Tank Seismic on Concreate

Engineering Structures 33 (2011) 2186–2200

Contents lists available at ScienceDirect

Engineering Structures

journal homepage: www.elsevier.com/locate/engstruct

The effect of earthquake frequency content on the seismic behavior of concreterectangular liquid tanks using the finite element method incorporatingsoil–structure interactionM.R. Kianoush ∗, A.R. GhaemmaghamiCivil Engineering Department, Ryerson University, Toronto, ON, Canada

a r t i c l e i n f o

Article history:Received 15 February 2010Received in revised form13 February 2011Accepted 18 March 2011Available online 17 April 2011

Keywords:SloshingRectangular tankDynamic analysisTank flexibilityFinite elementLiquid containing

a b s t r a c t

A three-dimensional soil–structure–liquid interaction is numerically simulated using the finite elementmethod in order to analyze the seismic behavior of partially filled concrete rectangular tanks subjected todifferent groundmotions. In this paper, the effect of earthquake frequency content on the seismic behaviorof fluid rectangular tank system is investigated using four different seismic motions. A simplemodel withviscous boundary is used to include deformable foundation effects as a linear elasticmedium. Thismethodis capable of considering both impulsive and convective responses of liquid-tank system. Six different soiltypes defined in the well-recognized seismic codes are considered. The sloshing behavior is simulatedusing linear free surface boundary condition. Two different finite element models corresponding withflexible shallow and tall tank configurations are studied under the effects of longitudinal, transversal andvertical ground motions. By means of changing the soil properties, comparisons are made on base shear,base moment and sloshing responses under different ground motions. It is concluded that the dynamicbehavior of the fluid–tank–soil system is highly sensitive to frequency characteristics of the earthquakerecord.

© 2011 Elsevier Ltd. All rights reserved.

1. Introduction

The dynamic interaction between fluid, structure and soil is asignificant concern inmany engineering problems. These problemsinclude systems as diverse as off-shore and submerged structures,biomechanical systems, suspension bridges and storage tanks. Theinteraction can drastically change the dynamic characteristics ofthe structure and consequently its response to transient and cyclicexcitation. Therefore, it is desired to accuratelymodel these diversesystems with the inclusion of fluid–structure interaction (FSI).

One of the critical lifeline structures which has becomewidespread during the recent decades is liquid storage tank. Thesestructures are extensively used in water supply facilities, oil andgas industries and nuclear plants for storage of a variety of liquidor liquid-like materials such as oil, liquefied natural gas (LNG),chemical fluids and wastes of different forms.

Problems associatedwith liquid tanks involvemany fundamen-tal parameters. In fact, the dynamic behavior of liquid tanks isgoverned by earthquake characteristics, the interaction betweenfluid and structure as well as soil and structure along their bound-aries. For example, it has been found that hydrodynamic pressure

∗ Corresponding author. Tel.: +416 979 5000x6455.E-mail addresses: [email protected] (M.R. Kianoush),

[email protected] (A.R. Ghaemmaghami).

0141-0296/$ – see front matter© 2011 Elsevier Ltd. All rights reserved.doi:10.1016/j.engstruct.2011.03.009

in a flexible tank can be significantly higher than the correspond-ing rigid container due to the interaction effects between flexiblestructure and contained liquid.

Even though there have been numerous studies done on thefluid–structure interaction effects in liquid containers, most ofthem are concerned with cylindrical tanks and the studies onseismic response of rectangular tanks are quite rare. Housner [1]developed the most commonly used analytical model for rigidtanks in which hydrodynamic pressure is separated into impulsiveand convective components using lumped mass approximation.This model has been adopted with some modifications in most ofthe current codes and standards.

Later, Yang [2], Veletos and Yang [3] studied the effects of wallflexibility on the pressure distribution in liquid and correspond-ing forces in the tank structure through an analytical model. Mi-nowa [4,5] investigated the effect of flexibility of tankwalls and hy-drodynamic pressure acting on the wall using both analytical andexperimental methods.

Studies on the three-dimensional fluid–structure interaction ina domain of a more general geometry, other than the cylindricalshape, can be found in the literature on the seismic design ofconcrete dams.

Chopra and Gupta [6] investigated the effects of fluid–structureand soil–structure interaction on the frequency function re-sponses of water–dam–foundation system. They concluded that

http://dx.doi.org/10.1016/j.engstruct.2011.03.009

http://www.elsevier.com/locate/engstruct

http://www.elsevier.com/locate/engstruct

mailto:[email protected]

mailto:[email protected]

http://dx.doi.org/10.1016/j.engstruct.2011.03.009

M.R. Kianoush, A.R. Ghaemmaghami / Engineering Structures 33 (2011) 2186–2200 2187

fluid–structure interaction has a significant effect on resonant fre-quencies of the dam under seismic ground motions.

Haroun [7] presented a very detailed analytical method in thetypical system of loading in rigid rectangular tanks. In addition,Haroun [8] carried out a series of experiments including ambientand forced vibration tests. Also, Haroun and Tayel [9] used thefinite element method (FEM) for analyzing dynamic response ofliquid tanks subjected to vertical seismic ground motions.

Veletsos and Tang [10] analyzed the liquid storage tankssubjected to vertical groundmotion on both rigid and flexible sup-portingmedia. Haroun and Abou-Izzeddine [11] conducted a para-metric study of numerous factors affecting the seismic soil–tankinteraction under vertical excitations.

Veletso et al. [12] found that the convective components ofresponse are insensitive to the flexibilities of the tank wall andsupporting soils, and may be computed considering both the tankand the supporting medium to be rigid. Kim et al. [13] furtherdeveloped analytical solutionmethods and presented the responseof filled flexible rectangular tanks using both two- and three-dimensional modeling. Park et al. [14] performed research studieson dynamic response of the rectangular tanks using both finiteelement method (FEM) and boundary element method (BEM).

Subhash and Bhattacharyya [15] developed a numerical schemeusing the finite element technique to calculate the sloshingdisplacement of liquid and pressure developed to such sloshing.Koh et al. [16] presented a coupled BEM–FEM, including freesloshingmotion, to analyze three-dimensional rectangular storagetanks subjected to horizontal ground motion. In addition, theyconducted a shaking table test on a three-dimensional rectangulartank under horizontal El-Centro ground motion record to verifytheir numerical method.

Dogangun et al. [17] investigated the seismic response of liquid-filled rectangular storage tanks using the finite element methodimplemented in the general purpose structural analysis computercode SAPIV.

Chen and Kianoush [18] used the sequential method to calcu-late impulsive hydrodynamic pressure in two-dimensional rect-angular tanks including wall flexibility effects. Also, Kianoushand Chen [19] investigated the dynamic behavior of rectangulartanks subjected to vertical seismic vibrations in a two-dimensionalspace. In addition, Kianoush et al. [20] introduced a new methodfor seismic analysis of rectangular containers in two-dimensionalspace in which the effects of both impulsive and convective com-ponents are accounted for in time domain.

