An axisymmetric membrane shell fully filled with an incompressible fluid is investigated and it is modeled as a half dro...
Engineering Structures 68 (2014) 111–120
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Engineering Structures j o u r n a l h o m e p a g e : : w w w . e l s e v i e r . c o m / l o c a t e / e n g s t r u c t
Nonlinear static analysis of an axisymmetric shell storage container in spherical polar coordinates with constraint volume Weeraphan Jiammeepreecha Jiammeepreecha a, Somchai Chucheepsakul a, , Tseng Huang b ⇑
a b
Department of Civil Engineering, Faculty of Engineering, King Mongkut’s University of Technology Thonburi, Bangkok 10140, Thailand Department of Civil Engineering, University of Texas at Arlington, Arlington, TX 76019, USA
a r t i c l e
i n f o
Article history: Received 17 January 2013 Revised 21 February 2014 Accepted 23 February 2014 Available online 27 March 2014 Keywords: Axisymmetric Axisymmetric membrane membrane shell Incompressible Incompressible fluid Half drop shell Hydrostatic Hydrostatic pressure Spherical polar coordinate
a b s t r a c t
An axisymmetric axisymmetric membrane membrane shell fully filled with an incompre incompressibl ssible e fluid is investigat investigated ed and it is modeled as a half drop shell storage container under hydrostatic pressure subjected to the volume constrain straintt condit condition ions s of the shell shell and contai contained ned fluid. fluid. The shell shell geomet geometry ry is simula simulated ted using using one-d one-dime imensi nsiona onall beam elements described in spherical polar coordinates. Energy functional of the shell expressed in the appropriate forms is derived from the variational principle in terms of displacements and the obtained nonlinear equation can be solved by an iterative procedure. Numerical results of the shell displacements with 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 in many many engineer engineering ing fields, fields, such as structural structural,, mechani mechanical, cal, aeronaut aeronautiical and offshore engineering [1–8] [1–8].. Examples of shell structures in structural structural and mechanic mechanical al engineer engineering ing are concrete concrete arch domes, domes, and liquid tanks, tanks, including including the develop developmen mentt of pressure pressure vessels vessels and underwater storage containers. Aircrafts, ship hulls, and submarin marines es are exam exampl ples es of the usage usage of shells shells in aeron aeronaut autica icall and offoffshore shore engineer engineering. ing. In the field of biomecha biomechanics nics,, axisymm axisymmetric etric shells are commonly used in the cornea and crystalline lens of a human human eye model model [9–14]. [9–14]. These structures are very effective in resisting external loading [15] [15].. The The geome geometri trical cally ly nonli nonlinea nearr analys analysis is of thin thin elasti elastic c shells shells of revrevolution olution subjected subjected to arbitrary arbitrary load has been presente presented d by Ball [16,17].. The analysis [16,17] analysis was based on Sanders Sanders nonlinear nonlinear thin shell theory and solved by finite difference formulation. Based on this method, the nonlinear terms were treated in the form of pseudoloads. The advantage of this method is the considerable reduction in execution time. However, many researchers have presented the use of finite finite element element formulat formulations ions for analysis analysis of axisymm axisymmetric etric shells [18–24] [18–24].. Delpa Delpak k and Peshkam Peshkam [20] [20] develo developed ped the variation variational al metho method d in order order to study study the geome geometri trical cally ly nonli nonlinea nearr behav behavior ior of a ⇑
Corresponding author. Tel.: +66 2 470 9146; fax: +66 2 427 9063.
E-mail address:
[email protected] (S.
[email protected] (S. Chucheepsakul). http://dx.doi.org/10.1016/j.engstruct.2014.02.014 0141-0296/ 2014 Elsevier Ltd. All rights reserved.
