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Introduction

Large-capacity ground-supported cylin-drical tanks are used to store a varietyof liquids, e.g. water for drinking andfire fighting, petroleum, chemicals, and liquefied natural gas. Satisfactoryperformance of tanks during strongground shaking is crucial for modernfacilities. Tanks that were inadequatelydesigned or detailed have suffered extensive damage during past earth-quakes [1–7].

Earthquake damage to steel storagetanks can take several forms. Large ax-ial compressive stresses due to beam-like bending of the tank wall can cause“elephant-foot” buckling of the wall(Fig. 1). Sloshing liquid can damagethe roof and the top of tank wall (Fig. 2). High stresses in the vicinity ofpoorly detailed base anchors can rup-ture the tank wall. Base shear canovercome friction causing the tank toslide. Base uplifting in unanchored orpartially anchored tanks can damagethe piping connections that are inca-pable of accommodating vertical dis-placements, rupture the plate-shelljunction due to excessive joint stresses,and cause uneven settlement of thefoundation.

Initial analytical studies [8, 9] dealtwith the hydrodynamics of liquids inrigid tanks resting on rigid founda-

tions. It was shown that a part of theliquid moves in long-period sloshingmotion, while the rest moves rigidlywith the tank wall. The latter part ofthe liquid – also known as the impul-sive liquid – experiences the same acceleration as the ground and con-tributes predominantly to the baseshear and overturning moment. Thesloshing liquid determines the heightof the free-surface waves, and hencethe freeboard requirement.

It was shown later [10–12] that theflexibility of the tank wall may causethe impulsive liquid to experience accelerations that are several timesgreater than the peak ground acceler-ation. Thus, the base shear and over-turning moment calculated by assum-ing the tank to be rigid can be non-conservative. Tanks supported on flex-ible foundations, through rigid basemats, experience base translation androcking, resulting in longer impulsiveperiods and generally greater effectivedamping. These changes may affectthe impulsive response significantly[13, 14]. The convective (or sloshing)response is practically insensitive toboth the tank wall and the foundationflexibility due to its long period of oscillation.

Tanks analysed in the above studieswere assumed to be completely an-chored at their base. In practice, com-plete base anchorage is not always fea-sible or economical. As a result, manytanks are either unanchored or onlypartially anchored at their base. Theeffects of base uplifting on the seismicresponse of partially anchored and

unanchored tanks supported on rigidfoundations were therefore studied[15]. It was shown that base upliftingreduces the hydrodynamic forces inthe tank, but increases significantly theaxial compressive stress in the tankwall.

Further studies [16, 17] showed thatbase uplifting in tanks supported directly on flexible soil foundationsdoes not lead to a significant increasein the axial compressive stress in thetank wall, but may lead to large foun-dation penetrations and several cyclesof large plastic rotations at the plateboundary. Flexibly supported unan-chored tanks are therefore less proneto elephant-foot buckling damage, butmore prone to uneven settlement ofthe foundation and fatigue rupture atthe plate-shell junction.

In addition to the above studies, nu-merous other experimental and nu-merical studies have provided valuableinsight into the seismic behaviour oftanks [18–27]. This paper deals onlywith the elastic analysis of fully an-chored, rigidly supported tanks. Theeffects of foundation flexibility andbase uplifting on the tank responsemay be found elsewhere [13–17].

Simple Procedure for Seismic Analysis of Liquid-Storage TanksPraveen K. Malhotra, Senior Res. ScientistFactory Mutual Research, Norwood, MA, USA

Thomas Wenk, Civil Eng.Swiss Federal Institute of Technology, Zurich, Switzerland

Martin Wieland, DrElectrowatt Engineering Ltd, Zurich, Switzerland

Summary

This paper provides the theoretical background of a simplified seismic designprocedure for cylindrical ground-supported tanks. The procedure takes into ac-count impulsive and convective (sloshing) actions of the liquid in flexible steel orconcrete tanks fixed to rigid foundations. Seismic responses – base shear, over-turning moment, and sloshing wave height – are calculated by using the site re-sponse spectra and performing a few simple calculations. An example is present-ed to illustrate the procedure, and a comparison is made with the detailed modalanalysis procedure. The simplified procedure has been adopted in Eurocode 8.

