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Sloshing behavior of floating-roof oil storage tanks
Computers & Structures Vol. 19, No. 1-2, pp. 18W92, 1984 Printed in the U.S.A.
SLOSHING
w45-7949184 $3.00 + .ofl 0 1984 Pergamon Press Ltd.
BEHAVIOR OF FLOATING-ROOF OIL STORAGE TANKS
F. SAKAI,~ M. NISHIMURA$and H. OGAWAS Steel Structure and Industrial Equipment Division, Kawasaki Heavy Industries Ltd., Minamisuna, Koto-ku, Tokyo 136, Japan
Ahatract-The subject of this paper is to investigate the sloshing behavior of floating-roofed oil storage tanks through theoretical analysis and model testing. The analysis employed theory of fluid-elastic vibration to study the interaction between a roof and the contained liquid. The tinite element method was applied, in which a technique based on the variational principle of boundary integrals was used to simplify the solution. The theory was verified by shake table experiments with three large models of single deck type and double deck type of floating roofs. From these results we have come to the following conclusions: (1) The existence of floating roofs hardly affects the first natural mode of sloshing. (2) The influence of the higher modes should be considered in determining stresses of double deck type floating roofs. (3) The local deformation of the lower deck plays a great role on the sloshing behavior.
INTRODUCTION
The damages of oil storage tanks due to the Earthquake in Niigata, Japan, in 1964[1] have inspired us to consider the importance of sloshing. Many studies on sloshing have been conducted since the earthquake, among which is a study by Yamamoto [2] who found that sloshing had caused the damage of the two floating-roofed oil storage tanks. The analysis was based on theory of potential flow. In seismic design of large tanks, long-period components of earthquake ground motion should be considered, since the natural periods of sloshing are comparatively long, of an order of five to ten seconds. However, such ground motions have not been studied fully yet, and practice of sloshing design has not been established. Shibata et a1.[3] proposed an approach called “the method of three wave response” from the results of their sequential studies, but the method for earthquake design is left unclear. Kaneda and Ogawa[4], Yumoto[5] and Shima and Nakamura[6] investigated experimentally the sloshing behavior in floating-roofed tanks. Their results were mainly on the reduction of liquid response by floating roofs, and therefore do not appear to be sufficient for fundamental or quantitative characterization of sloshing. The present authors[7j studed parametrically the sloshing in floating-roofed tanks as fluid-elastic vibration problems. The results of the first sloshing mode were in agreement with the experiments and the other studies. Not enough information was obtained for the higher modes. This was partly due to the assumption of a rectangular tank. This paper is a report on the sld%ing behavior of a cylindrical floating-roofed tank from the same viewpoint as the authors’ previous study[7j. In that
t&&ion Manager. $Research Engineer.
paper floating roofs are considered as uniform rectangular plates to simplify the problem. In this study two real types of floating roofs, one of which is a single deck type and and the other a double decl type, are dealt with. We carried out an exact fluid-elastic analysis and a large-sized model test and intended to obtain essential data for the seismic design through a more precise paramentric study. FUNDAMENTAL THEORY
Variational principle of boundary integral form Let us consider a floating-roofed tank shown in
Fig. 1, where the symbols are defined as follows: 0-z-2
direction of the ground motion coordinates radius of the tank : liquid depth P density of the contained liquid mass per unit area of the roof zt bending rigidity of the roof V fluid domain SW boundary surface of the wall boundary surface of the bottom sb contact surface between the roof and the s, liquid R given ground displacement
In addition, let t and rl be the given velocity of the tank and the elevation of the fluid respectively. The following assumptions are made: (1) The tank has a rigid flat bottom and a rigid cylindrical wall. (2) The contained liquid is a inviscid and incompressible fluid in an irrotational motion. (3) The floating roof deforms in bending only. (4) The floating roof is always in contact with the contained liquid. A Lagrangian of this system[8,9] is expressed by 183
F.
184
et al.
SAW1
in which t represents time and n the normal unit vector on the boundary outward from the field. Assuming that the velocity potential Q satisfies the following conditions, V2@= 0
in
V
(4)
eqn (3) is rewritten as
su
L/‘=
+-J{-gg)(D+$-$rj2}dS. (6)
-X
This equation is a Lagrangian variational functional in the form of boundary integral. Equation (3) which involves a volume integral is reduced into eqn (6) which contains only a surface integral. This is much easier to deal with, and the finite element method can be applied to the system more efficiently.
.Hr
Finite element formulation We consider the tank subjected to lateral ground motion. The continuity between the tank and the fluid requires the following conditions:
--x
Fig. I. A cylindrical floating-roof liquid storage tank.
