Thin-Walled Structures 96 (2015) 49–55

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Simplified theoretical solution of circular toroidal shell with ribs underuniform external pressure$

Qinghai Du a,b,n, Guang Zou a, Bowen Zhang a, Zhengquan Wan a

a China Ship Scientific Research Center, No. 222 East Shanshui Road, Wuxi, Jiangsu 214082, Chinab Hadal Science and Technology Research Center, College of Marine Science, Shanghai Ocean University (Shanghai Engineering Research Center of HadalScience and Technology), Shanghai 201306, China

a r t i c l e i n f o

Article history:Received 15 November 2014Received in revised form20 July 2015Accepted 24 July 2015

Keywords:Toroidal shellTheoretical solutionCurve-beam modelFEMExperiment

x.doi.org/10.1016/j.tws.2015.07.01931/& 2015 Elsevier Ltd. All rights reserved.

entation fund: Chinese NFSC-51109190.esponding author at: Hadal Science and Techarine Science, Shanghai Ocean University (Shf Hadal Science and Technology), Shanghai 2ail addresses: tsinghai.du@gmail.com, duqh@s

a b s t r a c t

The toroidal shell with stiffened ribs is a new-style structure in ocean engineering especially in under-water engineering. This paper attempts to provide a simple theoretical method to obtain the stress so-lution of toroidal shell with ribs for its strength assessment. Firstly according to the structural property oftoroidal shell with ribs and theory of curve-beam, a simple model for toroidal shell with ribs has beendeveloped; then coupled with theory of thin-shell and elastic beam, its stress and deformation have beensolved and can be expressed into analytic formulas; lastly by finite element method (FEM) and modelexperiment method, this simple theoretical solution has been verified to be reasonable and quite ac-curate. Thus this simple theoretical solution could be applied for analysis and design of pressureequipment in such toroidal structure type.

& 2015 Elsevier Ltd. All rights reserved.

Contents

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 492. Simple mechanical model for toroidal shell with ribs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

2.1. Structural characteristic of toroidal shell under pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 502.2. Establishment of simple mechanical model for toroidal shell with ribs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

3. Theoretical solution of toroidal shell with ribs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 523.1. Governing equation for the problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 523.2. Solution of the governing equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 523.3. Stress and deformation of toroidal shell with ribs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

4. Verification of theoretical solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 534.1. Finite element analysis and experiment of toroidal model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 534.2. Verification of the simple theoretical solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

5. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

1. Introduction

Compared to the cylindrical, spherical or conical shell, thetoroidal shell is rarely used in underwater structures because of

nology Research Center, Col-anghai Engineering Research01306, China.hou.edu.cn (Q. Du).

the difficulty of theoretical solution and the manufacture. How-ever the type of toroidal shell is still widely used as joint com-ponents or accessories in pressure vessel and piping industry,nuclear power industry and ocean engineering for its specificstructural shape to obtain kinds of performance. Especially Ross[1,2] recently suggested a new conceptual design of an underwatervehicle and an underwater space station with the main pressurehull in a toroidal shape, because of the advantages and dis-advantages of using a circular cylindrical shell as pressure hull.

Fig. 1. Geometry model and parameters of toroidal shell without ribs.

Du et al. / Thin-Walled Structures 96 (2015) 49–5550

Many researchers in the field of engineering mechanics havespent great efforts to solve this problem in order to obtain theanalysis and design method. Zhang [3], Clark [4] and Novozhilov[5] had obtained an elastic asymptotic solution of pure toroidalshells under axisymmetric loading. Moreover Xia [6] had obtainedits elastic general solution due to arbitrary loading. Maching [7],Sobel and Flügge [8,9] had explored the buckling and stability oftoroidal shells. Bushnel [10], Jordan [11] and Panagiotopoulos [12]had approached the stability analysis of the toroidal shells by finitedifference method (FDM) or FEM. The buckling of the segments oftoroidal shells had been solved by Stein and Mcelman [13] andHutchinson [14]. Galletly [15,16] had studied the stability of closedtoroidal shells with circular or non-circular cross-section. Blachut[17] had obtained three toroidal models experiments and revealedthe collapse pressure by different material, geometry and manu-factory, compared and confirmed with numerical results. Wang[18] had obtained the analysis of geometric nonlinear buckling andpost-buckling of toroidal shells by asymptotic approach.

