatskiy, S. B., 1966, “Thread Nets on a Surface,” Tr.V. I. Vses. konf. po teorii obolochek i plastin, Nauka, Moscow, pp. 749–753,in Russian.

�137� Ganiev, N. S., 1981, “A Local Stability of Glass-and-Plastic Shells of Revo-lution With Taking Into Consideration the Deformation Of Transverse Shear-ing,” Issledovoniya po teorii plastin i obolochek, Kazan, No. 16, pp. 106–109, in Russian.

�138� Vinson, J. R., 1993, The Behavior of Shells Composed of Isotropic and Com-posite Materials, Springer, Berlin, pp. 576.

�139� Vasiliev, A. N., and Ivanov, V. A., 1986, “Stability of Layered EllipsoidalShells of Revolution With Filling,” Prochnost i ustoychivost obolochek, Tr.semin. Kaz. fiz.-tehn. in-ta, Kazan, Vol. 19, Part 2, pp. 37–47, in Russian.

�140� Ganiev, N. S., Gumerova, H. S., and Nurullina, D. A., 1992, “Stability of anOrthotropic Ellipsoidal Shell of Revolution Under Internal Uniform Pres-sure,” Issledovaniya po teorii plastin i obolochek, No. 24, pp. 3–6, in Rus-sian.

�141� Ganiev, N. S., 1982, “Application of the Finite Difference Method to Inves-tigation of Stability of an Orthotropic Ellipsoidal Shell,” Kazan. him.-tehnol.in-t, dep v VINITI No. 5826–82Dep, 25.11.1982, pp. 8, in Russian.

�142� Ganeeva, M. S., and Kosolapova, L. A., 1984, “Nonaxisymmetrical Loss ofStability of Orthotropic Prolate Ellipsoidal and Spherical Shells.” Tr. semin.po teorii obolochek, Kazan, Kaz. fiz.-tehnol. in-t, Vol. 17, pp. 32–45, inRussian.

�143� Haliullin, G. H., 1979, “A Local Stability of Two-Layered Orthotropic Shellsof Revolution,” Kazan. him.-tehnol. in-t, Kazan, Ruk. dep. v VINITI25.12.1979, No. 4409-79Dep, pp. 16, in Russian.

�144� Alekseeva, O. V., Ganeeva, M. S., and Kosolapova, L. A., 1982, “LargeAxisymmetrical Displacements and Nonsymmetrical Loss of Stability ofNonshallow Shells of Revolution Weakened by an Opening,” KazISI, Kaz.fiz.-tehn. in-t A. N. SSSR, Kazan, Ruk. dep. v VINITI 22.10.1982, No. 5304-82Dep, pp. 18.

�145� Mikhasev, G. I., 1984, “Local Loss of Stability of a Thin Truncated EllipsoidUnder Combined Load,” Vestnik LGU, 19, pp. 85–90.

�146� Tovstik, P. E., 1982, “On a Question of Local Stability of Shells,” VestnikLGU, 13, pp. 72–78, in Russian.

�147� Ganiev, N. S., and Cherevatskiy, A. S., 1989, “On Calculation of CriticalLoads of Shells of Revolution With Application of a Theory of Local Stabil-ity,” Prikladnaya teoriya uprugosti, Saratov, pp. 49–55, �in Russian�.

�148� Wunderlich, W., Rensch, H. J., and Obrecht-Bochum, H., 1982. “Analysis ofElastic-Plastic Buckling and Imperfection-Sensitivity of Shells of Revolu-tion, Buckling of Shells,” Proceedings of a State-of-the-Art Colloq., Univer-sitat Stuttgart, Germany, May 6–7, pp. 137–174.

�149� Alekseeva, O. V., and Ganeeva, M. S., 1979, “Numerical Searching LargeAxisymmetrical Displacements of Thin Nonshallow Shells of Revolution,”Statika i dinamika obolochek, Tr. semin., Kazan, Vol. 12, pp. 92–102, inRussian.

�150� Ganeeva, M. S., and Alekseeva, O. V., 1980, “Numerical Searching Nonaxi-symmetrical Loss of Stability of Elastic Shells of Revolution,” Tr. semin. poteorii obolochek, Kazan, Kaz. fiz.-tehnol. in-t, Vol. 13, pp. 89–99, in Russian.

