Engineering Structures 28 (2006) 1450–1461www.elsevier.com/locate/engstruct

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Fundamental modes of tank-liquid systems under horizontal motions

Juan C. Virella, Luis A. Godoy∗, Luis E. Suarez

Department of Civil Engineering and Surveying, University of Puerto Rico Mayag¨uez, PR 00681-9041, Puerto Rico

Received 24 February 2005; received in revised form 12 September 2005; accepted 15 December 2005Available online 29 March 2006

Abstract

This paper reports results on the fundamental impulsive modes of vibration of cylindrical tank-liquid systems anchored to the foundathorizontal motion. The analyses are performed using a general purpose finite element (FE) program, and the influence of the hydrostatic presand the self-weight on the natural periods and modes is considered. The roof and walls are represented with shell elements and tmodeled using two techniques: the added mass formulation and acoustic finite elements. Tank height to diameter ratios from 0.40 toused, with a liquid level at 90% of the height of the cylinder. The effect of the geometry on the fundamental modes for the tank-liquid sstudied using eigenvalue and harmonic response analyses. Similar fundamental periods and mode shapes were found from these twoThe fundamental modes of tank models with aspect ratios(H/D) larger than 0.63 were very similar to the first mode of a cantilever beamthe shortest tank(H/D = 0.40), the fundamental mode was a bending mode with a circumferential waven = 1 and anaxial half-wave(m)

characterized by a bulge formed near the mid-height of the cylinder.c© 2006 Elsevier Ltd. All rights reserved.

Keywords: Added mass; Dynamics; Finite elements; Horizontal motion; Hydrostatic pressure; Tanks; Shells

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1. Introduction

Many above-ground steel tanks have suffered significdamage during past earthquakes, and this has motivatedinterest in understanding and predicting the seismic behaviotanks. In the early 1960s, Housner [1] considered cylindricarigid tanks anchored to the foundations and subjectedhorizontal translation, and decomposed the hydrodynaresponse as the contribution of an impulsive and a sloscomponent. The impulsive component was attributed topart of the liquid that moves with the tank, while thesloshing component, which was characterized by long-peoscillations, was formed by the liquid near the free surfaTo model these effects, Housner [1] developed equations tocompute the impulsive and sloshing liquid masses, alongtheir location above the tank base. The fundamental impulmodeconsisted of a cantilever beam type mode.

Veletsos and Yang [2] and Haroun and Housner [3] foundthat the pressure distributions due to the liquid for rigid andflexible anchored tanks were similar, but the magnitudehighly dependent on the flexibility of the wall. Housner [1],

∗ Corresponding author. Tel.: +1787 265 3815; fax: +1787 833 8260.E-mail address:lgodoy@uprm.edu(L.A. Godoy).

0141-0296/$ - see front matterc© 2006 Elsevier Ltd. All rights reserved.doi:10.1016/j.engstruct.2005.12.016

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Haroun and Housner [2], and Veletsos and co-workers [2,4,5], adopted the assumption that a cylindrical tank containliquid predominantly develops a cantilever beam type min response to a horizontal base motion. For the fundamecantilever beam mode, Veletsos [4] included the tank flexibilityby replacing the pseudo-acceleration function insteadthe ground acceleration in the relevant response equatMalhotra and Veletsos [5] stated that, because of the lardifferences in the natural periods of the impulsive and sloshresponses, these two actions can be considered uncoupledthough most of the response is affected by the motion ofliquid due to the impulsive component.

Other researchers were not convinced by this cantilemode and explored other more complex modes as posfundamental modes under horizontal excitation. Natchiet al. [6] proposed to use shell modal forms to model taliquid systems, stating that the cantilever beam type madopted by previous researchers [1–5] was obsolete. Theyconcluded that the fundamental modes of steel cylindrical tasubjected to earthquake excitations are not associated witfirst fundamental modes of a cantilever beam, but rather shbe modeled using a circumferential wavy pattern with cos(nθ),wheren is the circumferential wave number, with 5< n < 25.

J.C. Virella et al. / Engineering Structures 28 (2006) 1450–1461 1451

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Design codes for steel tanks, such as API 650 [7] andAWWA-D100 [8], based their seismic standards for anchoflexible tanks on the recommendations given by Veletsos4],together with a response spectrum analysis, in which cantilevebeam modes are assumed to dominate the response otank-liquid systems to horizontal excitation. To calculateactual seismic behavior of tank-liquid systems it is essentiaaccurately represent the predominant modes of vibrationa detailed study is performed in this paper to determine whmodes are dominant in predicting the response.

The natural frequencies and mode shapes of cylindtanks partially filled with liquid have been studied in ttechnical literature by means of experiments [9], analyticaland semi-analytical approaches [10,11], and finite element (FEtechniques [9,12]. However, those studies have not been ablestablish the fundamental modes of vibration for the predicof the response of the tank-liquid systems to a horizomotion.