Livaoglu [21] evaluated the dynamic behavior of fluid–rectangular tank foundation systemwith a simple seismic analysisprocedure. In this procedure, interaction effects were presented byHousner’s two mass approximations for fluid and the cone modelfor soil foundation system. Two different earthquake recordswere used to investigate the effect of earthquake frequency onsoil–structure–fluid interaction.

In the field of seismic behavior of structural buildings, manystudies have investigated the effect of vibration frequency ondynamic responses. Some of the major findings are reported byJennings and Kuroiwa [22], Foutch et al. [23], Mcverry [24] andChopra [25]. It is clear that peak responses of the structure occurwhen the fundamental frequency of the vibration reaches near thecorresponding natural frequencies of the structure.

Considering the previous studies done on this topic, it isclear that very limited research has been done on the soil–structure–fluid interaction (SSFI) effect on seismic behavior ofconcrete rectangular tanks.Wolf and Song [26,27] simplified foun-dation as an isotropic homogeneous elastic medium. The near fieldwas modeled using finite elements, and the far field was treatedby adding some special boundaries such as springs and dampers.The soil treated in most cases as a semi-infinite medium, and

this unbounded domain was large enough to include the effect ofsoil–structure interaction. This approach has been used by Cloughand Penzien [28] and Wilson [29].

When analyzing a soil–structure–liquid system using FEM, afoundationmodel that extends one tank length in the downstream,upstream and downward directions usually suffices in most cases.This approach permits different soil properties to be assigned todifferent elements, so that the variation of soil characteristics withdepth can be considered.

There are different boundary models available in frequencyor time domains. Earlier, Lysmer and Kuhlmeyer [30] developeda viscous boundary model using one-dimensional beam theory.This theory has been commonly used with the FE method. Later,more complex boundary types were used and developed suchas damping-solvent extraction [31,27], doubly asymptotic multi-directional transmitting boundary [26,27] and paraxial boundarymethods [32].

This study comprehensively investigates the dynamic behaviorof concrete rectangular tanks using the finite element method(FEM) in three-dimensional space in which the coupled fluid–stru-cture equations are solved using the direct integral method in timedomain. Contrary to the commonmethod of analysis in this subjectwhich is based on frequency domain and lumped mass approach,the governing equation of fluid domain in this study is discretizedusing the FEM which accounts for both impulsive and convectivebehavior.

Four different earthquake ground motions including 1994Northridge, 1940 El-Centro, 1971 San-Fernando and 1957 San-Francisco with different frequency contents are applied in FE anal-yses to account for the effect of frequency content. All records arescaled in such a way that the horizontal peak ground accelera-tion reaches 0.4 g. A good indicator of the frequency content ofthe ground motion is the ratio of peak ground acceleration (PGA)which is expressed in units of g to peak ground velocity (PGV) ex-pressed in units of m/s. Earthquake records may be classified ac-cording to the frequency content ratio into three categories, highPGA/PGV ratio when PGA/PGV > 1.2, intermediate PGA/PGV ra-tio when 1.2 > PGA/PGV > 0.8 and low PGA/PGV ratio whenPGA/PGV < 0.8. The Northridge record has low frequency con-tent, the El-Centro and San-Fernando earthquakes have intermedi-ate frequency contents and the San-Francisco record has the highfrequency content.

To account for the effect of soil–structure interaction, sixdifferent soil types which are adopted from common design codesare used in the FE modeling. It should be noted that the mainobjective of this study is to investigate the effect of earthquakefrequency content on the dynamic behavior of liquid tanks. As aresult, this study is mainly focused towards this part and a briefsummary is given on SSI effect.

This study has led to some new findings which are presentedwith the aid of two different tank configurations that are ana-lyzed under time history excitations incorporating soil–structureinteraction. One of the major advantages of proposed method is inaccounting for soil–structure interaction (SSI), fluid–structure in-teraction (SFI), fluid damping properties and considering impul-sive and convective components separately which have not beenconsidered in previous studies. It should be noted that this studyis based on analyzing a linear system of fluid–structure–soil andtherefore effect of material and wave nonlinearity is not consid-ered.

2. Analysis method for fluid–tank–soil system

In liquid domain, the hydrodynamic pressure distribution isgoverned by the pressure wave equation. Owing to the smallvolume of containers, the velocity of pressure wave assumed to

2188 M.R. Kianoush, A.R. Ghaemmaghami / Engineering Structures 33 (2011) 2186–2200

be infinity. Assuming that water is incompressible and neglectingits viscosity, the small-amplitude irrotational motion of water isgoverned by the three-dimensional wave equation:

∇2P(x, y, z, t) = 0 (1)

where P(x, y, z, t) is the hydrodynamic pressure in excess ofhydrostatic pressure.

The hydrodynamic pressure in Eq. (1) is due to the horizontaland vertical seismic excitations of the walls and bottom of thecontainer. For earthquake excitation, the appropriate boundarycondition at the interface of liquid and tank is governed by

∂P(x, y, z, t)∂n

= −ρan(x, y, z, t) (2)

where ρ is the density of liquid and an is the component ofacceleration on the boundary along the direction outward normaln. No wave absorption is considered in the interface boundarycondition.

Accounting to the small-amplitude gravity waves on the freesurface of the liquid, the resulting boundary condition is given as

1g

∂2P∂t2

+∂P∂z

= 0. (3)

In which z is the vertical direction and g is the gravitationalacceleration.

However, due to the large amplitude of sloshing under thestrong seismic excitations and turbulence effects in liquid tanks,more complicated boundary conditions on the surface of liquid areneeded to accurately model the convective motions. In a recentstudy done by Virella et al. [33], the influence of nonlinear wavetheory on the sloshing natural periods and their modal pressuredistribution for rectangular tanks with HL/Lx ratios ranging from0.4 to 1.65 was investigated. They concluded that the nonlinearityof the surface wave does not have a major effect on the pressuredistribution on the walls and on natural sloshing frequencies. Inthe present study, two different tank configurations namely ashallow and a tall tank are analyzed as will be discussed later.The ratios of HL/Lx are 0.37 and 1.26 for the shallow and talltanks, respectively. In this case, the linearized boundary conditionis appropriate particularly for practical applications. Neglecting thegravity wave effects leads to the free surface boundary conditionwhich is appropriate for impulsive motion of liquid. The relatedgoverning equation is given as

P(x, y,Hl, t) = 0 (4)

where Hl is the height of liquid in the container.Using finite element discretization and discretized formulation

of Eq. (1), thewave equation can bewritten as the followingmatrixform:

[G]{P} + [H]{P} = {F}. (5)

In which Gi,j =∑

Gei,j, Hi,j =

∑He

i,j and Fi =∑

F ei . The coef-

ficients Gei,j, H

ei,j and F e

i for an individual element are determinedusing the following expressions:

Gei,j =

1g

∫Ae

NiNjdA

Hei,j =

∫Ve

∂Ni

∂x∂Nj

∂x+

∂Ni

∂y∂Nj

∂y+

∂Ni

∂z∂Nj

∂z

dV

F ei =

∫Ae

Ni∂P∂n

dA

(6)

whereNi is the shape function of the ith node in the liquid element.Ae and Ve are the integration over side and area of the element, re-spectively.

