Seismic

F. G. Golzar

R. Shabani1e-mail: [email protected]

Department of Mechanical Engineering,

Faculty of Engineering,

Urmia University,

Urmia 15311-57561, Iran

S. TariverdiloDepartment of Civil Engineering,


Urmia University,

Urmia 15311-57561, Iran

G. RezazadehDepartment of Mechanical Engineering,


Urmia University,

Urmia 15311-57561, Iran

Sloshing Response of FloatingRoofed Liquid Storage TanksSubjected to Earthquakesof Different TypesUsing extended Hamiltonian variational principle, the governing equations for sloshingresponse of floating roofed storage tanks are derived. The response of the floating roofedstorage tanks is evaluated for different types of ground motions, including near-sourceand long-period far-field records. Besides comparing the response of the roofed andunroofed tanks, the effect of different ground motions on the wave elevation, lateralforces, and overturning moments induced on the tank is investigated. It is concluded thatthe dimensionless sloshing heights for the roofed tanks are solely a function of their firstnatural period. Also it is shown that while long-period far-field ground motions controlthe free board height, near-source records give higher values for lateral forces and over-turning moments induced on the tank. This means that same design spectrum could not beused to evaluate the free board and lateral forces in the seismic design of storage tanks.Finally, two cases are studied to reveal the stress patterns caused by different earth-quakes. [DOI: 10.1115/1.4006858]

Keywords: liquid storage tank, sloshing response, earthquake, floating roof

1 Introduction

Floating roofs are used in the petroleum industries for storageof liquid hydrocarbons in atmospheric storage tanks. Reducingthe evaporation of storage materials, floating roofs provide betterprotection against possible ignition of the vapors by sparks gen-erated by different sources, such as static electricity, earthquake,cigarette smoking, etc. [1]. In addition to economic benefit bypreventing the evaporation of valuable products, these roofs arealso helpful in reduction of the environmental pollution causedby evaporation. The conventional floating roofs can be catego-rized into two types: single deck and double deck. Single deckroofs are composed of a circumferential buoyant ring (pontoon)with relatively large stiffness surrounding an inner circular plate(deck) with small thickness and stiffness. Double deck roofs, onthe other hand consisting of upper and lower decks, are heavierand more rigid.Serious damages of floating roofs in past earthquakes could be

attributed to the sloshing of the contained liquid [2]. Therefore,it seems essential to take into account the interaction betweenfloating roof and contained liquid in any analysis of the systemsubject to ground motion excitation. As a pioneering approach,Sakai et al. [3] derived the equations of motion for a cylindricaltank with floating roof using variational principle to evaluatenatural frequencies and pressure distribution upon the roof. Theyvalidated the results experimentally for single deck and doubledeck models. Isshiki and Nagata [4] used Hamiltons principlefor plate and Kelvins principle for water to derive the extendedHamiltonian principle for fully coupled problem. Robinson andPalmer [5] carried out modal analysis and derived transfer func-tion for low amplitude oscillations of a floating plate. Amabili

and Kwak [6] used RayleighRitz method and Hankel transfor-mation to analyze the free vibrations of circular plates with uni-form boundary conditions. Matsui [7] employed FourierBesselseries to derive the linear equations of motion for a floating roofwith relatively high rigidity and compared the results with thoseof unroofed tank. Developing the frequency content of the waveelevation and roof pressure for a tank under seismic excitation,his results revealed the suppression of higher modes in wave ele-vation and magnification of higher modes in the pressureresponse. His results were verified by shaking table tests carriedout by Nagaya et al. [8]. Their experiments showed the strongdependency of damping ratio on roof type. Later, Matsui usedthe same method to analyze the motion of single deck floatingroof with pontoon [9]. Virella et al. [10] used linear and nonlin-ear finite element methods to analyze the response of harmoni-cally base excited rectangular tanks and used nondimensionalparameters to compare the results with those of the cylindricaltanks. They showed that the wall pressures were not muchinfluenced by the nonlinearities. In a similar way, Mitra and Sin-hamahapatra [11] investigated the response for seismic excita-tions. Response of base isolated cylindrical tanks, with andwithout roof, to earthquake excitations was subject of a compre-hensive experimental study by De Angelis et al. [12]. They veri-fied the results by simple numerical methods and showed thefavorable effects of base isolators and floating roofs on reducingthe wall pressures and oscillation amplitudes, respectively.Efforts were also put into mere nonlinear analysis. Utsumi andIshida [13,14] accounting for nonlinear sloshing investigated thepossibility of internal resonance under seismic and harmonicexcitations.The above mentioned papers have considered elemental