Barton and Parker [13] used finite element models of tank-liquid systems to study the seismic response of anchoredunanchored tanks, in which the tank was modeled with selements and the liquid was represented with liquid finiteelements and added liquid mass. For cylindrical tanks wheight/diameter larger than 0.5 under horizontal excitatiBarton and Parker [13] stated that those modes involvingdeformations of the cylinder with the form cos(nθ) andn > 1havevery small participation factors, and are not importantpredicting the response. Thus, only the cantilever beam m(i.e. n = 1) would be fundamental in predicting the horizonseismic response for tanks with height/diameter> 0.5. Thisconclusion is in disagreement with the findings reportedNatchigall et al. [6]. Because of these conflicting views owhich modes are dominant in the seismic response of twalled tanks, there is a need to investigate the fundamemodes of tank-liquid systems subjected to a horizontal bacceleration for the range of shell dimensions of interesthe oil industry, and to determine the influence of geometrparameters of the tanks on their behavior.

2. Tank models

The tanks considered in this paper have clamped or pincondition at the base, with a cone roof partially supporteda set of radial beams and columns, as shown inFigs. 1 and2. The radial beams are connected at their ends directly totank cylinder. In the finite element model the interior supportcolumns are represented by linear springs that take into accthe axial stiffness of the columns. The springs are connectto the ring beams and to the bottom of the tank. Since oanchored tanks are considered, the bottom of the tank isincluded in the model. The rafters stiffen the roof to suchextentthat predominantly cylinder modes result for the empttank, regardless of the roof geometry (Virella [14], Virella et al.[15]). The geometries considered in this work have aspect ratH/D = 0.40 (Model A), H/D = 0.63 (Model B), andH/D = 0.95 (Model C), whereH and D are indicated inFig. 1. In all computations, the liquid height was assumed

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Fig. 1. Section of a typical anchored tank with roof rafters.

Fig. 2. Typical model of cone roof tank with rafters.

be HL = 0.9H . For the three models considered, the tapethicknesses of the shells weredesigned using the API650 [7]provisions. The geometry of each of the three types of taconsidered is shown inFig. 3.

The finite element package ABAQUS [16] was used toperform the computations using S3R triangular elementsS4R quadrilateral elements. The S4R is a four-node, doucurved shell element with reduced integration, hour-gcontrol, and finite membrane strain formulation. The Sis a three-node, degenerated version of the S4R, with fimembrane strain formulation. The element S3R has conbending and membrane strain approximations, therefore amesh refinement is required to model pure bending situatiBoth elements S3R and S4R are discussed in more detaHibbit et al. [17]. The number of elements used in the finelement meshes for Models A, B, and C are listed inTable 1.

3. Tank-liquid models

The primary interest of this study is to evaluate the natperiods, mode shapes and dynamic response to horizground motions of cylindrical tanks partially filled with a liqui

1452 J.C. Virella et al. / Engineering Structures 28 (2006) 1450–1461

Fig. 3. Tank models with cone roof supported by rafters:t = shell thickness;tr = roof thickness;tr = 0.00635 m (roof with rafters). (a) Tank withH/D = 0.95;(b) tank with H/D = 0.63; (c) tank withH/D = 0.40.

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Table 1Number of finite element elements in the models with added liquid mass

Model H/D Number ofshell elements

A 0.40 10,439B 0.63 12,767C 0.95 15,871

The recommendation by Housner [1] on the convenience ofseparating the impulsive and convective actions to characterthe hydrodynamic response of horizontally excited tank-liqsystems is adopted here. Only those modes correspondinthe impulsive mode in which there is a coupling action betwthe tank and liquid are considered in this paper. The liquidrepresented by means of an added mass approach, andacoustic finite elements. A densityρ = 983 kg/m3 and a bulkmodulusK = 2.07 GPa (i.e. the properties of water) are uin the computations.

3.1. Model with added liquid mass

The finite element meshes for the structures of ModelsB, and C with added liquid mass are indicated inTable 1andFig. 4. Convergence studies were carried out and the finelement meshes listed inTable 1were found to provide goodresults for the purpose of the present study. The bottom pof the tank was not included in the model, since this study o

edton

with

d

,

tely

Fig. 4. Finite element mesh for Model A, with added liquid mass.

covers anchored tanks, with emphasis on the behavior ocylindrical shell and not on the foundation.

