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1 Copyright © 2009 by ASME

Proceedings of PVP2009

2009 ASME Pressure Vessels and Piping Division Conference

July 26-30 2009 Prague Czech Republic

PVP2009-77187

SLOSHING CHARACTERISTICS OF SINGLE DECK FLOATING ROOFS

IN ABOVEGROUND STORAGE TANKS

- NATURAL PERIODS AND VIBRATION MODES -

Shoichi YoshidaYokohama National University

Yokohama, Japan

Kazuyoshi SekineYokohama National University

Yokohama, Japan

Katsuki IwataJapan Oil, Gas and Metals

National CorporationKawasaki, Japan

ABSTRACTThe floating roofs are widely used to prevent evaporation

of content in large oil storage tanks. The 2003 Tokachi-Oki

earthquake caused severe damage to the floating roofs due to

liquid sloshing. The structural integrity of the floating roofs

for the sloshing is urgent issue to establish in the petrochemical

and oil refining industries. This paper presents the sloshing

characteristics of the single deck floating roofs in cylindrical

storage tanks. The hydrodynamic coupling of fluid and

floating roof is taken into consideration in the axisymmetric

finite element analysis. It is assumed that the fluid is

incompressible and inviscid, and the floating roof is linear

elastic while the sidewall and the bottom are rigid. The basic

vibration characteristics, natural periods and vibration modes,

of the floating roof due to the sloshing are investigated.

These will give engineers important information on the floating

roof design.

INTRODUCTIONThe floating roof is a steel cover that floats on the liquid

surface for the prevention of evaporation in aboveground

storage tanks. It is widely used in large-sized tanks to store

mainly crude oil and naphtha. One floating roof type is a

single-deck which is a deck with an annular pontoon that

provides buoyancy. Seven single deck floating roofs had

experienced sinking failures in the 2003 Tokachi-Oki

earthquake at a refinery in Tomakomai, Japan. These floating

roofs deformed to leak oil on them due to the liquid sloshing,

and they lost buoyancy to sink. A devastating full surface fire

broke out in one of the damaged tanks due to unclear ignition

of flammable vapors after sinking the floating roof into the

content. The fire was extinguished 44 hours later. This

accident became a cause to establish the seismic design method

of floating roofs which had not been carried out by that time

The behavior of floating roofs in sloshing becomes an urgent

issue to solve.

The linear solution based on the velocity potential theory

assuming imcompressive and inviscid fluid was established in

the 1970s regarding the sloshing response of free fluid surface in

cylindrical storage tanks [1]. The linear solution for the

sloshing response of the floating roof as a rigid circular plate was

also established after that [2]. With regard to the sloshing

response of the floating roof as an elastic circular plate, the

hydrodynamic coupling of fluid and floating roof has to be taken

into consideration, and Sakai introduced the solution using the

Ritz method first[3]. Matsui proposed the analytical solutions

for the sloshing of the floating roof as an elastic circular plate

with and without a ring attached to the perimeter of the plate

[4,5]. The studies of the experiment[6], the design criteria[7

and the nonlinear vibration[8] for the sloshing response of the

floating roofs were also reported.

Several three dimensional nonlinear finite element analyses

using the general purpose computer code were performed for the

simulation of the sloshing behavior of the floating roofs which

had damaged at the 2003 Tokachi-Oki earthquake. These

analyses were taken the hydrodynamic coupling of fluid and

floating roof into consideration, and were nonlinear. However

the theoretical backgrounds of the fluid motion were not

described in these reports.

The basic vibration characteristics of the floating roof for

the sloshing, such as natural periods and vibration modes, can be

obtained from the linear analysis. These will give engineers

important information on the design. Although Sakai[3] and

Proceedings of the ASME 2009 Pressure Vessels and Piping Division ConferencePVP2009

July 26-30, 2009, Prague, Czech Republic




Matsui[4] have already reported the linear analysis of the

sloshing response of the circular plate floating roof, the detailed

researches on the basic sloshing characteristics of the single

deck floating roofs have not been seen.