Fig. 1: Elephant-foot buckling of a tankwall (courtesy of University of California at Berkeley)

Peer-reviewed by international ex-perts and accepted for publicationby IABSE Publications Committee

198 Reports Structural Engineering International 3/2000

Method of Dynamic Analysis

The dynamic analysis of a liquid-filledtank may be carried out using the con-cept of generalised single-degree-of-freedom (SDOF) systems representingthe impulsive and convective modes ofvibration of the tank-liquid system.For practical applications, only the firstfew modes of vibration need to be con-sidered in the analysis (Fig. 3). Themass, height and natural period ofeach SDOF system are obtained by themethods described in [10–14]. For agiven earthquake ground motion, theresponse of various SDOF systemsmay be calculated independently andthen combined to give the net baseshear and overturning moment.

and first convective modes are consid-ered satisfactory in most cases. Thereis, however, some merit in slightly ad-justing the modal properties of thesetwo modes to account for the entireliquid mass in the tank.

Simple Procedure for SeismicAnalysis

The procedure presented here is basedon the work of Veletsos and co-work-ers [10, 12, 14] with certain modifica-tions that make the procedure simple,yet accurate, and more generally ap-plicable. Specifically, these modifica-tions include

– representing the tank-liquid systemby the first impulsive and first con-vective modes only

– combining the higher impulsive modalmass with the first impulsive modeand the higher convective modalmass with the first convective mode

– adjusting the impulsive and convec-tive heights to account for the over-turning effect of the higher modes

– generalising the impulsive periodformula so that it can be applied tosteel as well as concrete tanks of various wall thicknesses.

The impulsive and convective respons-es are combined by taking their nu-merical sum rather than their root-mean-square value.

Model Properties

The natural periods of the impulsive(Timp) and the convective (Tcon) responses are

(1)

(2)

where h is the equivalent uniformthickness of the tank wall, ρ the massdensity of liquid, and E the modulus ofelasticity of the tank material. The co-efficients Ci and Cc are obtained fromFig. 4 or Table 1. The coefficient Ci isdimensionless, while Cc is expressed ins/√m. For tanks with non-uniform wallthickness, h may be calculated by tak-ing a weighted average over the wettedheight of the tank wall, assigning thehighest weight near the base of thetank where the strain is maximal.

Fig. 2: Sloshing damage to upper shell of tank (courtesy of University of California at Berkeley)

Fig. 3: Liquid-filled tank modelled by gen-eralised single-degree-of-freedom systems

For most tanks (0.3 < H/r < 3, where H is the height of water in the tank andr the tank radius), the first impulsiveand first convective modes togetheraccount for 85–98% of the total liquidmass in the tank. The remaining massof the liquid vibrates primarily in high-er impulsive modes for tall tanks (H/r > 1), and higher convective modesfor broad tanks (H/r ≤ 1). The resultsobtained using only the first impulsive

H/r Ci Cc [s/√m] mi /ml mc /ml hi /H hc /H hi’/H hc’/H

0.3 9.28 2.09 0.176 0.824 0.400 0.521 2.640 3.414

0.5 7.74 1.74 0.300 0.700 0.400 0.543 1.460 1.517

0.7 6.97 1.60 0.414 0.586 0.401 0.571 1.009 1.011

1.0 6.36 1.52 0.548 0.452 0.419 0.616 0.721 0.785

1.5 6.06 1.48 0.686 0.314 0.439 0.690 0.555 0.734

2.0 6.21 1.48 0.763 0.237 0.448 0.751 0.500 0.764

2.5 6.56 1.48 0.810 0.190 0.452 0.794 0.480 0.796

3.0 7.03 1.48 0.842 0.158 0.453 0.825 0.472 0.825

Table 1: Recommended design values for the first impulsive and convective modes of vibra-tion as a function of the tank height-to-radius ratio (H/r). All coefficients are based on anexact model of the tank-liquid system [10, 12, 14].

