A Static Nonlinear Procedure for the Evaluation.pdf

Journal of Earthquake Engineering, 12:1–24, 2008Copyright © A.S. Elnashai & N.N. AmbraseysISSN: 1363-2469 print / 1559-808X onlineDOI: 10.1080/13632460701672714

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UEQE1363-24691559-808XJournal of Earthquake Engineering, Vol. 0, No. 0, Dec 2007: pp. 0–0Journal of Earthquake Engineering

A Static Nonlinear Procedure for the Evaluation of the Elastic Buckling of Anchored Steel

Tanks Due to Earthquakes

xxxxJ. C. Virella, L. E. Suarez, and L. A. Godoy

JUAN C. VIRELLA, LUIS E. SUÁREZ, and LUIS A. GODOY

Department of Civil Engineering and Surveying, University of Puerto Rico atMayagüez, Mayagüez, Puerto Rico

Ground-supported steel tanks experienced extensive damage in past earthquakes. The failure oftanks in earthquakes may cause severe environmental damage and economic losses. This studydeals with the evaluation of the elastic buckling of above-ground steel tanks anchored to the founda-tion due to seismic shaking. The proposed nonlinear static procedure is based on the capacityspectrum method (CSM) utilized for the seismic evaluation of buildings. Different from the standardCSM, the results are not the base shear and the maximum displacement of a characteristic point ofthe structure but the minimum value of the horizontal peak ground acceleration (PGA) thatproduces buckling in the tank shell. Three detailed finite element models of tank-liquid systems withheight to diameter ratios H/D of 0.40, 0.63, and 0.95 are used to verify the methodology. The 1997UBC design spectrum and response spectra of records of the 1986 El Salvador and 1966 Parkfieldearthquakes are used as seismic demand. The estimates of the PGA for the occurrence of first elasticbuckling obtained with the proposed nonlinear static procedure were quite accurate compared withthose calculated with more elaborate dynamic buckling studies. For all the cases considered, theproposed methodology yielded slightly smaller values of the critical PGA for the first elasticbuckling compared to the dynamic buckling results.

Keywords Nonlinear Analysis of Tanks; Seismic-Induced Buckling; Steel Tanks; NonlinearStatic Procedure; Buckling of Cylindrical Tanks

1. Introduction

There is ample evidence of cases of buckling failure or other severe damage of ground-supported steel storage tanks during earthquakes [Cooper and Wachholz, 1999; Jain et al.,2001; Suzuki, 2002]. Practically every major earthquake provides further evidence of thevulnerability of these structures. For instance, a recently published report by an EERIreconnaissance team [2003], describes the buckling of two ground supported steel storagetanks during the 2003 San Simeon earthquake in California. In addition to the loss of thestructure, the failure of the tanks may have other severe consequences, such as long-termcontamination of the soil when there is loss of liquid content and loss of the storage capacityof the plant for a long time. In some cases, water storage tanks are part of fire-fightingsystems, and so the interruption of the water supply could result in more damage due tothe fires that might follow an earthquake.

Early investigations on the seismic behavior of anchored liquid storage tanks allowedus to gain significant insight into the effect of the hydrodynamic fluid-structure interaction

Received 6 March 2006; accepted 27 August 2007.Address correspondence to Luis E. Suárez, Department of Civil Engineering and Surveying, University of

Puerto Rico at Mayagüez, Mayagüez, Puerto Rico, 00681-9041; E-mail: [email protected]

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2 J. C. Virella, L. E. Suárez, and L. A. Godoy

on the seismic response. Jacobsen [1949] was the first researcher that studied the seismicbehavior of rigid cylindrical tanks. Housner [1963], Haroun and Housner [1981], Veletsosand Yang [1977], and Veletsos [1984] reported that a circular cylindrical tank containingliquid develops a cantilever beam type mode when subjected to horizontal excitations.Housner [1963] evaluated the hydrodynamic response of a tank-liquid system as thecontribution of two different components, the convective mode and the impulsive mode.The liquid in the upper portion of the tank vibrates with a long period sloshing motionwhile the rest moves rigidly with the tank, with an impulsive mode.

Several studies have addressed the behavior and design of anchored and unanchoredsteel cylindrical tanks subjected to earthquakes, and clearly established the importance ofthe buckling behavior of the tanks in their seismic response [Rammerstorfer et al., 1990;Lau et al., 1996; Liu and Schubert, 1999; Handam, 2000]. The American LifelinesAlliance [2001] has collected different modes of failure that have been observed in tanksduring past earthquakes. These failure modes include shell buckling, roof damage, anchor-age failure, tank support system failure, foundation failure, hydrodynamic pressurefailure, connecting pipe failure, and manhole failure. Among the modes of failure stated,the most relevant for the present study is the shell buckling. Sakai and Isoe [1989] andRammerstorfer et al. [1988] pointed out that the most important types of damage for tankssubjected to earthquakes are elastic-plastic buckling and elastic buckling of the tank shell.

Ito et al. [2003] described the buckling patterns of thin-walled liquid storage tanks asbending buckling, shear and compression buckling, and nonlinear ovaling vibration. Thebending buckling occurs at the lower part of the cylindrical shell and it includes thediamond shape buckling and elephant foot buckling (an elastic-plastic buckling type),which is the most frequently encountered buckling mechanism in tanks duringearthquakes. The shear and compression buckling is an elastic type of buckling, occurringnear the middle of the tank. The nonlinear ovaling vibration is a buckling type which istypically encountered at the top of the tank. Rammerstorfer et al. [1990] described thisphenomenon as buckling due to external pressure.

Design codes for ground supported storage tanks, such as API 650 [1991] andAWWA-D100 [1984], prescribe the buckling evaluation of steel tanks using an allowablestress formulation based on test results of cylindrical thin shells subjected to uniform com-pression. This is clearly a different load configuration than the actual earthquake loadingthat induces buckling. Therefore, it would be of great importance to have a methodologyto evaluate the seismic buckling of tank structures for a specific seismic demand that cantake into account the hydrodynamic response of the tank-liquid system based on threedimensional models.

