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Thin–Walled Structures

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Full length article

Buckling behavior of the anchored steel tanks under horizontal and verticalground motions using static pushover and incremental dynamic analyses

M.S. Sobhana, F.R. Rofooeia,⁎, Nader K.A. Attarib

a Department of Civil Engineering, Sharif University of Technology, Tehran, Iranb Department of Structural Engineering, Road, Housing and Urban Development Research Center (BHRC), Tehran, Iran

A R T I C L E I N F O

Keywords:Static bucklingDynamic bucklingSteel tankStatic pushover analysisIncremental dynamic analysisVertical excitation

A B S T R A C T

This paper investigates the static and dynamic buckling of an anchored cylindrical steel tank subjected tohorizontal and vertical ground acceleration. The buckling capacity of the tank is estimated using static pushover(SPO) and incremental dynamic analyses (IDA). Appropriate load patterns due to the horizontal and verticalcomponents of ground excitations are utilized for SPO analyses. The buckling capacity curves and criticalbuckling loads computed using SPO analyses are compared to those obtained from IDA. A proper vertical tohorizontal acceleration ratio (av/ah) for SPO analysis is proposed that leads to good agreement between SPO andIDA results.

1. Introduction

Aboveground flexible cylindrical steel tanks are among the lifelinestructures widely used in various places such as water supply facilities,oil and gas refineries, etc. Different modes of failure and extensivedamages were observed in steel tanks during past major earthquakes.One of the common yet most damaging failure modes of steel tanksunder earthquake loading is shell dynamic buckling mode [1]. Dynamicbuckling of tank shell usually occurs in the forms of elephant footbuckling and diamond shaped buckling modes. The elephant footbuckling mode that is considered as an elasto-plastic type of instability,is described by an outward bulge of the tank shell near to its base. Thistype of buckling of steel tank wall is caused by the interaction of bothcircumferential tensile stress close to yield strength and axial compres-sive stress exceeding the critical stress. Due to cyclic nature of seismicloading, elephant foot buckling often extends around the circumferenceof the tank wall. On the other hand, the diamond buckling is a type ofelastic instability that is caused by severe axial compressive stresses[2,3].

In early 1960s, Housner conducted a research on the dynamicbehavior of the tank-liquid system [4]. He separated the hydrodynamicresponse of a rigid tank- liquid system into the liquid impulsive andconvective (sloshing) parts. The part of liquid which vibrates with thetank wall is known as the impulsive liquid, while the rest of the tankcontent that vibrates independently is considered as the convectiveliquid. Impulsive and convective responses of the tank-liquid system areidentified with a short and long natural period of vibration, respec-

tively. Impulsive and convective modes may be considered uncoupledas their corresponding modal frequencies are well separated [5].

Early experimental investigation on seismic behavior of steel tanksconducted by Clough [6], Manos and Clough [7]. Experimental studyon the buckling behavior of a small tank model made of polyester resinis reported by Shih and Babcock [8]. The tank model was subjected toone horizontal harmonic or simulated seismic base excitation. Theyobserved the occurrence of elasto-plastic buckling near the tank baseand the elastic buckling at the top of the tank wall.

Rotter [9] investigated the elastic-plastic buckling and collapse ofthin cylindrical shells subjected to axial compression load and internalpressure. He described the results and compared those with the existingdesign recommendations. Finally, based on a semi empirical work, heproposed a simple and practical equation for seismic tank design whichadopted by the European Standards ENV1998-4 [10]. Virella et al. [11]numerically investigated the dynamic buckling of ground supported,cylindrical, anchored steel tanks subjected to horizontal ground motion.They estimated the horizontal peak ground acceleration that inducesdynamic elastic buckling and material plasticity of tank wall. Kianoushand Chen [12] evaluated the importance of the vertical component ofbase excitation on the overall seismic response of rectangular concretetanks. They proved that the response of the tank wall due to verticalbase excitation should be considered in tank design. Virella et al. [13]studied the elastic buckling behavior of steel tanks under horizontalcomponent of ground motions using nonlinear static method. Theanalysis procedure was based on a capacity spectrum method that issimilar to the proposed method in ATC [14]. In this way, the minimum

http://dx.doi.org/10.1016/j.tws.2016.12.022Received 21 October 2016; Received in revised form 15 December 2016; Accepted 29 December 2016

⁎ Corresponding author.E-mail addresses: [email protected] (M.S. Sobhan), [email protected] (F.R. Rofooei), [email protected] (N.K.A. Attari).

Thin-Walled Structures 112 (2017) 173–183

Available online 04 January 20170263-8231/ © 2017 Elsevier Ltd. All rights reserved.

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peak horizontal acceleration that causes the elastic buckling at the topof the tank shell is calculated. The critical PGAs for three tank models,estimated by nonlinear static method in comparison to those obtainedfrom dynamic analysis are slightly smaller.

Amiri and Sabbagh-Yazdi [15] examined the effect of tank roof onnatural frequencies and mode shapes of three tall steel tanks using FEAand ambient vibration test results. The results of this research showedthat the influence of roof on the natural frequencies of the axial andvertical vibrational modes is insignificant, but it has remarkable effecton natural frequencies of circumferential modes. Recently, an excellentliterature survey is provided by Godoy [16] in the field of staticbuckling of steel tanks under static and quasi-static loads includinguniform pressure, wind, foundation settlement and fire.