L=L,-kL/’
n.;=Bcose
on
S,
Pa)
n.8=0
on
S,.
(7b)
A function @ satisfying eqn (4) and eqn (5) is assumed as follows:
(1)
(8)
in which J, is the Bessel function of the ftrst kind of the tirst order, A, the unknown constants and Q the positive roots of J&) = O(k = 1, 2, . . . , K), and the dot represents differentiation with respect to time t.
,7th
I
I
I
I
I
10-s
lo-”
10-8
10-z
lol 10-7
lo-6
131
lo-’
Ti
Fig. 2. Relationship between the bending rigidity and natural frequency.
Sloshing behavior of floating-roof oil storage tanks
Next, we have the following approximate sion:
expres-
185
of eqn (11) yields the following matrix equations: P.~+QQB++T.A+&J.B+~F.~=O
V=~BLI:a c0se i=l
0
(12)
-S.A+T’.h=O
in which Z$ is the unknown nodal displacements and f; the interpolation functions (i = 1, 2, . . , Z). Substituting eqn (8) and eqn (10) for the Lagrangian L, we obtain
(13)
in which the matrices P, Q, S, T, U and the vector F are derived from the element integration. Solving A from eqn (13) and substituting obtain the following vibration form:
it into eqn (12), we equation of general
(P+T’W’.T).fl+(Q+pgU)~B=
-pF.j
(14)
in which the first terms P and Q of the coefficient
matrices are the mass and stiffness matrices of the floating roof respectively, and the second terms are the matrices corresponding to the virtual (or added) mass and stiffness of the fluid respectively. ANALYSJS
OF A UNIFORM
ROOF
Although a floating roof in general is not uniform all over the structure, we introduce in the present investigation a simplified assumption that the bending rigidity is uniform, that is D, (radial) = D, (circumferential) = D. Non -dimensionalized parameters We consider the following non-dimensionalized parameters: A=; m
m=-
non-dimensionalized
liquid depth
non-dimensionalized
mass of the roof
non-dimensionalized of the roof
bending rigidity
non-dimensionalized frequency
natural
pa fi=s f=f
(11) If A and B are the unknown vectors, the variation
i J g rtg
rZ=s;,
non-dimensionalized the roof
displacement
Fig. 3. Ratio of the equivalent rigidity of the contained liquid or of the floating roof to the total equivalent rigidity.
of
F. SALAI et al.
186
p=P MS~
J
DOUBLE DECKTYPE
!T non-dimensionalized u
dynamic
pressure
where S, is the velocity response spectrum. As will be shown later, d is the most important parameter for the effect of a floating roof on the sloshing behavior. On the other hand, Z? and +i have little influence in real tanks, so we set fl= 1 and fi = lo-‘, as we will discuss later.
Behaviors of natural vibration
Figure 2 shows the relationship between the natural frequency and the rigidity of a floating roof. It reveals the following points: (1) The first natural frequency is constant regardless of the roof rigidity. (2) As the roof rigidity reaches a certain level, it effects the natural frequencies of higher orders. The higher the order of natural frequency is, the greater the effect is. The critical values of d are around 10m3 for the second order, 10m4for the third and low5 for the fourth. Figure 3 shows the ratio of the equivalent rigidity of the contained liquid or of the roof rigidity to the total equivalent rigidity. Each curve in it corresponds to an individual mode. It is seen that Fig. 3 indicates the same trend as that given in Fig. 2. Figure 4 shows the variation of the elevation shape with the roof rigidity. We can find the following points from the figure: (1) As the roof rigidity increases, the elevation shape is restrained by the roof and that of the first mode becomes very close to a straight line. (2) As far as elevation is concerned, the Iirst mode is predominant. The elevation of the second mode is less than one fifth of that of the first one, provided that the values of S, are the same with respect to the first and second mode. (3) The influence of the roof rigidity on the elevation is small. Therefore, we can estimate the elevation by neglecting the roof.
Usually there are two type floating roofs: the single deck and the double deck, which are shown in Fig. 5. While the former can be treated as an almost uniform plate, the latter must be treated as an orthogonally anisotropic plate with variable rigidity. Also in the latter, the bottom plate is subjected directly to dynamic pressures and deforms locally. The influence of the local deformation is one of the important points. Here we analyze the following double deck models to investigate the characteristics of a double deck type roof: (1) Global rigidities are 6,= 10m3 and 6, = 105m 10-I. (2) Bottom plate rigidities are 6,, = 6,,, = 10e6. (3) Circumferential stiffeners are located at six equally divided points on the radius. Natural frequency
Figures 6 shows the variation of the natural frequency of each order with 4. Figure 6(a) is the case of a “single plate analysis” neglecting the local deformation of a bottom plate and Fig. 6(b), the case of a “double plate analysis” when that deformation is considered. They indicate the following points: (1) The first natural frequency obtained by the two analyses is the same. (2) The natural frequency of higher orders is much lower by the double plate analysis than by the single plate analysis. Mode shapes of elevation
The elevation shapes of the first five modes for 6, = 10m5are shown in Fig. 7. It indicates that: (1) For the lower orders, the global deformation of a double deck type roof coincides with the deformation by the single plate analysis. (2) The higher the order of mode is, the more significant the influence of the local deformation of a bottom plate is, and for the mode shape of the fifth order the global deformation can hardly be recognixed. Therefore, for the higher orders the single plate
Fig. 4. Variation of elevation shape with roof rigidity.