�151� Ross, C. I. F., Youster, R., and Sadler, J. R., 2001, “The Buckling of PlasticHemi-ellipsoidal Dome Shells Under External Hydrostatic Pressure,” OceanEng., 28�7�, July, pp. 789–803.

�152� Grigorenko, Y. M., Gulyayev, V. I., and Gotzylyak, E. A., 1982, “A Numeri-cal Investigation of Stability of Component Shells of Revolution Under In-ternal Pressure,” Nadezhn. i dolgovechnost mashin i sooruzheniy, Kiev, pp.15–23, in Russian.

�153� Mushtari, H. M., 1939, “Some Generalizations of Thin Shell Theory WithApplication to a Problem of Stability of Elastic Equilibrium,” PMM, 2�4�,pp. 18–21, in Russian.

�154� Brown, K. W., and Kraus, H., 1976, “Stability of Internally-Pressurized Ves-sels With Ellipsoidal Heads,” ASME J. Pressure Vessel Technol., 98, pp.157–161.

�155� Galletly, G. D., 1978, “Elastic and Elastic-Plastic Buckling of InternallyPressurized Ellipsoidal Shells,” ASME J. Pressure Vessel Technol., 100, pp.335–343.

�156� Magnucki, K., Wegner, T., and Szyc, W., 1988, “On Buckling of EllipsoidalCups Under Internal Pressure,” Ing.-Arch., 58�5�, pp. 339–342.

�157� Bakirova, A. Z., and Surkin, R. G., 1977, “On Stability of Ellipsoidal andSpheroidal Shells of Revolution Under External Uniformly Distributed Pres-
sure,” Prochnost i ustoych. obolochek, Tr. semim. Kaz. fiz.-tehn. in-ta i Kaz.
NOVEMBER 2007, Vol. 60 / 353


�

�

�

�

�

�

�

�

�

�

�

�

�

�

�

�

�

�

�

�

�

�

�

�

�

�

3

Downlo

fil. AN. SSSR, Vol. 9, pp. 5–16, in Russian.158� Roshe, R. L., Alix, M., and Autrusson, B., 1984, “Design Rules Against

Buckling of Dished Heads,” Fifth International Conference Pressure VesselTechnology, San Francisco, CA, Sept. 9–14, 1984, Vol. 1, pp. 274–289.

159� Magnucki, K., and Szyc, W., 1999, “StabilityProblems of Pressure VesselEllipsoidal Heads,” Arch Budowy Maszyn, 46�1�, pp. 43–55.

160� Magnucki, K., and Szyc, W., 2000, “Stability of Ellipsoidal Heads of Cylin-drical Pressure Vessels,” Appl. Mech. Eng., 5�2�, pp. 389–404.

161� Bentebba, M. T., Gaydaychuk, V. V., and Gotsulyak, E. A., 1985, “Stabilityof Torispherical Shells Under External and Internal Pressure,” Mashinostro-enie, 12, pp. 25–28, in Russian.

162� Gulyayev, B. I., Bazhenov, V. A., Gotsulyak, E. A., and Gaidaichuk, V. V.,1990, “An Analysis of Shells of Complex Form,” Kiev: Budivelnik, pp. 192,in Russian.

163� Zhadrasinov, N. T., 1982, “Investigation of Nonlinear Stability of ComponentShells of Revolution,” Soprot. materialov i teoriya soor., �Kiev�, 40, pp.110–114, in Russian.

164� Aylward, R. W., and Galletly, G. D., 1978, “Elastic Buckling and First Yield-ing in Thin Torispherical Shells Subjected to Internal Pressure,” AppliedMechanics Division, Liverpool, No. 4, pp. 321–336.

165� Galletly, G. D., 1981, “Plastic Buckling of Torispherical and EllipsoidalShells Subjected to Internal Pressure,” Proc. Inst. Mech. Eng., 195�26�, pp.329–345.

166� Shield, R. T., and Drucker, D. G., 1961, “Design of Thin-Walled Torispheri-cal and Toriconical Pressure Vessels Heads,” Trans. ASME �2�, pp. 292–297.

167� Soric, J., and Smojver, I., 1995, “On the Effect of Fibre Directions in Lami-nated Composite Torispherical Shells,” Eng. Comput., 12, pp. 85–94.