In the above formulation, matrices [H] and [G] are constantsduring the analysis while the force vector {F}, pressure vector{P} and its derivatives are the variable quantities. In the couplingsystem of liquid–structure the pressures are applied to thestructure surface as the loads on the container walls. The generalequation of fluid–structure neglecting the soil interaction can bewritten in the following form:

[M]{U} + [C]{U} + [K ]{U} = {f1} − [M]{Ug} + [Q ]{P}

= {F1} + [Q ]{P}

[G]{P} + [C ′]{P} + [H]{P} = {F} − ρ[Q ]

T ({U} + {Ug})

= {F2} − ρ[Q ]T{U} (7)

where [M], [C] and [K ] are mass, damping and stiffness matricesof structure. {U} is the acceleration vector of nodes in the structuredomain, {Ug} is the ground acceleration vector applied to thesystem and [Q ] is the coupling matrix. The term [C ′

] is the matrixrepresenting the damping of liquid which is dependent on theviscosity of liquid and wave absorption in liquid domain andboundaries and is rigorously determined. As previously discussed,thematrix [Q ] transfers the liquid pressure to the structure as wellas structural acceleration to the liquid domain.

The direct integration scheme is used to find the displacementandhydrodynamic pressure at the endof time increment i+1 giventhe displacement and hydrodynamic pressure at i.

Descriptions regarding the coupling matrix and direct inte-gration method can be found in the studies done by Kianoushet al. [20].

Under free oscillations, the motion of free liquid surface de-cays due to damping forces created by viscous boundary layers.Basically, the damping factor depends on the liquid height, liquidkinematic velocity and tank dimensions. From this point of view,evaluation of damping characteristic for a fluid–tank system needsmore considerations. However, due to lack of sufficient data in thisfield, the classical damping scheme is used in the finite elementmodel. Considering impulsive and convective parts of liquid do-main, damping matrix can be given as

[Cf ] = a[G] + b[H]. (8)

In which a and b are computed by Rayleigh damping method.In this equation, coefficient a is calculated based on fundamentalfrequency of liquid sloshing to present the convective part of theresponse while, coefficient b is computed based on impulsive fun-damental frequency of the tank–liquid system which is obtainedby modal analysis. The fundamental frequencies of tank–liquidsystem are calculated using both finite element method and an-alytical equations adopted from ACI 350.3-06 [34]. The convectivefrequencies are exactly the same but a significant difference is seenfor impulsive values. In this study, the values corresponding to fi-nite element method is used to calculate the damping coefficientswhich will be discussed later.

Sloshing in a tank without any anti-sloshing device is usuallydamped by viscous forces. One more complete investigationswhich have been done by Mikishev and Dorozhkin [35] showedthat in a storage tank with rational dimensions viscous damping isless than 0.5%.

In the proposed FE procedure, Rayleigh damping as mentionedpreviously is used in the direct step-by-step integration method.The stiffness proportional damping equivalent to 5% of criticaldamping is assumed as structural damping. For sloshing behaviorof liquid 0.5% of critical damping is applied. Damping valuesassociated with horizontal and vertical motions are dependent onearthquake shear wave velocity and the ratio of HL/R in which R isthe equivalent radius of a cylindrical tank with the same plan areaas the rectangular tank. These values canbedrawnbasedon studiesdone by Veletsos and Tang [10] and Veletsos and Shivakumar [36]


Fig. 1. Viscous boundary condition in the three-dimensional finite element model [37].

for each tank configuration. In this study, the value of 5% of criticaldamping is applied for liquid domain of shallowand tall tankmodelas a conservative damping ratio.

Whenmodeling a dynamic problem involving soil–structure in-teraction, particular attention must be given to the soil bound-ary conditions. Ideally, infinite boundary condition should besurrounding the excited zone. Propagation of energy will occurfrom the interior to the exterior region. Since the exterior regionis nonreflecting, it absorbs all of the incoming energy. Yet, in afinite element analysis, we are constrained into applying finitesize boundaries for the foundations. Those boundaries in turn willreflect the elastic waves which is contrary to the physics of theproblem.

In this study, a viscous boundary method is used in three-dimensional space which was successfully employed in the FEmodeling of the elevated liquid tanks done by Livaoglu andDogangun [37]. The detailed formulation can be found in the workdone by Lysmer and Kuhlmeyer [30].

These viscous boundaries can be used with the FE mesh asshown in Fig. 1 for a generalized three-dimensional model [37].In this figure, An, At1 and At2 are the fields controlling the viscousdampers, σ and τ are the normal and shear stresses occurringin the boundaries of the medium, and the subscripts n and trepresent normal and tangent directions in the boundary. Whenthe viscous boundary is taken into consideration, an additionaldamping matrix is applied to the system as follows:

[C∗

i ] =

Anρvp 0 00 At1ρvs 00 0 At2ρvs

(9)

where vp and vs are dilatational and shear wave velocity of theconsidered medium. It should be noticed that in two-dimensionalanalysis, the third row and column are eliminated.

It is assumed that the tank structure is anchored to the foun-dation. Considering the effects of soil–structure interaction, Eq. (7)can be re-written as

([M] + [M∗

i ]){U} + ([C] + [C∗

i ]){U} + ([K ] + [K ∗

i ]){U}

= {F1} + [Q ]{P}

[G]{P} + [C ′]{P} + [H]{P} = {F2} − ρ[Q ]

T{U}

(10)

where [C∗

i ] and [K ∗

i ] are the damping and stiffness matrices asso-ciated with the foundation–structure interaction.

The most common soil–structure interaction (SSI) approach,used for three-dimensional soil–structure systems, is based onthe ‘‘added motion’’ formulation by Clough and Penzien [28]. Thisapproximation is mathematically simple, theoretically accurate,

and is easy to usewithin an FEmethod for analyzing a linear systemin time domain. On this basis, to account for the soil–structureinteraction, it is necessary to apply the inertial loads only to thestructure not to the soil foundation. The far-field boundaries aretreated using viscous dampers to absorb thewave energy reflectedby the system.

Most computer commercial programs automatically applythe seismic loading to all mass degrees of freedom within thecomputer model and cannot solve the SSFI problem. This lackof capability has motivated the development of the masslessfoundationmodel inwhich the inertia forceswithin the foundationmaterial are neglected. Most of the studies done on the behavior ofliquid storage tanks including soil–structure interaction are basedon this approximation.

A parametric study was carried out to determine the dimen-sions of the soil foundation in such away that the displacements ofnodes on the lateral boundaries are almost zero. It should be notedthat the stress and strain due to liquid–tank weight are consideredin the soil foundation.

3. Finite element implementation

In this study, two different model configurations associatedwith shallow and tall tanks are investigated in three-dimensionalspace. A schematic view of rectangular tank model is shownin Fig. 2. These tanks have also been used in some previousinvestigations by Kianoush and Chen [19], Chen and Kianoush [18]and Kim et al. [13]. The dimensions and properties of shallow andtall tank are as follows:Shallow Tank:

ρw = 2300 kg/m3 ρl = 1000 kg/m3 Ec = 26.44 GPaν = 0.17Lx = 15 m Lz = 30 m Hw = 6.0 m Hl = 5.5 mtw = 0.6 m.