The authors developed the theory for the axisymmetric

finite element analysis in the linear sloshing response of the

floating roofs in the cylindrical storage tanks, and a numericalexample was analyzed[9]. More precise modeling of the

floating roofs, such as a box shaped pontoon, could be possible

using the axisymmetric shell elements in the analysis. In this

paper, the free vibration analyses based on this theory in the

sloshing of the single deck floating roofs are carried out. The

basic sloshing characteristics are presented in comparison with

that of a free fluid surface and an elastic circular plate floating

roof.

NOMENCRATURE{d m} Overall nodal displacement vector

in global coordinate for m {d e-m} Nodal displacement vector in global coordinate for m

{e} Exciting vector

{ E } Overall exciting vector

{ f e--m} Equivalent load vector for m

g Acceleration of gravity

H Fluid height

I L Functional

[k a] Added stiffness matrix

[k L-m] Stiffness matrix of fluid element for m

[k s-m] Stiffness matrix of shell element for m

[ K a] Overall added stiffness matrix

[ K L-m] Overall stiffness matrix of fluid element for m

[ K s-m

] Overall stiffness matrix of shell element for m

m Circumferential wave number

[m s] Mass matrix of shell element

[ M a-m] Overall added mass matrix for m

[ M s] Overall mass matrix of shell element

[ N L] Shape function matrix of fluid element

[ N s] Shape function matrix of shell element

[ N s0] Shape function matrix of shell element

in global coordinate

p Dynamic pressure of fluid

p f Dynamic pressure on floating roof surface

p H Dynamic pressure at z=H

{ pm} Overall nodal dynamic pressure vector for m

{ pe-m} Nodal dynamic pressure vector for m

(r,θ ,z ) Cylindrical coordinate

R Tank radius

s Shell element coordinate

[ s] Coupling matrix

S 1 Floating roof boundary

S 2 Sidewall boundary

S 3 Bottom boundary

[S ] Overall coupling matrix

T Kinematic energy of fluid motion

T mi i-th natural period of single deck floating roof tank

T 0-mi i-th natural period of free fluid surface tank

T p-mi i-th natural period of elastic plate floating roof tank

u s Tangential displacement of shell element

uw Normal displacement of shell element

u z Vertical displacement

uθ Circumferential displacement of shell element{ue-m} Nodal displacement vector in local coordinate for m

V Fluid domain

W f External work done by floating roof

α Rotational angle of shell element

[ λ] Coordinate transformation matrix

{δ} Displacement vector within shell element

[θ m] Circumferential distribution matrix for m

ρ L Fluid density

(ξ ,η) Elemental coordinate of fluid element

φ Velocity potential

Subscript e Elemental equation

m Circumferential wane number

BASIC EQUATIONS Analytical Model and Finite Element

The fluid coupled floating roof tank system under

consideration is shown in Fig.1. In this figure, V is the fluid

domain, S 1 is the floating roof boundary, S 2 is the sidewal

boundary, S 3 is the bottom boundary， R is the tank radius and H

is the fluid height. It is assumed that the fluid is imcompressive

and inviscid, the floating roof is linear elastic and the sidewall

and the bottom are rigid.

A cylindrical coordinate system (r,θ ,z ) is used with the

center of the bottom being the origin. In the axisymmetricanalysis, displacements, stresses and dynamic pressures can be

represented by Fourier series expansion of circumferentia

coordinates, and then the equations are uncoupled between the

terms of Fourier series m, that is the circumferential wave

number, because of the orthogonality of trigonometric functions

It is known in the linear analysis that the lateral loading excites

the tank behavior expressed by only the term m=1, and the

vertical loading excites only the term m=0. Because of both the

nonlinearity of the fluid motion and the non axisymmetry of the

cylindrical tanks, however, the tank behavior expressed by the

term m=0 or 2m ≥ may occur[11].

S 1

H V

z

2 R

o r S 3

S 2

Deck Pontoon

Fig.1 Analytical model of floating roof tank




The finite element method applied to the hydrodynamic

coupling of fluid and solid can be categorized into the Eulerian

formulation and the Lagrangian formulation[10]. In the

Eulerian approach, the behavior of the fluid is expressed in

terms of a velocity potential or a dynamic pressure as the nodal

variable. In the Lagrangian approach, the behavior of the

fluid is expressed in terms of displacements, and it is easy toimplement into an existing structural analysis code. However,

this approach has a serious problem that the fluid element has

zero energy deformation modes due to the zero shear modulus.