Buckling analyses of structures can be performed by using static and dynamic buck-ling techniques. Computational dynamic buckling analyses considering geometric andmaterial nonlinearity have been performed for tank structures subjected to earthquakes.Using experimental or computational methods, Liu and Lam [1983], Natsiavas andBabcock [1987], Nagashima et al. [1987], Redekop et al. [2002], and Morita et al. [2003]reported the occurrence of buckling at the top of tanks induced by horizontal base excita-tions. In their experimental dynamic buckling studies, Natsiavas and Babcock [1987]considered a tall tank with an open top undergoing a horizontal harmonic base accelera-tion. Dynamic and static buckling experimental studies of tall tanks with roof were carriedout by Nagashima et al. [1987] and Morita et al. [2003]. Nagashima et al. [1987] consid-ered, in addition to the horizontal component, a vertical harmonic base acceleration, whileMorita et al. [2003] used both harmonic excitations along with a simulated earthquakeexcitation. The shell buckling at the top of steel tanks observed after earthquakes has beenfrequently attributed to the sloshing component of the hydrodynamic response of the

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tank-liquid system [Malhotra, 2000]. Both Ito et al. [2003] and Natsiavas and Babcock[1987] proved that this buckling mode comes mostly from the impulsive action of thehydrodynamic response of the liquid. Although the sloshing action contributes to theoccurrence of this type of buckling, it is not the main cause.

Because the seismic response is of dynamic nature, dynamic buckling studies can pre-dict more accurately the buckling behavior of a structure subject to an earthquake.However, dynamic buckling analyses are very expensive and thus the buckling evaluationmust frequently rely on static analyses. Fischer and Rammerstorfer [1982] and Redekopet al. [2002] used static buckling analyses for the seismic evaluation of steel tanks, andpresented the results in terms of the critical spectral acceleration that induces buckling.A simplified nonlinear analysis for the seismic design of liquid storage tanks was pro-posed by Malhotra [2000] based on the nonlinear static procedure for buildings [ATC-40,1996]. The design forces were obtained by considering two degrees of freedom beammodels, in which the impulsive and convective action were accounted for. Malhotra[2000] studied unanchored and partially unanchored tank-liquid systems and therefore theelastic acceleration-deformation spectrum (demand spectrum) was reduced based on theviscous damping of the tank structure and the hysteretic damping due to the plastic behaviorof the base plate.

This study presents a methodology to calculate the seismic buckling response ofabove ground steel tanks, using a three-dimensional model of the tank structure and a pro-cedure based upon the capacity and demand spectra. The seismic buckling response is dueto a horizontal earthquake excitation. Only the impulsive component of the hydrodynamicresponse of the tank-liquid system is considered, i.e., sloshing effects, are ignored. Anon-linear static procedure (NLSP) based on the capacity spectrum method (CSM) forbuilding structures [ATC-40, 1996] is proposed, for the evaluation of the first bucklingmode of the tank structure (buckling at the top). For any buckling seismic demand, theNLSP renders the spectral acceleration associated with the first buckling mode of the tank.Knowing this quantity, it is possible to obtain the horizontal peak ground acceleration at thecritical state and thus the results can be readily compared with dynamic buckling studies.

2. Nonlinear Static Procedure for Steel Tanks

A nonlinear static procedure (NLSP) based on the CSM can be implemented for the seis-mic evaluation of ground-supported steel tanks, provided that certain modifications areintroduced in the CSM to account for the differences between the seismic response oftanks and buildings. One of the main differences is that the fundamental mode for thehorizontal base excitation of tank-liquid systems does not always correspond to the first(i.e., highest period) natural mode, as it is usually the case for buildings. Usually the modeshapes of tank-liquid systems that are associated to the highest natural periods correspondto cos(m,n,q) modes that are characterized by a series of circumferential waves (n) andaxial half-waves (m). The fundamental modes for the seismic response of tank-liquidsystems are those with the largest participation factors to a horizontal base motion. Forhorizontal motions, the fundamental modes tend to cantilever beam mode shapes [Bartonand Parker, 1987]. A more detailed explanation of this concept is presented by Virellaet al. [2006a]. Thus, to apply the concepts of the CSM to tanks filled with liquid, theprocedure needs to be modified by using the fundamental mode instead of the first mode.

Notice also that the CSM applied to buildings is a sort of global collapse analysis; inour case, the CSM is used to study a local behavior. Thus, although there is a small changein periods when local buckling occurs near the top of the tank, this change is restricted tothose periods of higher modes. The fact that the buckling is localized does not diminish its

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importance because the tank’s buckling capacity is substantially reduced due to the intro-duction of an imperfection.

2.1. Development of the Capacity Curve

To develop the capacity curve for the seismic evaluation of steel tanks, a methodologysimilar to the load level 3 of the ATC-40 [1996] for buildings is applied, in which thelateral forces (Fx) vary according to the product of the story masses and the first modeshape of the structure. Certain modifications are introduced to adapt the methodology fortank-liquid systems.

These capacity curves were obtained by considering only the impulsive mode of thetank-liquid system, ignoring the sloshing effects. The force distribution for the level3 approach used in the tank structure is produced by the impulsive hydrodynamic pressuredistribution due to the fundamental mode of the tank-liquid-system under a horizontalexcitation, including the flexibility of the tank. In previous studies, such as those by Veletsosand Shivakumar [1997] and Fischer and Rammerstorfer [1982], an impulsive modalpressure distribution formulation was proposed for the seismic analysis of flexible tanks.Although in principle this formulation could be applied to obtain the capacity curves, inthe present study the impulsive pressure distribution for the fundamental mode proposedby Virella et al. [2006a] was used instead. The latter distribution was obtained from three-dimensional finite element models in which the liquid was represented with acoustic FEand the tank was modeled with shell elements. Notice that the mass of the liquid and theimpulsive pressure are correlated because the added mass of the liquid is calculated interms of the impulsive pressure. The calculation of the added mass and impulsive pressureis explained in Virella et al. [2005].

The capacity curve is obtained from a step-by-step nonlinear static buckling analysisin which geometric and material nonlinearity are considered. Material nonlinearity isconsidered by means of an elastic-plastic model with the von Mises yield criterion inwhich the maximum value reached by the von Mises stresses is the yield stress, sy = 248MPa. The pressure distribution used in the analysis is normalized for a base accelerationof 1 g.

The buckling load estimate corresponds to the sum of any initial loadls applied to thestructural system before the static buckling step, plus the loads applied in the buckling stepmultiplied by a load factor l, i.e.,

The initial loads are the hydrostatic pressure and the self-weight of the tank, which weredefined in ABAQUS [2002] in an initial step by means of a static analysis. Then, a step-by-step nonlinear analysis was performed, in which the hydrodynamic impulsive pressurewas introduced. The load factor, l, obtained from the buckling analysis then correspondsto the fraction of the impulsive pressure required to buckle the tank.