187
Sloshing behavior of floating-roof oil storage tanks
(a)
c i rcumferent St 1ffener
Single
Deck
i41
Plane
View (h)
Douhlc
Deck
Fig. 5. Two types of floating roofs.
, ..’
/’
6th
5th 4th 3rd
0’ ,; _A’
.A
2nd --__
q-0 __
__
_--
-_
I
I
1st
0.1 10-c
10-s
(a)
lo+ (6)
lo-’
single
lo-’
double
10-a
10-Z
lo-’
a,
1OP
lo-’
B,
analysis
plate
1a+
plate
analysis
Fig. 6. Natural frequency vs variation of a, of double deck types.
188
F.
SALAI
et al.
5th
Fig. 7. Natural mode shape.
analysis is in large error for the global deformation when compared with the double plate analysis. Pressure acting upon a bottom plate
Figure 8 shows the dynamic pressure acting upon a bottom plate for each mode when 4 is 10e3. It reveals the following facts: (1) The pressures have peaks at the stiffeners. Generally, the distributions are similar to those by the single plate analysis. (2) When the order of mode is the tifth or higher, the pressures obtained by the single plate analysis are higher and may be in large error when compared with those of the double plate analysis.
MODEL TEST Models
In order to verify the sloshing behavior described above, we have made a vibration test by using a cylindrical rigid tank with various floating roofs (2a = 2m, H = lm). The tank filled with water was fixed on a shaking table and subjected to a lateral simple harmonic excitation. The following three kinds of floating roofs are used: (1) Single deck type roofs (made of tin-plate sheets). (I) Plate thickness = 0.27 mm; (II) Plate thickness = 0.60 mm. (2) Double deck type roofs (made of brass sheets). Plate thickness = 0.10 mm.
Sloshing behavior of floating-roof oil storage tanks
189
2nd
3rd
4th
-30
I
\
Y \
40
./ -2 5f h
Fig. 8. Roof pressure of the natural modes (a, = 10m3).
Response of elevation The frequency response
curve of the water elevation when the tank has no floating roofs is shown in Fig. 9. The experimental values are compared with the theoretical curve with the damping coefficient of 0.5% of critical. The correspondence between both values is good concerning the resonant frequencies. Judging from the experimental results at resonance,
the damping coefficient seems to be a little less than 0.5%. Figure 10 shows one of the frequency response curves of the elevation when the tank has the double deck. The data near the first resonance were not plotted in the figure, but the correspondence between the experimental and theoretical values is very good in all cases. In the experiments, the resonances of the
190
F. SALAIet al.
200 Amp11tude
180 Expertmental
160
A
0.2 5ma
0
0.5 0..
Cl l.OO.m
140
120
Theory
_
c,-?,=0.50$
100 c” 2 ” 5
80
z 3 : ; m ; r;
60
40
20
a
0
I
0.5
J 0 -A
V 0.7
0.6
I 0.8
A
h
0.9
1.0
0
I
1.1
1.2
1.5
1.4
1.3
Frequency
Fig. 9. Elevation amplitude vs frequency curve (no roof, r = 1OOcm).
70
6.0
TImy -c,-o.50%~
C,,,=5.0%
5.0
;
---
c,=o.5
G v <
4‘
-c a 6 ;- 3.c L z
2.c
1.c
\6 ‘\
Fig. 10. Elevation amplitude vs frequency (double desk, r = 93.5 cm).
