Thin-Walled Structures 96 (2015) 286–294

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Thin-Walled Structures

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Equatorial bending of an elliptic toroidal shell

Alphose Zingoni n, Nosakhare Enoma, Nishalin GovenderDepartment of Civil Engineering, University of Cape Town, Rondebosch 7701, Cape Town, South Africa

a r t i c l e i n f o

Article history:Received 5 August 2015Received in revised form18 August 2015Accepted 18 August 2015

Keywords:Toroidal shellElliptic toroidShell of revolutionBending theory of shellsShell analysisDiscontinuity stressesContainment vessel

x.doi.org/10.1016/j.tws.2015.08.01731/& 2015 Elsevier Ltd. All rights reserved.

esponding author. Fax: þ27 21 650 3293.ail address: alphose.zingoni@uct.ac.za (A. Zing

a b s t r a c t

The exact differential equations for the axisymmetric bending of elliptic toroidal shells are difficult tosolve. In this paper, and by considering a semi-elliptic toroid, we present an approximate bending so-lution that is valid in regions adjacent to the horizontal equatorial plane. The formulation accuratelysimulates edge effects which may arise from loading and geometric discontinuities located in theequatorial plane of elliptic toroids. In particular, the developed closed-form results provide a very ef-fective means for evaluating the state of stress in the relatively narrow zones experiencing mid-side localeffects in complete elliptic toroidal vessels subjected to hydrostatic loading, and for calculating the de-formed shape of the shell midsurface.

& 2015 Elsevier Ltd. All rights reserved.

1. Introduction

The membrane theory of shells is a greatly simplified yet veryeffective basis for estimating stresses and deformations in thoseregions of the shell over which the loading and geometry do notchange too rapidly. However, and as is well-known, the theorybecomes inadequate at or in the vicinity of supports, concentratedloadings, shell junctions or discontinuities in shell geometry(thickness, slope, radii of curvature), loading and material prop-erties. Novozhilov [1] has called these locations “lines of distor-tion” in reference to the existence of a bending effect locally dis-turbing the membrane state of stress in these regions. Dis-continuity problems in shells of revolution have been the subjectof many investigations, and a good body of closed-form resultsexists for the more common types of shells and loading conditions[2–5].

The performance of containment shells is usually assessed withregard to their stress and deformation response in the linearelastic range [2,3], their vibration characteristics and dynamic re-sponse, as well as their nonlinear buckling and postbuckling be-haviour within the elastic and plastic ranges of material behaviour.Metal shells are particularly susceptible to buckling on account oftheir thin-ness (radius-to-thickness ratios typically in excess of500). Numerical studies have been carried out on the bucklingcapacity of vertical cylindrical steel tanks [6–11], horizontal

oni).

cylindrical and near-cylindrical vessels [12–14] and conical tanks[15–17]. The buckling capacity of multi-segmented shells underexternal water pressurisation has also been investigated [18], ashas the elastic buckling of certain unusual mathematical forms forshells [19,20].

Junction stresses in various shell assemblies and multi-seg-mented vessels have been the subject of intensive studies over thepast 15 years [21–25]. Mechanics phenomena around shell inter-sections and at shell-branching locations have also been of interest[26,27]. The presence of ring beams at shell junctions has a con-siderable influence on the behaviour of the shell; some effortshave also been directed towards understanding ring–shell inter-actions [28,29]. A more comprehensive review of recent studies onthe statics, dynamics and stability of various types of liquid-con-tainment shells under a variety of loading conditions may be seenin a recent survey [30].

Toroidal shells have mostly been studied with pressure-vesselapplications in mind, though liquid-containment applications havealso been of interest. The classical solutions for pressurised circularand elliptical toroids may be seen in texts on linear shell analysis[2–5]. Even where toroidal shells with uniform geometry aresubjected to internal pressure only, the membrane solution be-comes inadequate in the vicinity of the horizontal circles furthestfrom the equatorial plane, owing to the vanishing of the curvaturein one of the principal planes [31].

Sutcliffe [32] tackled the stress analysis of both circular andelliptical toroidal shells subjected to internal pressure. While ac-curate for the purpose, the formulation is somewhat cumbersome

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A. Zingoni et al. / Thin-Walled Structures 96 (2015) 286–294 287

for practical implementation. Galletly [33] considered the elasticbuckling of an elliptic toroidal shell subjected to uniform internalpressure, and confirmed that internally pressurised elliptical tor-oids, unlike circular toroids, may possibly buckle, depending onthe axes ratio of the elliptical cross-section. The study was alsoextended to plastic buckling [34], for which the post-bucklingbehaviour of the shell was noted to be stable.

Redekop [35] studied the buckling behaviour of an orthotropictoroidal shell of elliptical cross-section, while Yamada et al. [36]considered the free vibration response of a toroidal shell of ellipticsection. Xu and Redekop [37] also considered the free vibration ofelliptic toroidal shells, but with orthotropic properties. Zhan andRedekop [38] studied toroidal tanks with cross-sections made upof combinations of circular arcs of different radii (ovaloid shape),and observed the vibration, buckling and collapse behaviour ofthis type of toroidal vessel.