Since the objective is to obtain the basic sloshing

characteristics of floating roofs, the Eulerian formulation is

applied in this paper.

The theory of the axisymmetric FEA was developed for

the circumferential wave number m=1 in the previous paper

[9]. It is extended to 2m ≥ in order to explain the sloshing

characteristics in this paper. The fluid is modeled using 4-

nodes axisymmetric quadrilateral elements as shown in Fig.2.

Using the nodal dynamic pressure vector { pe-m}, the dynamic

pressure p within the fluid element is defined by Eq.(1);[ ]{ }∑

∞

=−=

1m

me L p N mcos p θ (1)

where [ N L] is the shape function matrix of the fluid element,

and the subscript e expresses the elemental equation and m

expresses the circumferential wave number.

r

ξ

η

η＝-1

η＝1

2a

2b

ξ =-1 ξ =1

o

r 0

o

i

kl

Fig.2 Axisymmric fluid element

The floating roof is modeled using the axisymmetric shell

elements as shown in Fig.3. In this element, the tangential

displacement u s and the circumferential displacement uθ are

assumed to be linear, and the normal displacement uw to be

cubic with regard to s, where s is the elemental meridional

coordinate. The strain-displacement relation based on

Kirchhoff-Love's assumption is given by the Novozhilov'sequation [12]. The displacements at the nodal point i are

defined by Eq.(2);

⎪⎪

⎭

⎪⎪

⎬

⎫

⎪⎪

⎩

⎪⎪

⎨

⎧

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

=

⎪⎪

⎭

⎪⎪

⎬

⎫

⎪⎪

⎩

⎪⎪

⎨

⎧

−

−

−

−

∞

=∑

mi

mwi

mi

m si

1m

i

wi

i

si

u

u

u

mcos000

0mcos00

00m sin0

000mcos

u

u

u

α θ

θ

θ

θ

α

θ θ (2)

where α is the rotational angle, and the subscript i expresses the

nodal point. Eq.(2) can be simply written as follows.

{ } [ ]{ }∑∞

=−=

1m

mimi uu θ (3)

Using the nodal displacement vector {ue-m}, the displacemen

vector {δ} within the shell element is defined by Eq.(4);

{ } [ ][ ]{ }∑∞

=−−=

1m

me sme u N θ δ (4)

where [ N s] is the shape function matrix of shell element. {ue-m}

and [θ e-m] are defined by the following equations.

{ } { }

{ }⎪⎭⎪⎬⎫

⎪⎩

⎪⎨⎧

=−

−−

m j

mi

me u

uu (5)

[ ] [ ]

[ ]⎥⎦

⎤⎢⎣

⎡=−

m

mme

θ

θ θ

0

0 (6)

L

uw

u s

s

r

φ

i

α=duw /ds

o

Fig.3 Axisymmric shell element

Equation of Fluid MotionThe fluid velocity of i-th direction vi can be calculated using

the velocity potentialφas follows;

ii

xv

∂

∂−= φ

(7)

where xi is the i-th coordinate. The dynamic pressure p is given

from the velocity potential using the Bernoulli's equation;

φ ρ & L p = (8)

where ρ L is the fluid density. The governing equation and the

boundary conditions expressed by the dynamic pressure p is

given as follows;

0 z

p p

r

1

r

p

r

1

r

p p

2

2

2

2

22

22 =

∂

∂+

∂

∂+

∂

∂+

∂

∂=∇

θ

(in V ) (9)

z Lu z

p&& ρ −=

∂

∂ (on S 1) (10)

0r

p=

∂

∂ (on S 2) (11)

0=∂

∂

z

p (on S 3) (12)

where u z is the vertical displacement of the floating roof. Eq.(9

is the equation of continuity written as the Laplace's equation

Eq.(10)~Eq.(12) are the boundary conditions which are obtained

from the coincidence of fluid velocity with structural velocity in

the normal direction on the boundary.