rotati rotationa onall shell. shell. The The formu formulat lation ion is based based on the secon second d variat variation ion of the total potential energy equation and implemented using finite element method, as in the research of Peshkam and Delpak [21] [21].. New finite formulatio formulations ns for analysis analysis of shells shells of revolutio revolution n have been been presente presented d by Delpak Delpak [18], [18], Teng and Rotter [19], [19], and Hong and Teng [22] Teng [22].. In addition, Polat and Calayir [23] [23] proposed proposed formulations lations for investigating investigating shells of revolutio revolution n based based on the total Lagra Lagrangi ngian an appro approach ach,, and and where where the mater material ial behav behavior ior was was assum assumed ed to be linea linearly rly elasti elastic. c. The The numeri numerical cal solut solutio ions ns were were obtained by using Newmark Newmark integration technique coupled with Newton–Raphso Newton–Raphson n iteration procedure. Wu [24] Wu [24] presented presented a vector form intrinsic intrinsic finite finite element element (VFIFE) (VFIFE) for the dynami dynamic c nonline nonlinear ar analysis of shell structures. This method was based on the theory of vector vector form form analys analysis. is. The The numeri numerical cal result results s are accura accurate te and effiefficient cient for for solvi solving ng the shell shell proble problems ms includ includes es geome geometric trical al andmaterial nonlinea nonlinearity. rity. Moreove Moreover, r, Sekhon Sekhon and Bhatia Bhatia [25] presented a method for generating stiffness coefficients and fixed edge forces for a spherical spherical shell element. element. The formulation formulation was based based on an approximate analytical solution of the differential equations. The numerica numericall results results are accurate accurate and computi computing ng time is reduced reduced consider considerably ably because because of the use of fewer elements. elements. Lang et al. [26] proposed the nonlinear static analysis of shells of revolution [26] using a ring element. The displacement displacement field in the circumferential direction of the ring element is defined by Fourier series. For analyzing a shell is fully filled with an incompressible fluid, Sharma et al. [27] presented presented the effect of internal fluid height level on
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the natural frequency of a vertical clamped-free cylindrical shell. It was found that the natural frequency of the cylindrical shell decreased when increasing the internal fluid height level. In the field of offshore work, the applications of an axisymmetric shell for investigating the behavior of an underwater spherical shell has received much attention from many researchers [5,6,28–31]. Yasuzawa [28] proposed the static and dynamic responses of an underwater half drop shell by the theory of thin shells of revolution and finite element method. It was found that the displacement and membrane stress distribution are uniform along the meridian except at the bottom part. Furthermore, the optimal dome shape of submerged spherical domes has been presented by Vo et al. [29] and Wang et al. [30]. They proposed the membrane analysis and minimumweight design of submerged sphericaldomes. Thenumerical solutions were obtained using shooting-optimization technique. It was found that the variation of the shell thickness of spherical domes can be accurately defined by the first-nine terms in the power series for practical applications. Recently, Jiammeepreechaet al. [31] adoptedthe finite element techniqueused by Goan [11] in finding the static equilibriumconfigurations of a deep water halfdrop shell with constraint volume.In order to solve thisproblem and obtain the correct solution, the meridian line should be divided into many finiteelements in two sub-regions.At the junction of two adjacent sub-regions, the function values, displacements and slopes are made all continuous. The problem can be solved by using the Lagrange multiplier technique; in the process four Lagrange multipliers are used. To alleviate this difficulty, this paper presents a new technique of discretization by using spherical polar coordinates. The advantage of this discretization technique is that the meridiancurve is not dividedinto many regions. The meridiancurve is 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 analysis 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 meridional curve may be considered as the generating curve. Therefore, the shell is simulated using one-dimensional beam elements and is described in spherical polar coordinates. Third-order shape functions are used in the finite element formulation. In this study, large displacements and rotations are considered. Thus, the strain energy density is expressed as a quadratic function of Lagrangian strains, and the material behavior is assumed to be linearly elastic. By using the strain–displacement relations, the strain energy density is given in terms of displacements and their derivatives. This problem is formulated in a variational form by using shell theory [15], and written in the appropriate forms [32] which are introduced in order to reduce the computation time. The principle of virtual work [33] and finite element method are used to solve the problem, and then the nonlinear equilibrium equations are derived. These equations are solved by an iterative process. The final deformed configuration of the shell is to be determined.
Fig. 1. Three states of the shell.
are assumed to be zero. Finally, this shell is subjected to several external loadings such as linearly hydrostatic pressure, uniform force or imposed displacement. The final deformed configuration is to be determined. This state is referred to as the deformed state or 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. ^ ^ be the unit Let ( X , Y , Z ) be the rectangular coordinates and ^i; j; k vectors along the coordinate axes. A surface may be defined by parametric parameters (h, /); that is X = X (h, /), Y = Y (h, /), and Z = Z (h, /) where (h, /) are the two surface parameters defining the position of a point on the meridian and longitude, respectively. These two surface parameters specify the orthogonal curvilinear
ð
2. Analytical model Consider the three states of a shell, shown in Fig. 1. At an undeformed state, the empty shell without any strains is designated as the initial unstrained state (IUS). When the initially unstrained axisymmetric shell is fully filled with an incompressible fluid and the internal pressure is assumed to be constant, this state is called the reference state or equilibrium state 1 (ES1), in which the geometric configuration is known. The initial strains and displaced configuration at the reference state are small and can be determined by traditional shell analysis [34]. It is noted that the initial unstrained state and the reference state are the same when the initial strains
Fig. 2. Shell reference surface.