In this paper, we will focus attention on the thin elliptic tor-oidal shell. Noting the lack of a convenient analytical solution forthe axisymmetric bending of an elliptic toroidal shell, we aim atdeveloping a practical means for estimating bending-disturbanceeffects that may arise in the mid-side locations (herein referred toas “equatorial” locations) of vertically elongated thin elliptic tor-oids, where the vertical semi-axis b of the ellipse is greater thanthe horizontal semi-axis a. Specifically, we aim to develop andpresent a set of closed-form expressions for interior shell stressesdue to axisymmetric bending moments and shearing forces ap-plied in the equatorial plane of the elliptic section as uniformlydistributed edge actions.

The formulation is intended for use in quantifying (i) thejunction effects in the vicinity of the equatorial plane of subseaelliptic–toroidal shell structures (where a horizontal plate deckmay be attached to the inner walls of the toroid to provide aninterior working platform extending right round the torus), or (ii)the edge effects in the vicinity of supports where the elliptic tor-oidal vessel is used as an elevated circular tank supported onclosely-spaced vertical columns attached at both the intrados andextrados of the torus. The relatively weak edge effects associatedwith partial filling of the tank may also be quantified on the basisof this solution. We will begin by defining the geometry of theelliptic toroid.

2. Geometrical preliminaries

Fig. 1 shows the relevant geometrical parameters of an elliptictoroidal shell. To generate the torus, an ellipse of semi-axes a(horizontal) and b (vertical) is rotated about a vertical axis Y Y−that lies at a distance A ( a> ) from the local vertical axis y y− ofthe ellipse. The equatorial plane (horizontal plane of symmetry) is

P

E E

a

A AY

Yy

y

br1,r2O1

O2φ

Fig. 1. Geometrical parameters of an elliptic toroidal shell.

denoted by E E− . In what follows, we will take the generatorcurve (or meridian) of the toroidal shell as the ellipse to the left ofthe axis Y Y− . Let P be any point on the generator meridian. Theradius of curvature of the ellipse at point P is denoted by r1 and thecorresponding centre of curvature by O1. For the three-dimen-sional toroidal surface, there would be two principal radii of cur-vature (being the maximum and minimum values of curvature) atany given point P , and these occur in planes perpendicular to eachother. The first principal radius of curvature of the toroidal surfaceat point P is the radius of curvature r PO1 1(= ) of the generatorellipse at that point, while the second principal radius of curvatureat point P , denoted by r2, is equal to the distance PO2, where O2 isthe point at which the surface normal at P intersects the axis ofrevolution Y Y− of the torus.

Point P itself may be defined by an angular coordinate ϕ, whichis the angle measured from the upward direction of the axis ofrevolution of the torus to the surface normal at point P . The range0 2ϕ π≤ ≤ covers all points on the toroidal surface, with 0 ϕ π≤ ≤describing points in the outer region of the torus, and 2π ϕ π≤ ≤describing points in the inner region of the torus; the coordinates

/2ϕ π= and 3 /2ϕ π= define points on the equatorial plane, whichof course correspond to the extrados and intrados of the toruswith respect to the axis Y Y− .

For the outer region of the torus 0 ϕ π( ≤ ≤ ), the principal radiiof curvature are given by [3]

ra b

a bsin cospositive

1a1

2 2

2 2 2 2 3/2( )ϕ ϕ=

+( )

( )

rA a

a bsin sin cospositive

1b2

2

2 2 2 2 1/2( )ϕ ϕ ϕ= +

+( )

( )

while for the inner region 2π ϕ π( ≤ ≤ ), these become

ra b

a bsin cosnegative

2a1

2 2

2 2 2 2 3/2( )ϕ ϕ= −

+( )

( )

rA a

a bsin sin cospositive

2b2

2

2 2 2 2 1/2( )ϕ ϕ ϕ= −

+( )

( )

The values of r1 and r2 at the extrados of the torus ( /2ϕ π= ) andthe intrados ( 3 /2ϕ π= ), which correspond to the outer and innersides of the elliptical section, will be required in due course.Evaluating these, we obtain

rba 3a12

= ( )

r A a 3b2 = + ( )

at the extrados, and

rba 4a1

2= − ( )

r A a 4b2 = − ( )

at the intrados.

3. Governing equation

Fig. 2 shows a bending element of an axisymmetrically-loadedshell of revolution in the ,ϕ θ{ } coordinate system. Here, themeridional angle ϕ identifies the position of a point along a given

Nθ

N

N +dN

M

M +dM

Q +dQ

Q

Mθ

r1

r2

R

θ

dθ

d

NθMθ

φ

φ

φ

φ φ

φ φ

φ φ

φφ

Fig. 2. Element of a shell of revolution under axisymmetric bending.

A. Zingoni et al. / Thin-Walled Structures 96 (2015) 286–294288

meridian of the shell (as already defined), while the coordinate θ isthe circumferential angle measured from an arbitrary verticalplane to the vertical plane containing the meridian in question, thetwo vertical planes intersecting at the axis of revolution of theshell. The element is in equilibrium under the loads acting over itssurface (external loading on the shell) and the forces and momentsacting along its four edges (internal actions in the shell), but in thefigure, the surface loading is not shown, since we will only beconcerned with edge effects.