The dynamic pressure on the floating roof surface p f is

defined by the following form;

z L H f gu p p ρ −= (13)

where p H is the dynamic pressure on z = H where is the liquid

surface before deformation, g is the acceleration of gravity.

The associate functional can be written as;

( )∫ −=2

1

t

t

f L dt W T I (14)

where T is the kinematic energy, W f is the external work done

by the floating roof. These terms are expressed in the

following forms.

∫ ⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

⎟ ⎠

⎞⎜⎝

⎛

∂

∂+⎟

⎠

⎞⎜⎝

⎛

∂

∂+⎟

⎠

⎞⎜⎝

⎛

∂

∂=

V

dV z

p p

r r

pT

2221

2

1

θ (15)

dr rd u pW

S

z L f θ ρ ∫−=

1

&& (16)

The variation δ I L=0 leads to the governing Eq.(9) and the

boundary conditions Eq.(10) ~Eq.(12).Substituting Eq.(1) and Eq.(4) into Eq.(15) and Eq.(16)

and because of the orthogonality of trigonometric functions, the

stationarity of the functional gives the elemental equation of

fluid motion with uncoupling form between the circumferential

wave numbers m as follows:

[ ]{ } [ ]{ } 0d s pk memem L =+ −−− && (17)

where {d e-m} is the nodal displacement vector of the shell

element in the global coordinate, and is transformed into the

elemental coordinate by a coordinate transformation matrix [ λ]

as follows:

{ } [ ]{ }meme d u −− = λ (18)

In Eq.(17), [k L-m] is the stiffness matrix of the fluid element,

and [ s] is the coupling matrix. These are expressed by thefollowing forms.

[ ] [ ] [ ] rdrdz z

N

z

N N N

r

m

r

N

r

N k

V

L

T

L L

T L2

2 L

T

Lm L ∫ ⎟

⎟

⎠

⎞

⎜⎜

⎝

⎛

⎥⎦

⎤⎢⎣

⎡

∂

∂⎥⎦

⎤⎢⎣

⎡

∂

∂++⎥

⎦

⎤⎢⎣

⎡

∂

∂⎥⎦

⎤⎢⎣

⎡

∂

∂=−

(19)

[ ] ( )[ ] ( )[ ]∫ ==

1

1

S

z soT

L L rdr u N N s η ρ (20)

Although [k L-m] has no physical meaning as stiffness, it is called

"stiffness matrix", since [k L-m] is multiplied by nodal vector

{ pe-m} in Eq.(17).

[ N L(η=1)] is the shape function matrix in the global

coordinate, and is given by substituting η=1 into [ N L

] expressed

in terms of the elemental coordinate(ξ ,η) as shown in Fig.2.

[ N s0] is the shape function matrix of the shell element in the

global coordinate. [ N s0(u z )] is the matrix which takes only the

term of u z from [ N s0]. [ N L(ξ =1)] is the same as [ N L(η=1)]

except for the substitution ξ =1 into [ N L]. The relation

between [ N s] and [ N s0] is given by

[ ] [ ] [ ][ ]λ λ sT

so N N = (21)

The overall equation of fluid motion is obtained by the

superposition of Eq.(17) for each element as follows.

[ ]{ } [ ]{ } 0d S p K mmm L =+− && (22)

The matrices described with the capital letter mean the overal

equations.

Equation of Motion of Coupling SystemThe elemental equation of the floating roof motion can be

written for each circumferential wave number m as follows;

[ ]{ } [ ]{ } { }memem sme s f d k d m −−−− =+&& (23)

where [m s], [k s-m] are the mass matrix and the stiffness matrix

The equivalent load vector { f e-m} is written as follows.

{ } [ ]∫=−

1S

f T

some rdr p N f (24)

Substituting Eq.(13) into Eq.(24), the following equation is

obtained;

{ } [ ] { } [ ]{ }meameT

L

me d k p s1

f −−− −=

ρ

(25)

where [k a] is the added stiffness matrix expressed by the

following equation.