The “Modified Riks Method” is used for the step-by-step nonlinear analyses inABAQUS [2002], from which the nonlinear response of the structure is obtained, byapplying increments of load and evaluating equilibrium. In the modified Riks algorithm[Riks, 1979], the load factor l is an additional unknown and the arc length along the staticequilibrium path of the load-displacement space is used to measure the progress of thesolution. As the arc length is found, the static equilibrium equations are solved step-by-step, considering load increments, whereas the stiffness of the system decreases

Applied loads to estimate buckling initial loads loads applied= + l ( iin ond stepsec ) (1)

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successively in buckling problems. The magnitudes of the applied loads in Eq. (1) arefound for each step, and so are the displacements. The process continues until a maximumpredefined number of total steps is reached, or a predefined displacement is found at a pre-scribed point in the shell. With the values of l and the displacement at a point of the shellwhere buckling occurred, a graph of the load multiplier l vs. the radial (i.e., out of plane)displacements (Δ) is prepared. We consider the critical load multiplier lcrit as the last loadmultiplier l found before a negative load increment Δl occurs. Figure 1(a) shows a l vs. Δcurve. The l vs. Δ curve is equivalent to the pushover or capacity curve (i.e., base shear vs.roof displacement) used for buildings in the CSM. Since the applied pressure distribution,i.e., the loads that multiplied l in Eq. 1 was normalized for a base acceleration of 1 g, thecritical load factor lcrit can be associated to the critical spectral acceleration for the tank, asexplained in the next section.

2.2. Procedure to Find the Performance Point

2.2.1. The Capacity Spectrum. The participation factor PFf and the modal mass coeffi-cient af for the fundamental mode (instead of those for the first mode) will be used totransform the capacity curve to modal coordinates. These two modal parameters areobtained from eigenvalue analyses of the tank-liquid systems [Virella et al., 2006a].

Because for the analysis the impulsive pressure is normalized at an acceleration of 1 g,the load factor l in the capacity curve is a multiplier of that acceleration. Then the loadfactor is analogous to the base shear to weight of the structure ratio for buildings (V/W).The l vs. Δ curve is then transformed into a spectral acceleration (Sa) vs. spectral displace-ment (Sd) curve as shown in Fig. 1. The spectral acceleration Sa can be expressed as:

Also, the spectral displacement is obtained from a critical node in the zone of buckling as

where Δcrit is the radial displacement at the critical node where buckling occurs and fcrit, fis the modal coordinate of the fundamental mode shape at the location of the criticalnode.

FIGURE 1 Capacity curve for tank structures. (a) l vs. Δ format. (b) Sa vs. Sd format.

(b) (a)

Radial displacement Crit Node, Δ

Δcrit

LoadFactor, λ

λcrit

Established for firstnegative Δλ obtained

Saf

=la

. (2)

SPFd

crit

f crit f

=Δf ,

(3)

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2.2.2. The Ground Response Spectrum. To define the seismic demand, and as recom-mended by Fischer and Rammerstorfer [1982], a response spectrum for a damping ratio of2% is used instead of the usual 5% damping response spectrum adopted for buildings (seeFig. 2a). In addition to a design spectrum typical of those prescribed in seismic codes,ground response spectra associated with accelerograms of historic earthquakes are used inthis article. The latter permits an easier comparison of the results obtained from the CSMwith those from nonlinear time history analysis. The response spectrum is transformed tothe spectral acceleration (Sa) vs. spectral displacement (Sd) format ADRS spectra asillustrated in Fig. 2.

2.2.3. Performance Point. The primary interest for the seismic evaluation of anchoredtanks is to determine the first buckling mode. Thus, contrary to the CSM for buildings,there is no need to reduce the elastic demand curve (i.e., the response spectrum). It shouldbe pointed out that steel tanks in the nuclear industry are designed for an elastic seismicspectrum [Sugiyama et al., 2003]. Then, the performance objective here is not to find thenonlinear deformation of tank-liquid systems, but to obtain instead the horizontal peakground acceleration for which these structures would buckle elastically during a seismicevent. In other words, instead of finding a performance point for reduced demand spectra,the quantity sought is the maximum demand that can be resisted elastically by the tank-liquid system until buckling occurs.

The evaluation of the response of the steel tanks is done first using a smooth designspectrum as seismic demand. In this study, the elastic design spectrum of the UBC code[1997] was chosen to show the implementation of the methodology. The 1997 UBC spec-trum is defined by two seismic coefficients, Ca and Cv, that depend on the seismic zoneand soil profile. To implement the CSM, the parameters Ca and Cv are varied until thedemand spectrum intersects the capacity spectrum at the point where the first buckling ofthe tank occurs. Proceeding in that way, and depending on which zone of the design spec-trum is intersected by the capacity spectrum, either the critical coefficients Ca or Cv can befound. If the seismic coefficient, Ca is obtained from the procedure, it is directly related tothe horizontal peak ground acceleration (PGA) that the tank-liquid system can resist. If onthe other hand the coefficient Cv is found, then there are several PGAs associated with thisparameter, depending on the seismic zone and soil profile.

It should be noticed that the spectral accelerations for high periods do not correlatewell with the PGA. However, practically all anchored steel tanks have relatively smallfundamental periods (i.e., less than 0.4). Thus we are always in the plateau of the designspectrum, which is defined in terms of Ca in the UBC-97 spectrum used in the article. Asit is well known, in many codes (such as the UBC-97) the plateau is assumed to be

FIGURE 2 Demand Spectrum for tank structures. (a) Sa vs. T format. (b) Sa vs. Sd format.

(a) (b)

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2.5 times the PGA. Moreover, in our analyses the natural periods of the tanks remainequal to the elastic periods, at least until first local buckling occurs.

When a response spectrum corresponding to the accelerogram of a historic or artificialearthquake is used, the PGA that causes the tank to buckle elastically can be obtained indi-rectly as follows. The performance point is obtained first as the critical spectral accelerationin the capacity spectrum associated with the occurrence of the first buckling in the system.The elastic response spectrum is then calculated by scaling the original accelerogram withan initial trial scale factor. By scaling up or down, this response spectrum in the ADRSformat until it intersects the demand spectrum precisely at the performance point, one canobtain the scale factor associated with the critical PGA. Multiplying the PGA of the origi-nal accelerogram by this scale factor one can obtain the minimum PGA that induces thefirst buckling in the tank-liquid system. This process is illustrated graphically in Fig. 3.