Þ
W. Jiammeepreecha et al. / Engineering Structures 68 (2014) 111–120
coordinate lines on the surface. Also, X = X (h, /), Y = Y (h, /), and Z = Z (h, /) are single valued, continuous, and differentiable functions. Furthermore, there shall be a one-to-one correspondence between pairs of two surface parameter ( h, /) values and points on the surface P . Due to symmetry, the shell surface may be generated by rotating a plane curve about the axis of symmetry. For the case of a shell of revolution, any meridional curve may be considered as the generating curve on the r – Z plane, as shown in Fig. 2. Therefore, only the change shape of the generating curve need be considered. Let r be the position vector of any point P . Therefore, the position vector r can be defined by using parallel circle radius r as
r r cos / i r sin / j Z k
¼
^
þ
^
þ
ð 1Þ
^
where r = r (h) and Z = Z (h). r is given by The total differential line element of dr r h dh r / d/, where subscripts (h, /) denote partial derivatives along the shell coordinates. The first fundamental form of the reference surface (S ) is defined by
¼
þ
dr dr E dh2
2
ð 2Þ
where E , F , and G are the metric tensor components of the reference surface (S ), and are given by
E
2
2
h
h
¼ r r ¼ r þ Z h
h
F
ð3aÞ
¼ r r ¼ 0 h
G
ð3bÞ
/
r / r /
¼ ¼
r 2
3c
ð Þ
The unit vector normal to the shell surface at point P can be determined by
r rZ cos /i rZ sin / j þ rr k ¼ jr r ¼ D r j h
^
/
n
h
h
^
^
h
/
¼ jr r j ¼ h
/
p ffiffi ffi ffi ffi ffi ffi ffi ffi ¼ q ffiffi ffi þffi ffi ffi ffi ffi F 2
EG
h
ð 4Þ
r r 2h
Z 2h
þ
2
2
dr dn ¼ e dh þ 2 f d h d/ þ gd/
^
ð 6Þ
where e , f , and g are the curvature tensor components of the reference surface (S ), and given by
¼ r n ¼ r Z r Z r þ Z h
e
hh
^
hh
hh
2 h
f
h
ð7aÞ
q ffiffi ffi ffi ffi ffi ffi ffi 2
h/
g
//
^
ð7bÞ rZ h
ð7cÞ
q ffiffi ffi þffi ffi ffi ffi ffi r 2h
Z 2h
Accordingly, the curvature j of a normal section of the surface can be determined by
j ¼
e dh2 E dh2
h
/
ffiffi ffiffi
v
ð9Þ
^
where u, v , and w are the displacement components along meridian, longitudinal, and normal directions, respectively. Since an axial
p ffiffiffiffi Þ
r / = G shell problem is considered herein, the term
ð
p ffiffiffi q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p ffiffiffiffi zero. Let A ¼ E ¼ r þ Z and B ¼ G ¼ r . Then R 2 h
2 h
v
is equal to
h
and R/ can
be written as
Rh
¼ A þ u A e w Ar þ
¼ þ
h
Bh u A
B
e u wh n A
h
þ
ð10aÞ
^
g r / w B B
ð10bÞ
The metric tensor components of the deformed surface ( S ) can be written as 2
¼ R R ¼ A þ u A e w þ
E
h
F
h
e u wh A
h
þ
2
¼ R R ¼ 0 h
ð11aÞ ð11bÞ
/
Bh g G ¼ R / R/ ¼ B þ u w
A
B
2
ð11cÞ
2.3. Strain–displacement relations Consider an infinitesimal line element of length ds0 in the initial unstrained state (IUS) and ds in the deformed state (ES2); the total Lagrangian strains can be expressed as follows: L
e
¼ ð Þ ð Þ ð Þ 1 ds ds0 2 2 ds0
2
ð12Þ
Since all the displacements are measured from the reference state (ES1), it is necessary to express the strain component in terms of arc length ds at the reference state (ES1) and separate it into two parts, as follows: 2
eL ¼ e0
ds
2
ds0
þ e ds
2
ð13Þ
2
ds0
in which 2
^
¼ r n ¼
r r u þ p þ nw ¼ r þ q ¼ r þ p E G
h
¼ r n ¼ 0
R
2
ð 5Þ
^ ^ h dh ^ Since dn n n/ d/, the second fundamental form of the reference surface (S ) is determined by
¼
When the shell is deformed, its reference surface (S ) transfers to a new surface (S ). The position vector R of a point on the surface at the deformed state (ES2) is established. It is expressed in terms of displacements measured from the configuration at reference state (ES1) as follows:
^
in which
D
2.2. Displacements and deformed surface
R/
þ 2F dh d/ þ G d/
¼
113
2
þ 2 f d h d/ þ g d/ þ 2F dh d/ þ G d/
2
ð 8Þ
In the case of the axisymmetric shell, the lines of principal curvature coincide with the coordinate lines. That means F = f = 0. Therefore, the principal curvatures can be expressed as j1 = e/E and j 2 = g /G.