[ ] [ ] [ ]rdr u N N g k z so

S

T so La )(

1

∫= ρ (26)

The overall equation of floating roof motion is obtained by

the superposition of Eq.(23) for each shell element as follows.

[ ]{ } [ ] [ ]( ){ } [ ] { }mT

L

mam sm s pS 1

d K K d M ρ

=++ −&& (27)

The dynamic pressure vector { pm} is derived from Eq.(22) as

follows.

{ } [ ] [ ]{ }m1

m Lm d S K p &&−−−= (28)

Substituting this equation into Eq.(27), the equation of motion of

the hydrodynamic coupling system is given by;

[ ] [ ]( ){ } [ ] [ ]( ){ } 0d K K d M M mam smma s =+++ −− && (29)

where [ M a-m] is the added mass matrix of the fluid and is

expressed by the following form.

[ ] [ ] [ ] [ ]S K S 1

M 1

m LT

L

ma−

−− = ρ

(30)

Eq.(29) is the eigenvalue problem, and both the mass term and

the stiffness term are symmetric matrices.

SINGLE DECK FLOATING ROOF

The single deck floating roof consists of the deck plate andthe pontoon of box shaped cross section as shown in Fig.4. The

pontoon consists of the inner rim, the outer rim, the upper

pontoon plate and the lower pontoon plate. Each member is

joined using welding. The pontoon is divided into a number of

compartments by the radial plates called "bulkhead". Each

compartment has liquid tight requirement. The bulkhead plates

are not able to taken into consideration in this analytical mode

because of the axisymmetric analysis.




API Standard 650 Appendix C gives the minimum

requirements for the floating roof design [13], and the

minimum thickness of the deck plate is 4.76 mm(0.1875 inch).

It is 4.5 mm in the floating roof regulated by Fire Service Law

Japan. The thickness of deck plate is generally adopted the

minimum value regardless of the floating roof diameter.

The sloshing characteristics of three single deck floatingroofs which are named as SD20, SD30 and SD40 are

investigated. The detail data is shown in Table 1. The roof

radius of SD20, SD30 and SD40 are 20 m, 30 m and 40 m,

respectively. Also their capacities are approximately 20,000

m3, 50,000 m3 and 100,000 m3, respectively. Fig.5 shows the

mesh division of SD30, and that of the other tanks are the same

as SD30. All displacement components except for the

rotational angle α are restrained to zero at the nodal point of the

deck center which is shown at point A in Fig.4.

A

B

C

D

E

A～B : DeckB～C : Inner rimD～E: Outer rim

B～D : Lower pontoonC～E : Upper pontoon

ar br

cr d r

Deck Pontoon

Fig.4 Analytical model of single deck floating roof

Table 1 Numerical examples of single deck floating roofFloating roof No. SD20 SD30 SD40

Inner rim radius a r (mm) 17000 26000 35000Pontoon width b r (mm) 3000 4000 5000

Inner rim height c r (mm) 450 450 450

Outer rim height d r (mm) 800 800 800

Deck thickness (mm) 4.5 4.5 4.5

Inner rim thickness (mm) 10 12 12

Outer rim thicness (mm) 10 12 12

Lower pontoon thickness (mm) 4.5 6 6

Upper pontoon thickness (mm) 4.5 6 6

Fluid height H (mm) 16000 18000 20000

Young's modulus (GPa)

Poisson's ratio

Roof plate density (kg/m3

Fluid density ρ Ｌ (kg/m

3

)Deck mesh division

Pontoon mesh division

Fluid height mesh division

Inner rim mesh division

Outer rim mesh divison

0.3

200

8000

850

6

70

20

4

30

Fig.5 Mesh division of SD30

SLOSHING CHARACTERISTICS

The FEA computer code based on the theory described inthe previous chapter has been developed by the authors. The

free vibration analyses of the floating roofs in Table 1 are carried

out, and the basic sloshing characteristics are obtained.

Natural PeriodThe natural periods for the sloshing of the single deck

floating roof of SD20, SD30 and SD40 are shown in Fig.6. In

this figure, T mi is the natural periods of the single deck floating

roof in the circumferential wave number m and the order i. T mdecreases with increasing m if the order i is identical, and is long

in order of SD40, SD30 and SD20 if i and m are identical.

