Engineering Structures 68 (2014) 111–120

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Engineering Structures

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Nonlinear static analysis of an axisymmetric shell storage containerin spherical polar coordinates with constraint volume

http://dx.doi.org/10.1016/j.engstruct.2014.02.0140141-0296/� 2014 Elsevier Ltd. All rights reserved.

⇑ Corresponding author. Tel.: +66 2 470 9146; fax: +66 2 427 9063.E-mail address: somchai.chu@kmutt.ac.th (S. Chucheepsakul).

Weeraphan Jiammeepreecha a, Somchai Chucheepsakul a,⇑, Tseng Huang b

a Department of Civil Engineering, Faculty of Engineering, King Mongkut’s University of Technology Thonburi, Bangkok 10140, Thailandb Department of Civil Engineering, University of Texas at Arlington, Arlington, TX 76019, USA

a r t i c l e i n f o a b s t r a c t

Article history:Received 17 January 2013Revised 21 February 2014Accepted 23 February 2014Available online 27 March 2014

Keywords:Axisymmetric membrane shellIncompressible fluidHalf drop shellHydrostatic pressureSpherical polar coordinate

An axisymmetric membrane shell fully filled with an incompressible fluid is investigated and it ismodeled as a half drop shell storage container under hydrostatic pressure subjected to the volume con-straint conditions of the shell and contained fluid. The shell geometry is simulated using one-dimensionalbeam elements described in spherical polar coordinates. Energy functional of the shell expressed in theappropriate forms is derived from the variational principle in terms of displacements and the obtainednonlinear equation can be solved by an iterative procedure. Numerical results of the shell displacementswith various water depths, thicknesses, and internal pressures are demonstrated.

� 2014 Elsevier Ltd. All rights reserved.

1. Introduction

Axisymmetric shells as structural elements are widely used inmany engineering fields, such as structural, mechanical, aeronauti-cal and offshore engineering [1–8]. Examples of shell structures instructural and mechanical engineering are concrete arch domes,and liquid tanks, including the development of pressure vesselsand underwater storage containers. Aircrafts, ship hulls, and sub-marines are examples of the usage of shells in aeronautical and off-shore engineering. In the field of biomechanics, axisymmetricshells are commonly used in the cornea and crystalline lens of ahuman eye model [9–14]. These structures are very effective inresisting external loading [15].

The geometrically nonlinear analysis of thin elastic shells of rev-olution subjected to arbitrary load has been presented by Ball[16,17]. The analysis was based on Sanders nonlinear thin shelltheory and solved by finite difference formulation. Based on thismethod, the nonlinear terms were treated in the form of pseudo-loads. The advantage of this method is the considerable reductionin execution time. However, many researchers have presented theuse of finite element formulations for analysis of axisymmetricshells [18–24]. Delpak and Peshkam [20] developed the variationalmethod in order to study the geometrically nonlinear behavior of a

rotational shell. The formulation is based on the second variation ofthe total potential energy equation and implemented using finiteelement method, as in the research of Peshkam and Delpak [21].New finite formulations for analysis of shells of revolution havebeen presented by Delpak [18], Teng and Rotter [19], and Hongand Teng [22]. In addition, Polat and Calayir [23] proposed formu-lations for investigating shells of revolution based on the totalLagrangian approach, and where the material behavior wasassumed to be linearly elastic. The numerical solutions wereobtained by using Newmark integration technique coupled withNewton–Raphson iteration procedure. Wu [24] presented a vectorform intrinsic finite element (VFIFE) for the dynamic nonlinearanalysis of shell structures. This method was based on the theoryof vector form analysis. The numerical results are accurate and effi-cient for solving the shell problems includes geometrical and mate-rial nonlinearity. Moreover, Sekhon and Bhatia [25] presented amethod for generating stiffness coefficients and fixed edge forcesfor a spherical shell element. The formulation was based on anapproximate analytical solution of the differential equations. Thenumerical results are accurate and computing time is reducedconsiderably because of the use of fewer elements. Lang et al.[26] proposed the nonlinear static analysis of shells of revolutionusing a ring element. The displacement field in the circumferentialdirection of the ring element is defined by Fourier series. For ana-lyzing a shell is fully filled with an incompressible fluid, Sharmaet al. [27] presented the effect of internal fluid height level on

Fig. 1. Three states of the shell.

112 W. Jiammeepreecha et al. / Engineering Structures 68 (2014) 111–120

the natural frequency of a vertical clamped-free cylindrical shell. Itwas found that the natural frequency of the cylindrical shell de-creased when increasing the internal fluid height level.