e0 ¼
1 ds ds0 2 2 ds
2
2
ð Þ ð Þ ð Þ
;
ð Þ ð Þ ð Þ
1 ds ds 2 2 ds
e ¼
2
ð14a-bÞ
It is apparent that e0 is the initial Eulerian strains component at the reference state (ES1) and e is the added strains component associated 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 e 0 can be expressed in terms of the metric tensor components as follows:
e0h ¼
1 E 0 1 2 E
;
e0/ ¼
1 G0 1 2 G
ð15a-bÞ
where E 0, F 0, and G0 are the metric tensor components at the initial unstrained state (IUS). It should be noted that F 0 is zero due to the symmetrical shell. Thus, Eq. (5) becomes
W. Jiammeepreecha et al. / Engineering Structures 68 (2014) 111–120
114
D0
¼
p ffiffi ffi ffi ffi ffi ¼ q ffiðffi ffi ffiffi ffi ffi ffi ffi ffi ffiÞðffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffiÞffi E 0 G0
D
2e0h 1
1
2e0/
ð16Þ
Similarly, the added strains e are related to the metric tensor components by
eh ¼
1 E 2 E
1
e/ ¼
;
1 G 2 G
1
ð17a-bÞ
Finally, the total Lagrangian strains can be expressed in a matrix form L
fe g ¼ ½T ðfe g þ fegÞ
ð18Þ
0
The first term in the integrand of Eq. (26) is irrelevant, so it is dropped hereafter. Substituting Eq. (21) into Eq. (26), the strain energy can be expressed in the appropriate forms [32]; their first and second derivatives with respect to g i can be obtained by only changing coefficients without re-calculating the matrices. h2
U
¼
½T ¼
1 1 2e0h
0
0
1 1 2e0/
1 A
e
e/ ¼
2
A
Bh u AB
w
1 1 uh 2 A
þ þ
Bg w 2
Ae w
1 Bh u 2 AB
2
Bg w 2
T
2
1 e u 2 A2
þ
þ 1 A w
h
f g ¼ b
¼ C
n2kn
¼ C
0
h1
h2
dU
ð20bÞ
E 0
¼1m
2
1 DV 3
¼
ð21Þ
h2
U
h1
1 2
¼ 13
DV
½C 0 feL gtD0 d/ dh
ð22Þ
m
m 1
ðfe g þ feg Þ½C ðfe g þ fegÞdh 0
0
T
½C ¼ 2p½T ½C 0 ½T tD0
ð25Þ
Since [T ] and [C 0 ] are symmetric metric, [C ] is also symmetric and C ij = C ji. Using the index notations, the strain energy of the shell can be written as h2
1 C ij ei0 e j0 2
U
h1
C ij i0 j
þ
ee
þ
ð28a-cÞ
i j m H nk
ð28dÞ
H ikl H jmn
1 i j H H g g 2 nk lm l m
ð28eÞ
þ L þ
i n
j km
þ L H
g m
d g k c 0k
c 1kn
þ k þ kn
1 1 n 2 kn
þ
1 2 n g dh 3 kn n
ð29Þ
2p
h2
Z Z Rh
1 C ij ei e j dh 2
r r r d/ dh
0
h1
h
/
ð30Þ
2p
h2
Z Z ð
þ þ Þd/ dh
v 1
0
h1
Be
Bh r n
^
v 2
ð31Þ
v 3
r r h
2
Ag r n B
A
u
B r r h A
^
Bh e
Be r n A
^
AB w
wh
Be 2 u A
r r h
^
Bh e
32a
eg
r r h A A A2 B Bh Bh r n uuh r r h B uwh A A2 eg Ag Be g r n w2 r n B wuh AB B A B g r r h wwh AB
v 2
3
Bh e
v 3
^
r n
^
^
A Bh uuh w A
eg 2 u w AB
g uh w2 B
Bh uw
^
u3
2
2
Þ
Bh 2 u wh A Bh e 2
A
uw2
32b
Þ
g uwwh B
eg 3 w AB
32c
Þ
where v 1, v 2, and v 3 contain the linear, quadratic, and cubic terms of the displacements in the volume change DV , respectively. Using the 1 , a 2 , and a 3 ; the unit index notations, designate R h , R / , and R by a ^ ^ ^ r / =B, and ^ n can be denoted by i1, i2, and i3 , respectively. vectors r h = A, 1 2 Then the vectors a , a , and a3 can be written as
ai
ð26Þ
R/ R
B r n uh
ð24Þ
in which
Z ¼
¼ C L L
ð Þ ð Þ þ ð Þ ð Þ þ þ ð ð ÞÞ þ ð Þ ð ¼ ð Þ þ ð Þ þ ð Þ þ þ ð Þ þ ð Þ þ ð Þ þ þ ð Þ þ ð Þ ð ¼ þ þ þ þ þ þ ð þ
ð23Þ
T
i j ij k n
kkn
in which v 1
T
T
j kn ;
Substituting Eqs. (1), (9) and (10) into Eq. (30) yields
¼
where [C 0 ] is the shell material property matrix, t is the shell thickness, E 0 is the Young’s modulus, and m is the Poisson’s ratio. Substitution of Eq. (18) into Eq. (22) yields
Z ¼
ð27Þ
Lik H jmn
1
g k g n dh
The volume change of the shell from reference state (ES1) to deformed state (ES2) can be expressed in terms of displacements as follows:
in which
½C 0
ij
i ij 0
¼ C e H
Z ¼ þ
The shell is assumed to be a linearly elastic material of constant thickness. Then the strain energy of the shell can be expressed as follows:
f g
2.5. Volume change of the shell
2.4. Strain energy of the shell
U
kn
2 kn
According to Eq. (28), the matrices c 1, k, n 1, and n2 are symmetric and the variation of strain energy dU can be obtained as follows:
ð20aÞ
where Lik and H ikl are column and symmetric matrices, respectively. These matrices depend on the reference surface ( S ) characteristics and are identified by Eq. (20). The strains e0, eLi , and eN i are constant, linear, and nonlinear in terms of displacements, respectively.