sloshing characteristics like coupled frequencies and modeshapes or/and case studies of roofed tanks subjected to certainearthquake excitations while a need for a more inclusive deduc-tion is felt. In this paper, by applying the Hamiltons variationalprinciple on the floating roof and contained liquid, the sloshingresponse of the floating roof storage tanks is investigated for

1Corresponding author.Contributed by the Pressure Vessel and Piping Division of ASME for publication

in the JOURNAL OF PRESSURE VESSEL TECHNOLOGY. Manuscript received July 18, 2011;final manuscript received April 22, 2012; published online September 10, 2012.Assoc. Editor: Spyros A. Karamanos.

Journal of Pressure Vessel Technology OCTOBER 2012, Vol. 134 / 051801-1CopyrightVC 2012 by ASME

Downloaded From: http://asmedigitalcollection.asme.org/ on 10/10/2013 Terms of Use: http://asme.org/terms

different ground motion records. The records include long-period far-field and near-source ground motions. Numerical sim-ulations are used to evaluate the wave elevation, and lateralforce and overturning moment imposed on the tank wall.Effects of the roof and different types of the ground motions onthe results are investigated thoroughly. Computations are carriedout for a range of commonly used dimensions for storage tanks.The results could have practical implications for designpurposes.

2 Mathematical Model

Figure 1 depicts a typical liquid storage tank with a floatingroof and the cylindrical coordinate used in the analysis. The roofis considered of double deck type with radius R, thickness h, uni-form bending rigidity D, and areal mass density M. Peripheralwall of the tank is reasonably considered rigid as the natural fre-quencies of shell vibrations are much higher than the sloshing fre-quencies. The tank is subjected to a horizontal unidirectionalseismic excitation xg.Fluid movement in the domain, (0 r R; 0 h

< 2p; 0 z H) for an incompressible and inviscid liquid,is described by the velocity potential U which satisfies the Lap-lacian equation (r2U 0) as well as the following boundaryconditions:

@U@z

z0

0 (1)

@U@r

rR

_xg cos h (2)

@U@z

zH

@g@t

(3)

which suggest the impermeability condition at the bottom and pe-ripheral wall and kinematic condition at the interface of plate andliquid, respectively. Imposing these conditions, the velocity poten-tial is obtained to be in the following form:

Ur; z; h r _xg cos hX1k1

AkJ1

ekrR

J1 ek

coshekzR

cosh

ekHR

cos h r _xg cos h U0 (4)

Where J1 denotes the Bessel function of the first kind of order 1.ek are eigenvalues of the velocity potential and are found by solv-ing the equation J01ek 0. Floating plate motion can be consid-ered as a weighted summation of its mode shapes in air

gr; h; t X1i1

BitXir cos h (5)

where Bi and Xi denote the generalized coordinates and radialmode shapes, respectively. The radial mode shapes consist of rigidand elastic modes in the following form [15]:

Xir rR; i 1

Xir aiJ1 ki rR

biI1 ki

r

R

h i; i 2; 3

(6)

where ji are the radial eigenvalues of the mode shapes and arefound by satisfying zero shear and zero moment conditions on thefree edge. J1 denotes the Bessel function of the first kind of order1 and I1 denotes the modified Bessel function of the first kind oforder 1. The mode shapes are normalized in order to have

Rr0

2ph0

MR aiJ1 jir

R

biI1 ji

r

R

h icos h

2rdrdh 1 (7)

where MR denotes the areal mass density of the roof. Bearing inmind that the roof and liquid stay in complete contact, the Lagran-gian of the coupled system, accounting for the kinetic and poten-tial energies, is written in form of the sum of roof Lagrangian LR[15] and liquid Lagrangian Li [3]:

L LRLlRr0

2ph0

1

2MR _g

2

12DR

@2g@r2

2 1

r

@g@r

1r2@2g

@h2

!224

2 @2g

@r2

1

r

@g@r 1r2@2g

@h2

rdrdh

Rr0

2ph0

qL 12

@U0@z

U0@g

@tU1

2gg2

rdrdh

(8)