The actions N N,{ }ϕ θ are direct forces per unit length of theedge, acting in the direction of the tangent to the shell meridian atany given point (henceforth called the meridional direction), andthe direction of the tangent to the horizontal circumferential circlepassing through that point (henceforth called the hoop direction),respectively. The actions M M,{ }ϕ θ are bending moments per unitlength of the edge, as seen in the vertical meridional section andthe horizontal hoop section respectively. The action Q ϕ is a shearforce per unit length of the edge, acting in the meridional section;there is no shear force in the horizontal section Q 0( = )θ , owing toaxisymmetry. All meridional actions are shown incremented in thedirection of increasing ϕ, but the hoop actions do not change withrespect to θ owing to axisymmetry.

Considering equilibrium of the shell element, and making useof strain–displacement relations as well as Hooke’s law, we mayexpress Nϕ, Nθ , Mϕ and Mθ in terms of displacements, and thenreduce the ensuing relationships to the well-known Reissner–Meissner differential equations for the axisymmetric bending ofgeneral shells of revolution [2,3]

⎡⎣⎢

⎛⎝⎜

⎞⎠⎟

⎤⎦⎥

⎡⎣⎢

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟

⎤⎦⎥

⎡⎣⎢

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟

⎤⎦⎥

Drr

d Vd

Drr

dd

Drr

dVd

dDd

D Drr

V r r Q

sin cos sin

cos sincossin

sin5a

2

1

2

22

1

2

1

1

2

2

1 2

ϕϕ

ϕ ϕϕ ϕ

ν ϕϕ

ϕ ϕϕ

ϕ

+ +

+ − − = − ( )( )

ϕ

⎜ ⎟ ⎜ ⎟

⎡⎣⎢⎢

⎛⎝⎜

⎞⎠⎟

⎤⎦⎥⎥

⎡⎣⎢

⎧⎨⎩⎛⎝⎜

⎞⎠⎟

⎫⎬⎭⎧⎨⎩

⎛⎝⎜

⎞⎠⎟

⎫⎬⎭

⎧⎨⎩⎛⎝

⎞⎠

⎛⎝

⎞⎠

⎫⎬⎭⎤⎦⎥⎥

rtr

d Q

d

rtr

rtr

drd

dd

rtr

dQd

rtr

d rd

dd

rtr

drd

r rr

drd

r rt

dd

rt

rt

Q r EV

cot

cotcot

cot1

sin 5b

22

1

2

222

1

2

1

2 22

1

2

1

222

2

1

2

2 1

1

21 2

2 22 1

( )

ϕϕ

ϕ ϕ ϕ

ϕ ϕ ϕ

νϕ

ν ϕ ϕ

ν ϕϕ ϕ

+ + +

+ +

+ + − +

− ( ) − =( )

ϕ ϕ

ϕ

In these equations, the variable V is the angular rotation of theshell meridian as seen in the meridional section, while Q ϕ is theshear force as already defined. The material properties E (Young'smodulus of elasticity) and ν (Poisson's ratio) are assumed to beconstant. The parameter t denotes shell thickness, while D is theflexural rigidity of the shell, given by

DEt

v12 1 6

3

2= ( − ) ( )

For thin shells of revolution which are not shallow (non-shal-low shells shall be taken as those in which bending phenomenaoccur in locations for which ϕ is at least /6π from the locations

0ϕ = and ϕ π= ), second-derivative terms of V and Q ϕ are muchbigger than first-derivative and zero-derivative terms. As an ap-proximation, we may therefore drop all first-derivative and zero-derivative terms on the left-hand sides of Eqs. (5), so that theequations become

d Vd

rD

Q7a

2

212

ϕ= −

( )ϕ

d Q

d

r

rEtV

7b

2

212

22ϕ

=( )

ϕ

In the present problem of the prolate (vertically-elongated)elliptical cross-section b a( > ), this approximation is particularlyjustified, since we intend to investigate bending phenomena inregions that are adjacent to the equatorial plane ( /2ϕ π= ± ),where the sides of the elliptical section remain fairly steep for aconsiderable distance on either side of the equatorial plane.

Combining Eqs. (7a) and (7b), we obtain the fourth-order dif-ferential equation

d Q

dQ4 0

8

4

44

ϕλ+ =

( )ϕ

ϕ

where λ is a slenderness parameter defined as follows:

r Et

Dr

r

r t43 1

94 1

4

22

2 14

22 2( )λ ν= = − ( )

For the elliptical toroidal shell, the principal radii of curvature r1and r2 are functions of ϕ (as given by Eqs. (1) and (2)), so theslenderness parameter λ is also a function of ϕ, implying that thesecond term of the above fourth-order differential equation has avariable coefficient. Given the complexity of this coefficient, theexact solution of Eq. (8) is rather difficult to obtain.

4. Approximate general solution

Now for a prolate elliptical profile, the radii of curvature r1 andr2 vary rather slowly in the neighbourhoods of the equatorial

Fig. 3. Bending and shearing edge actions on a semi-elliptic toroidal shell.

A. Zingoni et al. / Thin-Walled Structures 96 (2015) 286–294 289

plane, which are the regions of present interest. Based on thisobservation, we will introduce the further approximation that r1and r2 are practically constant in the narrow zones adjacent to theequatorial plane and within which bending disturbances areconfined. These constant values of r1 and r2 will be taken as thevalues at the location /2ϕ π= (for the outer part of the toroidalshell) and the location 3 /2ϕ π= (for the inner part of the toroidalshell), as already given by Eqs. (3) and (4).