1 L e 2
ij
h1
c
1 2
2p
1 kn
2
i i ei ¼ eLi þ eN i ¼ L k g k þ H kl g k g l
h2
1 n þ 12 k þ 16 n þ 12
c 1kn
¼ C e L
2
Let g u w uh wh . Then the added strains can be separated into two parts, a linear and nonlinear, and written in the following index form:
Z Z ¼
i j ij 0 k ;
c 0k
ð19Þ
where [T ] is the diagonal material-element matrix. It is formed in order 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 displacements as follows:
eh ¼ uh
h1
n1kn
#
c 0k g k
in which
in which
"
1 1 c 2 kn
Z þ
i j j
i j
i jk k
¼ a i ¼ a þ b g ^
i j
^
ð33Þ
Consequently, using the permutation symbols ( eijk) as follows:
W. Jiammeepreecha et al. / Engineering Structures 68 (2014) 111–120
Rh
R R ¼ e /
1 2 3 ijk ai a j a k
ð34Þ
Therefore, the volume change can be expressed in the appropriate forms [35] as follows: h2
Z ¼
1 K v 2 kn
þ
c v k g k
DV
h1
þ
1 N g g dh v 6 kn k n
ð35Þ
ð36aÞ
in which
¼ 23p e
c
v k
K
¼
v kn
ijl
a1i a j2 b3lk þ a1i b jk2 a3l þ b1ik a j2 a3l
2 p e a1 b 2 b3 3 ijl i jk ln
þ b
¼ þb
a
þ b
1 2 3 ik b jn l
2 3 2 3 blk þ b1in a j2 b3lk þ b1in b jk a þ a1i b jn al
ð36bÞ
2 p 2 3 eijl b1ik b jm bln 3
N v kn
1 2 3 ik j b ln
1 2 3 bim b jn blk
þ
1 2 3 ik b jn blm
2 3 b1im b jk bln
þ
1 2 3 bin b jk blm
þ
1 2 3 bin b jm blk
þ
g m
ð36cÞ
N where v c k, v K kn , and v kn 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), N the matrices v K kn and v kn are symmetric and directly obtained from the variation of volume change d (DV ) as follows: h2
Z Þ¼ þ
ðDV
d
h1
d g k v c k
K v kn
þ
1 N g dh v 2 kn n
ð37Þ
Assume that the internal pressure at reference state (ES1) is linearly proportional to the volumetric strain by the relation
¼ k DV V
0W
~
ð38Þ
0W
where ~k is the bulk modulus of the fluid, V 0W is the unstrained fluid volume, and DV 0W is the volume change of fluid from the unstrained fluid state to the reference state (ES1). Then the strain energy in the enclosed fluid is determined by the relation
1 k 2
C
¼
~
2 DV 0W DV V 0W V 0W
þ
ð39Þ
where DV is the volume change of fluid and shell which are unchangeable. Associated with the variation of volume change, the variation of strain energy due to internal fluid dC can be written as
¼ k
dC
~
DV 0W
þ DV dðDV Þ
V 0W
ð40Þ
¼ ð p þ kÞdðDV Þ 0
ð41Þ
Physically, k , which represents the change of pressure from the reference state (ES1) to the deformed state (ES2) is defined as DV
¼ k V
k
~
ð42Þ
0W
According to Eqs. (38) and (42), the constraint equation is considered by the relation DV
þ
V k p0 ~
k
¼ 0
¼ q gZ
pw
ð44Þ
w
w
where q w is the density of the sea water, g is the specific gravity, and Z w is the vertical distance from the sea water level. The virtual work done by linearly hydrostatic pressure dX can be expressed as follows: 2p
Z Z ¼
pw dw D d/ dh
dX
f g
0
h1
¼ 2p
h2
Z
pw dw D dh
ð45Þ
f g
h1
3. Equilibrium equation requirement Based on the principle of virtual work, the equilibrium equation of the shell can be obtained by setting the total virtual work of the shell to be zero, as follows:
dp
¼ dU þ dC þ dX ¼ 0
ð46Þ
Substituting Eqs. (29), (37), (41), and (45) into Eq. (46) gives h2
1 1 n 2 kn
1 2 n g 3 kn n
Z þ þ þ þ þ Z ð þ Þ þ þ c 0k
d g k
p0
c 1kn
c
k
kkn
1 N g v 2 kn n
K
v k
v kn
dh
2p
h2
h1
pw dw Dd h
f g
¼ 0 ð47Þ
Two highly nonlinear differential equations in terms of u(h) and w(h) are embedded in the above Euler’s equation. Before the hydrostatic pressure is applied, the shell is in equilibrium at the reference state (ES1). Thus, setting p w, k, and g n to be zero, Eq. (47) is also valid. This requires h2
Z
h1
d g k c 0k
c 0 v k
ð p Þdh ¼ 0
ð48Þ
This equation should be satisfied everywhere, and it can be used to predict the value of initial strains e 0.