168� Soric, J., 1995, “Imperfection Sensitivity of Internally Pressurized Torispheri-cal Shells,” Thin-Walled Struct., Vol. 23, pp. 57–66.

169� Vlaykov, G. G., 1980, “The Stress State of Thick-Walled Shells of Revolu-tion Under Nonuniform Heat Loading,” Prikl. Mekh., 16�8�, pp. 116–119, inRussian.

170� Horoshun, L. P., Kozlov, A. V., and Patlashenko, I. Y., 1988, “A Stress-StrainState of Heat-Sensitive Shells of Revolution of Changing Thickness,” Prikl.Mekh., 24�9�, pp. 38–44, in Russian.

171� Ivanov, Y. P., 1971, “On the Lowest Frequency of Vibration of a ConvexShell of Revolution,” Prikl. Mekh., 7�5�, pp. 131–134, in Russian.

172� Kosawada, T., Suzuki, K., and Takahashi, S., �1986�, “Axisymmetric FreeVibrations of Shells of Revolution Having General Meridional Curvature,”Trans. Jpn. Soc. Mech. Eng., Ser. C, 52�473�, pp. 209–215, in Japanese.

173� Suzuki, K., Miura, K., and Kosawada, T., 1988, “Asymmetric Vibrations ofShells of Revolution Having Meridionally Varying Curvature and Thick-ness,” Trans. Jpn. Soc. Mech. Eng., Ser. C 54�508�, pp. 2830–2836, in Japa-nese.

174� Suzuki, K., Takahashi, F., Kosawada, T., and Takahashi, S., 1987, “Axisym-metric Vibration of Thick Shells of Revolution Having Meridionally VaryingCurvature,” Trans. Jpn. Soc. Mech. Eng., Ser. C 53�495�, pp. 2222–2227, inJapanese.

175� Suzuki, K., Yachita, T., and Kosawada, T., 1988, “Asymmetric Vibrations ofThick Shells of Revolution Having Meridionally Varying Curvature,” Trans.Jpn. Soc. Mech. Eng., Ser. C 54�508�, pp. 2822–2828, in Japanese.

176� Hayek, S. I., and Boisvert, J. E., 2003, “Vibration of Prolate SpheroidalShells With Shear Deformation and Rotary Inertia: Axisymmetric Case,” J.Acoust. Soc. Am., 114�5�, pp. 799–811.

177� Filippov, S. V., 1976, “Free Axisymmetrical Vibrations of Shells in the Formof an Ellipsoid of Revolution,” Vestnik LGU, No. 1, pp. 117–121, in Russian.

178� Leissa, A. W., 1973, “Vibration of Shells,” U.S. Government Printing Office,Washington, D.C. �reprinted by the Acoustial Society of America, 1993�, pp.428.

179� DiMaggio, F., and Rand, R. H., 1966, “Axisymmetrical Vibrations of ProlateSpheroidal Shells,” J. Acoust. Soc. Am., 40, pp. 179–186.

180� Ross, C. I. F., and Johns, T., 1986, “Vibration of Hemi-Ellipsoidal Axisym-metric Domes Submerged in Water,” Proc. I. Mech. E., 200, pp. 338–398.

181� Todorovska-Azhievska, L., 1983, “Prilog kon Formiranyeto na Deformaziy-alnite Ravenki za Slobodni Neaksialno-Simmetrichni Oszilazii na RotazioniLushpi Optovareni so Normalen Ramnomeren Nadvoreshen Pritisok,” Zb. tr.Univ. zent. mat.-tehn. nauki. Mash. fak. Skope, 5, pp. 35–41, in Macedonian.

182� Gulyayev, V. I., Kirichuk, A. A., and Sadil, O. M., 1991, “Stability of Non-linear Vibrations of Ellipsoidal Shells Under Kinematic Excitation,” Di-namika i prochnost mashin, 52, pp. 20–24, in Russian.

183� Bespalova, E. I., Grigorenko Y. M., Kitaygorodskiy, A. B., and Shinkar, A.I., 1991, “Free Vibrations of Precast Anisotropic Shells of Revolution,” On-cologica, 27�5�, pp. 51–57, in Russian.