With this simplification, the slenderness parameter λ in Eq. (9)is now a constant, and the fourth-order differential equation (Eq.(8)) now has constant coefficients. This allows us to take thegeneral solution of Eq. (8) in the well-known form [3]

Q e C C e C Ccos sin cos sin 101 2 3 4( ) ( )λϕ λϕ λϕ λϕ= + + + ( )ϕ λϕ λϕ−

where C1, C2, C3 and C4 are constants of integration to be evaluatedfrom the boundary conditions of the shell. Based on the well-known behaviour of spherical and cylindrical shells, let us tenta-tively assume that for the semi-elliptical toroidal shell, the bend-ing disturbance emanating from either the outer edge or the inneredge is of a decaying character. Furthermore, let us assume thatany adjacent edges of the shell are sufficiently far apart to allowdecoupling of edge effects. This permits us to retain only two ofthe constants appearing in Eq. (10), and to rewrite the solution as

Q Ce sin 11λψ β= ( + ) ( )ϕ λψ−

where ψ is now the meridional angle measured from the normalat the edge of the shell to the normal at the point in question; Cand β are new constants of integration. For the shell edge locatedin the equatorial plane, the relationship between ϕ and ψ is simply

2;

32 12a, b

ψ π ϕ ψ ϕ π= − = − ( )

where the first equation refers to the outer edge of the semi-el-liptic toroidal shell, and the second refers to the inner edge (closerto the axis of revolution).

Equilibrium considerations of the shell element yield

N Q cot 0 13ϕ= − ≈ ( )ϕ ϕ

since cot 0ϕ ≈ in the vicinity of the toroidal shell edges( /2; 3 /2ϕ π π= ). For the hoop stress resultant, we obtain,

⎛⎝⎜

⎞⎠⎟N

rr

dQd r

drd

Qrr

dQd

114

2

1 1

2 2

1ϕ ϕ ϕ= − + ≈ −

( )θ

ϕϕ

ϕ

ignoring the second term in Q ϕ in relation to the first term in thefirst derivative of Q ϕ, particularly as the rate of variation of r2 withrespect to ϕ is also very small in the neighbourhood of theequatorial plane.

In evaluating the edge effects, compatibility conditions invol-ving the horizontal displacement δ of the shell edge (which is amovement perpendicular to the axis of revolution of the torus),and the meridional rotation V of the shell edge, will be required.From Eq. (7b), we may write V as follows:

⎛⎝⎜

⎞⎠⎟V Et

r

r

d Q

d1

1522

12

2

2ϕ=

( )ϕ

From strain–displacement and stress–strain considerations, thehorizontal displacement δ may be expressed in terms of the stressresultants Nϕ and Nθ as follows:

Etr N N

Etr N

1sin

1sin 162 2( )( ) ( )δ ϕ ν ϕ= − ≈ ( )θ ϕ θ

since N 0≈ϕ . Using result (14) to eliminate Nθ , we obtain

⎛⎝⎜

⎞⎠⎟Et r

rr

dQd

1sin

172

2

1( )δ ϕ

ϕ≈ −

( )ϕ

Using curvature–rotation relations and Hooke's law, the bend-ing moment Mϕ may be expressed in terms of the rotation V asfollows:

⎧⎨⎩⎫⎬⎭

⎛⎝⎜

⎞⎠⎟M D r

dVd r

V Dr

dVd

1cot

1181 2 1ϕ

ν ϕϕ

= − + ≈ −( )

ϕ

neglecting the second term (in V ) in relation to the first term (inthe first derivative of V ), particularly as the second term alsocontains the factor cot ϕ ( 0≈ in the neighbourhood of the equa-torial plane). Making use of Eq. (15) to eliminate V , and using theapproximation that, for the prolate elliptical profile, r2 and r1 arepractically constant in the narrow zone experiencing edge effects,we obtain

⎛⎝⎜

⎞⎠⎟M

DEtr

r

r

d Q

d 19122

12

3

3ϕ≈ −

( )ϕ

ϕ

Similarly,

⎧⎨⎩⎫⎬⎭

⎛⎝⎜

⎞⎠⎟M D r V r

dVd

Dr

dVd

M1

cot202 1 1

ϕ νϕ

νϕ

ν= − + ≈ − ≈( )

θ ϕ

(neglecting the first term in relation to the second).At this stage, we may now make use of the general solution for

Q ϕ (Eq. (11)) to eliminate Q ϕ from the above relationships. Theresults are as follows:

N 0 21≈ ( )ϕ

⎛⎝⎜

⎞⎠⎟N C

rr

e cos sin22

2

1λ λψ β λψ β≈ { ( + ) − ( + )}

( )θ

λψ−

⎛⎝⎜

⎞⎠⎟M

DEtr

r

rCe

2cos sin

23

3

1

22

12

λ λψ β λψ β≈ { ( + ) + ( + )}( )

ϕλψ−

M M 24ν≈ ( )θ ϕ

⎛⎝⎜

⎞⎠⎟

CEt

rr

r esin cos sin25

2

12( )δ λ ϕ λψ β λψ β≈ { ( + ) − ( + )}

( )λψ−

⎛⎝⎜

⎞⎠⎟V

CEt

r

re

2cos

26

222

12

λ λψ β≈ − ( + )( )