0%.
c,_,,=
10.0%
I Hz )
191
Sloshing behavior of floating-roof oil storage tanks
Damping coeficients We evaluate a damping coefficient 4’ from the experimental response curves. At the first resonance, c, is about 0.4% when the tank has no floating roof and about 0.5% when the tank has a roof. In the case without floating roof, c2 and [, are still about 0.4x, but when the tank has a roof cz and c3 are estimated to be 5 N 10%. The theoretical curves in Fig. 10 mentioned above are drawn with the damping coefficients from & and [,,, of 5% and 10%.
higher order were not observed as clearly as in the case without roof, and the resonant frequencies of the second and third orders can be recognized. Other measured values such as dynamic pressures and strains in the roof yield almost the same tendencies, but are abbreviated for convenience of the space. CONSIDERATIONS
Resonant frequency Table 1 shows the resonant frequencies obtained from the experiment. The frequencies of the tist order could be obtained clearly, but the second and
CONCLUSIONS
From the above-mentioned results, we have come to the following conclusions: (1) As far as the first mode of sloshing, the existence of floating roofs hardly has any influence. This fact seems to be valid for every type floating roof. Therefore, we can ignore the existence of floating roofs, when we investigate the overflow of contained liquids. (2) The roof rigidity has important effects on the behavior of the higher modes. We recommend the use of a corrected plate bending rigidity of about ten times the real one, to account for the increase of the rigidity due to the membrane action. (3) In real single deck type floating roofs, d seems to be in the range of 10m7to 10d6 and the existence of the roof can be ignored. (4) In double deck type floating roofs, the global rigidity seems to be round 10e3, and the local deformation of bottom plates has significant influence on the behavior, especially in the higher orders. Therefore, to get the more accurate values of natural frequency and of dynamic pressure, we should not
third ones, which were determined roughly with the aid of the measured values of phase lag, were not so clear. On the other hand, the natural frequencies calculated by the above-mentioned theory are shown in Table 2. It can be seen from the table that the first natural frequency by the theory coincides well with that by the experiment, but the frequencies of the higher order are not in good agreement, that is, the experimental values are much greater. This discrepancy seems to be caused by membrane action of the thin plates, which the floating roofs are composed of. Therefore, we consider the roof rigidity to be higher due to the membrane action and increased bending rigidities ten times for the plate of the single deck type and five times for the bottom plate of double deck type. This, of course, is only an estimation. Table 3 shows the corrected theoretical natural frequencies obtained this way. They are in fair agreement with the experimental values. It is also noted that the marks V in Fig. 10 represent the corrected natural frequencies.
Table 1. Resonant frequency of floating roofs (experimental values) Natural Single
Frequency
(Hz)
Deck
order
Double I 0.27mm
Deck
II 0.6Omm
1st
0.657",0.661
2nd
0.70-1.40
0.655-0.659 2.0%
3rd
2.0
5.os7.0
23.0
0.659%0.665
3.0
2.501.2.75 4.0
%5.0
Table 2. Natural frequency (theoretical values) Natural Single
Deck II
0.6Omn
Frequency
(Hz)
Double
Single Plate
Deck Double Plate
192
F.
SALAI ef al.
Table 3. Natural frequency (corrected theoretical values)
apply the single plate analysis but the double plate analysis. (5) For a tank with a floating roof, the experimental result shows that the higher mode responses are not so pronounced. The apparent damping coefficients are estimated as 5 N 10%. Judging collectively from the study, for the seismic design of both type floating roofs, we may have to check the stresses including those caused by the higher mode response. Acknowledgements-The authors are much indebted to Prof. Emeritus T. Okumura of the University of Tokyo and Prof. N. Akiyama of Saitama University for their generous advice, also thanking the Fire Defence Agency of the Ministry of Home Affairs for financial support of the experiment. REFERENCES
1 The JSCE Committee
on the investigation of the Ni igaata earthquake disaster, The Report on the Investigation of the damages due to the Niigata earthquake of 1964, JSCE (1966).
2. Y. Yamamoto, The liquid sloshing and the impulsive pressures of oil storage tanks due to earthquakes. JHPI 3 (1965). 3. K. Sogabe, T. Shigeta and H. Shibata, A fundamental study on the aseismic design of liquid storages. Rep. Inst. of Ind. of Sci., Vol. 26, No. 7. The Univ. of Tokyo (1977). 4. K. Kaneda and S. Ogawa, The vibration test of a floating-roofed rank. JHPI 3 (1965). 5. G. Yumoto, The preventive mechanism of the oscillation of the liquid and floating roof in tanks during earthquakes. J. JSSE 7 (1968). 6. Y. Shima and H. Nakamura, A fundamental experiment for the prevention of liquid sloshing due to the vibration of a water tank. Proc. Ann. Meet. JSCE 1974. 7. F. Sakai, Y. Tsukioka, A. Masago and M. Sakai, A theoretical and experimental study on sloshing of floating roof. JHPI 15 (1977). 8. F. Sakai, Some considerations on variational principles in perfect fluids. Proc. 11th Nat. Symp. on Matrix Meth. Analysis of JSSC (1977). 9. F. Sakai, Vibration analysis of fluid-solid systems. Proc. U.S.-Japan Seminar on Interdisciplinary Finite Element Analysis, Cornell Univ., New York (1978).