3. Numerical Examples

3.1. Tank Models

The NLSP is applied for the seismic evaluation of three tanks with different geometries.The detailed description of the geometries and models used can be found in Virella et al.[2005]. The finite element models of these tanks are identified as Models A, B, and C, andthey correspond to tanks with height to diameter ratios (H/D) of 0.40, 0.63, and 0.95,respectively (see Fig. 4). These cylindrical tanks have a clamped condition at the base,with a cone roof supported by roof rafters. The tapered thicknesses of the tanks illustratedin Fig. 4 were chosen for serviceability conditions using the API 650 provisions [1991],without taking any seismic design considerations into account. The bottom plate of thetank was not modeled since only anchored tanks are considered in this study and ourprimarily interest lies on the buckling of the cylinder shell. Because the tanks are assumedto be anchored, this condition will prevent the deflections of the plate and thus it is notnecessary to include it in the FE model.

The liquid height of 90% the height of the cylinder was used for all the tank models.The rationale for considering that the tanks were filled up to a height of 0.9 times theheight H of the cylinder is based on the fact that design codes for such as API 650 [1991]always require a freeboard, so that if a sloshing wave is formed, it will not come in contactwith the tank roof. The amplitude of the sloshing wave needs to be computed for a specific

FIGURE 3 Location of performance point.

Spectral Displacement, Sd

Performance point P.P.Demand spectra reduced toincrease to reach Sa-crit

P.PSa-crit

Sd crit

Sa elastic

SpectralAcceleration, Sa

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seismic demand, but in previous studies [Virella et al., 2005, 2006a, b] we have foundadequate to use 10% of the height of the cylinder as freeboard.

The finite element package ABAQUS [2002] was used to perform the computations.The finite element meshes of the tank models consisted of S3R triangular elements for theroof, S4R quadrilateral elements for the cylinder, and B31 beam elements for the rafters inthe roof. The S4R element is a four node, doubly curved shell element with reduced

FIGURE 4 Tank models with cone roof supported by rafters; t = shell thickness, roofthickness tr = 0.00635 m (for roof with rafters). (a) Tank with H/D = 0.40; (b) Tank withH/D = 0.63; (c) Tank with H/D = 0.95.

t = 0.0079 m

t = 0.0079 m

t = 0.0095 m

t = 0.0127 m

t = 0.0079 m

D = 30.48 m

2.416 m

2.416 m

2.416 m

2.416 m

2.425 m

H = 12.191m

2.858 mtr

slope = 3V:16H

(a)

D = 30.48 m

2.416 m2.416 m2.416 m

2.416 m

2.425 m

2.416 m2.416 m2.416 m

t = 0.0127 m

t = 0.0159 m

t = 0.0191 m

t = 0.0191 m

t = 0.0127 m

t = 0.0095 m

t = 0.0095 m

t = 0.0079 m

H = 19.337 m

2.858 mtr

slope = 3V:16H

(b)

(c)

tr

D = 30.48 m

2.416 m2.416 m2.416 m2.416 m

2.416 m2.416 m2.416 m2.416 m2.425 m

2.416 m2.416 m2.416 m

slope = 3V:16H

H = 29.103 m

t = 0.0254 m

t = 0.0254 m

t = 0.0222 m

t = 0.0191 m

t = 0.0079 m

t = 0.0095 m

t = 0.0095 m

t = 0.0286 m

t = 0.0191 m

t = 0.0127 m

t = 0.0159 m

t = 0.0127 m

2.858 m

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integration, hourglass control and finite membrane strain formulation. The S3R element isa three node degenerated version of the S4R with finite membrane strain formulation. TheB31 is a two node linear beam element in space. All of these elements are described inmore detail in Hibbit et al. [2002]. The finite element meshes consisted of 9,262 elementsfor Model A and 10,942 elements for Models B and C.

3.2. Capacity Curves

The capacity curves were obtained using the level 3 loading distribution as describedpreviously. The fundamental modes for the impulsive component of the hydrodynamicresponse of tank-liquid systems subjected to horizontal base motions were obtained in aprevious study by Virella et al. [2006a]. Figures 5–7 show the fundamental impulsive

FIGURE 5 Fundamental mode for Model A. (a) 3D view; (b) Deformed shape in themeridian with maximum displacements.

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Normalized U-radial

Hei

ght [

m]

H HL

Smooth deformed shapeof cylinder

Xg

XgU-radial variation

with circumference

mode: n = 1

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θ

2

Meridianin Figure

(b)

(a)

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modes for Models A, B, and C, and Table 1 presents the modal parameters that character-ize the fundamental mode of each of the tank-liquid systems.

The pressure distributions for the fundamental impulsive mode of the three tank-liquid systems at the meridian in the direction of the excitation were obtained from Virellaet al. [2006a] and they are presented in Fig. 8.

A step-by-step static nonlinear analysis was performed for all the models to find theelastic buckling. For Model A, buckling occurred at a region extending from about halfthe height up to the top of the cylinder, while for Models B and C buckling occurred at aregion in the shell near the top of the cylinder. Figure 9 shows the buckling modes found.The node used to calculate the capacity curves for the systems is also indicated in thefigure. The capacity curves for the tank-liquid systems are presented in Fig. 10. As indi-cated in the figure, elastic buckling occurred for load factors equal to λ = 0.556 for ModelA, λ = 0.476 for Model B, and λ = 0.370 for Model C. Therefore, the load factor of thetank decreased with an increase in the H/D ratio.

The capacity curves are transformed into capacity spectra with the help of Eqs. (2)and (3). The modal parameters of the fundamental mode needed for the transformation arepresented in Table 1.

FIGURE 6 Fundamental mode for Model B. (a) 3D view; (b) Deformed shape in themeridian with maximum displacements.