Fig. 4: Impulsive and convective coefficientsCi and Cc

T C

H

h r Eimp i=

×

ρ

/

T C rcon c=

10

8

6

4

2

00 0.5 1 1.5 2 2.5 3

H/R

Ci

Cc

H

Rhi mi

mc

hc

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The impulsive and convective masses(mi and mc) are obtained from Fig. 5 orTable 1 as fractions of the total liquidmass (ml).

Seismic Responses

The total base shear is given by

(3)

where mw is the mass of tank wall, mrthe mass of tank roof, Se(Timp) the im-pulsive spectral acceleration (obtainedfrom a 2% damped elastic responsespectrum for steel and prestressedconcrete tanks, or a 5% damped elasticresponse spectrum for concrete tanks),and Se(Tcon) the convective spectral acceleration (obtained from a 0.5%damped elastic response spectrum).

The overturning moment above thebase plate, in combination with ordi-nary beam theory, leads to the axialstress at the base of the tank wall. Thenet overturning moment immediatelyabove the base plate (M) is given byEq. (4), where hi and hc are the heightsof the centroids of the impulsive andconvective hydrodynamic wall pres-sures (Fig. 6, Table 1), and hw and hrare the heights of the centres of gravityof the tank wall and roof, respectively.

The overturning moment immediatelybelow the base plate (M’) is dependent

on the hydrodynamic pressure on thetank wall as well as that on the baseplate. It is given by Eq. (5), where theheights hi’ and hc’ are obtained fromFig. 6 or Table 1.

If the tank is supported on a ring foun-dation, M should be used to design thetank wall, base anchors and the foun-dation. If the tank is supported on amat foundation, M should be used todesign the tank wall and anchors only,while M’ should be used to design thefoundation.

The vertical displacement of the liquidsurface due to sloshing (d) is given byEq. (6), where g is the acceleration dueto gravity.

Comparison with DetailedModal Analysis

Three steel tanks were selected forcomparing the results obtained fromthe proposed procedure with thosefrom a detailed modal analysis. Threeimpulsive and three convective modeswere used in the detailed analysis. Themodal analysis results were calculatedusing a combination of root-mean-square and algebraic-sum rules. Thenet impulsive and the net convective

responses were calculated first, usingthe root-mean-square rule, then nu-merically added to give the overall re-sponse. The base shear, for example,was obtained using Eq. (7), where Qi

1

and Qc1 are the base shear values for

the first impulsive and first convectivemodes, respectively. The responsespectra for the site are the same asthose used in the given example (Fig. 7).

The results (Table 2) show that the values of base shear and moment ob-tained from the proposed procedurewere 2–10% higher than those from

Fig. 7: Elastic design response spectra for0.5% and 2% damping

Fig. 5: Impulsive and convective masses asfractions of the total liquid mass in the tank

Fig. 6: Impulsive and convective heights asfractions of the height of the liquid in thetank

(4)

(5)

(6)

(7) Q Q Q Q Q Q Qi i i c c c= + + + + +( ) ( ) ( ) ( ) ( ) ( )1 2 2 2 3 2 1 2 2 2 3 2 d R

S Tg

e con=( )

r [m] H [m] h/r Q [MN] M [MNm] M’ [MNm] D [cm]

15 7.5 0.001 15.7 (14.9) 49.3 (44.6) 167 (164) 57 (51)

15 15 0.001 53.6 (52.4) 346 (334) 577 (557) 75 (66)

7.5 15 0.001 18.3 (16.5) 127 (123) 140 (136) 79 (67)

Table 2: Comparison of results from proposed procedure with those from detailed analysis(values in parentheses are from the modal analysis)

Tank contents Importance factor (γI) for

Class 1 Class 2 Class 3

Drinking water, non-toxic non-flammable chemicals 1.2 1.0 0.8

Fire-fighting water, non-volatile toxic chemicals, 1.4 1.2 1.0lowly flammable petrochemicals

Volatile toxic chemicals, explosive and highly 1.6 1.4 1.2flammable liquids

Table 3: Importance factor (γI) for tanks according to Eurocode 8 [28]