In the field of offshore work, the applications of an axisymmetricshell for investigating the behavior of an underwater spherical shellhas received much attention from many researchers [5,6,28–31].Yasuzawa [28] proposed the static and dynamic responses of anunderwater half drop shell by the theory of thin shells of revolutionand finite element method. It was found that the displacement andmembrane stress distribution are uniform along the meridian ex-cept at the bottom part. Furthermore, the optimal dome shape ofsubmerged spherical domes has been presented by Vo et al. [29]and Wang et al. [30]. They proposed the membrane analysis andminimum weight design of submerged spherical domes. The numer-ical solutions were obtained using shooting-optimization tech-nique. It was found that the variation of the shell thickness ofspherical domes can be accurately defined by the first-nine termsin the power series for practical applications. Recently, Jiammeep-reecha et al. [31] adopted the finite element technique used by Goan[11] in finding the static equilibrium configurations of a deep waterhalf drop shell with constraint volume. In order to solve this problemand obtain the correct solution, the meridian line should be dividedinto many finite elements in two sub-regions. At the junction of twoadjacent sub-regions, the function values, displacements and slopesare made all continuous. The problem can be solved by using theLagrange multiplier technique; in the process four Lagrange multi-pliers are used. To alleviate this difficulty, this paper presents anew technique of discretization by using spherical polar coordi-nates. The advantage of this discretization technique is that themeridian curve is not divided into many regions. The meridian curveis divided into number of sub-regions with equal arc length. Thus,the Lagrange multiplier technique is no longer used in this analysis.

The purpose of this paper is to present the nonlinear static anal-ysis of an axisymmetric membrane shell storage container sub-jected to hydrostatic pressure and constraint volume of the shell.Since the geometry of the shell is always axisymmetric, any merid-ional curve may be considered as the generating curve. Therefore,the shell is simulated using one-dimensional beam elements and isdescribed in spherical polar coordinates. Third-order shape func-tions are used in the finite element formulation. In this study, largedisplacements and rotations are considered. Thus, the strainenergy density is expressed as a quadratic function of Lagrangianstrains, and the material behavior is assumed to be linearly elastic.By using the strain–displacement relations, the strain energy den-sity is given in terms of displacements and their derivatives. Thisproblem is formulated in a variational form by using shell theory[15], and written in the appropriate forms [32] which are intro-duced in order to reduce the computation time. The principle ofvirtual work [33] and finite element method are used to solvethe problem, and then the nonlinear equilibrium equations are de-rived. These equations are solved by an iterative process. The finaldeformed configuration of the shell is to be determined.

Fig. 2. Shell reference surface.

2. Analytical model

Consider the three states of a shell, shown in Fig. 1. At an unde-formed state, the empty shell without any strains is designated asthe initial unstrained state (IUS). When the initially unstrained axi-symmetric shell is fully filled with an incompressible fluid and theinternal pressure is assumed to be constant, this state is called thereference state or equilibrium state 1 (ES1), in which the geometricconfiguration is known. The initial strains and displaced configura-tion at the reference state are small and can be determined by tra-ditional shell analysis [34]. It is noted that the initial unstrainedstate and the reference state are the same when the initial strains

are assumed to be zero. Finally, this shell is subjected to severalexternal loadings such as linearly hydrostatic pressure, uniformforce or imposed displacement. The final deformed configurationis to be determined. This state is referred to as the deformed stateor equilibrium state 2 (ES2).

2.1. Shell geometry at the reference state (ES1)

A portion of the shell at reference state (ES1) is shown in Fig. 2.Let (X, Y, Z) be the rectangular coordinates and ði; j; kÞ be the unitvectors along the coordinate axes. A surface may be defined byparametric parameters (h, /); that is X = X(h, /), Y = Y(h, /), andZ = Z(h, /) where (h, /) are the two surface parameters definingthe position of a point on the meridian and longitude, respectively.These two surface parameters specify the orthogonal curvilinear

W. Jiammeepreecha et al. / Engineering Structures 68 (2014) 111–120 113

coordinate lines on the surface. Also, X = X(h, /), Y = Y(h, /), andZ = Z(h, /) are single valued, continuous, and differentiable func-tions. Furthermore, there shall be a one-to-one correspondence be-tween pairs of two surface parameter (h, /) values and points onthe surface P.

Due to symmetry, the shell surface may be generated by rotat-ing a plane curve about the axis of symmetry. For the case of a shellof revolution, any meridional curve may be considered as the gen-erating curve on the r–Z plane, as shown in Fig. 2. Therefore, onlythe change shape of the generating curve need be considered. Let�r be the position vector of any point P. Therefore, the position vec-tor �r can be defined by using parallel circle radius r as

�r ¼ r cos /iþ r sin /jþ Zk ð1Þ

where r = r(h) and Z = Z(h).The total differential line element of �r is given by

d�r ¼ �rhdhþ �r/d/, where subscripts (h, /) denote partial derivativesalong the shell coordinates. The first fundamental form of the ref-erence surface (S) is defined by

d�r � d�r ¼ Edh2 þ 2F dhd/þ Gd/2 ð2Þ

where E, F, and G are the metric tensor components of the referencesurface (S), and are given by

E ¼ �rh � �rh ¼ r2h þ Z2

h ð3aÞ

F ¼ �rh � �r/ ¼ 0 ð3bÞ

G ¼ �r/ � �r/ ¼ r2 ð3cÞ

The unit vector normal to the shell surface at point P can bedetermined by

n ¼�rh � �r/

j�rh � �r/j¼ �rZh cos /i� rZh sin /jþ rrhk

Dð4Þ

in which

D ¼ j�rh � �r/j ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiEG� F2

p¼ r

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffir2h þ Z2

h

qð5Þ

Since dn ¼ nhdhþ n/d/, the second fundamental form of thereference surface (S) is determined by