The natural period for the sloshing of the free fluid surface

T 0-mi is given by the following equation [1];

⎟ ⎠

⎞⎜⎝

⎛ =−

R

H tanh

R

g

2T

mimi

mi0

ε ε

π (31)

where εmi is the i-th positive root of the following differentia

equation.

0=dr

)r ( dJ m (32)

J m(r ) is the Bessel function of the first kind of order m.

The natural period ratios T mi/T 0-mi of the floating roofs are

shown in Fig.7. In the circumferential wave number m=1, the

ratio T 1i/T 0-1i in the order i=1~3 exceeds 1.0 slightly, and that is

less than 1.0 in the order ≥i 4. Therefore, the natural period o

the floating roof T 1i is almost identical with that of the free fluid

surface T 0-1i in the low order of i, and T 1i is less than T 0-1i in the

high order of i. In the circumferential wave number ≥m 2

T mi/T 0-mi is less than 1.0 at i=1, and it increases to approach 1.0 a

i=2~4, then it decrease again with increasing i.




SD20

0

2

4

6

8

10

12

1 2 3 4 5 6 7 8 9 10

Order i

N a t u r a l

P e r i o d

T m i ( s )

m=1 m=2

m=3 m=4

m=5

(a) SD20

SD30

0

2

4

6

8

10

12

1 2 3 4 5 6 7 8 9 10

Order i

N a t u r a l P e r i o d

T m i ( s )

m=1 m=2

m=3 m=4

m=5

(b) SD30

SD40

0

2

4

6

8

10

12

1 2 3 4 5 6 7 8 9 10

Order i

N a t u r a l P e r i o d T

m i ( s )

m=1 m=2

m=3 m=4

m=5

(c) SD40

Fig.6 Natural period of sloshing motion for single deck floating

roof tank

This paper defines the rigidity ratio as the deck plate

thickness divided by the floating roof radius. The deck plate

thickness is 4.5 mm regardless of the roof radius as shown in

Table 1. The rigidity ratio is large if the roof radius is small,

and is large in order of SD20, SD30 and SD40. The natural

period ratio T mi/T 0-mi is small in order of SD20, SD30 and

SD40. It is found that if the rigidity ratio increases, the

natural period T mi decreases.

SD20

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.1

1 2 3 4 5 6 7 8 9 10

Order i

T m i / T 0 - m

i

m=1 m=2

m=3 m=4

m=5

(a) SD20

SD30

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.1

1 2 3 4 5 6 7 8 9 10

Order i

T m i / T 0 -

m i

m=1 m=2

m=3 m=4

m=5

(b) SD30

SD40

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.1

1 2 3 4 5 6 7 8 9 10

Order i

T m i / T 0 - m

i

m=1 m=2

m=3 m=4

m=5

(c) SD40

Fig.7 Relation of natural periods between single deck floating

roof tank and free fluid surface tank

Fig.8 shows the natural period ratios T mi/T p-mi of SD30

T p-mi is the natural period for the sloshing of the elastic circular

plate floating roof which has the same radius (ar +br ) and the

same material properties as the single deck floating roof SD30 in

Table 1 and its thickness is 4.5 mm. The mesh divisions along

both radial direction and vertical direction are 90 and 30 with

regular intervals, respectively.




In the circumferential wave number m=1, the ratio T 1i/T p-1i

of order i=1,2 is almost 1.0, and it decreases with increasing the

order i. In the circumferential wave number m ≥ 2, T mi/T p-mi is

less than 1.0 and it decreases with increasing m. In all cases

except for i=1,2 of m=1, the natural period of the single deck

floating roof T mi is less than that of the elastic circular plate

floating roof T p-mi. The difference between the two floatingroofs is only the existence of the pontoon. The rigidity of the

single deck is higher than that of the elastic circular plate.

Consequently, T mi is less than T p-mi especially in the high m and

i.

Fig.9 shows the natural period ratios T mi/T 0-mi of SD30.