Q m m m S T m S Ti w r e imp c e con= + + +( ) ( ) ( )×

1

0.8

0.6

0.4

0.2

00 0.5 1 1.5 2 2.5 3

H/R

0 0.5 1 1.5 2 2.5 3

4

3

2

1

0

H/R

h’c /H

h’i /H

hc/H

hi /H

1.4

1.2

1

0.8

0.6

0.4

0.2

0Timp 0.2 0.4 1 2 3 Tcon6

Period, T (s)

Spec

tral

Acc

eler

atio

n, S

e(g

)

mc /ml

mi /ml

ζ = 0.5%

ζ = 2%

M m h m h m h S T m h S Ti i w w r r e imp c c e con= + + +( ) ( ) ( )×

M m h m h m h S T m h S Ti i w w r r e imp c c e con' ( ' ) ( ) ' ( )= + + +×

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the detailed modal analysis. The valuesof sloshing wave height obtained fromthe proposed procedure were 12–18%higher than those from the detailed

modal analysis. The results of the pro-posed procedure are therefore con-servative but close to those from the detailed modal analysis.

Design According to Eurocode 8

The presented simple procedure wasused in Eurocode 8 [28] and integratedin its limit state design concept. Theserviceability and ultimate limit stateshave to be verified. The specificationof the corresponding seismic actions isleft to the national authorities. The level of seismic protection is estab-lished based on the risk to life and theeconomic and environmental conse-quences. This reliability differentiationis achieved by adjusting the return pe-riod of the design seismic event. Threetank reliability classes are defined cor-responding to situations with high(Class 1), medium (Class 2) and low(Class 3) risk. Depending on the tankcontents, an importance factor (γI) isassigned to each of the three classes(Table 3).

The seismic action effects have to bemultiplied by the selected importancefactor. For the reference case (γI = 1),the recommended return periods ofthe design seismic event are 475 yearsfor the ultimate limit state and 50–70years for the serviceability limit state.In the case of the largest importancefactor (γI = 1.6), the return period ofthe design event for the ultimate limitstate is about 2000 years. According toEurocode 8, the analysis has to assumelinear elastic behaviour, allowing onlyfor localised non-linear phenomenawithout affecting the global response,and to include the hydrodynamic re-sponse of the fluid. Particularly, itshould account for the convective andimpulsive components of fluid motionas well as the tank shell deformationdue to hydrodynamic pressure and interaction effects with the impulsivecomponent. The proposed proceduresatisfies these principles in a simpleand efficient way for the design offixed-base cylindrical tanks.

Future Research Needs

In regions of strong ground shaking, it is sometimes impractical to designtanks for forces obtained from elastic(no damage) response analysis. Elasticforces are so large that they are arbi-trarily reduced by factors of 3 or moreto obtain the design forces. When sub-jected to strong shaking, tanks there-fore respond in a non-linear fashionand experience some damage. How-ever, no generally acceptable methodsexist to perform a non-linear seismic

Example

A steel tank with a radius r of 10 m and total height of 9.6 m is fully anchored toa concrete mat foundation. The tank is filled with water to a height H of 8 m (H/r = 0.8). The total mass of water in the tank (ml) is 2.51 × 106 kg. Thetank wall is made of four courses, each 2.4 m high. The lower two courses are 1cm thick and the upper two courses 0.8 cm thick. The total mass of the tank wall(mw) is 43 × 103 kg, and the height of its centre of gravity (hw) is 4.53 m. Themass of the tank roof (mr) is 25 × 103 kg and the height of its centre of gravity(hr) is 9.6 m. The 0.5% and 2% damped elastic response spectra for the site areshown in Fig. 7.

Model Properties

First, the equivalent uniform thickness of the tank wall is calculated by the weighted average method. Using weights equal to the distance from the liquid surface

For steel, E = 2 × 1011 N/m2. For water, ρ = 1000 kg/m3. For H/r = 0.8, Ci = 6.77and Cc = 1.57 s/m0.5 (Table 1). Hence, from Eqs. (1) and (2),

For H/r = 0.8, mi /ml = 0.459 and mc /ml = 0.541 (Table 1). Hence,

kg

kg

Also from Table 1, hi /H = 0.404, hc /H = 0.583, hi’/H = 0.891, hc’/H = 0.954.Hence, hi = 3.23 m, hc = 4.66 m, hi’= 7.13 m, and hc’ = 7.63 m.