The model with added mass liquid was previously presenby Virella et al. [18], and is summarized here. The added mis obtained from a pressure distribution for the impulsive mof the tank-liquid system due to Veletsos and Shivakumar [19],which has a cosine distribution along the tank cylinder (Fig. 5). This pressure distribution for the rigid body horizonmotion of a rigid tank-liquid system is described as

Pi (η, θ, t) = ci (η)ρRxg(t) cosθ (1)

where Pi is the impulsive pressure;η is a non-dimensionavertical coordinate= z/HL; z is the vertical coordinate

J.C. Virella et al. / Engineering Structures 28 (2006) 1450–1461 1453

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Fig. 5. Impulsive pressure distribution around the tank circumferencecircumferential distribution; (b) pressure components in the direction of thexcitation.

measured from the tank bottom;R is the tank radius;xg(t) isthe ground acceleration; andt is the time. The functionci (η)

defines the impulsive pressure distribution along the cylinheight, and is computed as

ci (η) = 1 −∞∑

n=1

ccn(η) (2)

whereccn(η) is the function that defines the convective pressuredistribution along the cylinder height, computed as

ccn(η) = 2

λ2n − 1

cosh[λn(H/R)η]cosh[λn(H/R)] . (3)

The parameterλn is the nth root of the first derivative ofthe Bessel function of the firstkind and first order. The firsthree roots areλ1 = 1.841,λ2 = 5.311, andλ3 = 8.536. Thefunctionci (η) converges rapidly as the number of terms insummation increase, and hence it is sufficient to include thcoefficientsccn in Eq.(2). Thepressure distributions definedEq. (1) for each of the tank-liquid systems considered in tpaper and forθ = 0 are presented inFig. 6.

The added mass of the liquid on the tank shell is calculafrom the pressure distribution in Eq.(1). To obtain the lumpedmassmi at each node, the height of the cylinder is dividinto several segments and the area of the pressure below thcurve for the segment is divided by the normal acceleratiothe rigid wall xg. The lumpedmasses obtained from the addmass have the same vertical variation as the impulsive pressu

)

r

e

d

f

Fig. 6. Impulsive pressures forthe tank-liquid systems withχg = 1 m/s2.

Fig. 7. Model with normal mass along the cylinder height.

distribution from which they are derived (seeFig. 7), and theyhave a uniform distribution around the circumference oftank. Because this added mass is the liquid mass that mtogether with the tank, it acts normal to the cylindrical shell

For a radialsection of the tank (Fig. 7), the lumped massmi

at each location is computed using the rectangular rule. Fointerior node at the tank shell, the lumped massmi is computedas

mi = Pi �h

an(4)

wherePi is the pressure at nodei , �h is the constant distancebetween nodes, andan is the reference normal acceleratioNote thatan becomes the amplitude ofxg for θ = 0.

For nodes at the liquid surface and at the bottom of the tathe lumped mass is

ml = Pl

2

�h

an(5)

whereml is the mass at a boundary nodel andPl is the pressureat nodel .

Because the added masses are determined fromimpulsive pressure which is normal to the shell surface, tmust be added in such a way that they only add inertiathis direction. For this reason, the added masses are sometreferred to as the “normal masses”.

1454 J.C. Virella et al. / Engineering Structures 28 (2006) 1450–1461

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For each of the tank-liquid systems considered in this stthe mass distributions calculated as described previouslyverified by comparing them with the impulsive mass ratioproposed by Housner [1]. The impulsive massratio Mir for theparticular tank-liquid system is defined as

Mir = Mi

Mt, (6)

whereMi is the total impulsive mass andMt is the total liquidmass.

For our case, the total impulsive massMi can be calculateby first obtaining a mass resultantmres for the tank meridianin the direction of the ground excitation, and then integrataround the circumference to compute the total impulsive mof the tank. The mass resultantmres is calculated as the suof the individual massesmi at the different nodes along thθ = 0 meridian. The added mass components in the direcof the excitation are directly proportional to the impulspressure which varies with a cosine distribution aroundtank circumference (seeFig. 5(a)). Hence, to integrate the radmass in order to obtain the impulsive mass, we must prothe massmres along the direction of the excitation. Moreovmres wascalculated for theθ = 0 meridian. To calculatemresfor θ > 0, it must be multiplied by cosθ , in the same wayas the pressure was defined (seeFig. 5(b)). At the arc thatforms an angleθ with respect to the horizontal, the horizontacomponent of the impulsive mass component in the directiothe excitation is

dmi = (mrescosθ) cosθ(Rdθ). (7)

The total impulsive massMi is then

Mi = 4Rmres

∫ π/2

0cos2(θ) dθ = π Rmres. (8)

The total liquid mass is calculated using the followiexpression

Mt = π R2HLγL/g, (9)

whereγL is the liquidunit weight andg is the acceleration ogravity.

The impulsive mass ratio defined in Eq.(6) is

Mir = mresg

RHLγL. (10)

The impulsive massratio Mir can also be calculated usinthe expression proposed by Housner [1]:

Mi

Mt= tanh(1.7R/HL)

1.7R/HL. (11)

Table 2 shows that the differences between the impulsimass ratios computedwith the present analysis and with trecommendation by Housner [1] are smaller than 5%.