However all these previous works had been to solve the purecomplete toroidal shell or its segments in elastic or nonlinearanalysis, but then its critical pressure could not reach higher fornon-ring-shaped ribs. Du [19] had studied static structural char-acteristics of a ring-stiffened toroidal shell with certain parametersby FEM and membrane simply theory, and then he [20] showednonlinear structural properties of ring-stiffened toroidal shell bypresented nonlinear FEM and verified the feasibility used as mainpressure hull in underwater engineering. Zou and Du [21] ob-tained a theoretical method to calculate stress of ring-stiffenedtoroidal shell on two special positions that are on internal andexternal toroidal cycle. Du [22] also accomplished a steel weldingring-stiffened toroidal model experiment under hydrostatic ex-ternal pressure, obtain its static elastic stress state and collapsepressure, compared with numerical results and revealed thebuckling characteristics by nonlinear FEM.

In this paper the toroidal shell with ring-stiffened ribs will besolved in a theoretical method, which will be based on a simplecurve-beam model with elastic supports on it. Then the stress anddisplacement of toroidal shell will also be obtained through by thissolution of curve beam and thin shell theory. By numerical andmodel experimental method, this simple theoretical solution oftoroidal shell with ring-stiffened ribs would be compared, certifiedand confirmed correctly finally.

2. Simple mechanical model for toroidal shell with ribs

The toroidal shell with stiffened ribs is difficult to solve bytheoretical method, although all previous theoretical solution isjust for pure toroidal shell without ribs. However when it has beenset with series of stiffened ribs, the difficulty to solve it wouldbecome more and more than the former. Here it would be tried toaccomplish such multiplex problem by a simplified theoreticalmethod and shows out a simple curve-beam model.

2.1. Structural characteristic of toroidal shell under pressure

The toroidal shell is a special shells of revolution for the ex-istence of horizontal tangents at φ¼7π/2, which are called theturning points. The Gaussian curvature ratio changes its sign frompositive to negative (see from Fig. 1, where R represents the dis-tance of the center of the meridian circles to the axis of rotation, φis the tangential angle of shells, r is the curvature radius of theparallel circle on the toroidal shell to revolving direction, t is thewall thickness of shells, a is the radius of the meridian circle, θ isthe rotational angle of the meridian circle).

The toroidal shell has nonlinear geography structure while the

following characteristic of such type hull would expose this non-linearity property. Here toroidal shell without ribs will be solvedby FEM, including intensity and stability question. By all theseways, the structural characteristics of toroidal shell without ribscould have been discovered entirely.

The following toroidal model has dimensionless parameters asR/a¼2.22, a/t¼53, which are similar to those ever used in Ref.[20]. After geometric model being built and finite element meshed,the elastic statics analysis of toroidal shell under uniform externalpressure load has been carried out. Based on the numerical cal-culation results, the stress strength coefficients distributing alongsection circle are shown out as Figs. 2 and 3. Here stress con-centrated factor kφ and kθ will be defined as following by pressureand geometry sizes. Then membrane theory method for toroidalshell can be referred to our previous work [19].

ktpa

ktpa

2,

21

σ σ= =( )φ

φθ

θ

From Fig. 2 and Fig. 3, it can be found that the stress factor kφand kθ along section circle of toroidal shell is not linear ormonotone but changing sharply at the turning point B.

Fig. 4 shows that the stress factor kφ and kθ vary by R/a, wheremodels hold the same a/t. From this figure, it can be found thattoroidal shell would be trend into cylindrical shell where R/a isinfinity.

2.2. Establishment of simple mechanical model for toroidal shell with

Fig. 2. Distribution of kφ along the cross-section circle.

Fig. 3. Distribution of kθ along the cross-section circle.

Fig. 4. Varying of kφ and kθ by R/a.

Fig. 5. Beam model with elastic support under uniform pressure.

Fig. 6. Curve-beam model with elastic support under uniform pressure.

Du et al. / Thin-Walled Structures 96 (2015) 49–55 51

ribs

According to structural property of toroidal shell with series ofribs and uniform external pressure load, it can be found that thestructure shape and stress both vary along circle section betweentwo stiffened ribs.