In each m, the ratio T mi/T 0-mi of order i=1~4 is almost 1.0, and it

decreases with increasing the order i. The natural period of

the elastic circular plate floating roof T p-mi is less than that of

the free fluid surface T 0-mi.

SD30

0.80

0.85

0.90

0.95

1.00

1.05

1 2 3 4 5 6 7 8 9 10

Order i

T m i / T p - m

i

m=1 m=2

m=3 m=4

m=5

Fig.8 Relation of natural periods between single deck floating

roof tank and circular plate floating roof tank in SD30

SD30

0.80

0.85

0.90

0.95

1.00

1.05

1 2 3 4 5 6 7 8 9 10

Order i

T p - m

i / T 0 - m

i

m=1 m=2

m=3 m=4

m=5

Fig.9 Relation of natural periods between circular plate floating

roof tank and free fluid surface tank in SD30

Vibration Mode The vibration mode of the single deck floating roof

corresponding to the natural period T mi in SD30 are shown in

Fig.10. Also, the vibration mode of the elastic circular plate

floating roof corresponding to the natural period T p-mi in SD30

are shown in Fig.11. These are the deformation at θ =0, and

they distribute cosmθ along the circumferential direction. The

vibration mode of the single deck is almost the same as that of

the elastic circular plate except for the pontoon portion.

The vibration mode corresponding to T 53 which is thenatural period of the single deck of m=5 and i=3 is shown in

Fig.10(c). The vibration mode corresponding to T p-53 which is

the natural period of the elastic circular plate of m=5 and i=3 is

shown in Fig.11(c). In the mode of the single deck

corresponding to T 53, it dose not deform at the joint of deck and

pontoon which is the point B in Fig.4, and it locally deforms in

the pontoon bottom plate which is the range between the point B

and the point D in Fig.4. This is different from the mode of the

elastic circular plate corresponding to T p-53.

(a) m=1

(b) m=3

(c) m=5Fig.10 Vibration modes of single deck floating roof in SD30




The vibration modes which locally deforms in the pontoon

occur in high m and i of SD20 and SD40, too. As a result, T mi

is different from T p-mi in this case. T 53 is 2.7069 s and T p-53 is

2.9490 s in SD30, and the T mi/T p-mi curve is not smooth at m=5

and i=3 in Fig.7(b).

The pontoon of the single deck floating roof is divided

into a number of compartments by the bulkhead. The bulkhead plates are not taken into consideration in the model

because of the axisymmetric analysis. The vibration mode

which deforms locally in the pontoon is presumed not to appear

due to the bulkhead actually. This is a future subject.

(a) m=1

(b) m=3

(c) m=5

Fig.11 Vibration modes of circular plate floating roof in SD30

CONCLUSIONSThe sloshing characteristics of the single deck floating

roofs in the aboveground storage tank are investigated using

the axisymmetric finite element method. As a result, the

following conclusions are obtained.

(1)The natural period for sloshing of the single deck floating roo

T mi is almost the same as that of the free fluid surface T 0-mi in the

circumferential wave number m=1 and low order i. T mi is less

than T 0-mi in all m and i except for m=1 and low order i. Theratio T mi /T 0-mi decreases with increasing m.

(2)T mi is almost the same as that of the elastic circular plateT p-m

in both m=1 and low order i=1,2. T mi is less than T p-mi in all m

and i except for m=1 and low order i=1,2. The ratio T mi /T p-m

decreases with increasing m.

(3)The vibration modes which locally deforms in the pontoon

occur in high m and i.

ACKNOWLEDGEMENT

This research was sponsored by Japan Oil, Gas and Metals National Corporation(JOGMEC). Its financial support is

gratefully acknowledged.

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(10)Wilson, E.L. and Khalvati, M., Finite Elements for theDynamic Analysis of Fluid-Solid Systems, International

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(11)Chiba, M. and Tani, J., The Dynamic Stability of Liquid-FilledCylindrical Shells under Vertical Excitation(in Japanese),Transactions of Japan Society of Mechanical Engineers, SeriesC , Vol.51, No.471(1985), pp.2980-2987.

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