Since the added mass comes from the liquid pressureacts in the direction normal to the shell, the mass shoulassigned in the direction normal to the cylinder. Howebecause in most FE programs the same magnitude of themass is assigned to all the degrees of freedom, a sui ge

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Table 2Comparisons of impulsive mass ratios calculated with the present studwith the Housner equation(11)

Model H/D Mir Housner DifferenceMi/Mt (%)

A 0.40 0.4057 0.406 0.07B 0.63 0.5710 0.587 2.73C 0.95 0.7040 0.736 4.35

Fig. 8. Model with normal mass around the circumference.

schemeneeds to be introduced in order to represent this admass. It must be pointed out, though, that some strucanalysis programs allow for the addition of lumped masalong three global Cartesian axes, but it is not common tothe option of adding masses in cylindrical or polar coordinat

The added liquid mass in lumped form is attached to the snodes by means of rigid, massless links with small lengthsFig. 8). The links are rigid truss elements with supports orienin the local axes of the truss elements. The supports must pthe motion of the nodal masses only in the direction normto the shell. Hence, the motion of the support is restrictedthe global tangential direction (perpendicular to the elemeaxis) and in the vertical direction, but it is free to movethe global radial direction (i.e. the local axial direction). Ttotal impulsive massMi in a specific direction calculatewith Eq. (8) is twice the impulsive mass computed using themethodology of Ref. [1]. However, as the masses can only moin the radial direction, it can be shown that half of this toimpulsive mass is excited in a specific direction.

3.2. Model with liquid finite elements

A typical finite element mesh forthe tank (Model A in thiscase) in which the liquid is modeled with finite elementsshown inFig. 9(a). Acoustic three-dimensional finite elemebased on linear wave theory were used in this part of tstudy to represent the liquid. Such a model representsdilatational motion of the liquid, allowing a wave to bdescribed as a single pressuredegree of freedom at each poin space. The formulation is based on the Laplace equatioin the pressure domain and, although viscosity effects canaccounted for, an inviscid liquid was considered for the prestudy. The formulation of the acoustic elements is describe

J.C. Virella et al. / Engineering Structures 28 (2006) 1450–1461 1455

Fig. 9. Typical finite element mesh for Model A with liquid acoustic elements: (a) tank structure mesh; (b) liquid mesh.

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Table 3Number of finite elements in the models where the liquid is represented wacoustic elements

Model H/D Number ofelementsShell Acoustic

A 0.40 10,536 39,312B 0.63 12,216 56,784C 0.95 12,216 56,784

Refs. [16,17]. The liquid mesh has 39,312 elements for ModA and 56,784 elements for Models B and C (seeTable 3).The elements for the liquid are identified in ABAQUS [16] asAC2D4, which are solid, eight-node brick acoustic elemewith bilinear interpolation and with only one pressure unknoper node. The finite element mesh for the liquid of Model Ashown inFig. 9(b).

For the finite element model of the tank, triangular shelements (S3R) are used in the roof and quadrilateraltriangular elements are used for the cylinder. The numbeelements for each model is listed inTable 3. It was mentionedbefore that it was not necessary to include the bottom ofthe tank in the tank models, because this study emphathe behavior of the cylindrical shell. However, the bottomthe tank was discretized with finite elements in the mowith liquid acoustic elements in order to specify the boundcondition (i.e. contact interaction) between the bottom surfof the liquid and the tank. Quadrilateral (S4R) and triangu(S3R) shell elements were used for the bottom of the tank,pinned supports were placed at each node of the tank botto represent an anchored tank resting on a rigid base.

The location of each node on the constrained surfaces oliquid corresponds exactly to the location of a node on the tcylinder or base. Surface tied normal contact was considbetween the surfaces of the liquid and the tank walls andbottom. This contact formulation is based on a master–sapproach, in which both surfaces remain in contact throughthe simulation, allowing the transmission of normal forcesbetween them. The formulation for normal contact is descriin Refs. [16,17]. No sloshing waves were considered in thstudy, and thus no pressure was applied to the nodes afree liquid surface. The boundary conditions specified at thinterfaces of the liquid model are illustrated inFig. 10.

th

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ndm,

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d

the

Fig. 10. Conditions assumed for the three-dimensional tank-liquid finiteelement model.

Table 4Results of the free vibration analyses using the normal mass models

Model H/D Tmax (s) Tmin (s) α3 Number of modes

A 0.40 0.644 0.189 0.74 891B 0.63 0.859 0.199 0.81 796C 0.95 1.057 0.205 0.79 794

4. Fundamental periods and mode shapes

4.1. Free vibration analyses

To compute the natural periods and mode shapes fornormal mass models of the tank-liquid systems, the hydrospressure and the self-weight of the tank were initially applii.e. they were modeled as a pre-stress state. The naperiods and mode shapes were computed until the ratioα3of the horizontal componentM3 (i.e. global direction 3)of the accumulated effective mass to the total mass ofsystem(α3 = M3/(Mi + Mshell)) was larger than 70% (seeTable 4). The analysis uses the Lanczos solver [16], in whichall the natural frequencies and mode shapes are found wa specified range of frequencies. The fundamental modes widentified as those with the largest participation factors intranslational directions.