3.1. Constraint equation ~ approaches infinity and the Since the fluid is incompressible, k last term in Eq. (43) becomes zero. Thus, k in Eq. (43) may be interpreted as a Lagrange multiplier associated with the constraint volume (DV = 0). Finally, the constraint equation can be written as h2
Z h1
1 K v 2 kn
þ
c v k g k
þ
1 N g g dh v 6 kn k n
¼ 0
ð49Þ
4. Finite element method
Substituting Eq. (38) into Eq. (40) yields
dC
The linearly hydrostatic pressure acting on the normal surface of the shell is given by
h1
2.6. Strain energy due to internal fluid
p0
2.7. Virtual work done by linearly hydrostatic pressure
h2
115
ð43Þ
where V is the shell volume at reference state (ES1). It is noted that the numerical value of k is an unknown, which will be obtained by solving the entire problem as will be presented.
To solve the problem by using the finite element method, the shell is divided along the h coordinate into many finite ring elements. Consider a general single element with the local coordinate u, the shell global coordinate h and the angle a = h2 h1, as shown in Fig. 3. The local coordinate u is related to the global coordinate h by u = h h1, and the derivatives of any quantity with respect to u and h are equal. Therefore, using the C 1 continuity in finite element method [36], the displacements u(u) and w(u) within each element are approximated by a third-order polynomial of the local coordinate u
uu
2
3
ð Þ ¼ b þ b u þ b u þ b u wðuÞ ¼ b þ b u þ b u þ b u 1
5
2
6
3
7
2
4
8
3
ð50aÞ ð50bÞ
W. Jiammeepreecha et al. / Engineering Structures 68 (2014) 111–120
116
h2
Z f g ½ ðf g ð þ Þf gÞ Z þ ½ ½ þ½ þ ½ þ ½ ð þ Þ ½ þ ½ ½ f g þ f g ¼ dd
T
T
w
c 0
p0
c
k
v
dh
h1
h2
T
w
c 1
1 1 n 2
k
h1
po
1 2
K
k
v
N
v
1 2 n 3
w dh d
f
0
ð55Þ
in which
f f g ¼ 2pfdwg
h2
Z
T
pw w D dh
fg
h1
ð56Þ
Since the global degree of freedom { Q } is the same as the local degree of freedom {d}, the global equilibrium equation can be obtained by assembly process using Eq. (55). The results are Fig. 3. Deep water axisymmetric half drop shell.
fC g ð p þ kÞfVC g þ ½C þ ½K þ 12 ½N þ 13 ½N ð p þ kÞ ½VK þ 12 ½VN fQ g þ fF g ¼ f0g ð57Þ 0
0
where b i i 1; 2; . . . ; 8 are unknown coefficients. Their first derivatives with respect to u or h are as follows:
ð ¼
uh u
Þ
ð Þ ¼ b þ 2b u þ 3b u 2
wh u
3
4
2
ð51aÞ
2
ð Þ ¼ b þ 2b u þ 3b u 6
7
8
ð51bÞ
Consider the eight-unknown coefficient (bi) in Eq. (50). In finite element formulation the displacements u and w are expressed in terms of element nodal degrees of freedom { d} via the cubic polynomial shape function. Therefore, the displacement gradient vector { g } can be expressed as
f g g ¼ ½wfdg
1
1
2
0
Similarly, the constraint equation, Eq. (49), becomes
1 VK 2
f g f gþ ½ Q
T
VC
1 VN 6
þ ½
f g ¼ Q
0
ð58Þ
Finally, the equilibrium equation, Eq. (57), and the constraint equation, Eq. (58), are combined into a symmetrical matrix form, as follows:
ð52Þ
in which T
f g g ¼ b uðuÞ
wu
T
fdg ¼ b uð0Þ
2 6 ½ ¼6 4 w
uh u
ð Þ
wh u
ð Þ
w0
ð Þ uð aÞ
uh 0
ðÞ
ð53aÞ
ð Þc wa
wh 0
ðÞ
ðÞ
uh a
N 1
0
N 2
0
N 3
0
N 4
0
0
N 1
0
N 2
0
N 3
0
N 4
N 1 u
0
N 2 u
0
N 3 u
0
N 4 u
0
0
N 1 u
0
N 2 u
0
N 3 u
0
N 4 u
;
;
;
;
;
;
;
wh a
ðÞ
;
ð Þc ð53bÞ
3 77 5
ð59Þ ð53cÞ
Since an axially symmetrical shell is considered, the boundary conditions at the top are
u
where [w] is the cubic polynomial shape function. This function and its derivatives can be expressed as follows:
2
¼ u 2 ua þ ua
N 2
3
2
3
¼ 3 ua 2 ua
N 3
2
¼ ua þ ua
N 4
;
N 2 u
;
;
N 3 u ;
3
2
;
;
3
2
2
N 1 u
;
¼ a
u a
u2 a2
þ
¼ 1 4 ua þ 3 ua
¼ 6a
u a
ð54a-bÞ
2
2
ð54c-dÞ
wh
¼ 0
ð60Þ
¼ 0
;
w
¼ 0
;
uh
¼ 0
;
wh
¼ 0
ð61Þ
The system of nonlinear equations in Eq. (59), which is constrained by both boundary conditions Eqs. (60) and (61), can be solved numerically by an iterative procedure.