λψ−

5. Boundary conditions and generalised edge effects

Fig. 3 shows the edges of the semi-elliptical toroidal shellsubjected to uniformly distributed bending moments and hor-izontal shearing forces, where M H,e e1 1{ } are the axisymmetricactions applied at the outer edge, and M H,e e2 2{ } are the ax-isymmetric actions applied at the inner edge. The followingtreatment applies equally to the two edges. In the formulation, we

A. Zingoni et al. / Thin-Walled Structures 96 (2015) 286–294290

will therefore denote the applied edge bending moments and edgeshearing forces simply by Me and He, with ψ denoting the mer-idional angle from the edge in question (outer or inner).

When Me only is applied at a given edge (in the absence of He),we want to choose the constants C and β such that the followingboundary conditions are satisfied:

M M 27ae0( ) = ( )ϕ ψ=

Q 0 27b0( ) = ( )ϕ ψ=Applying these conditions to Eqs. (11) and (23), we obtain

0 28aβ = ( )

⎛⎝⎜

⎞⎠⎟C

Etr

D

r

rM

2 28be

13

12

22λ

=( )

Substituting these values of β and C into Eqs. (21)–(26), weobtain the interior stress resultants, bending moments and de-formations due to the edge bending moment Me as follows:

N 0 29≈ ( )ϕ

⎛⎝⎜

⎞⎠⎟N

EtrD

rr

e M2

cos sin30

e1

21

2λλψ λψ= ( − )

( )θ

λψ−

M e Mcos sin 31eλψ λψ= ( + ) ( )ϕ λψ−

M M 32ν≈ ( )θ ϕ

⎛⎝⎜

⎞⎠⎟

r

De M

2sin cos sin

33e

12

2δ λϕ λψ λψ= ( ) ( − )

( )λψ−

⎜ ⎟⎛⎝⎞⎠V

rD

e Mcos34e

1

λλψ= − ( )

( )λψ−

At the edge 0ψ( = ), we have

⎛⎝⎜

⎞⎠⎟

r

DM

2 35e e

12

2δ λ=

( )

⎜ ⎟⎛⎝⎞⎠V

rD

M36e e

1

λ= −

( )

When He only is applied at a given edge (in the absence of Me), wewant to choose the constants C and β such that the followingboundary conditions are satisfied:

M 0 37a0( ) = ( )ϕ ψ=

Q H 37be0( ) = ( )ϕ ψ=Applying these conditions to Eqs. (11) and (23), we obtain

4 38aβ π= − ( )

C H2 38be= − ( )

Substitution of these values of β and C into Eqs. (21)–(26) givesthe interior stress resultants, bending moments and deformationsdue to the edge shear He as follows:

N 0 39≈ ( )ϕ

⎛⎝⎜

⎞⎠⎟N

rr

e H2 cos40

e2

1λ λψ= − ( )

( )θ

λψ−

⎛⎝⎜

⎞⎠⎟M

DEtr

r

re H

4sin

41e

3

1

22

12

λ λψ= − ( )( )

ϕλψ−

M M 42ν≈ ( )θ ϕ

⎛⎝⎜

⎞⎠⎟Et

rr

r e H2

sin cos43

e2

12( )δ λ ϕ λψ= − ( )

( )λψ−

⎛⎝⎜

⎞⎠⎟V Et

r

re H

2cos sin

44e

222

12

λ λψ λψ= ( + )( )

λψ−

At the edge 0ψ( = ), we have

⎛⎝⎜

⎞⎠⎟Et

rr

r H2

45ee

2

12δ

λ= −( )

⎛⎝⎜

⎞⎠⎟V Et

r

rH

2

46e e

222

12

λ=( )

6. Shell stresses due to edge actions

The bending-related shell stresses σϕ (in the meridional direc-tion) and σθ (in the hoop direction) are obtained by combining thedirect stresses due to the stress resultants Nϕ and Nθ (which areconstant across the shell thickness) with the flexural stresses dueto the bending moments Mϕ and Mθ (which vary linearly across theshell thickness from a positive value on one side of the shellmidsurface to a negative value of the same magnitude on theopposite side of the shell midsurface). Thus,

Nt

M

t

M

t

6 647a2 2

σ = ± ≈ ±( )ϕ

ϕ ϕ ϕ

⎛⎝⎜

⎞⎠⎟

Nt

Mt

Nt

M

t6 6

47b2 2σ ν= ± = ±

( )θ

θ θ θ ϕ

where the first equation makes use of the fact that N 0≈ϕ (Eqs.(29) and (39)) and the second equation makes use of the relation-ship M Mν≈θ ϕ (Eqs. (32) and (42)). In the notation ± , the uppersign refers to the inner surface of the shell (with respect to theglobal axis of revolution of the torus), while the lower sign refersto the outer surface.