Table 4displays the range of the natural periods and the tnumber of modes found from the free vibration analysesthe tank-liquid systems. For Model A, 891 modes with natu

1456 J.C. Virella et al. / Engineering Structures 28 (2006) 1450–1461

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e

f

totedheeresssis.urefte

one

Fig. 11. Variation of natural period with the normalized modal participatiofactor in the horizontal axis 3 for Model A.

periods between 0.64 s and 0.19 s were found with the condα3 ≥ 0.7. The natural periods ranged from 0.86 s and 0.2(796 modes) for Model B and between 1.0 s and 0.20 s (modes) for Model C. Typical cylinder modes (see Ref. [15])characterized by circumferential wave numbers(n) and axialhalf-wave numbers(m) resulted for all the modes of eachthe tank-liquid systems.

Fig. 11 illustrates, for Model A, the variation of the naturperiods with the participation factor P3 (in absolute value)along the axis 3 horizontal direction. The participation factonormalized with respectto its maximum value.Fig. 11 showsthat the participation factor(P3) is much larger for a certainmode, the fundamental mode, indicating that the responsof the system to a horizontal motion will be predominandetermined by this mode. Similar results were obtainedModels B and C, and thus the same approach described bwas used to determine the fundamental modes for these taliquid systems.

Fig. 12 illustrates the variation of the natural period withe circumferential wave numbern, for modes with axial half-wavenumberm = 1. The figure also shows the fundamenperiod calculated previously for each of the three models.modes with the largest natural periods inFig. 12 had smallparticipation factorsP3, and therefore their contribution to theresponse of the tank-liquid systems to horizontal motionsbe neglected. The modes for Models A, B, and C with the thlargest participation factorsP3 are listed inTable 5. This tableshows that, for Models B and C, there are modes (the secand third) that will contribute to the response to horizonground motions that are not cantilever beam modes (i.e.n > 1).However, the fundamental mode still dominates the responsα3 for the first mode of Model A is 94% of the totalα3 obtainedadding those of the three modes considered. For other moα3 for the first mode is even higher: 98% for Model B a97% for Model C. Therefore, for practical purposes, onlyfundamental mode needs to be considered in order to prthe response of cylindrical tank-liquid systems subjectedhorizontal base motions. These results compared wellthe recommendations by Barton and Parker [13], who statedthat, for cylindrical tank-liquid systems with aspect ratiosH/Dlarger than 0.5, the modes characterized by circumferewave numbersn > 1 and axial half-wave numbersm ≥ 1are not important for determining the response to horizobase motions. The present work showed that even for t

ns4

re-

le

ne

dl

as

ls,

ict

h

al

lks

Fig. 12. Variation in thenatural period T withn for m = 1 for Models A, Band C.

Fig. 13. Damping ratios for the two frequency ranges considered indynamic analyses: (a) frequency range 1; (b) frequency range 2.

with an aspect ratioH/D = 0.40, which falls outside the rangconsidered by Barton and Parker [13], modes withn > 1 andm ≥ 1 have very small participation factors in the direction othehorizontal motion.

4.2. Steady state harmonic response analyses

A linear, steady state harmonic analysis of the tank-liquidsystems under uniaxial horizontal motion was carried outfind the periods and deformed shapes that will be exciby this input. To account for the geometric non-linearity, thydrostatic pressure and the self-weight of the tank wincluded in an initial static step; in this way, the stiffneof the system was modified prior to the dynamic analyThe tank-liquid system was next loaded with the pressdistribution given by Eq.(1), and a ground acceleration o1g (g = 9.81 m/s2) was considered. The linear, steady staharmonic response to a horizontal harmonic motion in

J.C. Virella et al. / Engineering Structures 28 (2006) 1450–1461 1457

66

Table 5Modes of the tank-liquid systems relevantfor the response to horizontal ground motions

Mode Model A Model B Model CT (s) α3 n m T (s) α3 n m T (s) α3 n m

1 0.2116 0.64 1a 1 0.2395 0.79 1a 1 0.3001 0.77 1a 12 0.2021 0.02 1 5 0.2362 0.01 28 7 0.2960 0.01 203 0.1957 0.02 1 4 0.2046 0.01 1 7 0.2998 0.01 25

a Fundamental mode.

tion

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atetioaan

in

ingo

1

ethtios

ithachtwod.

wa

desestdsicwn inhe

entedhethental

e

entaloughp of

asthe

nts to

nd innse

rthee

ods

ntsehe

entaldinger

n iningded

Fig. 14. Variation of the normalized radial displacement with the excitaperiod for Model A with added mass: –•– Node A; –�– Node B (seeFig. 15).

direction was obtained by direct integration. The tank-liqsystems with the liquid accounted for by added masses wexcited along the axis 3horizontal direction, while the modewith acoustic liquid elements were excited along the axhorizontal direction. The dynamic response as a function oexcitation period was obtained. The periods of the tank-liquidsystems excited by the ground motion will be those associwith peaks in a plot of a maximum response versus excitaperiod. The radial displacement response (U-radial) yieldsdeformed shape associated with a mode of vibration of the tliquid system.