Tall Tank:

ρw = 2300 kg/m3 ρl = 1000 kg/m3 Ec = 20.77 GPaν = 0.17Lx = 9.8 m Lz = 28 m Hw = 12.3 m Hl = 11.2 mtw = 1.2 m.

It is assumed that the tank is anchored at its base and the effects ofuplift pressure are not considered.

In this study, an eight-node isoparametric element with threetranslational degrees of freedom in each node is used in the finiteelement procedure to model the tank walls and the base slab.


Fig. 2. Schematic configuration of a rectangular liquid tank.

Fig. 3. Finite element model of a three-dimensional rectangular tank: (a) tall tank model, (b) shallow tank model, and (c) viscous boundary condition.

The liquid domain ismodeled using eight-node isoparametric fluidelements with pressure degree of freedom in each node. The finiteelement (FE) model configurations for both shallow and tall tanksare shown in Fig. 3.

The longitudinal, transversal and vertical components recordedfor 1994 Northridge, 1940 El-Centro, 1971 San-Fernando and1957 San-Francisco earthquakes are used as excitations of thetank–liquid system. The components are scaled in such a waythat the peak ground acceleration in the longitudinal directionreach 0.4 g, as shown in Fig. 4. Based on the ratio of peak groundacceleration (PGA) to peak ground velocity (PGV), the Northridgerecord is considered as low frequency ground motion; El-Centro and San-Fernando records are categorized as intermediatefrequency excitations and San-Francisco record is considered ashigh frequency content excitation.

4. Mesh sensitivity and error estimation

The first step in a finite element analysis is to estimate thepossible error by selecting a mathematical model to represent theobject being analyzed. The mathematical target for this study wasselected as themaximumpressure at the bottomof rigid tank usingequations derived by Haroun [7].

The analysis goal is to compute the exact solution of bottompressure named PEX and then calculate its value using the FEMwhich is PFE . The term PEX depends only on the definition ofthe mathematical model and not on the method used for findingan approximate solution. Therefore, it does not depend on meshquality, type, and size of elements. The difference between PEX andthe physical property it represents is called the modeling error. Asa result, discretization error is defined by


Fig. 4. Scaled longitudinal components of earthquake records: (a) 1994 Northridge (b) 1940 El-Centro (c) 1971 San-Fernando (d) 1957 San-Francisco.

Fig. 5. Finite element discretization error: (a) Rigid shallow tank model (b) Rigid tall tank model.

e =(PEX ) − (PFE)

(PEX ). (11)

The variations of ratio of FE bottom pressure to the analyticalpressure with the number of mesh divisions are shown in Fig. 5(a)and (b) for both two-dimensional rigid shallow and tall tankmodels, respectively. It is found that the discretization errors areequal to 1.66% and 0.03% for proposed shallow and tall tankmodelsrespectively when subjected to an earthquake ground motion. Onthis basis, the selectedmesh pattern for this analysis is appropriateto investigate the fluid behavior. Since both structure and soilmedium are linear, a very refined mesh is not necessary for theanalysis.

5. Spectrum analysis

Prior to conducting the time history analyses, the fundamentalperiods of impulsive and convective behaviors as well as theirrelated mass ratios are calculated using both finite element andanalytical method for shallow and tall tank models. The analyticalfundamental frequencies and related mass ratios are obtained inaccordance with ACI 350.3-06 [34].

A comparison between these results is shown in Table 1which indicates that the FE results are in agreement withanalytical values. Further details on the calculation of sloshingfrequencies and masses are given by Patkas and Karamanos [38]and Karamanos et al. [39]. Using the design spectral accelerations,the values of base shear and base moment can be easily calculatedfor unit length of wall. A comparison between spectral results withthose calculated in accordance with ACI 350.3-06 [34] and the FEtime history will be discussed later.

6. Time history analysis

Two rectangular concrete liquid containermodels given in Fig. 2are used basically for the example analyses in time domain. Itshould be noted that both longitudinal and transversal compo-nents of earthquake perpendicular to longer and shorter tankwalls (X and Y directions) respectively are applied simultaneouslyin three-dimensional modeling and are referred to as horizontalexcitation.

Since the sloshing height variation is a matter of concern indesign of rectangular tanks, its value is measured at three differentlocations (points A, B and C shown in Fig. 3) of three-dimensionalmodel. Points A, B and C are located at the fluid surface at the


Fig. 6. Time history of base shear response of a shallow tank model under longitudinal excitation: (a) Northridge (b) El-Centro (c) San-Fernando (d) San-Francisco.

Fig. 7. Time history of sloshing height due to all components of earthquake for a shallow tank model: (a) Northridge (b) El-Centro (c) San-Fernando (d) San-Francisco.

middle of longer length, middle of shorter length and the cornerof tank wall, respectively.

Finally, seismic analyses are performed using both horizontaland vertical components of ground acceleration and the results arecompared with those calculated by spectral analyses.

The results of this study are presented in two parts. In the firstpart, a detailed discussion on the effect of earthquake frequencycontent on the seismic behavior of liquid tanks supported on a rigidfoundation is described.

In the second part the combined effects of SSI and earthquakefrequency content on the dynamic behavior of liquid tanks isdiscussed.

6.1. Effect of earthquake frequency content on the dynamic behaviorof liquid tanks

Seismic behavior of shallow tank model with rigid baseThe transient base shear and base moment for flexible shallow

tankmodel due to horizontal and vertical excitations are calculated


Fig. 8. Time history of base moment response of a tall tank model under longitudinal excitation: (a) Northridge (b) El-Centro (c) San-Fernando (d) San-Francisco.

Table 1Modal analysis results.

Tank type FEM AnalyticalFundamental period (s) Effective mass ratio Fundamental period (s) Effective mass ratioTi Tc mi/m mc/m Ti Tc mi/m mc/m

Shallow tank 0.15 8.58 0.18 0.72 0.10 8.56 0.21 0.75Tall tank 0.33 5.30 0.60 0.33 0.22 5.50 0.59 0.44

by the proposed method. Four different earthquake records areapplied to investigate the effect of frequency content on theresponse of the tank–liquid system. The base shear response of thetank due to longitudinal ground motion measured at the middlecross-section of the longer wall is presented in Fig. 6.

The absolute maximum values of resulting base shear andbase moment due to all ground excitations are presented inTable 2. In addition, the results are presented in brackets interms of normalized values with respect to those of the El-Centro record. These results show that the responses due tohigh frequency content earthquake of San-Francisco are highlymagnified because of the similarity between impulsive dynamiccharacteristics of the tank–liquid system and earthquake record.Also, the least response values are obtained under low frequencyNorthridge earthquake which are almost one-third of the valuesdue to San-Francisco earthquake. Unlike the impulsive behavior,the convective response values are amplified due to intermediatefrequency content earthquake of El-Centro. In this case, theabsolute peak values of convective base shear andbasemoment arepresented for all records in Table 2. It should be noticed that whenthe impulsive response reaches its peak, the convective response isat the beginning stage and has not yet fully developed. As a result,the convective component does not have amajor effect on the totalstructural response in time-domain analysis.