�d�r � dn ¼ edh2 þ 2f dhd/þ gd/2 ð6Þ

where e, f, and g are the curvature tensor components of the refer-ence surface (S), and given by

e ¼ �rhh � n ¼rhZhh � rhhZhffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

r2h þ Z2

h

q ð7aÞ

f ¼ �rh/ � n ¼ 0 ð7bÞ

g ¼ �r// � n ¼rZhffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

r2h þ Z2

h

q ð7cÞ

Accordingly, the curvature j of a normal section of the surfacecan be determined by

j ¼ edh2 þ 2f dhd/þ g d/2

Edh2 þ 2F dhd/þ Gd/2 ð8Þ

In the case of the axisymmetric shell, the lines of principal cur-vature coincide with the coordinate lines. That means F = f = 0.Therefore, the principal curvatures can be expressed as j1 = e/Eand j2 = g/G.

2.2. Displacements and deformed surface

When the shell is deformed, its reference surface (S) transfers toa new surface (S�). The position vector R of a point on the surface atthe deformed state (ES2) is established. It is expressed in terms ofdisplacements measured from the configuration at reference state(ES1) as follows:

R ¼ �r þ �q ¼ �r þ�rhffiffiffi

Ep uþ

�r/ffiffiffiffiGp v þ nw ð9Þ

where u, v, and w are the displacement components along meridian,longitudinal, and normal directions, respectively. Since an axialshell problem is considered herein, the term ð�r/=

ffiffiffiffiGpÞv is equal to

zero. Let A ¼ffiffiffiEp¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffir2h þ Z2

h

qand B ¼

ffiffiffiffiGp¼ r. Then Rh and R/ can

be written as

Rh ¼ Aþ uh �eA

w� ��rh

Aþ e

Auþwh

� �n ð10aÞ

R/ ¼ Bþ Bh

Au� g

Bw

� ��r/

Bð10bÞ

The metric tensor components of the deformed surface (S�) canbe written as

E� ¼ Rh � Rh ¼ Aþ uh �eA

w� �2

þ eA

uþwh

� �2ð11aÞ

F� ¼ Rh � R/ ¼ 0 ð11bÞ

G� ¼ R/ � R/ ¼ Bþ Bh

Au� g

Bw

� �2

ð11cÞ

2.3. Strain–displacement relations

Consider an infinitesimal line element of length ds0 in the initialunstrained state (IUS) and ds� in the deformed state (ES2); the totalLagrangian strains can be expressed as follows:

eL ¼ 12ðds�Þ2 � ðds0Þ2

ðds0Þ2ð12Þ

Since all the displacements are measured from the referencestate (ES1), it is necessary to express the strain component in termsof arc length ds at the reference state (ES1) and separate it into twoparts, as follows:

eL ¼ e0ds2

ds20

þ eds2

ds20

ð13Þ

in which

e0 ¼12ðdsÞ2 � ðds0Þ2

ðdsÞ2; e ¼ 1

2ðds�Þ2 � ðdsÞ2

ðdsÞ2ð14a-bÞ

It is apparent that e0 is the initial Eulerian strains component atthe reference state (ES1) and e is the added strains componentassociated with the surface deformation from the reference state(ES1) to the deformed state (ES2).

For the case of a symmetrical shell, the shearing strains c0h/ = 0.Hence, the initial Eulerian strains e0 can be expressed in terms ofthe metric tensor components as follows:

e0h ¼12

1� E0

E

� �; e0/ ¼

12

1� G0

G

� �ð15a-bÞ

where E0, F0, and G0 are the metric tensor components at the initialunstrained state (IUS). It should be noted that F0 is zero due to thesymmetrical shell. Thus, Eq. (5) becomes

114 W. Jiammeepreecha et al. / Engineering Structures 68 (2014) 111–120

D0 ¼ffiffiffiffiffiffiffiffiffiffiE0G0

p¼ D

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið1� 2e0hÞð1� 2e0/Þ

qð16Þ

Similarly, the added strains e are related to the metric tensorcomponents by

eh ¼12

E�

E� 1

� �; e/ ¼

12

G�

G� 1

� �ð17a-bÞ

Finally, the total Lagrangian strains can be expressed in a matrixform

feLg ¼ ½T�ðfe0g þ fegÞ ð18Þ

in which

½T� ¼1

1�2e0h0

0 11�2e0/

" #ð19Þ

where [T] is the diagonal material-element matrix. It is formed inorder to transform the strain components from the reference state(ES1) to the initial unstrained state (IUS). By substituting Eq. (11)into Eq. (17), the added strains can be expressed in terms of dis-placements as follows:

eh ¼1A

uh �e

A2 wþ 12

1A

uh �e

A2 w� �2

þ 12

e

A2 uþ 1A

wh

� �2

ð20aÞ

e/ ¼Bh

ABu� g

B2 wþ 12

Bh

ABu� g

B2 w� �2

ð20bÞ

Let fggT ¼ bu w uh whc. Then the added strains can be sep-arated into two parts, a linear and nonlinear, and written in the fol-lowing index form:

ei ¼ eLi þ eN

i ¼ Likgk þ

12

Hiklgkgl ð21Þ

where Lik and Hi

kl are column and symmetric matrices, respectively.These matrices depend on the reference surface (S) characteristicsand are identified by Eq. (20). The strains e0, eL

i , and eNi are constant,

linear, and nonlinear in terms of displacements, respectively.