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25.0

–3.0

Normalized U-radial

Hei

ght [

m]

H HL


Xg

U-radial variationwith circumference

mode: n = 1

3

θ

2

Meridianin Figure

Xg

(b)

(a)

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3.3. Demand Spectra

This section presents the elastic design spectrum and the ground response spectra from earth-quakes records used for the seismic buckling evaluation of the steel tanks. The design spectrumis the 1997 UBC elastic spectrum presented in Fig. 11. This spectrum is for a site with a zonefactor (Z) of 0.30, a soil profile type SB (rock), an importance factor (I) of 1.0, and a dampingratio of 2%. The seismic coefficients Ca and Cv are equal to 0.39 and 0.37, respectively.

FIGURE 7 Fundamental mode for Model C. (a) 3D view; (b) Deformed shape in themeridian with maximum displacements.

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35.0

–5.0

Normalized U-radial

Hei

ght [

m]

H HL


Xg


mode: n = 1

3

θ

2

Meridianin Figure

Xg

(b)

(a)

5.03.01.0–1.0–3.0

TABLE 1 Modal parameters for the fundamental modes of the tank-liquid systems

Model H/D Tf[s] af PFf fcrit,f PFf* fcrit,f

A 0.40 0.212 0.64 1888 0.0003803 0.718B 0.63 0.200 0.79 1.5545 0.67349 1.047C 0.95 0.196 0.77 1.582 0.8678 1.373

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In addition, the accelerograms from the 1986 El Salvador earthquake and from the1966 Parkfield earthquake in California shown in Fig. 12 were also used for the seismicbuckling evaluation of the tanks. The 1986 El Salvador earthquake had a momentmagnitude of 5.6, an epicentral distance of 4.3 km, a focal depth of 7.3 km, and a totalduration of 9.04 s. The accelerogram used was recorded in rock at the GeotechnicalInvestigation Center (CIG) station, and it has a PGA of 0.69 g. The 1966 Parkfieldearthquake had a magnitude of 5.5, a focal depth of 6 km, an epicentral distance of27 km, and a total duration of about 30 s. This accelerogram was also recorded in rockand has a PGA of 0.27 g.

The 2% damping ratio response spectrum was computed for both accelerograms. Theresponse spectra were smoothed by using a running average with 5 points. The smoothspectra are presented in Fig. 13.

FIGURE 8 Pressure distributions at the meridian with maximum displacements for thefundamental mode. (a) Model A, (b) Model B, (c) Model C.

–2.0

Normalized pressure Normalized pressure1.00.0–1.0

0.0

2.0

4.0

6.0

8.0

10.0

12.0

14.0

Hei

ght [

m]

Hei

ght [

m]

HHL

H HL

Xg


mode: n = 1

3

θ

2

Meridianin Figure

Xg

30.0

5.0

15.0

10.0

20.0

25.0

–3.0

HHL

Xg


mode: n = 1

3

θ

2

Meridianin Figure

Xg

(b)(a)

Normalized pressure(c)

1.00.0–1.0–2.0

–3.0 1.00.0–1.0–2.0


mode: n = 1

θMeridianin Figure

Xg

Xg

32

0.0

5.0

15.0

20.0

Hei

ght [

m]

25.0

10.0

30.0

35.0

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Notice that the methodology does not have restrictions regarding the specific groundmotion that can be used in the analysis (regarding their duration, frequency content, etc.).However, for the validation of the present approach there was a limitation in the resultsavailable that could be used for a comparison. The results of dynamic buckling analysesperformed by Virella et al. [2006b] were used for comparison. These analyses were basedon two earthquake ground motions: El Salvador (1986) and Parkfield (1966).

3.4. Nonlinear Static Procedure

This section discusses the evaluation of the elastic buckling for Models A, B, and C for theseismic demand defined in the previous section. First, the performance point is found forthe 1997 UBC elastic spectrum, and then results are presented by considering the responsespectra from the earthquake accelerograms.

The first sets of results are obtained for the 1997 design spectrum shown in Fig. 11.Both the capacity and demand spectrum were plotted in the ADRS format and the iterativeprocedure mentioned in a previous section was applied to find the performance point. Theperformance point for Models A, B, and C subjected to the seismic demand represented bythe 1997 UBC design spectrum is presented in Fig. 14. It can be observed that the criticalspectral acceleration (whose values are shown in the figures) diminishes with the H/Dratio. The coordinates (Sa, Sd) of the performance point, the physical radial displacementΔ, and the seismic coefficient Ca for the three tanks are presented in Table 2. Notice thatthe seismic coefficient Ca decreases with the H/D ratio of the tanks while the radialdisplacement at the zone of buckling increases with H/D.

The next sets of results correspond to the response spectra of the actual earthquakerecords shown in Fig. 13. The performance points for the tank models A, B, and C

FIGURE 9 First buckling mode for the tank-liquid systems, from step-by-step static non-linear analyses. (a) Model A, (b) Model B, (c) Model C.

(c)

(a) (b)

Criticalnode

Criticalnode

Criticalnode

340

345

350

355


subjected to the 1986 El Salvador and 1966 Parkfield records are displayed in Figs. 15 and16. The peak ground acceleration PGA that will induce the first buckling mode is illus-trated in the figures. It was found that in these cases also the PGA associated with the firstelastic buckling mode decreases with H/D. Tables 3 and 4 summarize the results of theseismic evaluation of the tank-liquid systems subjected to the El Salvador and Parkfieldearthquake records. Note that as shown in Tables 3 and 4, the ratio between the maximumdisplacement at the buckling zone and the shell thickness (Δcrit /tshell) increases with H/D.

FIGURE 10 Capacity curves for the tank-liquid systems. (a) Radial displacements fromoriginal configuration; (b) Radial displacements from static equilibrium position.

0.00

0.10

0.20

0.30

0.40

0.50

0.60

– 0.002

Radial displacement [m]

Loa

d fa

ctor

, λL

oad

fact

or, λ

Model C, λcrit = 0.370

Model B, λcrit = 0.476

Model A, λcrit = 0.556

(a)

0.0230.0180.0130.0080.003

0.00

0.10

0.20

0.30

0.40

0.50

0.60

Radial displacement [m]

Model C, λcrit = 0.370

Model B, λcrit = 0.476

Model A, λcrit = 0.556

(b)

0.000 0.005 0.010 0.015 0.020 0.025

FIGURE 11 The 1997 UBC elastic spectrum for 2% damping ratio.