It is found that the effect of earthquake frequency content issignificant on the structural response of liquid–tank models andmay cause a considerable increase in time-domain peak responsevalues.

Considering the combined effect of vertical and horizontalground motions, the impulsive response almost remains un-changed for all earthquakes. The convective part increases due to

both Northridge and El-Centro vertical components but remainsunchanged under applying vertical components of San-Fernandoand San-Francisco ground motions.

In addition to the structural response, the fluid dynamicbehavior is thoroughly investigated. The time history diagrams ofsurface sloshing height are shown in Fig. 7.

As previously mentioned, these values are measured at pointsA, B and C. The maximum sloshing height which occurs at pointC have values of 325 mm, 838 mm, 311 mm and 138 mmunder Northridge, El-Centro, San-Fernando and San-Franciscoearthquakes, respectively.

As presented in Table 2, the vertical acceleration has the mostsignificant effect on sloshing behavior of fluid domain under the El-Centro earthquake and causes an increase of about 14% in sloshingheight whereas, the sloshing height remains unchanged afterapplying vertical motions of the Northridge and San-Fernandoearthquakes.

Unlike the convective behavior, the impulsive behavior is lesssensitive to vertical motions as seen for all earthquake records.Since the impulsive response values are so much higher thanthose of convective, it can be concluded that the effect of verticalexcitation is insignificant on overall seismic behavior of shallowtank model.

7. Seismic behavior of tall tank model with rigid base

Fig. 8 presents the diagrams of impulsive structural responsein terms of base moment time history for tall tank model.The absolute maximum values of the resulting base shear andbase moment of the impulsive and convective components arepresented in Table 3. As previously seen in shallow tankmodel, the


Table 2Summary of maximum dynamic responses of a shallow tank model.

Impulsive response Convective responseH H + V (H + V )/H H H + V (H+V )/H

NorthridgeBase shear (kN/m) 102 (0.53) 104 (0.54) 1.02 16 (0.44) 17 (0.42) 1.10Base moment (kN m/m) 273 (0.57) 284 (0.59) 1.04 36 (0.61) 37 (0.58) 1.03Sloshing (mm) – – – 326 (0.44) 325 (0.39) 1.00

El-CentroBase shear (kN/m) 191 (1.00)a 193 (1.00) 1.01 36 (1.00) 40 (1.00) 1.11Base moment (kN m/m) 479 (1.00) 478 (1.00) 1.00 59 (1.00) 64 (1.00) 1.08Sloshing (mm) – – – 734 (1.00) 838 (1.00) 1.14

San-FernandoBase shear (kN/m) 236 (1.24) 236 (1.22) 1.00 12 (0.33) 12 (0.30) 1.00Base moment (kN m/m) 657 (1.37) 645 (1.35) 0.98 30 (0.51) 30 (0.47) 1.00Sloshing (mm) – – – 311 (0.42) 311 (0.37) 1.00

San-FranciscoBase shear (kN/m) 323 (1.69) 337 (1.75) 1.04 5 (0.14) 5 (0.13) 1.00Base moment (kN m/m) 821 (1.71) 860 (1.80) 1.05 13 (0.22) 13 (0.20) 1.00Sloshing (mm) – – – 128 (0.17) 138 (0.16) 1.08

a The values in the bracket are normalized responses with respect to El-Centro results.

Fig. 9. Time history of sloshing height due to all components of earthquake for a tall tank model: (a) Northridge (b) El-Centro (c) San-Fernando (d) San-Francisco.

Table 3Summary of maximum dynamic responses of a tall tank model.

Impulsive response Convective responseH H + V (H + V )/H H H + V (H+V )/H

NorthridgeBase shear (kN/m) 700 (0.83) 655 (0.78) 0.94 279 (5.58) 266 (4.43) 0.95Base moment (kN m/m) 3295 (0.77) 2994 (0.70) 0.91 1131 (5.36) 1087 (4.71) 0.96Sloshing (mm) – – – 1968 (2.14) 1957 (1.73) 0.99

El-CentroBase shear (kN/m) 846 (1.00)a 839 (1.00) 0.99 50 (1.00) 60 (1.00) 1.2Base moment (kN m/m) 4261 (1.00) 4251 (1.00) 0.99 211 (1.00) 231 (1.00) 1.09Sloshing (mm) – – – 920 (1.00) 1134 (1.00) 1.23

San-FernandoBase shear (kN/m) 814 (0.96) 908 (1.08) 1.11 48 (0.96) 49 (0.82) 1.02Base moment (kN m/m) 4059 (0.95) 4119 (0.97) 0.98 245 (1.16) 244 (1.06) 1.00Sloshing (mm) – – – 973 (1.06) 965 (0.85) 0.99

San-FranciscoBase shear (kN/m) 429 (0.51) 483 (0.58) 1.13 10 (0.2) 11 (0.18) 1.10Base moment (kN m/m) 2369 (0.56) 2606 (0.61) 1.10 56 (0.26) 60 (0.26) 1.08Sloshing (mm) – – – 211 (0.23) 254 (0.22) 1.20

a The values in the bracket are normalized responses with respect to El-Centro results.

effect of frequency content is significant on the structural responseof the tank. In this case, the structural response is amplified under

the El-Centro earthquake which has the nearest frequency to theimpulsive fundamental frequency of the tank–liquid system.


Fig. 10. Impulsive pressure distribution along height of a three-dimensional talltank model measured at the middle section of longer wall under longitudinalexcitations for different earthquake records.

Considering the free surface motion, the absolute maximumvalues of convective base shear and base moment are calculatedand shown in Table 3. It is obvious that applyingNorthridge groundmotion results in the highest convective responses. Under applyingthree components of groundmotion, the structural response of thetank is affected by the frequency of the earthquake. In comparisonwith the El-Centro record, theNorthridge vertical component leadsto a decrease in peak values of base shear and moment while theSan-Francisco earthquake results in an increase in these values.

In comparison with shallow tank, the tall tank is more sensitiveto vertical component of ground motion and its response variesfrom one earthquake to another.

The time history diagrams of sloshing height due to allcomponents of earthquake are shown in Fig. 9 for all records.Similar to shallow tank, the maximum sloshing height occurs atpoint C which is almost 20% higher than its value measured atpoint A. Applying vertical acceleration may lead to an increaseor decrease in sloshing height or in some cases may not affectthe sloshing height as seen for Northridge and San-Fernando

earthquakes. This variation is affected by the characteristics of theground motion and tank configuration.

The impulsive pressure distribution along tank wall measuredat the middle section of longer wall is presented in Fig. 10 forall records under longitudinal ground motions. Higher valuesare obtained due to the El-Centro earthquake as compared toother earthquake records. This is consistent with other structuralresponse values as presented in Table 3.