2.4. Strain energy of the shell

The shell is assumed to be a linearly elastic material of constantthickness. Then the strain energy of the shell can be expressed asfollows:

U ¼Z h2

h1

Z 2p

0

12feLgT ½C 0�feLgtD0 d/dh ð22Þ

in which

½C 0� ¼ E0

1� m2

1 mm 1

� �ð23Þ

where [C0] is the shell material property matrix, t is the shell thick-ness, E0 is the Young’s modulus, and m is the Poisson’s ratio. Substi-tution of Eq. (18) into Eq. (22) yields

U ¼Z h2

h1

12ðfe0gT þ fegTÞ½C�ðfe0g þ fegÞdh ð24Þ

in which

½C� ¼ 2p½T�T ½C 0�½T�tD0 ð25Þ

Since [T] and [C0] are symmetric metric, [C] is also symmetric andCij = Cji. Using the index notations, the strain energy of the shellcan be written as

U ¼Z h2

h1

12

Cijei0e

j0 þ Cijei

0ej þ 1

2Cijeiej

� �dh ð26Þ

The first term in the integrand of Eq. (26) is irrelevant, so it isdropped hereafter. Substituting Eq. (21) into Eq. (26), the strain en-ergy can be expressed in the appropriate forms [32]; their first andsecond derivatives with respect to gi can be obtained by only chang-ing coefficients without re-calculating the matrices.

U ¼Z h2

h1

c0kgk þ

12

c1kn þ

12

kkn þ16

n1kn þ

112

n2kn

� �gkgn

� �dh ð27Þ

in which

c0k ¼ Cijei

0Ljk; c1

kn ¼ Cijei0Hj

kn; kkn ¼ CijLikLj

n ð28a-cÞ

n1kn ¼ Cij Li

kHjmn þ Li

mHjnk þ Li

nHjkm

� �gm ð28dÞ

n2kn ¼ Cij Hi

klHjmn þ

12

HinkHj

lm

� �glgm ð28eÞ

According to Eq. (28), the matrices c1, k, n1, and n2 are symmet-ric and the variation of strain energy dU can be obtained as follows:

dU ¼Z h2

h1

dgk c0k þ c1

kn þ kkn þ12

n1kn þ

13

n2kn

� �gn

� �dh ð29Þ

2.5. Volume change of the shell

The volume change of the shell from reference state (ES1) to de-formed state (ES2) can be expressed in terms of displacements asfollows:

DV ¼ 13

Z h2

h1

Z 2p

0Rh � R/ � R� �rh � �r/ � �r

d/dh ð30Þ

Substituting Eqs. (1), (9) and (10) into Eq. (30) yields

DV ¼ 13

Z h2

h1

Z 2p

0ðv1 þ v2 þ v3Þd/dh ð31Þ

in which

v1 ¼ Bhð�r � nÞ �Be

A2 ð�r � �rhÞ� �

uþ �AgBð�r � nÞ � Be

Að�r � nÞ þ AB

� �w

þ ðBð�r � nÞÞuh þ � BAð�r � �rhÞ

� �wh ð32aÞ

v2 ¼ �Bhe

A3 ð�r � �rhÞ �BeA

� �u2 þ �Bhe

A2 ð�r � nÞ þeg

A2Bð�r � �rhÞ þ Bh

� �uw

þ Bh

Að�r � nÞ

� �uuh þ � Bh

A2 ð�r � �rhÞ � B� �

uwh

þ egABð�r � nÞ � Ag

B� Be

A

� �w2 þ � g

Bð�r � nÞ þ B

� �wuh

þ gABð�r � �rhÞ

� �wwh ð32bÞ

v3 ¼ �Bhe

A2

� �u3 þ eg

AB

� �u2wþ �Bh

A

� �u2wh þ

gB

� �uwwh

þ Bh

A

� �uuhwþ � g

B

� �uhw2 þ �Bhe

A2

� �uw2 þ eg

AB

� �w3 ð32cÞ

where v1, v2, and v3 contain the linear, quadratic, and cubic terms ofthe displacements in the volume change DV, respectively. Using theindex notations, designate Rh, R/, and R by �a1, �a2, and �a3; the unitvectors �rh=A, �r/=B, and n can be denoted by i1, i2, and i3, respectively.Then the vectors �a1, �a2, and �a3 can be written as

�ai ¼ aij ij ¼ ai

j þ bijkgk

� �ij ð33Þ

Consequently, using the permutation symbols (eijk) as follows:

W. Jiammeepreecha et al. / Engineering Structures 68 (2014) 111–120 115

Rh � R/ � R ¼ eijka1i a2

j a3k ð34Þ

Therefore, the volume change can be expressed in the appropri-ate forms [35] as follows:

DV ¼Z h2

h1

vckgk þ

12

vKkn þ

16

vNkn

� �gkgn

� �dh ð35Þ

in which

vck ¼

2p3

eijl a1i a

2j b

3lk þ a1

i b2jka

3l þ b1

ika2j a

3l

� �ð36aÞ

vKkn ¼

2p3

eijl a1i b

2jkb

3ln þ b1

ika2j b

3ln þ b1

ikb2jna

3l þ a1

i b2jnb

3lk þ b1

ina2j b

3lk þ b1

inb2jka

3l

� �ð36bÞ

vNkn ¼

2p3

eijl b1ikb

2jmb3

ln þ b1imb2

jnb3lk þ b1

inb2jkb

3lm þ b1

inb2jmb3

lk

�þb1

ikb2jnb

3lm þ b1

imb2jkb

3ln

�gm ð36cÞ

where vck, vK

kn, and vNkn are linear, quadratic, and cubic in terms of

displacement gradients g, respectively. However, the values of ai

and bij can be derived by Eq. (32). According to Eqs. (36b), (36c),the matrices vK

kn and vNkn are symmetric and directly obtained from

the variation of volume change d(DV) as follows:

dðDVÞ ¼Z h2

h1

dgk vck þ vK

kn þ12

vNkn

� �gn

� �dh ð37Þ

2.6. Strain energy due to internal fluid

Assume that the internal pressure at reference state (ES1) is lin-early proportional to the volumetric strain by the relation

p0 ¼ �~kDV0W

V0Wð38Þ

where ~k is the bulk modulus of the fluid, V0W is the unstrained fluidvolume, and DV0W is the volume change of fluid from the un-strained fluid state to the reference state (ES1). Then the strain en-ergy in the enclosed fluid is determined by the relation

C ¼ 12

~kDV0W þ DV

V0W

� �2

V0W ð39Þ

where DV is the volume change of fluid and shell which areunchangeable. Associated with the variation of volume change,the variation of strain energy due to internal fluid dC can be writtenas

dC ¼ ~kDV0W þ DV

V0W

� �dðDVÞ ð40Þ

Substituting Eq. (38) into Eq. (40) yields

dC ¼ �ðp0 þ kÞdðDVÞ ð41Þ

Physically, k, which represents the change of pressure from thereference state (ES1) to the deformed state (ES2) is defined as

k ¼ �~kDVV0W

ð42Þ

According to Eqs. (38) and (42), the constraint equation is con-sidered by the relation

DV þ V~k� p0

k ¼ 0 ð43Þ

where V is the shell volume at reference state (ES1). It is noted thatthe numerical value of k is an unknown, which will be obtained bysolving the entire problem as will be presented.

2.7. Virtual work done by linearly hydrostatic pressure

The linearly hydrostatic pressure acting on the normal surfaceof the shell is given by

pw ¼ qwgZw ð44Þ

where qw is the density of the sea water, g is the specific gravity,and Zw is the vertical distance from the sea water level. The virtualwork done by linearly hydrostatic pressure dX can be expressed asfollows:

dX ¼Z h2

h1

Z 2p

0pwfdwgDd/dh ¼ 2p

Z h2

h1

pwfdwgD dh ð45Þ

3. Equilibrium equation requirement

Based on the principle of virtual work, the equilibrium equationof the shell can be obtained by setting the total virtual work of theshell to be zero, as follows:

dp ¼ dU þ dCþ dX ¼ 0 ð46Þ

Substituting Eqs. (29), (37), (41), and (45) into Eq. (46) givesZ h2

h1

dgk c0kþ c1

knþkknþ12

n1knþ

13

n2kn

� �gn

� ��

�ðp0þkÞ vckþ vK

knþ12

vNkn

� �gn

� ��dhþ2p

Z h2

h1

pwfdwgDdh¼0 ð47Þ

Two highly nonlinear differential equations in terms of u(h) andw(h) are embedded in the above Euler’s equation. Before the hydro-static pressure is applied, the shell is in equilibrium at the refer-ence state (ES1). Thus, setting pw, k, and gn to be zero, Eq. (47) isalso valid. This requiresZ h2

h1

dgkðc0k � p0vc

kÞdh ¼ 0 ð48Þ

This equation should be satisfied everywhere, and it can be usedto predict the value of initial strains e0.