Seismic Responses

The impulsive spectral acceleration for Timp = 0.123 s, obtained from the 2%damped elastic response spectrum (Fig. 4), is Se(Timp) = 0.874 g. The convectivespectral acceleration for Tcon = 4.96 s, obtained from the 0.5% damped response spectrum in Fig. 4, is Se(Tcon) = 0.07 g.

The base shear obtained from Eq. (3) is

The overturning moment above the base plate, obtained from Eq. (4), is

and the overturning moment below the base plate, obtained from Eq. (5), is

The maximum vertical displacement of the liquid surface due to sloshing, obtained from Eq. (6), is

h =

× × + × × + × × + × ×× + × + × + ×

=0 01 2 4 6 8 0 01 2 4 4 4 0 008 2 4 2 0 008 0 8 0 4

2 4 6 8 2 4 4 4 2 4 2 0 8 0 40 00968

. . . . . . . . . . .. . . . . . .

. m

Timp = ×

×

× ×=6 77

8 1000

0 00968 10 2 100 123

11.

. /. s

Tcon = × =1 57 10 4 96. . s

mi = × × = ×0 459 2 51 10 1 15 106 6. . .

mc = × × = ×0 541 2 51 10 1 36 106 6. . .

Q = + + × × × + × × × =( . . . ) . . . . .1 15 0 043 0 025 10 0 874 9 81 1 36 10 0 07 9 81 116 6 MN

M = × + × + × × × × + × ×( . . . . . . ) . . . .1 15 3 23 0 043 4 53 0 025 9 6 10 0 874 9 81 1 36 10 4 666 6

= 40 MNm

M' ( . . . . . . ) . . . . . .= × + × + × × × × + × × × ×1 15 7 13 0 043 4 53 0 025 9 6 10 0 874 9 81 1 36 10 7 63 0 07 9 816 6

= 81 MNm

d = × =10 0 07 0 7. . m

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analysis of tanks. Therefore, the dam-age sustained by tanks subjected toground motions of different intensitiescannot be quantified easily. There is aneed for practical methods of non-lin-ear analysis and design of liquid-stor-age tanks.

Unlike ductile building systems, tankslack a mechanism to dissipate largeamounts of seismic energy in a ductilemanner. Methods of improving theseismic performance of tanks by in-creasing their ability to dissipate seis-mic energy need to be examined. The tank could either be anchored toits foundation with energy dissipating devices [29] or seismically isolated byspecial bearings [30, 31].

References

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[2] U.S. Department of Commerce. Earthquakedamage to water and swage facilities, San Fernan-do Earthquake of February 9, 1971. Proceedingsof the National Oceanic and Atmospheric Administration, Washington, Vol. 2, 1973, pp.135–138.

[3] GATES, W. E. Elevated and ground-support-ed steel storage tanks. Reconnaissance report, Imperial County, California Earthquake of Octo-ber 15, 1979. Earthquake Engineering ResearchInstitute, Oakland, 1980.

[4] MANOS, G. C.; CLOUGH, R. W. Tank dam-age during the May 1983 Coalinga Earthquake.Journal of Earthquake Engineering and Struc-tural Dynamics, Vol. 1, No. 4, 1985, pp. 449–466.

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[11] HAROUN, M. A.; HOUSNER, G. W. Seis-mic design of liquid-storage tanks. Journal ofTechnical Councils, ASCE, New York, Vol. 107,No. 1, 1981, pp. 191–207.

[12] VELETSOS, A. S. Seismic response and de-sign of liquid storage tanks. Guidelines for theSeismic Design of Oil and Gas Pipeline Systems,ASCE, New York, 1984 pp. 255–370.