5. Numerical example and results
2
u a2
¼ 2 ua þ 3 ua
N 4 u ;
6
;
The supported condition is considered to be fully fixed at the sea bed. Therefore
u
u2 u3 N 1 ¼ 1 3 2 þ 2 3 a a
¼ 0
ð54e-f Þ
2
2
ð54g-hÞ
Substituting Eq. (52) into the matrices c 0k , c 1kn , kkn , n1kn , n2kn , v c k, v K kn , and v N in Eq. (47) yields kn
In order to present the finite element formulation of the membrane shell theory, one has to study the behaviors of the axisymmetric 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 variable to h coordinate is used. This independent variable is generally for a spherical shell having a constant Gaussian curvature. In the case
W. Jiammeepreecha et al. / Engineering Structures 68 (2014) 111–120
of discretization by using h coordinate, the meridian curve is not divided into many regions. Therefore, the size of the global matrix is reduced in parts of the four Lagrange multipliers when compared with the previous work of Jiammeepreecha et al. [31]. To validate the accuracy of the present solutions, consider a half drop shell, as shown in Fig. 3, submerged at a water depth of H = 1745 mm. The shell geometry and material are: a = 220 mm, t = 2.5 mm, E 0 = 757 kgf/mm2, and m = 0.36, and the specific weight of external fluid is cw = 1.0 106 kgf/mm3. The present results show the tangential and normal displacements for a linearly distributed hydrostatic pressure along the sea depth and a constant hydrostatic pressure at the sea bed. As shown in Fig. 4, it can be seen that the results of constant hydrostatic pressure are in close agreement with Yasuzawa’s results [28], except the normal displacement near the support. In this study, the hydrostatic pressure is varied along the sea depth, while there is no information on the hydrostatic pressure in Yasuzawa’s [28] work. However, the numerical results from this study were verified with Roark’s formula for a spherical subjected to uniform external pressure [37], and were found to be conformable. The input parameters employed in this analysis are tabulated in Table 1.
117
Table 2
Convergence of deflection of half drop shell at the apex. Number of elements
wapex ( 103 m)
8 12 16 20 24 28
0.019425 0.020044 0.020356 0.020544 0.020669 0.020759
5.1. Half drop shell behavior subjected to hydrostatic pressure Table 2 shows the convergence of the apex displacement for half drop shell with constraint volume. It can be seen that the higher mesh models gives the more accurate result. However, the difference of apex movement is less than 0.50% between the model with 24 and 28 elements. In this paper the model with 24 elements is assumed to be sufficient for accurate results. Based onthe results of the first part of the study, the shell at the deformed state (ES2) subjected to hydrostatic pressure is shown in Fig. 5. The present results show very good agreement with the previous work of Jiammeepreecha et al. [31].
Fig. 5. Configuration of the half drop shell at deformed state (ES2).
5.2. Effects of hydrostatic pressure on half drop shell Linearly varying hydrostatic pressure has no effect on the displacement response of the half drop shell, as shown in Figs. 6 and 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 the hydrostatic pressure; that is, the value of k increases under large hydrostatic pressure and decreases when the hydrostatic pressure becomes 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-thickness ratio (a/t ratio) from 25 to 200, the tangential and normal displacements of the half drop shell are shown in Figs. 9 and 10,
Fig. 4. Comparison of displacement responses with Yasuzawa’s results [28].
Table 1
Input parameter data. Parameter
Value
Young’s modulus, E 0 (N/m2)
2.04 1011 0.30 40 5 0.20 50 103 1025
Poisson’s ratio, m Sea water level, H (m) Radius of shell, a (m) Thickness of shell, t (m) Initial internal pressure, p 0 (N/m2) Density of sea water, q w (kg/m3)
Fig. 6. Effects of linearly varying hydrostatic pressure on tangential displacement of half drop shell.
W. Jiammeepreecha et al. / Engineering Structures 68 (2014) 111–120
118
Fig. 7. Effects of linearly varying hydrostatic pressure on normal displacement of half drop shell.
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. 8. Effects of linearly varying hydrostatic pressure on the change of pressure k .
respectively. It can be seen that the radius-to-thickness ratio has a significant effect on the displacements on a deep water half drop shell. On the contrary, changing of the radius-to-thickness ratios may have little effect on the values of k , as shown in Fig. 11. Furthermore, the results show that the point of intersection is on the same location, as shown in Fig. 10. This intersection point location 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 internal pressure ( p0) from 50 103 to 50 106 N/m2, the tangential and normal displacements of the half drop shell are shown in Figs. 12
Fig. 12. Effects of initial internal pressure on tangential displacement of half drop shell.
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 drop shell.