3.1. Constraint equation

Since the fluid is incompressible, ~k approaches infinity and thelast term in Eq. (43) becomes zero. Thus, k in Eq. (43) may be inter-preted as a Lagrange multiplier associated with the constraint vol-ume (DV = 0). Finally, the constraint equation can be written asZ h2

h1

vckgk þ

12

vKkn þ

16

vNkn

� �gkgn

� �dh ¼ 0 ð49Þ

4. Finite element method

To solve the problem by using the finite element method, theshell is divided along the h coordinate into many finite ring ele-ments. Consider a general single element with the local coordinateu, the shell global coordinate h and the angle a = h2 � h1, as shownin Fig. 3. The local coordinate u is related to the global coordinate hby u = h � h1, and the derivatives of any quantity with respect to uand h are equal. Therefore, using the C1 continuity in finite elementmethod [36], the displacements u(u) and w(u) within each ele-ment are approximated by a third-order polynomial of the localcoordinate u

uðuÞ ¼ b1 þ b2uþ b3u2 þ b4u3 ð50aÞ

wðuÞ ¼ b5 þ b6uþ b7u2 þ b8u3 ð50bÞ

Fig. 3. Deep water axisymmetric half drop shell.

116 W. Jiammeepreecha et al. / Engineering Structures 68 (2014) 111–120

where biði ¼ 1;2; . . . ;8Þ are unknown coefficients. Their first deriv-atives with respect to u or h are as follows:

uhðuÞ ¼ b2 þ 2b3uþ 3b4u2 ð51aÞ

whðuÞ ¼ b6 þ 2b7uþ 3b8u2 ð51bÞ

Consider the eight-unknown coefficient (bi) in Eq. (50). In finiteelement formulation the displacements u and w are expressed interms of element nodal degrees of freedom {d} via the cubic poly-nomial shape function. Therefore, the displacement gradient vector{g} can be expressed as

fgg ¼ ½w�fdg ð52Þ

in which

fggT ¼ buðuÞ wðuÞ uhðuÞ whðuÞ c ð53aÞ

fdgT ¼ buð0Þ wð0Þ uhð0Þ whð0Þ uðaÞ wðaÞ uhðaÞ whðaÞ cð53bÞ

½w� ¼

N1 0 N2 0 N3 0 N4 0

0 N1 0 N2 0 N3 0 N4

N1;u 0 N2;u 0 N3;u 0 N4;u 0

0 N1;u 0 N2;u 0 N3;u 0 N4;u

266664

377775 ð53cÞ

where [w] is the cubic polynomial shape function. This function andits derivatives can be expressed as follows:

N1 ¼ 1� 3u2

a2 þ 2u3

a3 ; N1;u ¼6a�u

aþu2

a2

� �ð54a-bÞ

N2 ¼ u� 2u2

aþu3

a2 ; N2;u ¼ 1� 4uaþ 3

u2

a2 ð54c-dÞ

N3 ¼ 3u2

a2 � 2u3

a3 ; N3;u ¼6a

ua�u2

a2

� �ð54e-fÞ

N4 ¼ �u2

aþu3

a2 ; N4;u ¼ �2uaþ 3

u2

a2 ð54g-hÞ

Substituting Eq. (52) into the matrices c0k , c1

kn, kkn, n1kn, n2

kn, vck, vK

kn,and vN

kn in Eq. (47) yields

fddgTZ h2

h1

½w�Tðfc0g � ðp0 þ kÞfvcgÞdh

�

þZ h2

h1

½w�T ½c1� þ ½k� þ 12½n1� þ 1

3½n2�

�

�ðpo þ kÞ ½vK � þ 12½vN �

� ��½w�dhfdg

�þ ffg ¼ 0 ð55Þ

in which

ffg ¼ 2pfdwgTZ h2

h1

pwfwgD dh

� �ð56Þ

Since the global degree of freedom {Q} is the same as the localdegree of freedom {d}, the global equilibrium equation can be ob-tained by assembly process using Eq. (55). The results are

fC0g � ðp0 þ kÞfVCg

þ ½C1� þ ½K� þ12½N1� þ

13½N2� � ðp0 þ kÞ ½VK� þ 1

2½VN�

� �� �fQg

þ fFg ¼ f0g ð57Þ

Similarly, the constraint equation, Eq. (49), becomes

fQgT fVCg þ 12½VK� þ 1

6½VN�

� �fQg

� �¼ 0 ð58Þ

Finally, the equilibrium equation, Eq. (57), and the constraintequation, Eq. (58), are combined into a symmetrical matrix form,as follows:

ð59Þ

Since an axially symmetrical shell is considered, the boundaryconditions at the top are

u ¼ 0; wh ¼ 0 ð60Þ

The supported condition is considered to be fully fixed at thesea bed. Therefore

u ¼ 0; w ¼ 0; uh ¼ 0; wh ¼ 0 ð61Þ

The system of nonlinear equations in Eq. (59), which is con-strained by both boundary conditions Eqs. (60) and (61), can besolved numerically by an iterative procedure.

5. Numerical example and results

In order to present the finite element formulation of the mem-brane shell theory, one has to study the behaviors of the axisym-metric half drop shell storage container installed in deep water,as shown in Fig. 3. A computer program developed by Goan [11]is modified to solve the problem, and the independent variableto h coordinate is used. This independent variable is generally fora spherical shell having a constant Gaussian curvature. In the case

Table 2Convergence of deflection of half drop shell at the apex.