[13] VELETSOS, A. S.; TANG, Y. Soil-structureinteraction effects for laterally excited liquid-stor-age tanks. Journal of Earthquake Engineeringand Structural Dynamics, Vol. 19, No. 4, 1990,pp. 473–496.

[14] VELETSOS, A. S. et al. Seismic response ofanchored steel tanks. Proceedings of the ThirdSymposium on Current Issues Related to Nu-clear Power Plant Structures, Equipment andPiping (Gupta, A. K. Ed.), North Carolina StateUniversity, 1990, pp. 2–15.

[15] MALHOTRA, P. K.; VELETSOS, A. S.Seismic response of unanchored and partially an-chored liquid-storage tanks. Report TR-105809,Electric Power Research Institute, Palo Alto,1995.

[16] MALHOTRA, P. K. Base uplifting analysisof flexibly supported liquid-storage tanks. Jour-nal of Earthquake Engineering and StructuralDynamics, Vol. 24, No. 12, 1995, pp. 1591–1607.

[17] MALHOTRA, P. K. Seismic response ofsoil-supported unanchored liquid-storage tanks.Journal of Structural Engineering, ASCE, NewYork, 1997, Vol. 123, No. 4, pp. 440–450.

[18] WOZNIAK, R. S.; MITCHELL, W. W. Ba-sis of seismic design provisions for welded steeloil storage tanks. Sessions on Advances in Stor-age Tank Design, American Petroleum Institute,Washington, 1978.

[19] CLOUGH, R. W.; NIWA, A. Static tilt testsof a tall cylindrical liquid-storage tank. ReportEERC 79–06, University of California, Berke-ley, 1979.

[20] MANOS, G. C.; CLOUGH, R. W. Furtherstudy of the earthquake response of a broadcylindrical liquid-storage tank model. ReportEERC 82–07, University of California, Berke-ley, 1982.

[21] AULI, W.; FISCHER, F. D.; RAMMERS-TORFER, F. G. Uplifting of earthquake-loadedliquid-filled tanks. Proceedings of the PressureVessels and Piping Conference, New Orleans,Vol. 98, No. 7, 1985, pp. 71–85.

[22] LEON, G. S.; KAUSEL, E. A. M. Seismicanalysis of fluid storage tanks. Journal of Struc-tural Engineering, ASCE, New York, Vol. 112,No. 1, 1996, pp. 1–18.

[23] BARTON, D. C.; PARKER, J. V. Finite ele-ment analysis of the seismic response of anchoredand unanchored liquid storage tanks. Journal ofEarthquake Engineering and Structural Dynam-ics, Vol. 15, No. 3, 1987, pp. 299–322.

[24] PEEK, R.; JENNINGS, P. C. Simplifiedanalysis of unanchored tanks. Journal of Earth-quake Engineering and Structural Dynamics,Vol. 16, No. 7, 1988, pp. 1073–1085.

[25] NATSIAVAS, S. Response and failure of fluid-filled tanks under base excitation. PhD the-sis, California Institute of Technology, Pasadena,1987.

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[27] WUNDERLICH, W.; SEILER, C. Nonlin-ear treatment of liquid filled storage tanks underearthquake excitation by a quasistatic approach.Advances in Computational Structural Mechan-ics (Topping, B. H. V., Ed.), Chapter 12.2, Saxe-Coburg Publications & Civil-Comp Press, 1998.

[28] Eurocode 8: Design provisions of earth-quake resistance of structures, Part 4: Silos, tanksand pipelines. European Committee for Stan-dardization, Brussels, 1998.

[29] MALHOTRA, P. K. Seismic strengtheningof liquid-storage tanks with energy-dissipatinganchors. Journal of Structural Engineering, Vol.124, No. 4, 1998, pp. 405–414.

[30] MALHOTRA, P. K. New method for seis-mic isolation of liquid-storage tanks. Journal ofEarthquake Engineering and Structural Dynam-ics, Vol. 26, No. 8, 1997, pp. 839–847.

[31] BACHMANN, H.; WENK, T. Softening in-stead of strengthening for seismic rehabilitation.Structural Engineering International, IABSE,Zurich, Vol. 10, No. 1, 2000, pp. 61–65.