To introduce damping to the models, a Rayleigh dampmatrix [C] was used:

[C] = β[K ] + α[M] (12)

where β is the stif fness proportional coefficient andα isthe mass proportional coefficient. The Rayleigh dampcoefficients were computed by considering two rangesfrequencies. The frequency ranges are shown inFig. 13(a) and(b). The values ofβ are 0.003351 for the frequency rangeand 0.001273 for the frequency range 2. The coefficientα is0.3175 for frequency range 1 and 0.8042 for frequency rang2. By using the two frequency bands, the damping ratio ofsystem was kept from 3% to 4% for the ranges of the excitafrequencies considered in the dynamic analyses, as can bein Fig. 13.

The variation in the radial displacement (U-radial) wexcitation period for Model A with the added mass approis shown inFig. 14. The displacements shown are those atnodes of the cylinder where maximum displacements occurreThese nodes, identified as A and B, are shown inFig. 15.The highest peaks inFig. 14 occur at the excitation periodassociated with the fundamental mode. A similar behavior

e

1e

dn

k-

g

f

eneen

s

obtained for Models B and C, and the fundamental mowere also identified from the excitation period with the highradial displacement peak. Thefundamental modes resultein bending modes(n = 1). Similar fundamental modewere obtained from the free vibration and from the harmonresponse analyses for the added mass models as shoTable 6, with differences of less than 3% in all cases. Tfundamental modes for each of the tank-liquid system, in whichthe liquid was represented with added masses, are presin Figs.15–17. The maximum radial displacements at tmeridian inthe direction of the excitation did not occur neartop of the tank for any model, as is the case for the fundamemodeof a cantilever beam. As shown inFig. 15(b), for ModelA (the shortesttank with H/D = 0.40) the fundamental modis a bending mode(n = 1) with an axial half-wave(m)

characterized by a bulge near the mid-height of the cylinder.However, as shown inFigs. 16(b) and17(b), for Models B andC which have aspect ratios larger than 0.63, the fundammodes tend to the first mode of a cantilever beam, even ththe maximum radial displacements do not occur at the tothe tank.

In the cases studied so far, the liquid in the tank waccounted for by means of the added mass formulation. Insequel, the analyses are repeated using acoustic elemerepresent the liquid.

The variation in the radial displacement with the excitatioperiod for Model A, using acoustic elements, is presenteFig. 18. The mode that contributes the most to the respoof the tank-liquid system is identified by the peak inFig. 18.Notice from Figs. 14and 18 that the fundamental period foModel A obtained with the added mass approach and withmodels with acoustic elements are quite similar; in fact, thdifferences are smaller than 3% for the three tank models.

Evidently, the response to the ground motion with periequal to the fundamental modes (indicated inFigs. 14 and18) are much larger than that computedfor other excitationperiods. Thus, for both models (with liquid acoustic elemeand added mass) and regardless of theH/D ratio, the responsto a horizontal harmonic excitation is dominated by tfundamental mode.

The deformed shapes associated with the fundammodes for the models with acoustic elements were benmodes(n = 1) that tend to the first mode of a cantilevbeam, similar to those presented inFigs.15–17for the addedmass models. For instance, the deformed shape showFig. 19 for the fundamental mode of Model C computed usacoustic elements is similar to that computed with the admass model and displayed inFig. 17.

1458 J.C. Virella et al. / Engineering Structures 28 (2006) 1450–1461

.

Fig. 15. Fundamental mode for Model A with liquid added mass: (a) 3D view;(b) deformed shape in the meridian with maximum displacements

ts.

Fig. 16. Fundamental mode for Model B with liquid added mass: (a) 3D view;(b) deformed shape in the meridian with maximum displacemen

s.

Fig. 17. Fundamental mode for Model C with liquid added mass: (a) 3D view;(b) deformed shape in the meridian with maximum displacement

s)

Table 6Fundamental periods for the tank-liquid systems obtained by using two formulations to represent the liquid

Model H/D Periods with added mass formulation Periods with acoustic FEFree vibration (s) Harmonic response (s) Harmonic response (

A 0.40 0.212 0.212 0.210B 0.63 0.239 0.240 0.244C 0.95 0.300 0.303 0.296

J.C. Virella et al. / Engineering Structures 28 (2006) 1450–1461 1459

tion

tatio

nt

nce

thentsdurpe

tsThod

infor

tans

tan

eselsthe

for

sode

fortaltsos

. Thesosorhell

sesation

fnk

odsper

to

hatotntalhatntal

calo

ortatic

Fig. 18. Variation of the normalized radial displacement with the excitaperiod for Model A with liquid acoustic elements: –•– Node A; –�– Node B(seeFig. 15).

The fundamental periods obtained from the steady sharmonic response analyses and from the free vibra(eigenvalue) analyses using different formulations to represethe liquid are presented inTable 6. The results are very similarfor the three models of the tanks: in all cases, the differeare smaller than 3%.