Similar to simplification of cylindrical shell with series stiffenedribs (Fig. 5), the curve beam model will be adopted to simplifytoroidal shell with ribs to solve. It should be noticed that in Fig. 6,QQ′ is an arbitrary arc along toroidal direction but between twoneighbor ribs AC and A′C′. This arc QQ′ is curve beam in plane QO′′

Du et al. / Thin-Walled Structures 96 (2015) 49–5552

Q′, then we will use curve beam QQ′ as basic mechanical model asshown in Fig. 6.

3. Theoretical solution of toroidal shell with ribs

Based on the above curve-beam model, the strength of toroidalshell with stiffened ribs will be tried to be solved, and it wouldadopt some simplified model or some assumption in its derivation.Then finally the stress and displacement of toroidal shell can beexpressed into detail formulas.

3.1. Governing equation for the problem

The toroidal shell is not axisymmetric after its ribs set up.However its mechanical model can be simplified into curve-beammodel to solve and obtain stress and deformation.

The differential equation of curve beam is as follows:

⎜ ⎟⎡⎣⎢

⎛⎝

⎞⎠

⎤⎦⎥w

parD

wEtrDa

wpD

rar Etr

Dadud2

12

cos2 2

2 4

24

3 4

2μ φ

φ + ¨ + = − − +

( )

Define:

⎧

⎨

⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

⎡⎣⎢

⎤⎦⎥

m

n

k r

r a

2

1 cos

parD

Etr

Da

pD

ar

R

2

2

1 24

2

cos

2

4

2

3( ) φ

=

=

= − −

= +

μ

φ

Here E and m is Young’s modulus and Poisson’s ratio, while D is

bending stiffness of shell which is D Et

12 1

3

2=μ( − )

. Then

w mw n w kpD

ndud

23

21

2

φ + ¨ + = +

( )

The boundary of curve beam with elastic support is as follow-ing

⎧⎨⎪⎩⎪

w

w k w kdud

kEFrDa

cos ,0

,2 4

02 2 2

3

2

θ θ φφ

= =

− = − =( )

w0, 0 5θ = = ( )

3.2. Solution of the governing equation

The homogeneous plus particular solution of Eq. (2) can beexpressed as

wkn

C C

C C

ch cos sh cos

ch sin sh sin 6

12 1 1 2 2 1 2

3 1 2 4 1 2

α ψ α ψ α ψ α ψ

α ψ α ψ α ψ α ψ

= + +

+ + ( )

Considering about the axisymmetric of curve beam at θ¼0 andθ0cosφ. which means the constant coefficient C2 and C3 must bezero. Thereby the above formula (6) can be simplified as following

wkn

C Cch cos sh sin7

12 1 1 2 4 1 2α ψ α ψ α ψ α ψ= + +

( )

Meanwhile it must be noticed that u is displacement of internalcurve beam direction in toroidal shell, and based on its distribu-tion along circle section plus that axisymmetric of toroidal shell. Inderivation, we assume that u can be expressed by the function ofφ, that is

u C sh sin 8u 1 2α φ α φ= ( )

According to the boundary (4) and (5) of curve beam and thenespecially in our derivation some minor items are ignored. So thecoefficients C1, C4 and Cu can be solved as following formulas.

⎧

⎨⎪⎪⎪

⎩⎪⎪⎪

Cu u u u u u

F

Cu u u u u u

F

2 ch sin sh cos

2 ch sin sh cos

9

k

n

dud

k

n

dud

1

1 1 2 2 1 2 1

5

4

2 1 2 1 1 2 1

5

12

12

( )

( )

( )

( )

ε

ε

= −− +

= −− −

( )

φ

φ

Cu u u u u u

F

sh cos ch sin

10u

k

n2 1 2 1 1 2 1

5

12 ( )γ ε

=−

( )

Here

F u uF u u F u u

F u u

11 ,

, , , ,

, ,

11 1 2

2 1 2 1 2 3 1 3 1 2

4 1 4 1 2

εβ

ε ε ε ε

ε ε

=+ ( )

= ( ) = ( )

= ( )

Fu u

F

1 ch2 cos 21

21 2

5

( )γ=

− −

FDn u u u u

par tF

6 sh2 sin 2k

n

dud

2

2 1 1 2

25

12( )( )

=− −

φ

FDn u u u u u u

par tF

12 ch sin sh cosk

n

dud

3

1 1 2 2 1 2

25

12( )( )

=− −

φ

FEt u u u u u u

pa F

2 ch sin sh cosk

n

dud

4

1 1 2 2 1 2

25

12( )( )

=− +

φ

F u u u ush2 sin 25 2 1 1 2= +

u u R at

Fcos , cos , cos

21 1 0 2 2 0

0α θ φ α θ φ β φ θ= = = ( + )

n m n mmn

12

,12

,1 2α α γ= ( − ) = ( + ) =

Here F is the across area of stiffened ribs.