Number of elements wapex (�10�3 m)

8 0.01942512 0.02004416 0.02035620 0.02054424 0.02066928 0.020759

W. Jiammeepreecha et al. / Engineering Structures 68 (2014) 111–120 117

of discretization by using h coordinate, the meridian curve is notdivided into many regions. Therefore, the size of the global matrixis reduced in parts of the four Lagrange multipliers when comparedwith the previous work of Jiammeepreecha et al. [31].

To validate the accuracy of the present solutions, consider a halfdrop shell, as shown in Fig. 3, submerged at a water depth ofH = 1745 mm. The shell geometry and material are: a = 220 mm,t = 2.5 mm, E0 = 757 kgf/mm2, and m = 0.36, and the specific weightof external fluid is cw = 1.0 � 10�6 kgf/mm3. The present resultsshow the tangential and normal displacements for a linearly dis-tributed hydrostatic pressure along the sea depth and a constanthydrostatic pressure at the sea bed. As shown in Fig. 4, it can beseen that the results of constant hydrostatic pressure are in closeagreement with Yasuzawa’s results [28], except the normal dis-placement near the support. In this study, the hydrostatic pressureis varied along the sea depth, while there is no information on thehydrostatic pressure in Yasuzawa’s [28] work. However, thenumerical results from this study were verified with Roark’s for-mula for a spherical subjected to uniform external pressure [37],and were found to be conformable. The input parameters em-ployed in this analysis are tabulated in Table 1.

Fig. 5. Configuration of the half drop shell at deformed state (ES2).

5.1. Half drop shell behavior subjected to hydrostatic pressure

Table 2 shows the convergence of the apex displacement forhalf drop shell with constraint volume. It can be seen that the high-er mesh models gives the more accurate result. However, the dif-ference of apex movement is less than 0.50% between the modelwith 24 and 28 elements. In this paper the model with 24 elementsis assumed to be sufficient for accurate results.

Based on the results of the first part of the study, the shell at thedeformed state (ES2) subjected to hydrostatic pressure is shown inFig. 5. The present results show very good agreement with the pre-vious work of Jiammeepreecha et al. [31].

Fig. 4. Comparison of displacement responses with Yasuzawa’s results [28].

Table 1Input parameter data.

Parameter Value

Young’s modulus, E0 (N/m2) 2.04 � 1011

Poisson’s ratio, m 0.30Sea water level, H (m) 40Radius of shell, a (m) 5Thickness of shell, t (m) 0.20Initial internal pressure, p0 (N/m2) 50 � 103

Density of sea water, qw (kg/m3) 1025

5.2. Effects of hydrostatic pressure on half drop shell

Linearly varying hydrostatic pressure has no effect on the dis-placement response of the half drop shell, as shown in Figs. 6and 7. Fig. 8 describes the values of k versus the sea water level.It can be seen that the change of pressure from the reference state(ES1) to the deformed state (ES2) is linearly proportional to thehydrostatic pressure; that is, the value of k increases under largehydrostatic pressure and decreases when the hydrostatic pressurebecomes small.

5.3. Effects of radius-to-thickness ratio on half drop shell

Using the main data in Table 1 and varying the radius-to-thick-ness ratio (a/t ratio) from 25 to 200, the tangential and normal dis-placements of the half drop shell are shown in Figs. 9 and 10,

Fig. 6. Effects of linearly varying hydrostatic pressure on tangential displacement ofhalf drop shell.

Fig. 7. Effects of linearly varying hydrostatic pressure on normal displacement ofhalf drop shell.

Fig. 8. Effects of linearly varying hydrostatic pressure on the change of pressure k.

Fig. 10. Effects of thickness variation on normal displacement of half drop shell.

Fig. 11. Effects of thickness variation on the change of pressure k.

Fig. 12. Effects of initial internal pressure on tangential displacement of half dropshell.

118 W. Jiammeepreecha et al. / Engineering Structures 68 (2014) 111–120

respectively. It can be seen that the radius-to-thickness ratio has asignificant effect on the displacements on a deep water half dropshell. On the contrary, changing of the radius-to-thickness ratiosmay have little effect on the values of k, as shown in Fig. 11. Fur-thermore, the results show that the point of intersection is onthe same location, as shown in Fig. 10. This intersection point loca-tion is independent of the radius-to-thickness ratio.

5.4. Effects of initial internal pressure on half drop shell

Using the main data in Table 1 and varying the initial internalpressure (p0) from 50 � 103 to 50 � 106 N/m2, the tangential andnormal displacements of the half drop shell are shown in Figs. 12

Fig. 9. Effects of thickness variation on tangential displacement of half drop shell.Fig. 13. Effects of initial internal pressure on normal displacement of half dropshell.

Fig. 14. Effects of initial internal pressure on the change of pressure k.

W. Jiammeepreecha et al. / Engineering Structures 68 (2014) 111–120 119

and 13, respectively. When a low value of initial internal pressureis applied to the shell, the shell has a large effect versus the highinitial internal pressure. Fig. 14 shows the effects of the increaseof initial internal pressure on the values of k. It can be seen thatthe values of k decrease when the initial internal pressure becomeslarge. However, the values of k are unchangeable under high initialinternal pressure.