7.1. Effect of soil–structure interaction on the dynamic behavior ofliquid tank

Response of shallow tank with flexible foundationAs mentioned before, in order to consider the effects of

deformability of tank foundation on impulsive and convectiveresponse of tank structure, additional FE models with flexiblefoundation boundary condition are investigated in this study.To evaluate the dynamic response of liquid tank supported onflexible foundation, six soil types recommended in the literatureand design codes are considered. The soil properties are shown inTable 4. In this table, S1, S2, S3, S4, S5 and S6 soil types representhard rock, rock, very dense soil and soft rock, stiff soil, soft clay soiland soils vulnerable to potential failure or collapse under seismicloading, such as liquefiable soils and quick and highly sensitiveclays, respectively.

A comparison among peak values of impulsive base shear ofshallow tank supported on different soil types is presented inFig. 11 under different ground motions.

It is clear that the effect of soil–structure interaction (SSI) onthe structural response of liquid tanks is highly dependent onthe earthquake frequency content. SSI has led to an increase ofabout 47%, 64% and 52% in peak base shear response in comparisonwith rigid foundation condition under Northridge, El-Centro andSan-Fernando records, respectively. However, a decrease of about

Fig. 11. Comparisons of peak base shear responses of shallow tank models for different soil types: (a) Northridge (b) El-Centro (c) San-Fernando (d) San-Francisco.


Fig. 12. Comparisons of peak sloshing heights of shallow tank models for different soil types: (a) Northridge (b) El-Centro (c) San-Fernando (d) San-Francisco.

Table 4Properties of the soil types considered in this study.

Soiltypes

E(kN/m2) G(kN/m2) γ (kg/m3) ν vs(m/s) vp (m/s)

S1 7000,000 2692,310 2000 0.30 1149.10 2149.89S2 2000,000 769,230 2000 0.30 614.25 1149.16S3 500,000 192,310 1900 0.35 309.22 643.68S4 150,000 57,690 1900 0.35 169.36 352.56S5 75,000 26,790 1800 0.40 120.82 295.95S6 35,000 12,500 1800 0.40 82.54 202.18

13% is seen for base shear peak value due to the San-Franciscoearthquake.

The response amplification has occurred in different soil typesas the input earthquake changes. For moderate frequency contentrecords, the highest increase is seen in S4 and S5 soil typeswhile under low frequency content record, the highest values areobtained for S6 soil type which represents a very soft soil.

Under high frequency content record of San-Francisco, the peakstructural responses decrease as the soil stiffness decreases. On theother hand, the lowest peak values are obtained for S6 soil type.

For all cases, it is found that the soil type has an insignificanteffect on the convective behavior of liquid tank. A comparisonamongmaximum sloshing height measured at the right top cornerof the liquid domain for different soil types and earthquake recordsis shown in Fig. 12 under horizontal excitation. It is concludedthat the foundation deformability does not significantly affect thesloshing height of the liquid tank.

8. Response of tall tank with flexible foundation

Fig. 13 shows the effect of different soil types under differentground motions on base moment of tall tank models. Exceptfor Northridge earthquake, a significant reduction in structuralresponse is seen for tanks supported on S6 soil type. Thesereduction values are about 11%, 26%, 28% and 47% for Northridge,El-Centro, San-Fernando and San-Francisco records compared torigid base, respectively. However, an amplification of structuralresponse occurs in S4 soil type under Northridge earthquakes incomparison with rigid foundation condition.

In this case, the structural responses under moderate and highfrequency content earthquakes of El-Centro, San-Fernando andSan-Francisco decrease as the foundation soil becomes softer.

It is clear that the response amplification or reduction patterndue to deformable foundation is highly dependent on the nature ofearthquake.

To clarify the changes of structural response due to soil stiffnessvariation, the deviations of the base shear forces in time due to theEl-Centro earthquake are illustrated and compared among S1, S5and S6 soil types for tall tank models in Fig. 14. These time historyresponses have different frequency characteristics and describedifferent behavior of the tanks.

In addition, the power spectral density (PSD) functions for fourgroundmotions are shown in Fig. 15. The center of themass of PSDfunctions are located at 2.04 Hz, 3.20 Hz, 3.99 Hz and 6.21 Hz forNorthridge, El-Centro, San-Fernando and San-Francisco records,respectively. These values correspond with natural periods of0.49 s, 0.31 s, 0.25 s and 0.16 s, respectively. According to the PSDconcept, the significant amount of energy of the ground motion isconcentrated at the center of the mass of the PSD function. As aresult, themaximum response of the structure is expected to occurunder the ground motion record which has the nearest center ofmass location to the fundamental natural period or frequency ofthe structure.

The modal analysis results of tank–liquid–foundation systemare shown in Table 5 for both shallow and tall tanks. It is clear thatthe SSI significantly affects the impulsive fundamental periods. Onthe other hand, as the soil stiffness decreases, the impulsive periodincreases. This phenomenon has also been reported by Larkin [40].However, convective fundamental periods are almost independentof the flexibility of the foundation. This justifies the insensitivityof the sloshing height to the variation of foundation properties.This might be due to the long period nature of sloshing waves aspresented in Table 5.

It is clear that the variation in structural response is notonly dependent on the soil properties but also dependent on theearthquake characteristics.

9. Comparison with other methods

The seismic impulsive and convective responses of liquid–tankmodels are obtained in this study as discussed previously


Fig. 13. Comparisons of peak base moment responses of tall tank models for different soil types: (a) Northridge (b) El-Centro (c) San-Fernando (d) San-Francisco.

Fig. 14. Time history of impulsive base shear for a tall tank model with flexible foundation under horizontal excitation of the El-Centro earthquake for different soil types:(a) S1 (b) S5 (c) S6.

Table 5Modal analysis results of soil–structure–liquid system.

Impulsivefundamental period(s)

Convectivefundamental period(s)

S1 S2 S3 S4 S5 S6 S1 S2 S3 S4 S5 S6

Shallowtank

0.15 0.14 0.16 0.22 0.29 0.41 8.58 8.58 8.58 8.57 8.55 8.49

Tall tank 0.33 0.34 0.38 0.52 0.63 0.87 5.13 5.13 5.13 5.13 5.12 5.09

considering four earthquake records with different frequencycontents and accounting for foundation deformability, wallflexibility and fluid damping properties.

In order to verify the proposed FEmethod as well as to considerthe effect of fluid damping properties, two different conditionswith zero and non-zero fluid damping ratios were used forrigid tank models and the results were compared with analyticalsolutions derived by Haroun [7]. These results were presented interms of impulsive hydrodynamic pressure over the tank heightin previous research done by the authors [41]. The FE pressuredistribution is found to be in agreement with analytical resultswhen the fluid damping is ignored.