The variation in the impulsive pressure with the period ofground motion for tank models with liquid acoustic elemewasobtained. The pressure was normalized and it is calculateat the point of the cylinder where the maximum pressoccurred. In all cases, the pressure showed a pronouncedat the same period than for the maximum radial displacemen(see Fig. 18); therefore, both responses are in phase.impulsive hydrodynamic pressure for the fundamental mfollows the meridianal distribution shown inFig. 20, with acosine circumferential variation similar to that illustratedFig. 5. The impulsive hydrodynamic pressure distributionModels A and C is illustrated inFig. 20. The pressure isshown in the direction of the excitation and for the fundamenmode along the meridian at which the maximum respooccurs. The pressure distribution for an anchored rigidobtained from Eq.(1) is also shown inFig. 20. The twodistributions are similar in shape, with the smallest differencoccurring for Model A, and increasing successively for ModB and C. These results indicate that the difference between

ten

s

eak

ee

lek

impulsive pressure distributions for the fundamental modesflexible and rigid tanks increased with their aspect ratiosH/D.However, even for the tallest tank (Model C, where differenceare larger), the impulsive pressures for the fundamental mof the flexible and rigid tanks have similar shapes.

4.3. Comparisons with previous studies

The fundamental periods obtained in this paperthe impulsive modes of cylindrical tanks under horizontranslation were compared with those proposed by Veleand Shivakumar [19] and Sakai et al. [20]. Both groups ofresearchers identified cantilever beam fundamental modesequation for the fundamental impulsive mode by Veletand Shivakumar [19] considers a constant shell thickness. Fthe tanks considered in this paper, which have variable sthickness (seeFig. 3), the average thickness of the shell courbelow the liquid surface was computed and used in the equof Veletsos and Shivakumar [19]. For tanks withvariableshellthickness, Sakai et al. [20] used thethickness at an elevation o1/3 of the liquid height measured from the bottom of the tato compute the fundamental impulsive period.

A comparison between the impulsive fundamental pericomputed from the free vibration analyses in this paand those from Refs. [19] and [20] is displayed in Fig. 21.The fundamental periods from this study are very similarthose predicted by Veletsos and Shivakumar [19], with themaximum difference being smaller than 3.5%. Notice teven for Model A, in which the fundamental mode is nquite similar to a cantilever beam type mode, the fundameperiod compared extremely well (0.15% difference) with tproposed by Veletsos and Shivakumar. Smaller fundameperiods are predicted by the formulation of Sakai et al. [20],with differences between 9% and 14%.

5. Conclusions

The conclusions stated in this paper apply for cylindritank-liquid systems with height/diameter ratios from 0.40 t0.95 with roof, anchored to the foundation, with a perfectas-designed geometry, in which the influence of the hydrospressure and the self-weight is considered.

Fig. 19. Fundamental mode for Model C with acousticliquid elements: (a) 3D view; (b) side view.

1460 J.C. Virella et al. / Engineering Structures 28 (2006) 1450–1461

s fo

emtelThin

ere

de

nttioatis oosd batct

unthlatheddtinarhe

true

verl

e

ionsible

atiosnks

itethe

for

n

oralhors

histo

the

s-of):

d

Fig. 20. Pressure distribution at the meridian with maximum displacementthe fundamental mode: (a) Model A; (b) Model C.

It was verified that the response of a tank-liquid systsubjected to a horizontal ground motion can be accurapredicted by considering just the fundamental mode.fundamental mode for the tank-liquid systems is a bendmode (n = 1), regardless of the height-to-tank diametratios (H/D) considered in this study. This conclusion agrewith that stated previously by Barton and Parker [13]. Thelargest natural periods are associated with cylinder mocharacterized by circumferential wavesn > 1 and axial half-wavesm = 1. However, these modes are not the fundamemodes for predicting the response to a horizontal base moin the sense that they do not have the largest modal participfactors. It was proved that, for the range of shell dimensioninterest for the oil industry, the fundamental modes are not thassociated with circumferential wavy patterns, as proposeNatchigall et al. [6]. Nevertheless, it must be pointed out ththis study considers tanks with perfect geometry, i.e. the effeof imperfections in the shell were not taken into account.

Similar fundamental periods and mode shapes were fofrom the free vibration (eigenvalue) analyses and fromharmonic response analyses, using the added mass formuand the model with liquid acoustic finite elements. Tdifferences were smaller than 3% in all cases. Thus, the amass models can provide a good approximation for calculathe response of tanks filled with liquid, as the results compvery well with the more sophisticated models in which t

r

yeg

s

s

aln,onfey

s

deion

edge

Fig. 21. Variation in fundamental period(TD) with liquid height to tankdiameter ratio(HL/D).

liquid is represented by acoustic finite elements. This isregardless ofthe aspect ratio(H/D) of the tank.

The fundamental modes of tanks with aspect ratiosH/Dlarger than 0.63 are similar to the first mode of a cantilebeam. For the shortest tank(H/D = 0.40), the fundamentamode is a bending mode(n = 1) with an axial half-wave(m) characterized by a bulge formed near the mid-height of thcylinder.