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0.0 0.5 1.0 1.5 2.0 2.5T, s

Sa, g

360

365

xxxx 15

The critical horizontal PGAs for all the tank models, calculated with the NLSP arepresented in Fig. 17. The results obtained by using the 1997 UBC design spectrum areincluded in the same figure. The seismic coefficient Ca is related to the PGA (is thespectral acceleration corresponding to a period zero). Regardless of the seismic eventconsidered, the PGA that induces the first elastic buckling is very close for Models B(H/D = 0.63) and C (H/D = 0.95). For Model A (H/D = 0.40), the critical PGA is signifi-cantly smaller for the Parkfield record than for the other two ground motions.

3.5. Comparisons with Dynamic Buckling Analyses

The PGAs that cause elastic buckling obtained from the NLSP is an approximation to thetrue values. Therefore, it is important to verify the accuracy of the proposed numericalprocedure. In this study, the accuracy of the results were verified by a comparison withfull dynamic time history buckling analysis using finite element models that account forgeometric and material nonlinearities and the presence of liquid. To obtain the values ofthe critical PGA in Fig. 18 for a given tank model, between 80 and 100 computer hours arerequired in a Pentium IV computer with a 3.0 GHz processor. This lengthy process wascarefully undertaken to obtain the most accurate possible results until an experimental testprogram can be implemented. These comprehensive dynamic analyses can be used toprovisionally validate the proposed NLSP. In theory, the method could also be verified bycomparing its results with those from experimental works by other researchers. However,this is not a straightforward task because there are very few experimental studies on thedynamic buckling of liquid storage steel tanks subjected to earthquakes.

The dynamic buckling analysis of the tank liquid systems was described in detail inan article by Virella et al. [2006b]. The critical PGA obtained from the analysis was

FIGURE 12 Earthquake records used in this study. (a) 1986 El Salvador earthquake,PGA = 0.69 g; (b) 1966 Parkfield earthquake, PGA = 0.27 g.

0.0 2.0 4.0 6.0 8.0 10.0

time, s–0.80

–0.60

–0.40

–0.20

0.00

0.20

0.40

0.60

Acc

eler

atio

n, g

(a)

Acc

eler

atio

n, g

–0.30

–0.20

–0.10

0.00

0.10

0.20

0.30

0.0 5.0 10.0 10.5 20.0 20.5 30.0

time [s]

(b)

370

375

380

385


calculated by Virella et al. [2006b] for the same tank-liquid systems considered in thisarticle. The accelerograms for the 1986 El Salvador and 1966 Parkfield earthquakes wereused for the comparison.

As shown in Figs. 18 and 19, the PGA obtained from the proposed NLSP and thevalues predicted by the more accurate dynamic buckling analyses are reasonably similarfor all the tank-liquid systems. Notice also that the PGAs for the first elastic bucklingpredicted by the procedure proposed in this paper are either about the same or smaller thanthose calculated from dynamic buckling analyses. This occurred for all the tank-liquidsystems and for both accelerograms. Therefore, it seems that the modified CSM procedurenot only predicted well the elastic buckling acceleration of the tanks, but also providedconservative estimates when the differences were larger, at least for the tank geometriesinvestigated in this paper.

Notice that the PGA for the first elastic buckling mode obtained for both the UBC 97design spectrum and the response spectra of the two accelerograms, the PGA for the firstbuckling mode decreased with the H/D ratio of the tank. This trend is the same as thatobserved for other types of buckling modes induced by dynamic loads, as reported byRammerstorfer et al., [1990, p. 275].

The deformed shapes for the first dynamic buckling mode for models A, B, and Csubjected to the seismic demand from the 1986 El Salvador accelerogram are presented inFig. 20. The static and dynamic buckling analyses yielded similar buckling modes as it

FIGURE 13 Response spectra from earthquake accelerograms for 2% damping ratio.(a) 1986 El Salvador earthquake record; (b) 1966 Parkfield earthquake record.

0.0

0.5

1.0

1.5

2.0

2.5

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

T, s

Sa, g

(a)

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

T, s

0.0

0.2

0.4

0.6

0.8

1.0

1.2

Sa, g

(b)

390

395

400

405

xxxx 17

FIGURE 14 Performance point for the 1997 UBC elastic spectrum obtained with theNLSP. (a) Model A, (b) Model B, (c) Model C.

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.91.0

0.00 0.05 0.10 0.15 0.20

Sd, m

Sa, g

Sa, g

Sa, g

P.P. - first bucklingPSAcrit = 0.87g

(a)

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.00 0.02 0.04 0.06 0.08 0.10 0.12

Sd, m

P.P. - first bucklingPSAcrit. = 0.60g

(b)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.00 0.02 0.04 0.06 0.08 0.10

Sd, m

P.P. - first bucklingPSAcrit. = 0.48g

(c)

TABLE 2 Results for the tank models for the seismic demand from the 1997 UBC elastic spectrum

Model H/D Critical Sa [g] Critical Sd [cm] Δ-radial [cm] Ca [g]

A 0.40 0.87 0.98 0.70 0.35B 0.63 0.60 1.66 1.76 0.24C 0.95 0.48 1.54 2.11 0.19


can be verified by comparing Figs. 9 and 20. For all the tank-liquid systems, bucklingoccurred near the top of the cylinder as indicated in these figures.

A few comments may help to better understand the significance of the resultspresented. The buckling at the top is very common on anchored tanks and it usually occursbefore the better known elephant foot buckling. As a matter of fact, the computer modelused in our study does take into account the possibility of occurrence of elephant footbuckling, but this only occurs in an advanced post-critical state and the buckling at the topoccurs first. The first buckling occurrence, which is the object of study of this paper, is not

FIGURE 15 Performance point for the 1986 El Salvador accelerogram, obtained with theNLSP. (a) Model A, (b) Model B, (c) Model C.

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0.000 0.025 0.050 0.075 0.100 0.125 0.150

Sd, m

Sa, g

Sa, g

Sa, g

Capacity spectrum

Reduced spectrumPGA = 0.33g

P.P.- first buckling

(a)

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

Sd, m



Capacity curve

(b)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.00 0.02 0.04 0.06 0.08 0.10

0.00 0.02 0.04 0.06 0.08 0.10

Sd, m


Capacityspectrum


(c)

410

415

xxxx 19

FIGURE 16 Performance point for the 1966 Parkfield accelerogram, obtained with theNLSP. (a) Model A, (b) Model B, (c) Model C.