Note that the twisting effects on the strain energy of the roof havenot been accounted for. DR denotes the bending rigidity of theroof and is calculated by

DR EI1 t2 (9)

where E is the Youngs modulus of elasticity and I is the secondmoment of inertia of the roof cross section. Substituting Eq. (5)into Eq. (8), we will have

Fig. 1 Cylindrical liquid storage tank with a double deck float-ing roof

051801-2 / Vol. 134, OCTOBER 2012 Transactions of the ASME


L R

r0

X1i1

_BiXi

! X1j1

_BjXj

!1

2MRrdr

Rr0

X1i1

BiX00i

! X1j1

BjtX00j r !

12DR

rdr

Rr0

1

r

X1i1

BitX0t 1

r2

X1i1

BiXi

!

1

r

X1j1

BjX0j

1

r2

X1j1

BjXj

12DR

rdr

Rr0

X1i1

BiX00i

!1

r

X1j1

BjX0j

1

r2

X1j1

BjXj

!DRrdr

Rr0

X1j1

ekRAj

J1ejr=RJ1ej tanh ej

H

R

" #

X1l1

AlJ1elr=RJ1el

" # 12qL

rdr

Rr0

X1i1

_BiXi

" #r _xg

X1j1

AjJ1eKr=RJ1eK

" #qLrdr

Rr0

qL1

2gX1i1

BiXi

" # X1j1

BjXj

" #rdr

2ph0

cos2 h d h

(10)

Referring to the principle of variation, time integral of variation ofLagrangian vanishes in any desired time interval

t2t1

dLdt t2t1

CBdBj CAdAjdt 0 (11)

where coefficients CB and CA are defined as

CBRr0

X1i1

BiXi

!XjMRrdr

Rr0

X1i1

BiX00i

!X00j DRrdr

Rr0

1

r

X1i1

BiX0i

1

r2

X1i1

BiXi

!1

rX

0j

1

r2Xj

DRrdr

Rr0

"1

r

X1i1

BiX0i

!X00j

1

r

X1i1

BiX00i

!X0j

1

r2

X1i1

BiXi

!X00j

1r2

X1i1

BiX00i

!Xj

#DRrdr

Rr0

rxgX1j1

_AjJ1ejr=RJ1ej

" #XjqLrdr

Rr0

X1i1

BiXi

!XjqLgrdr (12)

CARr0

ekR

J1 ekr=R J1 ek tanh ek

H

R

X1l1

AlJ1 elr=R J1 el

" #12qL

rdr

Rr0

X1l1

elRAlJ1 elr=R J1 el tanh el

H

R

" #J1 ekr=R J1 ek

12qL

rdr

Rr0

X1i1

_BiXi

" #J1 ekr=R J1 ek

qLrdr

In order to satisfy Eq. (11), both coefficients are equated to zeroyielding the following matrix equations:

MR P B DRQ qLgU fBg qLT _A qLxg Ff g 0

(13)

Tt _B S Af g 0 (14)The significance of compatibility condition between roof and liq-uid is noticed by Eq. (14) as it, evidently, is a restatement of Eq.(3). The shape factor A can be rolled out by substituting its equiv-alent form from Eq. (14) into Eq. (13) so as to have

MR P qL T S1

Tt B DRQ qLgU fBg

qLxg Ff g 0 (15)

Fig. 2 Sketch of radial and vertical pressure distributions onthe tank wall

Table 1 Seismic data used in this study

Input ground motion Duration (s) PGA (g) Epicentral distance (km)

Kobe (1995) 40 0.4862 8.7Imperial Valley (1940) 40 0.3130 13Tokachi-oki (2003) 290 0.0800 227Tohoku (2011) 300 0.1397 364

Fig. 3 Time histories of different ground motions: (a) Kobe,(b) Imperial Valley, (c) Tokachi-oki, (d) Tohoku

Journal of Pressure Vessel Technology OCTOBER 2012, Vol. 134 / 051801-3


Fig. 4 Velocity response spectra of different ground motions: (a) Kobe, (b) Imperial Valley, (c) Tokachi-oki,(d)Tohoku