The differences between the impulsive pressure distributfor the fundamental modes calculated considering flexand rigid tanks increase with the aspect ratio(H/D) of thetanks. However, even for the tank with the largest aspect rconsidered here (Model C,H/D = 0.95) where the differenceare largest, the impulsive pressures of flexible and rigid tahave similar shapes.

The fundamental periods obtained with the detailed finelement models used in this paper compared well withrecommendations of Veletsos and Shivakumar [19], withdifferences smaller than 3.5% in all cases. The formulationthe fundamental period due to Sakai et al. [20] predicts smallerfundamental periods than those found in this paper, resulting idifferences between 9% and 14%.

Acknowledgments

J.C. Vi rella was supported by a PR-EPSCoR post-doctfellowship grant EPS-0223152 for this research. The autthank Dr. C.A. Prato (National University of C´ordoba,Argentina) for his contribution during the early stages of twork. Partial support from Mid America Earthquake Centerthis research is gratefully acknowledged.

References

[1] Housner GW. The dynamic behavior of water tanks. Bulletin ofSeismological Society of America 1963;53(2):381–9.

[2] Veletsos AS, Yang JY. Earthquakeresponse of liquid storage tankadvances in civil engineering through mechanics. In: Proceedingsthe second engineering mechanics specialty conference. Raleigh (NCASCE; 1977. p. 1–24.

[3] Haroun MA, Housner GW. Earthquakeresponse of deformable liquistorage tanks. Journal of Applied Mechanics 1981;48(2):411–8.

J.C. Virella et al. / Engineering Structures 28 (2006) 1450–1461 1461

.ms:

ge

ofe.

oi

l

cieith

996

t.

:

s:

se ofg

h.D.of

al

al,

ion

ons

:

suake

gs984.

[4] Veletsos AS. Seismic response and design of liquid storage tanksGuidelines for the seismic design of oil and gas pipeline systeIn: Technical council on lifeline earthquake engineering. New YorkASCE; 1984. p. 255–370, 443–61.

[5] Malhotra P, Veletsos AS. Uplifting response of unanchored liquid storatanks. Journal of Structural Engineering 1994;120(12):3525–47.

[6] Natchigall I, Gebbeken N, Urrutia-Galicia JL. On the analysisvertical circular cylindrical tanks under earthquake excitation at its basEngineering Structures 2003;25:201–13.

[7] American Petroleum Institute. API Standard 650. Steel tanks forstorage. 8th ed. 1988.

[8] American Water WorksAssociation. AWWA Standard for welded steetanks for water storage. AWWA D100. 1984.

[9] Mazuch T, Horacek J, Trnka J, Vesely J. Natural modes and frequenof a thin clamped-free steel cylindrical storage tank partially filled wwater: FEM and measurement. Journal of Sound and Vibration 1193(3):669–90.

[10] Gupta RK. Free vibrations of partially filled cylindrical tanks. Departmenof Engineering, Launceston (Australia): University of Tasmania; 1994

[11] Han RPS, Liu JD. Free vibration analysis of a fluid-loaded variablethickness cylindrical tank. Journal of Sound and Vibration 1994;176(2)235–53.

[12] Goncalves PB, Ramos NRSS. Free vibration analysis of cylindrical tankpartially filled with liquid. Journal of Sound and Vibration 1996;195(3)

.

l

s

;

429–44.[13] Barton DC, Parker JV. Finite element analysis of the seismic respon

anchored and unanchored liquid storage tanks. Earthquake Engineerinand Structural Dynamics 1987;15(3):299–322.

[14] Virella JC. Buckling of steel tanks subject to earthquake loadings. Pthesis. Department of Civil Engineering and Surveying, UniversityPuertoRico at Mayag¨uez; 2004.

[15] Virella JC, Godoy LA, Su´arez LE. Influence of the roof on the naturperiods of empty tanks. Engineering Structures 2003;25:877–87.

[16] Hibbit HD, Karlsson BI, Sorensen P. ABAQUS standard user manuversion 6.4. Rhode Island: Hibbit, Karlsson and Sorensen, Inc.; 2002.

[17] Hibbit HD, Karlsson BI, Sorensen P. ABAQUS theory manual, vers6.4. Rhode Island: Hibbit, Karlsson and Sorensen, Inc.; 2002.

[18] Virella JC, Godoy LA, Su´arez LE. Effect of pre-stress statesthe impulsive modes of vibration of cylindrical tank-liquid systemunder horizontal motion. Journal of Vibration and Control 2005;11(9)1195–220.

[19] Veletsos AS, Shivakumar P. Tanks containing liquids or solids. In: BeskoDE, Poulos A, editors. Computer analysis and design of earthqresistant structures: A Handbook, 3; 1997. p. 725–73 [chapter 15].

[20] Sakai F, Ogawa H, Isoe A. Horizontal, vertical, rocking fluid-elasticresponse and design of cylindrical liquid storage tanks. In: Proceedinof the 8th world conference on earthquake engineering. vol. V. 1p. 263–70.