3.3. Stress and deformation of toroidal shell with ribs

Based on the above solution of curve-beam displacement, thestress of toroidal shell can be solved through by strain–displace-ment relation and stress–strain relation. So the stresses can beobtained by following formulas.

⎧⎨⎪⎩⎪

Tt

Mt

E

6

11

2σ

σ ε μσ

= ±

= + ( )

θθ θ

φ φ θ

Here

adud

w1ε

φ= ( − )φ

M D κ μκ= ( + )θ θ φ

wr a

w u,1

2 2κ κ=¨

= ( ″ − ′)θ φ

Table 2Local thickness at special key points on toroidal model.

Position A C M M'

tlt (mm) 9.8 10.4 10.1 10.3

Du et al. / Thin-Walled Structures 96 (2015) 49–55 53

dd

dd

dd

, ,2

2ψ φ φ() = ()″ = ()′ =••

Therefore, the deformation and main stress at the end of curve-beam can be calculated by formula (12) and (13).

wkn

112f

12 1( )ε= −

( )

⎧⎨⎪⎪

⎩⎪⎪

⎛⎝⎜

⎞⎠⎟

pat

D w ua t

Ea

kn

u

12

6

113

e

e e

2 2 2

12 1

σ ε μ

σ ε μσ

= − ( ± ) ∓ ( ‵‵ − ‵)

= − ( − ) − ‵ +( )

θ

φ θ

while the deformation and main stress at the middle of curve-beam can also be calculated by formula (14) and (15)

wkn

paEt 14max

12

2

4ε= −( )

⎧

⎨⎪⎪

⎩⎪⎪

⎛⎝⎜⎜

⎛⎝⎜

⎞⎠⎟

⎞⎠⎟⎟

pat

D w ua t

Ea

ukn

paEt

12

6

15

m

m m

3 2 2

12

24

σ ε μ

σ ε μσ

= − ( ∓ ) ± ( ″ − ′)

= ′ − − +( )

θ

φ θ

Then from Eqs. (9) and (10), the normal displacement w andinternal displacement u can be obtained too, so the stresses at thekey position of curve beam may be solved totally from Eqs. (12)–(15).

Meanwhile based on the structural analysis of pure toroidalshell (shown as Fig. 2 and Fig. 3), it can be easily found that the keypoints are at φ¼0 and π. Therefore in this paper the stress of thesetwo positions should be shown out and checked in following part.

Fig. 7. Toroidal model by measured lead was put into pressure tank.

4. Verification of theoretical solution

To verify the curve beam model and this simplified theoreticalsolution, a steel welding model of toroidal shell with stiffened ribsis developed into being solved by numerical method and experi-ment too. Finite element method (FEM) is adopted to obtain thestress under some special pressure load, while experiment modelis built and tested in external uniform pressure load from low tohigh pressure as model is still in elastic state. Therefore by thesetwo methods, the theoretical solution could be verified andchecked.

4.1. Finite element analysis and experiment of toroidal model

Here a steel welding model of toroidal shell with stiffened ribsis experimental to verify and confirm presented theoreticalmethod, while its nominal geometric parameters and materialproperties are shown as Table 1, while the local thickness (tlt) atspecial points such as point are provided at Table 2. The FEA modelis respectively built according to its nominal parameters fromTables 1 and 2, external uniform pressure p is applied on thetoroidal shell, and meanwhile its rigid displacement will berestricted.

In the experimental model, some special points were set as

Table 1Nominal parameters of toroidal model size and material.

R (mm) a (mm) t (mm) tf�bf (mm�mm) E (MPa) l

381 131.5 10 10�40 2.06�105 0.3

measured points, put the strains, stuck and connected by wirelines, which will lead to machine for recording strain and pressuredata during the process of hydrometrics pressure experiment,which details can be seen in Fig. 7 and those special points areannotated on model such as points A, C, M and M’. Here strain andstress measured in the second cycle step during hydrometricspressure test will be used in the following section of this paper.The test cycle steps are provided as Fig. 8.