Table 2 The data used in this study

Contained liquid height, H (model A)Contained liquid height, H (model B)

15 m20 m

Roof areal mass density,MR 120 kg=m2

Roof bending rigidity, DR (model A)Roof bending rigidity, DR (model B)

1:612 105 kNm3:022 105 kNm

Roof thickness, h 800 mmLiquid density, qL 800 kg=m

3

Damping ratio, unroofed tankDamping ratio, roofed tank

0:0010:01

Effective cross-sectional coefficient (model A)Effective cross-sectional coefficient (model B)

1050 mm2

1200 mm2

Poissons ratio, 0:3Earth acceleration of gravity, g 9:81 m=s2

Number of modes adopted for liquid sloshing 29Number of modes adopted for roof vibration 5

Fig. 5 Lowest two natural periods (T1, T2) of assumed models: (a) model A: H5 15, (b) model B: H520



As the equation suggests, there exists both dynamic and staticcouplings between the roof and oscillating fluid, resulting in acoupled set of equations. The number of equations is the same asthe number of modes considered in evaluating roof displacements.Elements of matrices P;Q;U;F; S, and T are specified as follows:

Pij R0

XiXjrdr

Qij R0

X00i X

00j rdr

R0

1

rX

0i

1

r2Xi

1

rX

0j

1

r2Xj

rdr

R0

X00i

1

rX

0j

1

r2Xj

X00j

1

rX

0i

1

r2Xi

rdr

Uij R0

XiXjrdr

(16)

Tij Rr0

XiJ1 ejr=R J1 ej rdr

Fi Rr0

rXirdr

Sjj R2ej tanh

ejHR

1 1

ej2

In Eq. (15), the mass matrix is composed of two parts: Matrix Paccounting for the presence of floating plate and matrix TS1Tt, rep-resenting the added inertial effect of the liquid. Likewise, stiffnessmatrix is composed of matrix Q accounting for the plate stiffness andmatrix U representing the added stiffness due to fluid covibration.Linearized Bernoullis equation is used to evaluate the pressure

acting on the wall

Fig. 6 Response of model A to Kobe earthquake: (a) maximumsloshing height, (b) maximum lateral force, (c) maximum over-turning moment

Fig. 7 Response of model B to Kobe earthquake: (a) maximumsloshing height, (b) maximum lateral force, (c) maximum over-turning moment



p qL@U@t

qLgg (17)

The first term in the above equation stems from the liquid move-ment and the second is the hydrostatic pressure imposed by theelevated liquid. Figure 2 shows the pressure profiles on front andtop cross sections. The net lateral force and overturning momentare evaluated by adding up the projection of all radial forces alongthe excitation direction using the following formulae:

Fx Hz0

2ph0

pRdhdz (18)

Mx Hz0

2ph0

zpRdhdz (19)

3 Numerical Results

To investigate the sloshing response of the tank, following foursets of ground motion data were utilized as base excitation: 090component of 1995 Kobe earthquake at Japan recorded at Nishi-Akashi station (KOBE/NIS000), 180 component of 1940 ImperialValley earthquake recorded at El Centro station (IMPVALL/I-ELC180), EW component of 2003 Tokachi-oki earthquakerecorded at Tomakomai station (HKD1290309260450EW), andEW component of 2011 Tohoku earthquake (eastern Japan earth-quake) recorded at Chiba station (CHB0091103111446). Table 1shows the duration, peak round acceleration, and epicentral dis-tance of earthquake records. Time histories and velocity responsespectra (VRS) of the records are shown in Figs. 3 and 4.Kobe and Imperial Valley ground motions are classified as near-

source earthquakes and have rather similar frequency content con-taining spectral peaks at low periods. On the other hand, Tokachi-oki and Tohoku are classified as long-period, far-field records, withbroad band frequency content. Near-source earthquakes act in ashorter time interval but impose larger accelerations to the

Fig. 8 Response of model A to Imperial Valley earthquake: (a)maximum sloshing height, (b) maximum lateral force, (c) maxi-mum overturning moment