SLOSHING
w45-7949184 $3.00 + .ofl 0 1984 Pergamon Press Ltd.
BEHAVIOR OF FLOATING-ROOF OIL STORAGE TANKS
F. SAKAI,~ M. NISHIMURA$and H. OGAWAS Steel Structure and Industrial Equipment Division, Kawasaki Heavy Industries Ltd., Minamisuna, Koto-ku, Tokyo 136, Japan
Ahatract-The subject of this paper is to investigate the sloshing behavior of floating-roofed oil storage tanks through theoretical analysis and model testing. The analysis employed theory of fluid-elastic vibration to study the interaction between a roof and the contained liquid. The tinite element method was applied, in which a technique based on the variational principle of boundary integrals was used to simplify the solution. The theory was verified by shake table experiments with three large models of single deck type and double deck type of floating roofs. From these results we have come to the following conclusions: (1) The existence of floating roofs hardly affects the first natural mode of sloshing. (2) The influence of the higher modes should be considered in determining stresses of double deck type floating roofs. (3) The local deformation of the lower deck plays a great role on the sloshing behavior.
INTRODUCTION
The damages of oil storage tanks due to the Earthquake in Niigata, Japan, in 1964[1] have inspired us to consider the importance of sloshing. Many studies on sloshing have been conducted since the earthquake, among which is a study by Yamamoto [2] who found that sloshing had caused the damage of the two floating-roofed oil storage tanks. The analysis was based on theory of potential flow. In seismic design of large tanks, long-period components of earthquake ground motion should be considered, since the natural periods of sloshing are comparatively long, of an order of five to ten seconds. However, such ground motions have not been studied fully yet, and practice of sloshing design has not been established. Shibata et a1.[3] proposed an approach called “the method of three wave response” from the results of their sequential studies, but the method for earthquake design is left unclear. Kaneda and Ogawa[4], Yumoto[5] and Shima and Nakamura[6] investigated experimentally the sloshing behavior in floating-roofed tanks. Their results were mainly on the reduction of liquid response by floating roofs, and therefore do not appear to be sufficient for fundamental or quantitative characterization of sloshing. The present authors[7j studed parametrically the sloshing in floating-roofed tanks as fluid-elastic vibration problems. The results of the first sloshing mode were in agreement with the experiments and the other studies. Not enough information was obtained for the higher modes. This was partly due to the assumption of a rectangular tank. This paper is a report on the sld%ing behavior of a cylindrical floating-roofed tank from the same viewpoint as the authors’ previous study[7j. In that
t&&ion Manager. $Research Engineer.
paper floating roofs are considered as uniform rectangular plates to simplify the problem. In this study two real types of floating roofs, one of which is a single deck type and and the other a double decl type, are dealt with. We carried out an exact fluid-elastic analysis and a large-sized model test and intended to obtain essential data for the seismic design through a more precise paramentric study. FUNDAMENTAL THEORY
Variational principle of boundary integral form Let us consider a floating-roofed tank shown in
Fig. 1, where the symbols are defined as follows: 0-z-2
direction of the ground motion coordinates radius of the tank : liquid depth P density of the contained liquid mass per unit area of the roof zt bending rigidity of the roof V fluid domain SW boundary surface of the wall boundary surface of the bottom sb contact surface between the roof and the s, liquid R given ground displacement
In addition, let t and rl be the given velocity of the tank and the elevation of the fluid respectively. The following assumptions are made: (1) The tank has a rigid flat bottom and a rigid cylindrical wall. (2) The contained liquid is a inviscid and incompressible fluid in an irrotational motion. (3) The floating roof deforms in bending only. (4) The floating roof is always in contact with the contained liquid. A Lagrangian of this system[8,9] is expressed by 183
F.
184
et al.
SAW1
in which t represents time and n the normal unit vector on the boundary outward from the field. Assuming that the velocity potential Q satisfies the following conditions, V2@= 0
in
V
(4)
eqn (3) is rewritten as
su
L/‘=
+-J{-gg)(D+$-$rj2}dS. (6)
-X
This equation is a Lagrangian variational functional in the form of boundary integral. Equation (3) which involves a volume integral is reduced into eqn (6) which contains only a surface integral. This is much easier to deal with, and the finite element method can be applied to the system more efficiently.
.Hr
Finite element formulation We consider the tank subjected to lateral ground motion. The continuity between the tank and the fluid requires the following conditions:
--x
Fig. I. A cylindrical floating-roof liquid storage tank.
L=L,-kL/’
n.;=Bcose
on
S,
Pa)
n.8=0
on
S,.