4.2. Verification of the simple theoretical solution

Based on results from the elastic FEA and experiment of tor-oidal model with ribs, the stress of special key points can becompared with those results obtained by theoretical solution. Hereit is noticed that the thicknesses at those different tested pointsare different, and so the measured thickness referred from Table 2will be used when FE analysis and theoretical solution are exe-cuted at those special points. Table 3 and Table 4 have shown outstress concentration coefficients corresponding to those compar-ison results among these three different methods. Here the defi-nitions of kφ and kθ are same as previous formula (1), which meansthat the basic stress is membrane toroidal stress of the non-stif-fened toroidal shell under external pressure p.

From Table 3, it can be obviously found that this theoreticalsolution for the middle of two near ribs is both confirmed well byFEM and experiment results, and the maximum error is less than5%.

Fig. 8. The hydrostatic pressure test cycle steps of toroidal model.

Table 3Comparison data of toroidal model in the middle of two near ribs.

φ kφ kθ

Outside Inside Outside Inside

φ¼0 (Point M) Test (kexp) �1.68 �1.77 �1.27 �0.87Theory (kth) �1.70 �1.75 �1.29 �0.90Δ % 1.2 1.1 1.6 3.5FEM (kf) �1.72 �1.75 �1.28 �0.85Δ % 2.4 1.1 0.8 2.3

φ¼π (Point M’) Test (kexp) �1.38 �0.45 �1.95 �1.79Theory (kth) �1.41 �0.44 �1.99 �1.81Δ% 2.2 2.2 2.1 1.1FEM (kf) �1.40 �0.45 �1.98 �1.80Δ % 1.4 0 1.5 0.6

Table 4Comparison data of toroidal model at the end of ribs.

φ kφ kθ

Outside Inside Outside Inside

φ¼0 (Point A) Test (kexp) �1.02 �1.48 �0.24 �1.81Theory (kth) �1.04 �1.53 �0.22 �1.84Δ % 2.0 3.4 8.3 1.7FEM (kf) �1.01 �1.54 �0.23 �1.86Δ % 1.0 4.1 4.2 2.8

φ¼π (Point C) Test (kexp) �1.25 �2.01 0.17 �1.88Theory (kth) �1.29 �2.05 0.18 �1.94Δ% 3.2 2.0 5.9 3.2FEM (kf) �1.29 �2.04 0.16 �1.90Δ % 3.2 1.5 5.9 1.1

Du et al. / Thin-Walled Structures 96 (2015) 49–5554

From Table 4, it can be also found that theoretical solution forthe end of two near ribs is confirmed by FEM and experimentresults, and the maximum error is less than 10%. Here the errormay be caused by the quality of strain stuck on the shell, wherethe strains are difficult to stick on ideal position because of stif-fened ribs and welding.

Therefore, the curve beam model for toroidal shell with stif-fened ribs is confirmed to be reasonable and theoretical solution oftoroidal shell with ribs should be correct.

5. Conclusion

In this paper, the simple curve-beam model with elastic sup-ports has been applied for the solution of toroidal shell with stif-fened ribs under external pressure. By solving stress and dis-placement of curve-beam, the stress and displacement of toroidalshell with ribs have been obtained accordingly. Analysis results ofsimple theoretical solution have been certified by comparing withthose of FEM and one steel welding model experiment. Thereforethe theoretical solution has been confirmed correctly and could beadopted to design for underwater engineering, which could beconvenient to provide stress factor curve for analysis and design.

Acknowledgments

This work is supported by the Program of National NaturalScience Foundation of China, China “Theoretical and ExperimentalResearch about Structure of Pressure Toroidal Shell in Engineer-ing” (Grant no. 51109190) and it was completed when the firstauthor worked in China Ship Scientific Research Center. Since July2014, the first author has moved to Shanghai Ocean University andhe wishes to thank Prof. Weicheng Cui, who is the director of hadalscience and technology research center, Shanghai Ocean Uni-versity, for his help in revising this manuscript.

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[21] Guang Zou, Xingning Peng, Qinghai Du, Theoretical solution and essentialresearch of the ring- stiffened toroidal Shell, J. Ship Mech. 16 (1–2) (2012)83–92 (In Chinese).

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