Fig. 9 Response of model B to Imperial Valley earthquake: (a)maximum sloshing height, (b) maximum lateral force, (c) maxi-mum overturning moment



structures compared to long-period, far-field earthquakes. In thisstudy, recordings of the aforementioned earthquakes are used asbase excitation for two ranges of tank dimensions with the geomet-rical and material properties shown in Table 2.In numerical simulations, Newmarks method is employed to

solve Eq. (15) where the equations can also be modified for thecase of unroofed tank by omitting plate related parameters. Eventhough the equations were derived for an ideally undamped sys-tem, the actual structures show a rate of energy dissipation [8]which may be attributed to various sources like viscous interactionbetween liquid and tank wall(and roof) and also frictional contactbetween roof and wall. In order to account for these terms, damp-ing matrix C is added to Eq. (15) to yield the following more prac-tical equation:

MR P qL T S1

Tt B C _B

DRQ qLgU fBg qLxg Ff g 0(20)

The stiffness proportional damping is adopted with the modaldamping ratios shown in Table 2. The selected values for theroofed (0.01) and unroofed (0.001) cases are in accordance withthe experimental findings of Nagaya et al. [8] and theoreticalassumptions of Matsui [7] where it was also stated that the stiff-ness proportional damping was appropriate for double deck roofswhile Rayleigh damping was more appropriate for single deckroofs.Figure 5 shows the variation of the first and second natural peri-

ods with respect to radius for the two models. It is observed thatwhile the fundamental natural period of the tank is not much

Fig. 10 Response of model A to Tokachi-oki earthquake: (a)maximum sloshing height, (b) maximum lateral force, (c) maxi-mum overturning moment

Fig. 11 Response of model B to Tokachi-oki earthquake: (a)maximum sloshing height, (b) maximum lateral force, (c) maxi-mum overturning moment



altered by the presence of the roof, second natural period of thesystem is lowered substantially. This observation is in accordancewith the results of Sakai et al. [3] and Matsui [7]. Higher modesperiods also decrease noticeably though not shown in the figure.Figures 6 and 7 show the responses of the models to Kobe exci-

tation. Presence of the roof has suppressed the sloshing of liquidthroughout whole radius range, though the absolute value of thesloshing height is quite negligible relative to the radius (Figs. 6(a)and 7(a)). However, the values of the force and moment show anascending trend for increase in the tank radius for both models(Figs. 6(b), 6(c), 7(b), and 7(c)). On the other hand, presence ofthe roof has hardly affected the resulting lateral force andmoments on the tank wall.Figures 8 and 9 demonstrate how the models respond to the Im-

perial Valley excitation. Descriptions given for Kobe results are

also valid here: suppressed sloshing heights (but with a differenttrend) (Figs. 8(a) and 9(a)) and an ascending trend in force andmoment diagrams. Therefore, it could be concluded that the pres-ence of the floating roof does not have appreciable effect on thedynamics of the storage tanks in the case of excitation by near-source ground motions.Figures 10 and 11 show the responses of the models to

Tokachi-oki earthquake. Tanks with different radii exhibit differ-ent responses to the excitation. While for major parts of the rangethe floating roof has a suppressing effect on the wave elevation,there are also increases occurred for some subranges. For slendertanks (tanks with smaller radius), roof has a negative (increasing)effect on the force and moment but as the radius of the tank isincreased positive (reducing) effect of the roof emerges (Figs.10(b), 10(c), 11(b), and 11(c)). As the fundamental period of the

Fig. 12 Response of model A to Tohoku earthquake: (a) maxi-mum sloshing height, (b) maximum lateral force, (c) maximumoverturning moment

Fig. 13 Response of model B to Tohoku earthquake: (a) maxi-mum sloshing height, (b) maximum lateral force, (c) maximumoverturning moment



Fig. 14 Dimensionless sloshing heightsinduced by (a) Kobe, (b) Imperial Vally, (c) Tokachi-oki, (d) Tohoku earth-quakes (U, unroofed; R, roofed)

Fig. 15 Dimensionless lateral forces induced by (a) Kobe, (b) Imperial Vally, (c) Tokachi-oki, (d) Tohoku earth-quakes (U, unroofed; R, roofed)



system falls out of the rich area in the VRS diagram of the earth-quake, the absolute value of the sloshing height for both roofedand unroofed tanks decreases and the force and moment diagramslose their increasing trend and remain relatively constant.The responses of models to Tohoku excitation are depicted in