(7b)
A function @ satisfying eqn (4) and eqn (5) is assumed as follows:
(1)
(8)
in which J, is the Bessel function of the ftrst kind of the tirst order, A, the unknown constants and Q the positive roots of J&) = O(k = 1, 2, . . . , K), and the dot represents differentiation with respect to time t.
,7th
I
I
I
I
I
10-s
lo-”
10-8
10-z
lol 10-7
lo-6
131
lo-’
Ti
Fig. 2. Relationship between the bending rigidity and natural frequency.
Sloshing behavior of floating-roof oil storage tanks
Next, we have the following approximate sion:
expres-
185
of eqn (11) yields the following matrix equations: P.~+QQB++T.A+&J.B+~F.~=O
V=~BLI:a c0se i=l
0
(12)
-S.A+T’.h=O
in which Z$ is the unknown nodal displacements and f; the interpolation functions (i = 1, 2, . . , Z). Substituting eqn (8) and eqn (10) for the Lagrangian L, we obtain
(13)
in which the matrices P, Q, S, T, U and the vector F are derived from the element integration. Solving A from eqn (13) and substituting obtain the following vibration form:
it into eqn (12), we equation of general
(P+T’W’.T).fl+(Q+pgU)~B=
-pF.j
(14)
in which the first terms P and Q of the coefficient
matrices are the mass and stiffness matrices of the floating roof respectively, and the second terms are the matrices corresponding to the virtual (or added) mass and stiffness of the fluid respectively. ANALYSJS
OF A UNIFORM
ROOF
Although a floating roof in general is not uniform all over the structure, we introduce in the present investigation a simplified assumption that the bending rigidity is uniform, that is D, (radial) = D, (circumferential) = D. Non -dimensionalized parameters We consider the following non-dimensionalized parameters: A=; m
m=-
non-dimensionalized
liquid depth
non-dimensionalized
mass of the roof
non-dimensionalized of the roof
bending rigidity
non-dimensionalized frequency
natural
pa fi=s f=f
(11) If A and B are the unknown vectors, the variation
i J g rtg
rZ=s;,
non-dimensionalized the roof
displacement
Fig. 3. Ratio of the equivalent rigidity of the contained liquid or of the floating roof to the total equivalent rigidity.
of
F. SALAI et al.
186
p=P MS~
J
DOUBLE DECKTYPE
!T non-dimensionalized u
dynamic
pressure
where S, is the velocity response spectrum. As will be shown later, d is the most important parameter for the effect of a floating roof on the sloshing behavior. On the other hand, Z? and +i have little influence in real tanks, so we set fl= 1 and fi = lo-‘, as we will discuss later.
Behaviors of natural vibration
Figure 2 shows the relationship between the natural frequency and the rigidity of a floating roof. It reveals the following points: (1) The first natural frequency is constant regardless of the roof rigidity. (2) As the roof rigidity reaches a certain level, it effects the natural frequencies of higher orders. The higher the order of natural frequency is, the greater the effect is. The critical values of d are around 10m3 for the second order, 10m4for the third and low5 for the fourth. Figure 3 shows the ratio of the equivalent rigidity of the contained liquid or of the roof rigidity to the total equivalent rigidity. Each curve in it corresponds to an individual mode. It is seen that Fig. 3 indicates the same trend as that given in Fig. 2. Figure 4 shows the variation of the elevation shape with the roof rigidity. We can find the following points from the figure: (1) As the roof rigidity increases, the elevation shape is restrained by the roof and that of the first mode becomes very close to a straight line. (2) As far as elevation is concerned, the Iirst mode is predominant. The elevation of the second mode is less than one fifth of that of the first one, provided that the values of S, are the same with respect to the first and second mode. (3) The influence of the roof rigidity on the elevation is small. Therefore, we can estimate the elevation by neglecting the roof.
Usually there are two type floating roofs: the single deck and the double deck, which are shown in Fig. 5. While the former can be treated as an almost uniform plate, the latter must be treated as an orthogonally anisotropic plate with variable rigidity. Also in the latter, the bottom plate is subjected directly to dynamic pressures and deforms locally. The influence of the local deformation is one of the important points. Here we analyze the following double deck models to investigate the characteristics of a double deck type roof: (1) Global rigidities are 6,= 10m3 and 6, = 105m 10-I. (2) Bottom plate rigidities are 6,, = 6,,, = 10e6. (3) Circumferential stiffeners are located at six equally divided points on the radius. Natural frequency
Figures 6 shows the variation of the natural frequency of each order with 4. Figure 6(a) is the case of a “single plate analysis” neglecting the local deformation of a bottom plate and Fig. 6(b), the case of a “double plate analysis” when that deformation is considered. They indicate the following points: (1) The first natural frequency obtained by the two analyses is the same. (2) The natural frequency of higher orders is much lower by the double plate analysis than by the single plate analysis. Mode shapes of elevation
The elevation shapes of the first five modes for 6, = 10m5are shown in Fig. 7. It indicates that: (1) For the lower orders, the global deformation of a double deck type roof coincides with the deformation by the single plate analysis. (2) The higher the order of mode is, the more significant the influence of the local deformation of a bottom plate is, and for the mode shape of the fifth order the global deformation can hardly be recognixed. Therefore, for the higher orders the single plate
Fig. 4. Variation of elevation shape with roof rigidity.