Using the design earthquake response spectrum which iscommonly used in practice, the impulsive and convective baseshear and base moment of both shallow and tall tank models

are calculated. The mapped design spectral response accelerationat short periods are 1.50 g, 2.20 g, 2.63 g and 2.75 g for El-Centro, San-Francisco, San-Fernando and Northridge earthquakes,respectively. The mapped MCE spectral response accelerations ata period of 1 s are 0.60 g, 0.90 g, 0.92 g and 1.0 g for the El-Centro, San-Francisco, San-Fernando and Northridge earthquakes,respectively. A comparison between the FE and spectral resultsbased on the approximation used by ACI 350.3-06 [34] are shownin Figs. 16 and 17 for shallow and tall tank models under differentground motions. It should be noted that in the ACI method, thehydrodynamic pressure is the product of the first fundamentalimpulsive or the convective mass and the period-dependentspectral amplification. On the other hand, the equivalent forcesexerted by the fluid can be obtained using the following equation:

Pi = CiWi (12)Pc = CcWc . (13)

In which Pi and Pc are impulsive and convective forces, Ci and Ccare the seismic impulsive and convective coefficients and Wi andWc are impulsive and convective weights.

Although higher values are obtained using ACI method, thesame trend due to earthquake frequency content is seen. For bothshallow and tall tank models the ACI method results in higherimpulsive response. The maximum difference in results betweenthe FE and design code method is about 22%. Overall, it is clear


Fig. 15. PSD function of: (a) Northridge (b) El-Centro (c) San-Fernando (d) San-Francisco.

Fig. 16. Impulsive and convective structural responses of a shallow tank model: (a) peak impulsive base shear (b) peak impulsive base moment (c) peak convective baseshear (d) peak convective base moment.

that the effect of earthquake frequency content can be predictedaccurately using both methods. However, the values calculatedbased on the ACI code are more conservative.

Considering the SSI effect, different conclusions are reportedin the literature. Veletsos and Tang [42] and Veletsos et al. [12]

showed that soil–structure interaction may reduce significantlythe critical responses of shallow tanks, but may increase those oftall, stiff tanks that have high fundamental natural frequencies.Also, it was shown that the soil–structure interaction has anegligible effect on the convective components.


Fig. 17. Impulsive and convective structural responses of a tall tank model (a) peak impulsive base shear (b) peak impulsive base moment (c) peak convective base shear(d) peak convective base moment.

Moreover, Haroun and Abou-Izzeddine [11] showed thatinteraction of the tank and foundation soil magnifies the tankresponse, and that the magnification is a factor of both shear wavevelocity of the soil as well as the geometric properties of the tank.Livaoglu [21] concluded that the displacements and base shearforces generally decrease, with decreasing soil stiffness. However,the sloshing responsewas not practically affected by soil–structureinteraction in frequency domain which is similar to current FEresults.

Also, Cho et al. [43] obtained the same results as Livaoglu [21]showed that structural response values are reduced as soil stiffnessdecreases. The liquid sloshing height andmotionwere not affected.

All of these studies show that the convective term of dynamicresponse of liquid tank is not sensitive to soil properties. The sameresults are obtained using the FEM in this study. However, forboth tank configurations, the soil–structure interaction effect ishighly dependent on the earthquake frequency content and mayamplify or reduce the structural response. On the other hand, aunique pattern which is able to reflect the effect of soil–structureinteraction on tank behavior cannot be concluded from the resultsof this study.

10. Conclusions

In this study, a finite element method is introduced thatcan be used for the analysis of dynamic behavior of partiallyfilled rectangular fluid container under horizontal and verticalground excitations in three-dimensional space. The liquid sloshingis modeled using an appropriate boundary condition and thedamping effects due to impulsive and convective components ofthe stored liquid are modeled using the Rayleigh method. Thesoil foundation is modeled as an elastic homogeneous mediumwith viscous boundary condition applied on the truncated zone tosimulate thewave energy absorption. Two different configurationsincluding shallow and tall tank models are considered toinvestigate the effect of geometry on the response of theliquid–structure system in time domain. Effect of wall flexibilityon the dynamic response of system is taken into account. Fourdifferent ground motions with the same peak ground acceleration

are applied to investigate the effect of earthquake frequencycontent on the seismic behavior of liquid–tank system.

The effect of foundation deformability on the overall dynamicresponse of the system is investigated by comparing the resultsamong six different soil types.

The results are presented in terms of the maximum structuralbase shear and basemoment obtained from timehistory analysis ofthe considered system as well as pressure distribution and surfacesloshing heights for different seismic excitations. It is obvious thatthe records with frequency characteristics close to those of liquidtanks highly magnify the responses of the system. Assuming arigid foundation, the high frequency earthquakes result in thehighest impulsive response in shallow tank model, whereas theintermediate frequency earthquakes highly amplify the tall tankresponse. Due to significant difference between impulsive andconvective fundamental frequencies, a different trend is observedfor the convective response under the same earthquakes. It shouldbe noticed that due to the high magnitude of impulsive responseand a significant time lag between peak impulsive and convectiveresponse, the overall seismic behavior of the tank is governed bythe impulsive component.

In this study, the FE results are compared with those obtainedby spectral analysis based on design earthquake response spectra.Although the spectral values are higher than the FE results, thesame trend due to earthquake frequency content is seen for bothmethods.

Considering the effect of SSI, the results show that the maxi-mum impulsive base shear and base moment obtained from thetime history analysis of the considered systemmay increase or de-crease as the soil stiffness changes which is a result of dynamicpressure variation in the middle of the wall due to the rocking mo-tion of the foundation. This phenomenon is highly dependent onearthquake frequency content and tank configuration.

A unique trend is seen under low frequency content earthquakefor both shallow and tall tank configurations. In this case, thestructural responses increase as the soil stiffness increases.

In addition, the convective response is almost independent ofvariations of flexibility of the foundation and seems to be relatedto geometric configurations of tank, earthquake characteristics andliquid properties.


It is clear that the dynamic behavior of liquid concrete tanksdepends on a wide range of parameters such as seismic propertiesof earthquake, tank dimensions and fluid–structure interactionwhich should be considered in current codes of practice. This studyshows that the proposed FEmethod can be used in the time historyanalysis of rectangular liquid tanks.

The present study is done based on the three-dimensionalanalysis of rectangular tanks in time domain using four differentearthquake records and six different soil types. Although theliquid–tank design procedures are based on simplified frequency-basedmethods, the current FE time history results which are morerealistic can be used to verify these methods in future works.

References

[1] Housner GW. The dynamic behavior of water tanks. Bull Seismol Soc Amer1963;53(2):381–7.

[2] Yang JY. Dynamic behavior of fluid–tank systems. Ph.D. thesis, Department ofCivil Engineering, Houston (TX): Rice University; 1976.

[3] Veletsos AS, Yang JY. Earthquake response of liquid storage tanks-advancesin civil engineering through mechanics. In: ASCE proceedings of the secondengineering mechanics specially conference, Raleigh (NC). 1977. p. 1–24.

[4] Minowa C. Dynamic analysis for rectangular water tanks. Recent Adv LifelineEarthq Eng Japan 1980;135–42.

[5] Minowa C. Experimental studies of seismic properties of various type watertanks. In: Proceedings of eighth WCEE. San Francisco. 1984. p. 945–52.

[6] Chopra AK, Gupta S. Hydraudynamic and foundation interaction effects inearthquake response of a concrete gravity dam. J Struct Div 1981;107(8):1399–412.

[7] Haroun MA. Stress analysis of rectangular walls under seismically inducedhydrodynamic loads. Bull Seismol Soc Amer 1984;74(3):1031–41;Haroun MA, Ellaithy MH. Seismically induced fluid forced on elevated tanks. JTech Topics Civ Eng 1984;111:1–15.