�184� Gulyayev, V. I., Kirichuk, A. A., and Mrue, Z. A., 1988, “Stability of ForcedNonlinear Vibrations of a Closed Ellipsoidal Shell,” Soprotiv. materialov iteoriya soor., 53, pp. 3–6, in Russian.

�185� Lin, C. W., Kitipornchai, S., and Liew, K. M., 1996, “Modeling the Vibrationof a Variable Thickness Ellipsoidal Dish With Central Point Clamp or Con-centric Surface Clamp,” J. Acoust. Soc. Am., 99�1�, pp. 362–372.

�186� Kairov, A. S., 1999, “Influence of a Meridional Form and Attached Bodies toFree Vibration of Shells of Revolution,” Teor. i prikl. mehanika �Kiev�, 29,pp. 117–122, in Russian.

�187� Suzuki, K., Kosawada, T., Uehara, T., and Kumagai, H., 1991, “Free Vibra-tions of a Vessel Consisting of Circular Plates and a Shell of RevolutionHaving Varying Meridional Curvature,” J. Sound Vib., 144�2�, pp. 263–279.

�188� Penzes, L. E., 1969, “Free Vibrationof Thin Orthotropic Oblate SpheroidalShells,” J. Acoust. Soc. Am., Vol. 45, Issue 2 pp. 500–505.

�189� Gerasimov, A. V., and Lyukshin, P. A., 1978, “An Analysis of a Shell ofRevolution With Filling,” Teoriya uprugosti i plastichnosti, Tomsk, pp. 23–25, in Russian.

�190� Shulga, N. A., and Meish, V. F., 2003, “Forced Vibrations of Three-layeredSpherical and Ellipsoidal Shells Under Axisymmetrical Loads,” Meh. ko-mpozit. mater., 39�5�, pp. 659–670, in Russian.

�191� Dzama, M. A., and Egarmin, N. E., 1991, “A Procession of Elastic WavesUnder Rotation of Some Classes of Axisymmetrical Shells,” Izv. AN SSSR.MTT, 1, pp. 170–175, in Russian.

�192� Gulyayev, V. I., Kravchenko, A. G., and Lizunov, P. P., 1989, “Vibrations ofan Oblate Ellipsoidal Shell Under Complex Rotation,” Teoret. i priklad. me-hanika �Harkov�, 20, pp. 81–84, in Russian.

�193� Hayek, S. I., and Boisvert, J. E., 2003, “Vibration of Prolate SpheroidalShells With Shear Deformation and Rotary Inertia: Axisymmetric Case,” J.Acoust. Soc. Am., 114�5�, pp. 2799–2811.

�194� Michelitsch, T. M., Gao H., and Levin, V. M., 2003, “On the Dynamic Po-tential of Ellipsoidal Shells,” Q. J. Mech. Appl. Math., 56�4�, pp. 629–648.

�195� Kubenko, V. D., and Kovalchuk, P. S., 1998, “Nonlinear Problems of Vibra-tions of Thin Shells �A review�,” J. Ship Res., 34�8�, pp. 3–31, in Russian.

�196� Healey, J. J., 1965, “Hydrostatic Test of Two Prolate Spheroidal Shells,” J.Ship Res., 9�2�, pp. 77–78.

�197� Hyman, B. J., and Healey, J. J., 1967, “Buckling Prolate Spheroidal ShellsUnder Hydrostatic Pressure,” Raketn. tehnika i kosmonavtika, 5�8�, pp. 111–120, in Russian.

�198� Bakirova, A. Z., and Surkin, R. G., 1980, “An Experimental Investigation ofStability of Prolate Shells of Revolution Under External Uniform Pressure,”Tr. semin. po teorii obolochek, Kazan, Kaz. fiz.-tehnol. in-t, Vol. 13, pp.142–151, in Russian.

�199� Rachkov, V. I., 1966, “Experimental Investigations of Stability of EllipsoidalBottoms Under External Pressure,” Tr. VI. Vses. konf. po teorii obolochek iplastin, Baku, Nauka, Moscow, pp. 650–655, in Russian.

�200� A prospect of exhibition “Progressive research and constructive working outin ship building,” A. part II. N5.35, A. method and experimental equipmentfor the research of shell strength, L., TsNII “Rumb,” pp. 3, in Russian.