0.0

0.2

0.4

0.6

0.8

1.0

Sd, m

Sa, g

Sa, g

Sa, g

Capacityspectrum


P.P. - first buckling

(a)

0.0

0.2

0.4

0.6

0.8

1.0

Sd, m

P.P.- first bucling


Capacity curve

(b)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.000 0.010 0.020 0.030 0.040

0.00 0.02 0.04 0.06 0.08

0.00 0.02 0.04 0.06 0.08

Sd, m


Capacity spectrum


(c)

TABLE 3 Horizontal peak ground acceleration and radial displacements for the tank models using the NLSP and the 1986 El Salvador accelerogram

Model Accelerogram PGA [g] Scale factor Critical PGA [g] Δcrit. [cm] Δcrit/tshell

A 0.69 0.48 0.330 0.70 0.89B 0.69 0.30 0.210 1.76 1.85C 0.69 0.28 0.195 2.11 2.22


necessarily the damaged shape that is usually observed on tanks after an earthquakebecause after the elastic buckling occurs the post-buckling behavior and plasticity takeplace and the final deformed shape of the structure may be different from the initial elasticbuckling deformed shape.

Notice also that unlike silo structures, which are built with a constant thickness,storage tanks are constructed with a tapered wall. The wall thickness at the bottom of thestructure can be about twice the thickness at the top courses. As a consequence, bucklingwill usually initiate at the thinner zones of the cylindrical shell, before occurring at thelower stiffer parts.

4. Conclusions

This article introduced a method for the evaluation of the buckling of steel tanks subjectedto a horizontal seismic base motion. The method is based upon a static buckling analysis

TABLE 4 Horizontal peak ground acceleration and radial displacements for the tank models using the NLSP and the 1966 Parkfield accelerogram

Model Accelerogram PGA [g] Scale factor Critical PGA [g] Δcrit. [cm] Δcrit/tshell

A 0.269 0.85 0.228 0.70 0.89B 0.269 0.81 0.217 1.76 1.85C 0.269 0.68 0.182 2.11 2.22

FIGURE 17 Horizontal peak ground acceleration PGA obtained with the NLSP.

0.00

PGA

[g]

or

Ca

0.050.10

0.15

0.20

0.250.30

0.35

0.40

H/D

Static buckling - NLSPEl Salvador (1986)

Static buckling - CSMParkfield (1966)

Static buckling - NLSP1997 UBC spectrum

0.00 0.20 0.40 0.60 0.80 1.00 1.20

FIGURE 18 Comparison of the NLSP and dynamic buckling results, for the accelero-gram from the 1986 El Salvador earthquake.

PGA

[g]

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.00 0.20 0.40 0.60 0.80 1.00 1.20

H/D

Dynamic bucklingEl Salvador (1986)

Static buckling-NLSPEl Salvador (1986)

420

425

xxxx 21

FIGURE 19 Comparison of the NLSP and dynamic buckling results, for the accelero-gram from the 1966 Parkfield earthquake.

0.00

PGA

[g]

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0.00 0.20 0.40 0.60 0.80 1.00 1.20

H/D

Dynamic bucklingParkfield (1966)

Static buckling-NLSPParkfield (1966)

FIGURE 20 First buckling mode for the tank-liquid systems obtained from dynamicbuckling analyses [Virella et al., 2006a]. (a) Model A, (b) Model B, (c) Model C.

(a) (b)

(c)


instead of the much more expensive dynamic buckling analyses. The proposed method isbased on concepts similar to those used in the CSM applied for the seismic evaluation ofbuildings. For a given response spectrum, the method allows one to calculate the criticalhorizontal PGA that induces buckling in the tank shell. The results were compared withthose obtained from full dynamic buckling analyses to assess the accuracy of the results.

Three typical steel tanks with decreasing height to diameter ratios, identified asmodels A, B, and C, were used to provide numerical examples of the implementation ofthe proposed procedure. The PGAs that induce the first elastic buckling in the tank shellsobtained with the NLSP were similar but lower than those from the dynamic bucklingstudies, in all cases considered. Therefore, the method is able to provide conservativeestimates of the critical peak ground acceleration for the tank geometries considered inthis article. In addition, similar first buckling modes were observed by using both staticand dynamic buckling analyses. In the three tank models, buckling occurred at a regionnear the top of the cylinder. The numerical results show that the PGA for the first elasticbuckling mode decreases with the H/D ratio, regardless of the seismic demand considered.

Acknowledgments

J. C. Virella was supported by a PR-EPSCOR post-doctoral fellowship grantEPS-0223152 for this research. L. A. Godoy thanks the support of the Mid-AmericaEarthquake Center. We would also like to thank the two anonymous reviewers for theircontribution which improved the explanation of several concepts presented in the article.

References

ABAQUS Standard User Manual [2002] Version 6.4. Hibbit, Karlsson and Sorensen, Inc.,Pawtucket, Rhode Island.

American Lifelines Alliance [2001] “Seismic fragility formulations for water systems (ASCE)Part 1 - Guidelines, Part 2 - Appendices,” pp. 60–79.

API 650 [1991] “Welded steel tanks for oil storage,” American Petroleum Institute, Standard 650,Washington, D.C.

ATC-40 [1996] “Seismic evaluation and retrofit of concrete buildings,” Applied Technology Council,Redwood City, California.

AWWA D100 [1984] “AWWA standard for welded steel tanks for water storage,” American WaterWorks Association, Denver, Colorado.

Barton, D. C. and Parker, J. V. [1987] “Finite element analysis of the seismic response of anchoredand unanchored liquid storage tanks,” Earthquake Engineering and Structural Dynamics 15(3),299–322.

Cooper, T. W. and Wachholz, T. P. [1999] “Optimizing post-earthquake lifeline system reliability,”Proceedings of the 5th U.S. Conference on Lifeline Earthquake Engineering (ASCE), Seattle,Washington, 16, 878–886.

EERI Reconnaissance Team [2003] “The San Simeon, California, earthquake, December 22, 2003,”EERI Learning from Earthquakes Reconnaissance Report, Earthquake Engineering ResearchInstitute, Oakland, California, 23–44.

Fischer, D. F. and Rammerstorfer F. G. [1982] “The stability of liquid-filled cylindrical shells underdynamic loading.” In: Buckling of Shells, Ramm, E., Ed., Springer, New York, 569–597.

Handam, F. H. [2000] “Seismic behavior of cylindrical steel storage tanks,” Journal of ConstructionSteel Research 53, 307–333.