[8] Haroun MA. Vibration studies and tests of liquid storage tanks. Earthq EngStruct Dyn 1983;11:190–206.

[9] HarounMA, TayelMA. Response of tanks to vertical seismic excitations. EarthqEng Struct Dyn 1985;13:583–95.

[10] Veletsos AS, Tang Y. Dynamics of vertically excited liquid storage tanks. ASCEJ Struct Eng 1986;112(6):1228–46.

[11] Haroun MA, Abou-Izzeddine W. Parametric study of seismic soil–tankinteraction. II: vertical excitation. ASCE J Struct Eng 1992;118(3):783–97.

[12] Veletsos AS, Tang Y, Tang HT. Dynamic response of flexibly supported liquidstorage tanks. ASCE J Struct Eng 1992;118:264–83.

[13] Kim JK, Koh HM, Kwahk IJ. Dynamic response of rectangular flexible fluidcontainers. ASCE J Eng Mech 1996;122(9):807–17.

[14] Park JH, Koh HM, Kim J. Liquid–structure interaction analysis by coupledboundary element — finite element method in time domain. In: Brebbia CA,Ingber. MS, editors. Proceedings of the 7th international conference on bound-ary element technology. Southampton, (UK): Computational Mechanics Pub-lication BE-TECH/92, Computational Mechanics Publications; 1992. p. 89–92.

[15] Subhash S, Bhattachryya SK. Finite element analysis of fluid–structureinteraction effect on liquid retaining structures due to sloshing. Comput Struct1996;59(6):1165–71.

[16] KohHM, Kim JK, Park JH. Fluid–structure interaction analysis of 3D rectangulartanks by a variationally coupled BEM–FEM and comparison with test results.Earthq Eng Struct Dyn 1998;27:109–24.

[17] Doganun A, Durmus A, Ayvaz Y. Earthquake analysis of flexible rectangulartanks using the lagrangian fluid finite element. Euro J Mech-A/Solids 1997;16:165–82.

[18] Chen JZ, Kianoush MR. Seismic response of concrete rectangular tanks forliquid containing structures. Canad J Civ Eng 2005;32:739–52.

[19] Kianoush MR, Chen JZ. Effect of vertical acceleration on response of concreterectangular liquid storage tanks. Eng Struct 2006;28(5):704–15.

[20] Kianoush MR, Mirzabozorg H, Ghaemian M. Dynamic analysis of rectangularliquid containers in three-dimensional space. Canad J Civ Eng 2006;33:501–7.

[21] Livaoglu R. Investigation of seismic behavior of fluid-rectangular tank–soil/foundation systems in frequency domain. Soil Dyn Earthq Eng 2008;28(2):132–46.

[22] Jennings PC, Kuroiwa JH. Vibration and soil–structure interaction tests ofa nine-story reinforced concrete building. Bull Seismol Soc Amer 1968;58:891–916.

[23] Foutch DH, Housner GW, Jennings PC. Dynamic responses of six multistorybuildings during the San Fernando earthquake. Report No. EERL 75-02,California Institute of Technology; 1975.

[24] Mcverry GH. Frequency domain identifications of structural models fromearthquake records. Report No. EERL 79-02, California Institute of Technology;1979.

[25] Chopra AK. Dynamics of structures. Upper Saddle River (NJ): Prentice-Hall;1995.

[26] Wolf JP, Song CH. Doubly asymptotic multi-directional transmitting boundaryfor dynamic unbounded medium-structure-interaction analysis. Earthq EngStruct Dyn 1995;24(2):175–88.

[27] Wolf JP, Song CH. Finite element modeling of unbounded media. In: The 11thworld conference on earthquake engineering, 1996, 70, p. 1–9.

[28] Clough R, Penzien J. Dynamics of structures. 2nd ed. New York: McGraw-Hill,Inc; 1993.

[29] Wilson EL. Three-dimensional static and dynamic analysis of structures—aphysical approachwith emphasis on earthquake engineering. 3rd ed. Berkeley(CA, USA): Computers and Structures, Inc.; 2002.

[30] Lysmer J, Kuhlmeyer RL. Finite dynamic model for infinite media. ASCE EngMech Div J 1969;95:859–77.

[31] Song CH, Wolf JP. Dynamic stiffness of unbounded medium based on solventdamping extraction method. Earthq Eng Struct Dyn 1994;23:1073–86.

[32] AnradeWP. Implementation of second order absorbing boundary condition infrequency domain computations, Ph.D. thesis. Texas: The University of Texasof Austin; 1999.

[33] Virella JC, Prato CA, Godoy LA. Linear and nonlinear 2D finite element analysisof sloshing modes and pressure in rectangular tanks subjected to horizontalharmonic motions. J Sound Vibration 2008;312(3):442–60.

[34] ACI 350.3-06. Seismic design of liquid-containing concrete structures (ACI350.3-06) and commentary (350.3R-06). American Concrete Institute (ACI)Committee 350, Environmental Engineering Concrete Structures, FarmingtonHills, (MI). 2006.

[35] Mikishev GN, Dorozhkin NY. An experimental investigation of free oscillationsof a liquid in containers. Izvestiya Akademii Nauk SSSR, Otdelenie Tekhnich-eskikh Nauk, Mekhanika I Mashinostroenie 1961;4:48–83.

[36] Veletsos AS, Shivakumar P. Tanks containing liquid or solids. A handbook incomputer analysis and design of earthquake resistant structures. Computa-tional Mechanical Publications 1997;725–74.

[37] Livaoglu R, Dogangun A. Effect of foundation embedment on seismic behaviorof elevated tanks considering fluid–structure-soil interaction. Soil Dyn EarthqEng 2007;27(9):855–63.

[38] Patkas LA, Karamanos SA. Variational solutions for externally induced sloshingin horizontal-cylindrical and spherical vessels. ASCE J Eng Mech 2007;133(6):641–55.

[39] Karamanos SA, Patkas LA, Platyrrachos MA. Sloshing effects on the seismicdesign of horizontal-cylindrical and spherical industrial vessels. ASME J PressVessel Tech 2006;128(3):328–40.

[40] Larkin T. Seismic response of liquid storage tanks incorporating soilstructure interaction. ASCE J Geotech Geoenvironmental Eng 2008;134(12):1804–14.

[41] Ghaemmaghami AR, Kianoush MR. Effect of wall flexibility on dynamicresponse of concrete rectangular tanks under horizontal and vertical groundmotions. ASCE J Struct Eng 2010;136(4):441–51.

[42] Veletsos AS, Tang Y. Soil–structure interaction effects for laterally excitedliquid storage tanks. Earthq Eng Struct Dyn 1990;19:473–96.

[43] Cho KH, Kim MK, Lim MY. Seismic responses of base-isolated liquid storagetanks considering fluid–structure-soil interaction in time domain. Soil DynEarthq Eng 2004;24:839–52.

Tank Seismic on Concreate

Documents

Transcript of Tank Seismic on Concreate