For the stresses due to the edge bending moment Me, wesubstitute the results for Mϕ and Nθ (as given by Eqs. (30) and (31))into the above expressions, and then eliminate the parameters Dand λ using Eqs. (6) and (9) respectively, leading to the results:

Mt

e6

cos sin48a

e2σ λψ λψ= ± ( + ) ( )ϕ

λψ−

⎛⎝⎜

⎞⎠⎟

⎡⎣⎢

⎤⎦⎥

ErD

rr

M e

Mt

e

Mt

e

2cos sin

6cos sin

23 1 cos sin

3 cos sin48b

e

e

e

12

1

2

2

22 1/2{ }( )

σλ

λψ λψ

ν λψ λψ

ν λψ λψ

ν λψ λψ

= ( − )

± ( + )

= − ( − )

± ( + )( )

θλψ

λψ

λψ

−

−

−

For the stresses due to the edge shearing force He, we substitute

A. Zingoni et al. / Thin-Walled Structures 96 (2015) 286–294 291

the results for Mϕ and Nθ (as given by Eqs. (40) and (41)) intorelations (47), and then eliminate the parameters D and λ usingEqs. (6) and (9) respectively, leading to the results

⎛⎝⎜⎜

⎞⎠⎟⎟

⎛⎝⎜

⎞⎠⎟

D

Et r

r

re

HH

tr e

24sin

6

3 1

1sin

49a

ee

3

31

22

12

2 1/4 3/22

1/2

{ }( )( )

σ λ λψ

νλψ

= ∓ ( )

= ∓− ( )

ϕλψ

λψ

−

−

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜⎜

⎞⎠⎟⎟

⎛⎝⎜

⎞⎠⎟

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥ 49b

trr

e HD

Et r

r

re H

Ht

r e

2cos

24sin

21

3 1 cos3

3 1

sin

e e

e

2

1

3

31

22

12

3/2 21/2 2 1/4

2 1/4{ } { }

( )( )

( )( )

σ λ λψ ν λ λψ

ν λψ ν

νλψ

= − ( ) ∓ ( )

= − − ±−

θ λψ λψ

λψ

− −

−

7. Analytical observations

7.1. Effect of Me

The magnitudes of the stresses due to Me (both meridional andhoop) are not dependant on the principal radii r1 and r2 (and hencethe parameters a and b) of the elliptic torus. For a given appliedMe, the magnitudes of the induced shell stresses only depend onthe thickness t of the shell. This total lack of dependence on theelliptic parameters a and b is a surprising result.

It means that for the same moment Me applied at the inner andouter edges of the semi-elliptic toroidal shell, the peak values ofthe induced shell stresses will be the same. However, the rate ofdecay of the stresses with distance from the respective shell edge,as well as the wavelength of the oscillations of the variation, willnot be the same, because the value of the shell slendernessparameter λ (which governs both the rate of decay and the wa-velength of the oscillations) at the outer edge differs from that atthe inner edge – see Eq. (9).

7.2. Effect of He

The magnitudes of the stresses due to He (both meridional andhoop) not only depend on the shell thickness t , but more sig-nificantly, are directly proportional to r2 . Now r A a2 = + for theouter edge of the semi-elliptic toroidal shell, and r A a2 = − for theinner edge (see Eqs. (3b) and (4b)). It therefore follows that, for thesame horizontal shear force He applied at the inner and outeredges of the semi-elliptic toroidal shell, the peak values of theinduced shell stresses on the outer side of the torus will differfrom those on the inner side, since the r2 values differ; the outerside will experience larger stresses than the inner side.

We also observe that only the parameters A and a of the elliptictoroid (but not b) have an influence on the peak values of thestresses due to He. The rate of decay of stresses with distance fromthe shell edge, as well as the wavelength of the oscillations, willalso be different between the outer and inner sides, owing to thedifferent values of λ.

For He, we may write the stresses for the outer and inner edgesmore explicitly as follows:

Outer edge

⎛⎝⎜

⎞⎠⎟

Ht

A a e6

3 1

1sin

50a

e

2 1/4 3/21/2

{ }( )σ

νλψ= ∓

−( + )

( )ϕ

λψ−

⎛⎝⎜

⎞⎠⎟

⎡

⎣

⎢⎢⎢⎤

⎦

⎥⎥⎥

Ht

A a e2 1 3 1

cos 3

3 1sin

50b

e 3/21/2 2

1/4

2 1/4

{ }

{ }

( )

( )

σ ν

λψ ν

νλψ

= − ( + ) −

±− ( )

θλψ−

Inner edge

⎛⎝⎜

⎞⎠⎟

Ht

A a e6

3 1

1sin

51a

e

2 1/4 3/21/2

{ }( )σ

νλψ= ∓

−( − )

( )ϕ

λψ−

⎛⎝⎜

⎞⎠⎟

⎡

⎣

⎢⎢⎢⎤

⎦

⎥⎥⎥

Ht

A a e2 1 3 1

cos 3

3 1sin

51b

e 3/21/2 2

1/4

2 1/4

{ }

{ }

( )

( )

σ ν

λψ ν

νλψ

= − ( − ) −

±− ( )

θλψ−

7.3. Comparisons with cylindrical and spherical shell solutions

Not surprisingly, the derived theoretical results for the elliptictoroidal shell are quite similar to those for the bending of a circularcylindrical shell, and for the bending of a non-shallow sphericalshell when use is made of the Geckeler approximation [2,3]. Bothof these problems also result in a fourth-order governing differ-ential equation of the same form as Eq. (8), with a general solutionof the same form as Eq. (10) or Eq. (11).