Haroun, M. A. and Housner, G. W. [1981] “Earthquake response of deformable liquid storagetanks,” Journal of Applied Mechanics (ASME) 48(2), 411–418.

Hibbit H. D., Karlsson, B. I., and Sorensen, P. [2002] “ABAQUS Theory Manual, Version 6.4,”Pawtucket, Rhode Island.

430

435

440

445

450

455

460

465

470

475

xxxx 23

Housner, G. W. [1963] “The dynamic behavior of water tanks,” Bulletin of the Seismological Societyof America 53(2), 381–389.

Ito, T., Morita, H., Hamada, K., Sugiyama, A., Kawamoto, Y., Ogo, H., and Shirai, E. [2003]“Investigation on buckling behavior of liquid storage tanks under seismic excitation (1st report:Investigation on elephant foot bulge),” Proceedings of the Pressure Vessels and Piping Confer-ence (ASME), Cleveland, Ohio, 466, 193–201.

Jacobsen, L. S. [1949] “Impulsive hydrodynamics of fluid inside a cylindrical tank and of fluidsurrounding a cylindrical pier,” Bulletin Seismological Society of America 39(3), 189–203.

Jain S. K., Lettis, W. R., Murty, C. V. R., and Bardet, J. [2002] “Bjuh, India Earthquake of January 26,2001, Reconnaissance Report,” Earthquake Spectra, Supplement A to volume 18, 257–295.

Lau, D. T., Marquez, D., and Qu, F. [1996] “Earthquake resistant design of liquid storage tanks,” Proceed-ings of the Eleventh World Conference on Earthquake Engineering, Acapulco, Mexico, Paper no. 1293.

Liu, W. K. and Lam, D. [1983] “Nonlinear analysis of liquid filled tank” Journal of EngineeringMechanics (ASCE) 109(6), 1344–1357.

Liu, He. and Schubert, D. H. [1999] “Seismic behavior of anchored, partially anchored andunanchored liquid-storage tanks,” Proceedings of the 8th Canadian Conference on EarthquakeEngineering, Vancouver, Canada, 329–334.

Malhotra, P. K. [2000] “Practical nonlinear seismic analysis of tanks,” Earthquake Spectra 16 (2),473–492.

Morita, H., Ito, T., Hamada, K., Sugiyama, A., Kawamoto, Y., Ogo, H., and Shirai, E. [2003]“Investigation on buckling behavior of liquid storage tanks under seismic excitation: 2nd report –Investigation on the nonlinear ovaling vibration at the upper wall,” Proceedings of the PressureVessels and Piping Conference (ASME), Cleveland, Ohio, 466, 227–234.

Nagashima, H., Kokubo, K., Takayanagi, M., Saitoh, K., and Imaoka, T. [1987] “Experimentalstudy on the dynamic buckling of cylindrical tanks (Comparison between static buckling anddynamic buckling),” JSME International Journal 30 (263), 737–746.

Natsiavas, S. and Babcock, C. D. [1987] “Buckling at the top of a fluid-filled tank during base exci-tation,” Journal of Pressure Vessel Technology (ASME) 109, 374–380.

Rammerstorfer, F. G., Scharf K., Fisher F. D., and Seeber, R. [1988] “Collapse of earthquakeexcited tanks,” Res Mechanica Journal 25, 129–143.

Rammerstorfer, F. G., Scharf, K., Fisher, F. D., and Seeber, R. [1990] “Storage tanks under earth-quake loading,” Applied Mechanics Reviews 43(11), 261–282.

Redekop, D., Mirfakhraei, P., and Muhammad, T. [2002] “Nonlinear analysis of anchored tankssubject to equivalent seismic loading,” Proceedings of the Pressure Vessels and Piping Confer-ence (ASME), Vancouver, Canada, 442, 157–163.

Riks, E. [1979] “An incremental approach to the solution of snapping and buckling problems,”International Journal of Solids and Structures 15, 524–551.

Sakai, F. and Isoe, A. [1989] “Computation and experiment on base-uplift behavior of cylindrical oilstorage tanks during earthquakes,” Proceedings of the Pressure Vessel and Piping Conference(ASME), Honolulu, Hawaii, 157, 9–14.

Sugiyama, A., Setta, K., Kawamoto, Y., Hamada, K., Morita, H., Ito, T., Ogo, H., and Shirai, E.[2003] “Investigation on buckling behavior of liquid storage tanks under seismic excitation(3rd report; proposed design procedure considering dynamic response reduction),” Proceedingsof the Pressure Vessels and Piping Conference (ASME), Cleveland, Ohio, 466, 235–242.

Suzuki, K. [2002] “Report on damage to industrial facilities in the 1999 Kocaeli earthquake,Turkey,” Journal of Earthquake Engineering 6(2), 275–296.

Uniform Building Code [1997] “Structural Engineering Design Provisions” 2, International Confer-ence of Building Officials, Whittier, California.

Veletsos, A. S. [1984] “Seismic response and design of liquid storage tanks, Guidelines for theseismic design of oil and gas pipeline systems,” Technical Council on Lifeline EarthquakeEngineering (ASCE), New York, 255–370, 443–461.

Veletsos, A. S. and Shivakumar, P. [1997] “Tanks containing liquids or solids, in: Computer analysisand design of earthquake resistant structures: A Handbook,” In: Computational MechanicsPublications Beskos, D. E., Anagnostopoulos, S. A., Eds., Southampton, U.K., 3, 725–773.

480

485

490

495

500

505

510

515

520

525

530


Veletsos, A. S. and Yang, J. Y. [1977] “Earthquake response of liquid storage tanks-Advances inCivil Engineering through Mechanics,” Proceedings of the Second Engineering MechanicsSpecialty Conference (ASCE), Raleigh, NC, 1–24.

Virella, J. C., Godoy, L. A., and Suárez, L. E. [2006a] “Fundamental modes of tank-liquid systemsunder horizontal motions,” Engineering Structures 28(10), 1450–1461.

Virella, J. C., Godoy, L. A., and Suárez, L. E. [2006b] “Dynamic buckling of anchored steel tankssubjected to horizontal earthquake excitation,” Journal of Constructional Steel Research 62,521–531.

Virella, J. C., Suárez, L. E., and Godoy, L. A. [2005] “Effect of pre-stress states on the impulsivemodes of vibration of cylindrical tank-liquid systems under horizontal motions,” Journal ofVibration and Control 11(9), 1195–1220.

535

540

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