W. Jiammeepreecha et al. / Engineering Structures 68 (2014) 111–120
ee ¼
ds
ds
119
0
ðA 2Þ
ds0
:
In the present study, the initial engineering strains ee are assumed to be small, and the small displacement theory is used to calculate the initial strains. Furthermore, by neglecting the quadratic term in Eq. (A.1), e L0 ee . The initial engineering strains e e can be computed by using the membrane theory equilibrium equation of the shell; that is
N h r 1
þ N r ¼ p /
2
ðA 3Þ
0
:
where p0 is the internal pressure. For the case of a reference surface (S ) of a spherical shell having a constant Gaussian curvature, the total tension forces are N h = N / = N and the principal curvatures are 1/ r 1 = 1/r 2 = 1/a. Therefore, the tension force N is given by
Fig. 14. Effects of initial internal pressure on the change of pressure k .
and 13, respectively. When a low value of initial internal pressure is applied to the shell, the shell has a large effect versus the high initial internal pressure. Fig. 14 shows the effects of the increase of initial internal pressure on the values of k. It can be seen that the values of k decrease when the initial internal pressure becomes large. However, the values of k are unchangeable under high initial internal pressure.
¼ 12 p a
N
ðA 4Þ
0
:
The initial engineering strains e e can be determined by
r ee ¼ 0 ð1 mÞ E
p a ¼ E N 0 t ð1 mÞ ¼ 2E 0 t ð1 mÞ 0
ðA 5Þ :
Let a0 be the radius of a spherical shell at the initial unstrained state (IUS). The initial Lagrangian strain e L0 becomes
6. Conclusions The nonlinear static responses of a deep-water axisymmetric half drop shell storage container with constraint volume condition by using membrane theory are presented in this paper. The problem is formulated by using the variational principle and the finite element method in terms of displacements, which are expressed in the appropriate forms. The change of pressure from the reference state to the deformed state may be explained as a Lagrange multiplier. In the present study, small displacement theory and traditional shell analysis are used to calculate the initial strains and displaced configuration of the half drop shell, respectively. The numerical results indicate that the Lagrange multiplier represents the parameter for adjusting the internal pressure in order to sustain the shell in equilibrium position under the constraint volume condition. The numerical results show that the effect of radius-tothickness ratio has a major impact on the displacements on the shell, whereas changing the hydrostatic pressure has no effect. However, by varying linearly hydrostatic pressure, the change of pressure from the reference state to the deformed state is linearly proportional to the hydrostatic pressure. For a large value of initial internal pressure, the change of pressure is unchangeable.
eL0 ¼ ee ¼
1 a2 a20 2 a20
2
e0 ¼
1 ds ds0 2 2 ds
2
2 0
ðA 7Þ
2
:
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 a sin h cos / i a sin h sin / j a cos h k
¼
^
þ
^
þ
ðB 1Þ
^
:
in which
r a sin h
¼ Z ¼ a cos h
;
;
r h
¼ a cos h r ¼ a sin h Z ¼ a sin h Z ¼ a cos h hh
;
h
hh
;
ðB 2a-cÞ ðB 3a-cÞ :
:
The metric tensor components of the reference surface (S ) are 2
2
h
F 0
¼
;
G
;
2
¼ r ¼ a
2
sin 2 h
ðB 4a-cÞ :
Also,
q ffi ffi ffi ffi ffi ffi ffi ffi ¼ þ ¼
D
r r 2h
Z 2h
a2 sin h
ðB 5Þ :
p ffiffiffiffi p ffiffiffi Let A ¼ E and B ¼ G , then
A
¼ a
Appendix A. Derivation of initial Eulerian strain
2
ð Þ ð Þ ¼ 1 a a 2 a ð Þ
¼ þ Z ¼ a
The first and second authors gratefully acknowledge financial support by the Thailand Research Fund (TRF) and King Mongkut’s University of Technology Thonburi (KMUTT) through the Royal Golden Jubilee Ph.D. program (Grant No. PHD/0134/2552).
:
Finally, the initial Eulerian strain e0 component at the reference state (ES1) can be determined by
E r 2h
Acknowledgements
ðA 6Þ
;
B
¼ a sin h
Bh
¼ a cos h
;
ðB 6a-cÞ :
Therefore, the unit vector normal is given by The initial engineering strains e e and initial Lagrangian strains
eL0 are related by [11] 2
eL0 ¼
1 ds ds0 2 2 ds0
^
2
ð Þ ð Þ ¼ e þ 1 e 2 ð Þ
in which
sin / j þ rr k ¼ rZ cos /i rZ D ¼ sin h cos /i þ sin h sin / j þ cos hk
n e
2 e
ðA 1Þ :
h
^
^
h
^
h
^
^
^
ðB 7Þ :
The curvature tensor components of the reference surface ( S ) are
W. Jiammeepreecha et al. / Engineering Structures 68 (2014) 111–120
120
e
¼ r Z r Z ¼ a r þ Z h
hh
hh
h
2
2
h
h
f
¼ 0
;
q ffiffi ffi ffi ffi ffi ffi ffi
;
g
¼
rZ h
2
a sin h
q ffiffi ffi þffi ffi ffi ffi ffi ¼ r 2h
Z 2h
ðB 8a-cÞ :
Since F = f = 0, the coordinate lines (h, /) are also lines of principal curvature. Accordingly, the principal curvatures can be determined by
j1 ¼
e E
¼ 1a
j2 ¼
;
g G
¼ 1a
ðB 9a-bÞ :
r h and r ^n are given From Eqs. (B.1) and (B.7), the quantities r
by
r r h
¼ 0
;
r n
¼ a ^
ðB 10a-bÞ :
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