Looking at the expressions for Nθ , V , δ and Mϕ (Eqs. (14), (15),(17) and (18) respectively), and replacing r d1 ϕ (for the curvedmeridian) by dx (for the cylindrical shell), one can transform theresults for the elliptic–toroidal shell to those for a cylindrical shell.However, the cylindrical-shell model will not exactly replicate thebehaviour of the toroid, since dx (parallel to the axis of revolution)is only an approximation for the real r d1 ϕ, an approximation thatis very good in the neighbourhood of /2ϕ π= ± (errors are of theorder of /62ψ , where ψ is the angle from the edge), and that be-comes better as r1 becomes larger, and exact as r1 approaches in-finity. When r1 reaches infinity, the r2 for the toroidal shell becomesthe radius of the cylinder. If one wanted to use the cylinder so-lution to approximate the behaviour of the elliptic toroidal shell inthe equatorial zones, the equivalent cylindrical shells for the outerand inner edges of the semi-elliptic toroid are the two cylinderswhich are tangential to the elliptic toroid at the extrados(r A a2 = + ) and the intrados (r A a2 = − ).

Alternatively, one may replace the elliptic toroidal shell withthe equivalent spherical shell in the vicinity of the equatorialplane, and then use the Geckeler approximation. The equivalentspherical shells for the outer and inner edges of the semi-elliptictoroid are the two spheres which are tangential to the elliptictorus at the extrados and the intrados; these spheres are of radiiA a+ and A a− respectively.

For the outer edge of the elliptic toroid, the spherical-shellsolution is a better approximation of the stress distribution in theelliptic toroid than the cylindrical-shell solution (the sphericalsurface and the elliptic–toroidal surface have the same value of r1at their tangent circle), but for the inner edge of the elliptic toroid,the cylindrical-shell approximation is better than the spherical-shell solution, owing to the inability of the sphere to model themeridional curvature of the elliptic torus on its inner side (the

A. Zingoni et al. / Thin-Walled Structures 96 (2015) 286–294292

spherical surface and the elliptic–toroidal surface have differentmagnitudes of r1 at their tangent circle, and the signs of the cur-vature are also different).

Fig. 5. Application of Me2 at the inner edge: (a) analytical results; (b) FEM results.

8. Numerical results

As an example, consider a semi-elliptical toroidal shell with theparameters

a 10 m= ; b 20 m= ; A 30 m= ; t 0.05 m= ; E 200 10 N/m9 2= × ;0.3ν =These parameters are possible proportions for a subsea marine

observatory of circular plan shape, where the overall structuraldiameter is 80 m and the height is 40 m. The principal radii ofcurvature and shell slenderness parameters at the two edges fol-low from Eqs. (3), (4) and (9):

Outer edge: r 40 m1 = ; r 40 m2 = ; 36.3568λ =Inner edge: r 40 m1 = − ; r 20 m2 = ; 51.4163λ =A bending moment M 100 kNm/me = is first applied at both the

outer shell edge and the inner shell edge. The variations of mer-idional and hoop stresses with the angular coordinate ψ from theshell edge are given by Eqs. (48), which apply for both the outeredge and the inner edge, but using the λ value applicable for theedge in question. The arc length s (in metres) from the shell edgeis simply given by s r 401 ψ ψ= = , where angle ψ is in radians.Based on the theoretical formulation that has been developed inthis paper, Fig. 4(a) shows plots of meridional and hoop stressesversus distance s from the outer edge, while Fig. 5(a) shows plotsof meridional and hoop stresses versus distance s from the inneredge.

As a means for validating the theoretical results, a finite-ele-ment modelling of the semi-elliptic toroidal shell using the FEMprogramme ABAQUS [39] was performed, for an edge loading of

Fig. 4. Application of Me1 at the outer edge: (a) analytical results; (b) FEM results.

100 kNm/m. Two-node axisymmetric shell elements were em-ployed throughout, with the element lengths kept at 0.05 m (veryfine mesh) throughout in order to ensure accurate modelling ofthe rapidly varying bending-disturbance stresses in the edgezones. The FEM results are shown in Fig. 4(b) for the outer edgeand Fig. 5(b) for the inner edge.

For the same shell, a horizontal shear force H 100 kN/me = isnext applied at both the outer shell edge and the inner shell edge.The variations of meridional and hoop stresses with the angularcoordinate ψ from the shell edge are given by Eq. (50) for the outeredge and Eq. (51) for the inner edge. Based on the theoreticalformulation that has been developed in this paper, Fig. 6(a) showsplots of meridional and hoop stresses versus distance s from theouter edge, while Fig. 7(a) shows plots of meridional and hoopstresses versus distance s from the inner edge. The correspondingFEM results are shown in Figs. 6(b) and 7(b).