187
Sloshing behavior of floating-roof oil storage tanks
(a)
c i rcumferent St 1ffener
Single
Deck
i41
Plane
View (h)
Douhlc
Deck
Fig. 5. Two types of floating roofs.
, ..’
/’
6th
5th 4th 3rd
0’ ,; _A’
.A
2nd --__
q-0 __
__
_--
-_
I
I
1st
0.1 10-c
10-s
(a)
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lo-’
single
lo-’
double
10-a
10-Z
lo-’
a,
1OP
lo-’
B,
analysis
plate
1a+
plate
analysis
Fig. 6. Natural frequency vs variation of a, of double deck types.
188
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et al.
5th
Fig. 7. Natural mode shape.
analysis is in large error for the global deformation when compared with the double plate analysis. Pressure acting upon a bottom plate
Figure 8 shows the dynamic pressure acting upon a bottom plate for each mode when 4 is 10e3. It reveals the following facts: (1) The pressures have peaks at the stiffeners. Generally, the distributions are similar to those by the single plate analysis. (2) When the order of mode is the tifth or higher, the pressures obtained by the single plate analysis are higher and may be in large error when compared with those of the double plate analysis.
MODEL TEST Models
In order to verify the sloshing behavior described above, we have made a vibration test by using a cylindrical rigid tank with various floating roofs (2a = 2m, H = lm). The tank filled with water was fixed on a shaking table and subjected to a lateral simple harmonic excitation. The following three kinds of floating roofs are used: (1) Single deck type roofs (made of tin-plate sheets). (I) Plate thickness = 0.27 mm; (II) Plate thickness = 0.60 mm. (2) Double deck type roofs (made of brass sheets). Plate thickness = 0.10 mm.
Sloshing behavior of floating-roof oil storage tanks
189
2nd
3rd
4th
-30
I
\
Y \
40
./ -2 5f h
Fig. 8. Roof pressure of the natural modes (a, = 10m3).
Response of elevation The frequency response
curve of the water elevation when the tank has no floating roofs is shown in Fig. 9. The experimental values are compared with the theoretical curve with the damping coefficient of 0.5% of critical. The correspondence between both values is good concerning the resonant frequencies. Judging from the experimental results at resonance,
the damping coefficient seems to be a little less than 0.5%. Figure 10 shows one of the frequency response curves of the elevation when the tank has the double deck. The data near the first resonance were not plotted in the figure, but the correspondence between the experimental and theoretical values is very good in all cases. In the experiments, the resonances of the
190
F. SALAIet al.
200 Amp11tude
180 Expertmental
160
A
0.2 5ma
0
0.5 0..
Cl l.OO.m
140
120
Theory
_
c,-?,=0.50$
100 c” 2 ” 5
80
z 3 : ; m ; r;
60
40
20
a
0
I
0.5
J 0 -A
V 0.7
0.6
I 0.8
A
h
0.9
1.0
0
I
1.1
1.2
1.5
1.4
1.3
Frequency
Fig. 9. Elevation amplitude vs frequency curve (no roof, r = 1OOcm).
70
6.0
TImy -c,-o.50%~
C,,,=5.0%
5.0
;
---
c,=o.5
G v <
4‘
-c a 6 ;- 3.c L z
2.c
1.c
\6 ‘\
Fig. 10. Elevation amplitude vs frequency (double desk, r = 93.5 cm).