6. Conclusions

The nonlinear static responses of a deep-water axisymmetrichalf drop shell storage container with constraint volume conditionby using membrane theory are presented in this paper. The prob-lem is formulated by using the variational principle and the finiteelement method in terms of displacements, which are expressed inthe appropriate forms. The change of pressure from the referencestate to the deformed state may be explained as a Lagrange multi-plier. In the present study, small displacement theory and tradi-tional shell analysis are used to calculate the initial strains anddisplaced configuration of the half drop shell, respectively. Thenumerical results indicate that the Lagrange multiplier representsthe parameter for adjusting the internal pressure in order to sus-tain the shell in equilibrium position under the constraint volumecondition.

The numerical results show that the effect of radius-to-thickness ratio has a major impact on the displacements on theshell, whereas changing the hydrostatic pressure has no effect.However, by varying linearly hydrostatic pressure, the change ofpressure from the reference state to the deformed state is linearlyproportional to the hydrostatic pressure. For a large value of initialinternal pressure, the change of pressure is unchangeable.

Acknowledgements

The first and second authors gratefully acknowledge financialsupport by the Thailand Research Fund (TRF) and King Mongkut’sUniversity of Technology Thonburi (KMUTT) through the RoyalGolden Jubilee Ph.D. program (Grant No. PHD/0134/2552).

Appendix A. Derivation of initial Eulerian strain

The initial engineering strains ee and initial Lagrangian strainseL

0 are related by [11]

eL0 ¼

12ðdsÞ2 � ðds0Þ2

ðds0Þ2¼ ee þ

12e2

e ðA:1Þ

in which

ee ¼ds� ds0

ds0ðA:2Þ

In the present study, the initial engineering strains ee are as-sumed to be small, and the small displacement theory is used tocalculate the initial strains. Furthermore, by neglecting the qua-dratic term in Eq. (A.1), eL

0 � ee. The initial engineering strains ee

can be computed by using the membrane theory equilibrium equa-tion of the shell; that is

Nh

r1þ N/

r2¼ p0 ðA:3Þ

where p0 is the internal pressure. For the case of a reference surface(S) of a spherical shell having a constant Gaussian curvature, the to-tal tension forces are Nh = N/ = N and the principal curvatures are 1/r1 = 1/r2 = 1/a. Therefore, the tension force N is given by

N ¼ 12

p0a ðA:4Þ

The initial engineering strains ee can be determined by

ee ¼rE0ð1� mÞ ¼ N

E0tð1� mÞ ¼ p0a

2E0tð1� mÞ ðA:5Þ

Let a0 be the radius of a spherical shell at the initial unstrainedstate (IUS). The initial Lagrangian strain eL

0 becomes

eL0 ¼ ee ¼

12

a2 � a20

a20

ðA:6Þ

Finally, the initial Eulerian strain e0 component at the referencestate (ES1) can be determined by

e0 ¼12ðdsÞ2 � ðds0Þ2

ðdsÞ2¼ 1

2a2 � a2

0

a2 ðA:7Þ

Appendix B. Characteristic quantities of the reference surface

Referring to the position vector in Eq. (1), the reference surface(S) of a spherical shell having a radius a can be defined by

�r ¼ a sin h cos /iþ a sin h sin /jþ a cos hk ðB:1Þ

in which

r ¼ a sin h; rh ¼ a cos h; rhh ¼ �a sin h ðB:2a-cÞ

Z ¼ a cos h; Zh ¼ �a sin h; Zhh ¼ �a cos h ðB:3a-cÞ

The metric tensor components of the reference surface (S) are

E ¼ r2h þ Z2

h ¼ a2; F ¼ 0; G ¼ r2 ¼ a2 sin 2h ðB:4a-cÞ

Also,

D ¼ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffir2h þ Z2

h

q¼ a2 sin h ðB:5Þ

Let A ¼ffiffiffiEp

and B ¼ffiffiffiffiGp

, then

A ¼ a; B ¼ a sin h; Bh ¼ a cos h ðB:6a-cÞ

Therefore, the unit vector normal is given by

n ¼ �rZh cos /i� rZh sin /jþ rrhkD

¼ sin h cos /iþ sin h sin /jþ cos hk ðB:7Þ

The curvature tensor components of the reference surface (S)are

120 W. Jiammeepreecha et al. / Engineering Structures 68 (2014) 111–120

e ¼ rhZhh � rhhZhffiffiffiffiffiffiffiffiffiffiffiffiffiffiffir2h þ Z2

h

q ¼ �a; f ¼ 0; g ¼ rZhffiffiffiffiffiffiffiffiffiffiffiffiffiffiffir2h þ Z2

h

q ¼ �a sin2 h

ðB:8a-cÞSince F = f = 0, the coordinate lines (h, /) are also lines of princi-

pal curvature. Accordingly, the principal curvatures can be deter-mined by

j1 ¼eE¼ �1

a; j2 ¼

gG¼ �1

aðB:9a-bÞ

From Eqs. (B.1) and (B.7), the quantities �r � �rh and �r � n are givenby

�r � �rh ¼ 0; �r � n ¼ a ðB:10a-bÞ

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