Comparing the analytical results versus their FEM counterparts,it is noted that the agreement is excellent throughout. The curvesare practically identical. The discrepancy is generally less than 1%for both the outer edge of the semi-elliptic toroid and the inneredge, and for both the application of Me and the application of He.This is clearer from Table 1, where peak values of the analyticalplots are compared with peak values of the FEM plots; theoreticalvalues and FEM values are almost identical. Owing to the finenessof the FEM mesh that was adopted (further refinement of themesh did not change the results by more than 0.1%), the FEM re-sults may be taken as exact for practical purposes. The achievedagreement of over 99% between the analytical and FEM resultsleads us to conclude that the proposed theoretical formulation forthe equatorial bending of prolate elliptic toroidal shells, whileapproximate, is a very accurate and effective means for quantifyingedge effects in regions bordering the extrados and intrados of suchshells. This is a finding of significant practical importance.

Fig. 6. Application of He1 at the outer edge: (a) analytical results; (b) FEM results.

Fig. 7. Application of He2 at the inner edge: (a) analytical results; (b) FEM results.

A. Zingoni et al. / Thin-Walled Structures 96 (2015) 286–294 293

Comparing the two edges of the semi-elliptical toroid, andconsistent with the analytical predictions, it is seen that the effectsof He are larger on the outer side of the toroid (owing to the largervalue of r2 there). It is also observed that the bending disturbancedecays more quickly on the inner side of the toroid (owing to thelarger value of λ there).

9. Concluding remarks

The exact differential equations for the axisymmetric bendingof elliptic toroidal shells are complex and difficult to solve analy-tically, since the coefficients of these equations are themselvescomplicated functions of the independent variable ϕ. For the es-timation of bending effects in the mid-level regions of the prolateelliptical toroid ( b a> ), a simplication of the Reissner–Meissnerequations has been proposed in this paper. This takes advantage of(i) the rapidly decaying character of edge effects, (ii) the slow rateof change of r1 and r2 (principal radii of curvature of the shell) inthese regions of the toroid, and (iii) the fact that cot 0ϕ ≈ in thevicinity of the equatorial plane.

The relevant formulation for this approach has been developed,and closed-form expressions derived for shell stresses in the edgezones due to arbitrary bending moments Me and horizontalshearing forces He applied at the outer and inner edges of the shell.The obtained closed-form results are surprisingly simple, yet cap-able of modelling mid-side edge effects with very high accuracy.For the example that was considered, errors were less than 1%.

By examining the final form of the analytical results, it has beenobserved that the magnitude of the stresses due to an edgebending moment Me applied at the equatorial level is practicallyindependent of the characteristic parameters a, b and A of theelliptic toroid, while that of the stresses due to a horizontal edgeshear He is dependant on a and A (but not b). Thus the effects of agiven action He are larger on the outer side of the toroid (owing tothe larger value of r2 there). It has been found that the effects ofboth Me and He (which constitute the bending disturbance) decaymore quickly on the inner side of the toroid, owing to the largervalue of λ there. This finding is not obvious, since surfaces of ne-gative Gaussian curvature (such as we find on the inner side of thetorus) may exhibit more extensive propagation of edge effects [1].

Although arbitrary values of M 100 kNm/me = andH 100 kN/me = were assigned to the toroidal edges for the pur-poses of studying the ensuing stress distributions within the shell,in a real problem, the actual values of Me and He depend on theloading acting over the surface of the shell and the boundaryconditions prevailing at the shell edge; Me and He are usuallyevaluated by imposing conditions of compatibility of deformationsat the shell edge, when the effects of the membrane solution for aparticular type of loading are considered simultaneously with thegeneral effects of Me and He, as developed in the present paper.

In particular, for elliptic toroidal vessels subjected to external orinternal hydrostatic liquid pressure, the theoretical formulationthat has been developed will be useful in providing an accurateestimate of stresses and deformations in the vicinity of (i) anymid-side discontinuities of the shell (such as discontinuities inshell thickness), (ii) axisymmetric vertical supports located at themid-sides of the vessel, or (iii) the junction of the thin shell wallwith an external ring stiffener or an internal plate deck located inthe equatorial plane. A calculation of the deformed shape of theshell midsurface prior to the onset of elastic buckling, which thisformulation will facilitate, can afford valuable insights on the likelymode of first buckling.

Table 1Comparison of analytical (ANA) versus finite-element method (FEM) results.

inner surfaceσ ( )ϕ N/mm2 outer surfaceσ ( )ϕ N/mm2 inner surfaceσ ( )θ N/mm2 outer surfaceσ ( )θ N/mm2

ANA FEM ANA FEM ANA FEM ANA FEM

Me1 (outer edge) 240.0 240.3 �240.0 �240.3 204.2 204.1 60.2 59.9Me2 (inner edge) 240.0 240.6 �240.0 �240.6 204.2 204.0 60.2 59.7He1 (outer edge) �85.0 �85.1 85.0 85.1 �145.4 �145.6 �145.4 �145.4He2 (inner edge) �60.2 �60.3 60.2 60.3 �102.8 �103.0 �102.8 �102.7

A. Zingoni et al. / Thin-Walled Structures 96 (2015) 286–294294

Acknowledgement

This work has been undertaken within the framework of a re-search programme funded by the National Research Foundation ofSouth Africa.

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Equatorial bending of an elliptic toroidal shellIntroductionGeometrical preliminariesGoverning equationApproximate general solutionBoundary conditions and generalised edge effectsShell stresses due to edge actionsAnalytical observationsEffect of MeEffect of HeComparisons with cylindrical and spherical shell solutions

Numerical resultsConcluding remarksAcknowledgementReferences