Figs. 12 and 13. Surface sloshing has been suppressed throughoutthe radius range especially at the local peaks (Figs. 12(a) and13(a)). Force and moment graphs for the roofed and unroofedcases do not differ significantly except for the dimensions that nat-ural period and earthquake peak coincide (Figs. 12(b), 12(c),13(b), and 13(c)).Bearing in mind that the fundamental period of the tank does

not differ much by the presence of the roof, (Fig. 5), the change inthe values of sloshing height (g), lateral force (Fx), and overturn-ing moment (Mx) may be associated with the drastic change in thehigher modes periods or the impulsive mode participation. Pres-ence of the roof alters the contribution of higher modes to theresponse by rearranging their periods at lower values, thus push-ing them out of strong content area. It can be said that if the sec-ond mode participation in the response is in-phase with thefundamental mode, its exclusion will decrease the response andvice versa.Responses of the two models are redrawn in terms of dimen-

sionless ratios versus fundamental periods to provide a compara-ble graph. Figure 14 shows the dimensionless sloshing heights(g=R) of the two models for all four excitations. It is observed thatwhile the diagrams of unroofed tanks differ with each other, thoseof the roofed tanks overlap over the whole area. This indicatesthat, regardless of excitation type (near-source or long-period, far-field), the dimensionless sloshing heights are solely a function offundamental period of the system (Fig. 15).Response of the models in terms of dimensionless forces and

moments are depicted in Figs. 16 and 17, respectively. The differ-ences between the diagrams of roofed tanks suggest that the firstnatural period is not the only parameter influencing the dimen-

sionless values of force and moment. It is once again seen that theroof does not have major effect on the relative force and momentresponse of the tanks subjected to short period, near-source excita-tions Kobe and Imperial Valley.Figures 17 and 18 compare the effects of different earthquakes

on the models. Short period earthquakes of Kobe and ImperialValley cause small sloshing heights due to both their short dura-tion and weak frequency content around the fundamental naturalfrequencies. On the contrary, sloshing heights produced by long-period, far-field earthquakes Tokachi-oki and Tohoku are rela-tively larger as a result of their longer duration and stronger fre-quency content around the first and second natural frequencies.Considering the lateral force and moment imposed on the tankwall, near-source ground motions compared to long-period far-field records produce larger values. This may be attributed to thelarger acceleration and consequently larger contribution of impul-sive mode for near-source ground motions.In order to reveal the pattern and intensity of stresses occurred

within the roof, two certain tank dimensions (small tank,H 15;R 25; large tank, H 20;R 40), one from eachtype, were selected to be investigated. Radial and circumferentialmoments are calculated using following formulae:

Mr DR @2g

@r2 1

r

@g@r

1r2

@2g

@h2

(21)

Mh DR 1r

@g@r

1r2

@2g

@h2

@

2g@r2

(22)

Radial and circumferential stresses (rr;rh) are then obtained bydividing the moments by the effective cross-sectional coefficientsgiven in Table 2 [9]. Figure 19 shows the radial and circumferen-tial stress patterns in the roof of small tank. It is observed thatpeak stresses caused by the near-source earthquakes Kobe and

Fig. 16 Dimensionless overturning moments induced by (a) Kobe, (b) Imperial Vally, (c) Tokachi-oki, (d) Tohokuearthquakes (U, unroofed; R, roofed)



Fig. 17 Comparison of responses of model A to differentearthquakes: (a) sloshing height, (b) lateral force, (c) overturn-ing moment

Fig. 18 Comparison of responses of model B to differentearthquakes: (a) sloshing height, (b) lateral force, (c) overturn-ing moment

Fig. 19 Roof stress patterns in model A (H5 15, R5 25): (a) maximum radial stress, (b) maximum circumferentialstress



Imperial Valley are far larger than the stresses caused by far-fieldearthquakes Tokachi-oki and Tohoku. Both types of stress reachtheir peak within the range r 0:35R 0:40R with the radialstress vanishing at the plate edge as anticipated.Figure 20 depicts the radial and circumferential stress patterns

in the roof of large tank. While the far-field earthquakes result instresses qualitatively similar to those occurred in the small tank,near-source inputs show different stress patterns on the roof. Inthis case, the peaks of the graphs have moved toward the innerradii (r 0:25R). It is again observed that the near-source earth-quakes result in more destructive stresses in the roof.