�201� Glushko, I. K., 1977, “Experimental Investigation of Stress-strain State of aMultilayered Elliptical Bottom,” Sb. tr. Vses. zaoch. politeh. in-ta, No. 107,pp. 160–171, in Russian.

�202� Bazilevich, L., 2000, “Investigation of Residual Stresses in an Ellipsoid ofRevolution,” Matem. problemi mehaniki neodnorodn. struktur: Sb., Vol. 2,In-t prikl. probl. meh. i mat. NAN Ukraini, Lvov, pp. 141–144, in Ukrainian.

�203� Solodilov, V. E., 1982, “Experimental Investigations of Vibrations of Shellsof Revolution Interacting With Liquid,” Tr. 2-go Vses. simpos. po fizikeakustiko-gidrodin. yavleniy i optoakustike, Cuzdal, Nauka, Moscow, pp. 321–323, in Russian.

�204� Ross, C. T. E., and Johns, T., 1983, “Vibrations of Submerged Hemi-ellipsoidal Domes,” J. Sound Vib., 91, pp. 363–373.

�205� Infimovskaya, A. A., and Ponomaryov, I. I., 1987, “Vibration of an Ellipsoi-dal Shell in a Contact With Acoustic Medium,” Tr. XIV Vses. konf. po teoriiplastin i obolochek, Oct. 20–23, Vol. 1, pp. 625–630, in Russian.

�206� Singer, J., 1982, “The Status of Experimental Buckling Investigations ofShells,” Buckling shells, Proceedings State-of-the Art Colloq., Univ. Stut-tgart, May 6–7, pp. 501–533.

�207� Zarutskiy, V. A., and Sivak, V. F., 1999, “Experimental Investigations ofDynamics of Shells of Revolution,” Prikl. Mekh., 35�3�, pp. 3–11, in Rus-sian.

�208� Kiselyov, A. P., 2006, “Evolution of FEM in the Investigations of Linear andNon-linear Deforming 2-D and 3-D Elastic Bodies,” DSc dissertation.

�209� Jawed, M. H., 2004, Design of Plate and Shell Structures, �ProfessionalEngineering Publishing �ASME�, New York�, pp. 476.



cccIt

A

Downlo

Sergei Nikolaevich Krivoshapko was born in Volgograd, Russia in July 1948. He completed the Volgogradtechnical secondary school (Civil Engineering) in 1967. After arriving in Moscow in 1967, he entered thePeoples’ Friendship University of Russia, received his Dipl-Eng in Civil and Industrial Engineering fromthe same University in 1972, and remained there as a post-graduate student until 1976. Afterwards, heworked for about nine years as a civil engineer. He took part in design and control of some constructionsof diverse purpose. Krivoshapko received his PhD degree in Engineering Mechanics in 1981 from thePeoples’ Friendship University of Russia. He began as an Assistant Professor there in the Engineeringfaculty in 1984. In 1995, he received his DSc (Eng) degree and became Professor of Engineering Mechan-ics. At present, he is the head of Department of Strength of Materials. Approximately 100 publicationsincluding three monographs, two reference books, two textbooks, two inventions and 11 manuals of strengthof materials and of shell theory resulted from his research. His primary research interests are geometricinvestigations and stress-strain analysis of thin elastic shells of complex form. In 2000, Krivoshapko was

hosen as a Honorary worker of higher vocational education of the Russian Federation. He is a member of a specialized scientificouncil for the defense of PhD and DSc theses in the PFU. He was a member of the organizing committees of several scientificonferences of the Engineering faculty of the PFU and of five International scientific conferences. Prof. Krivoshapko is a member of thenter-Regional Public Organization Spatial Construction (Russia) and editor-in-chief of Structural Mechanics of Engineering Construc-ions and Buildings.

pplied Mechanics Reviews NOVEMBER 2007, Vol. 60 / 355

aded 21 Jan 2008 to 203.88.129.4. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms_Use.cfm

[Kryvoshapko_S.N.]_Research_on_General_and_Axisymm(Bookos.org)[1].pdf

Documents

Transcript of [Kryvoshapko_S.N.]_Research_on_General_and_Axisymm(Bookos.org)[1].pdf