0%.
c,_,,=
10.0%
I Hz )
191
Sloshing behavior of floating-roof oil storage tanks
Damping coeficients We evaluate a damping coefficient 4’ from the experimental response curves. At the first resonance, c, is about 0.4% when the tank has no floating roof and about 0.5% when the tank has a roof. In the case without floating roof, c2 and [, are still about 0.4x, but when the tank has a roof cz and c3 are estimated to be 5 N 10%. The theoretical curves in Fig. 10 mentioned above are drawn with the damping coefficients from & and [,,, of 5% and 10%.
higher order were not observed as clearly as in the case without roof, and the resonant frequencies of the second and third orders can be recognized. Other measured values such as dynamic pressures and strains in the roof yield almost the same tendencies, but are abbreviated for convenience of the space. CONSIDERATIONS
Resonant frequency Table 1 shows the resonant frequencies obtained from the experiment. The frequencies of the tist order could be obtained clearly, but the second and
CONCLUSIONS
From the above-mentioned results, we have come to the following conclusions: (1) As far as the first mode of sloshing, the existence of floating roofs hardly has any influence. This fact seems to be valid for every type floating roof. Therefore, we can ignore the existence of floating roofs, when we investigate the overflow of contained liquids. (2) The roof rigidity has important effects on the behavior of the higher modes. We recommend the use of a corrected plate bending rigidity of about ten times the real one, to account for the increase of the rigidity due to the membrane action. (3) In real single deck type floating roofs, d seems to be in the range of 10m7to 10d6 and the existence of the roof can be ignored. (4) In double deck type floating roofs, the global rigidity seems to be round 10e3, and the local deformation of bottom plates has significant influence on the behavior, especially in the higher orders. Therefore, to get the more accurate values of natural frequency and of dynamic pressure, we should not
third ones, which were determined roughly with the aid of the measured values of phase lag, were not so clear. On the other hand, the natural frequencies calculated by the above-mentioned theory are shown in Table 2. It can be seen from the table that the first natural frequency by the theory coincides well with that by the experiment, but the frequencies of the higher order are not in good agreement, that is, the experimental values are much greater. This discrepancy seems to be caused by membrane action of the thin plates, which the floating roofs are composed of. Therefore, we consider the roof rigidity to be higher due to the membrane action and increased bending rigidities ten times for the plate of the single deck type and five times for the bottom plate of double deck type. This, of course, is only an estimation. Table 3 shows the corrected theoretical natural frequencies obtained this way. They are in fair agreement with the experimental values. It is also noted that the marks V in Fig. 10 represent the corrected natural frequencies.
Table 1. Resonant frequency of floating roofs (experimental values) Natural Single
Frequency
(Hz)
Deck
order
Double I 0.27mm
Deck
II 0.6Omm
1st
0.657",0.661
2nd
0.70-1.40
0.655-0.659 2.0%
3rd
2.0
5.os7.0
23.0
0.659%0.665
3.0
2.501.2.75 4.0
%5.0
Table 2. Natural frequency (theoretical values) Natural Single
Deck II
0.6Omn
Frequency
(Hz)
Double
Single Plate
Deck Double Plate
192
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Table 3. Natural frequency (corrected theoretical values)
apply the single plate analysis but the double plate analysis. (5) For a tank with a floating roof, the experimental result shows that the higher mode responses are not so pronounced. The apparent damping coefficients are estimated as 5 N 10%. Judging collectively from the study, for the seismic design of both type floating roofs, we may have to check the stresses including those caused by the higher mode response. Acknowledgements-The authors are much indebted to Prof. Emeritus T. Okumura of the University of Tokyo and Prof. N. Akiyama of Saitama University for their generous advice, also thanking the Fire Defence Agency of the Ministry of Home Affairs for financial support of the experiment. REFERENCES
1 The JSCE Committee
on the investigation of the Ni igaata earthquake disaster, The Report on the Investigation of the damages due to the Niigata earthquake of 1964, JSCE (1966).
2. Y. Yamamoto, The liquid sloshing and the impulsive pressures of oil storage tanks due to earthquakes. JHPI 3 (1965). 3. K. Sogabe, T. Shigeta and H. Shibata, A fundamental study on the aseismic design of liquid storages. Rep. Inst. of Ind. of Sci., Vol. 26, No. 7. The Univ. of Tokyo (1977). 4. K. Kaneda and S. Ogawa, The vibration test of a floating-roofed rank. JHPI 3 (1965). 5. G. Yumoto, The preventive mechanism of the oscillation of the liquid and floating roof in tanks during earthquakes. J. JSSE 7 (1968). 6. Y. Shima and H. Nakamura, A fundamental experiment for the prevention of liquid sloshing due to the vibration of a water tank. Proc. Ann. Meet. JSCE 1974. 7. F. Sakai, Y. Tsukioka, A. Masago and M. Sakai, A theoretical and experimental study on sloshing of floating roof. JHPI 15 (1977). 8. F. Sakai, Some considerations on variational principles in perfect fluids. Proc. 11th Nat. Symp. on Matrix Meth. Analysis of JSSC (1977). 9. F. Sakai, Vibration analysis of fluid-solid systems. Proc. U.S.-Japan Seminar on Interdisciplinary Finite Element Analysis, Cornell Univ., New York (1978).