In order to determine the effect of different modes on the bendingstresses, the frequency content of the stress oscillation at the locationof peak stresses for two cases is presented in the following figures.Figure 21 shows the FFTs of maximum stresses in small tank roofproduced by Kobe and Tokachi-oki earthquakes. It is seen that whilethe fundamental mode plays a significant role in the stresses causedby far-field earthquake Tokachi-oki, second mode is approximatelythe sole controlling parameter of stress oscillation in case of Kobeexcitation. This may be attributed to the strong content of the near-source earthquakes around the higher modes periods. The same ex-planation is also valid for the case of large tank (Fig. 22).

Fig. 20 Roof stress patterns in a floating roofed tank (H5 20, R5 40): (a) maximum radial stress, (b) maximumcircumferential stress

Fig. 21 Frequency content of stress at roof (H5 15, R5 25): (a) Kobe excitation, (b) Tokachi-oki excitation

Fig. 22 Frequency content of stress at roof (H5 20, R5 40): (a) Kobe excitation, (b) Tokachi-oki excitation



4 Conclusion

Using Hamiltonian variational principle, the governing equa-tions for sloshing response of floating roofed storage tanks arederived. The results show that the presence of the floating roofhardly affects the fundamental period of the system but noticeablyincreases higher natural frequencies. Evaluating the response oftanks with floating roofs to different types of ground motionrecords, it is shown that the dimensionless vertical deflection ofroof is solely a function of the tank fundamental period. Also it isshown that while long-period far-field ground motions give higherroof deflection, the near-source records give higher value for lat-eral forces and overturning moments induced on the tank wall.This indicates that the same design spectrum would not be aproper reference to evaluate the free board and lateral forces inthe seismic design of storage tanks. Finally, a comparison of stresspatterns in two certain tanks demonstrated the more destructiveeffects of near-source earthquakes in comparison to far-field,long-period earthquakes.

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[6] Amabili, M., and Kwak, M. K., 1996, Free Vibration of Circular PlatesCoupled With Liquids: Revising the Lamb Problem, J. Fluids Struct., 10(7),pp. 743761.

[7] Matsui, T., 2007, Sloshing in a Cylindrical Liquid Storage Tank With a Float-ing Roof Under Seismic Excitation, ASME J. Pressure Vessel Technol.,129(4), pp. 557566.

[8] Nagaya, T., Matsui, T., and Wakasa, T., 2008, Model Tests on Sloshing of aFloating Roof in a Cylindrical Liquid Storage Tank Under Seismic Excitation,Proceedings of the ASME Pressure Vessels and Piping Division Conference,Chicago, IL, Paper No. PVP2008-61675.

[9] Matsui, T., 2009, Sloshing in a Cylindrical Liquid Storage Tank With a SingleDeck Type Floating Roof Under Seismic Excitation, ASME J. Pressure VesselTechnol., 131(2), pp. 557566.

[10] Virella, J. C., Prato, C. A., and Godoy, L. A., 2008, Linear and Nonlinear 2DFinite Element Analysis of Sloshing Modes and Pressures in Rectangular TanksSubject to Horizontal Harmonic Motions, J. Sound Vib., 312, pp. 442460.

[11] Mitra, S., and Sinhamahapatra, K. P., 2007, Slosh Dynamics of Liquid-FilledContainers With Submerged Components Using Pressure-Based Finite ElementMethod, J. Sound Vib., 304, pp. 361381.

[12] De Angelis, M., Giannini, R., and Paolacci, F., 2009, Experimental Investiga-tion on the Seismic Response of a Steel Liquid Storage Tank Equipped WithFloating Roof by Shaking Table Tests, Earthquake Eng. Struct. Dyn., 39, pp.377396.

[13] Utsumi, M., and Ishida, K., 2008, Vibration Analysis of a Floating Roof TakingInto Account Nonlinearity of Sloshing, ASME J. Appl. Mech., 75, p. 041008.

[14] Utsumi, M., and Ishida, K., 2010, Internal Resonance of a Floating Roof Sub-jected to Nonlinear Sloshing, ASME J. Appl. Mech., 77(1), p. 011016.

[15] Meirovitch, L., 1997, Principles and Techniques of Vibrations, Prentice-Hall,Englewood Cliffs, NJ, Chap. VII.



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