In previous studies such as Virella et al. [11], Buratti and Tavano[17], Maheri and Abdollahi [18] and Djermane et al. [19], the dynamicbuckling of steel tanks under only horizontal component of groundacceleration was investigated. Also, the dynamic analyses of steel tankwere carried out for only one or two earthquake records.

This study presents the static and dynamic buckling analyses ofanchored shallow steel tanks under horizontal and vertical groundmotions using nonlinear static pushover (SPO) and incremental dy-namic analyses (IDA). On that regard, the elasto-plastic buckling of awidely used tank model in the petrochemical industry with a height todiameter ratio (H/D) of 0.40 is considered. Also, the influence ofvertical component of ground acceleration on buckling behavior of steeltanks is investigated. The material and geometric nonlinearities areincluded in the static and dynamic analyses that are carried out usingthe finite element package ABAQUS [20] under horizontal component(uni-directional) and both horizontal and vertical components (bi-directional) of ground motions. The buckling criterion is used toevaluate the critical PGA and load pattern for elasto-plastic bucklingof the tank shell. The buckling capacity curves and critical loads for thesteel tank obtained from the SPO analyses are compared to thoseobtained from the IDA and the necessary conclusions are made.

2. The considered steel tank

The geometry of the tank model considered in this study is similar tothe shallow tank (model A) used by Virella et al. [11,13]. The heightand diameter of the steel tank are 12 and 30 m, respectively. The heightto diameter ratio (H/D) of the tank is 0.40. Since the majority of reportsof earthquake damage to cylindrical steel tanks indicate that the tankscompletely filled with liquid have suffered more damage, thus thefilling level of 90% is assumed in this work [11]. In this study, thethickness of the tank shell is determined based on the seismic designrequirements of the standard API 650 [21]. The seismic design of tankis carried out assuming PGA=0.35 g. The height of each shell course is2 m with the thickness of first course equal to 10 mm and a thickness of8 mm for the rest of the shell courses. The minimum shell thicknessaccording to API 650 [21] is equal to 6 mm but due to designconsideration, a minimum thickness of 8 mm is considered in thisstudy. Since this study focuses on the buckling of the cylinder tank wall,the tank is modeled without a roof. However, the effect of roof's inplane stiffness on the overall tank response is taken into account by arigid body constraint at the upper edge of tank shell. This rigid body is acollection of top wall nodes whose motion is governed by the motion ofa single node, called the rigid body reference node [20]. The geometricconfigurations of the designed steel tank are shown in Fig. 1. Thedesigned tank model is called as “TK040″ to specify the tank height todiameter ratio.

3. Finite element model of the tank-liquid system

The nonlinear static and dynamic analyses of the tank-liquid systemwere carried out using the finite element analysis package ABAQUS[20]. The finite element meshes of steel tank consisted of four-node,

doubly curved quadrilateral shell elements (S4R). Each node of shellelement has three translational and three rotational degrees of freedom.The liquid is modeled using eight-node brick acoustic elements(AC3D8). The acoustic finite element model is based on the linearwave theory and considers the dilatational motion of the liquid. Theacoustic element has only one pressure unknown as degree of freedomat each node.

The tank-liquid interaction was considered using the definition“Surface tied normal contact constraint” between the interfaces ofliquid and tank. This constraint is formulated based on a master-slavecontact method, in which normal force is transmitted using tied normalcontact between both surfaces through the simulation. The sloshingwaves are considered in the liquid model. Assuming the small-ampli-tude gravity waves on the free surface of the liquid, the boundarycondition specified at free liquid surface can be presented as [22]:

Pt

g Pz

∂∂

+ ∂∂

=02

2 (1)

in which P is the hydrodynamic pressure at the free liquid surface.The considered anchored tank model has a fixed connection to theground at its base level. The boundary conditions specified for theliquid-tank finite element model are shown in Fig. 2.

Both geometric and material nonlinearities are considered in thestatic and dynamic finite element analyses. Considering the von Misesyield criterion for the plastic behavior of the shell elements, a yieldstress of 248 MPa and elastic modulus of E=200 GPa is assumed for thesteel material. The water density is considered to be 1000 kg/m3, andthe Rayleigh mass proportional damping was employed for the tankmodel assuming a damping ratio of 2.0%, for the fundamental vibrationmode of the tank-liquid system [11].

Due to the structural symmetry and to reduce the computationalcost, only half of the tank-liquid system is modeled and symmetry planeboundary conditions were employed. Various sensitivity analysesresulted in selection of finite element mesh sizes of the tank and itscontent in longitudinal and circumferential directions as 0.20 m and0.50 m, respectively, to achieve acceptable accuracy. The 3-D finiteelement mesh of TK040 model is illustrated in Fig. 3.

4. Finite element model verification

To verify the tank-liquid finite element model used in this study, the

Fig. 1. Geometric characteristics of the designed steel tank (model TK040).

Fig. 2. Boundary conditions of the 3-D tank-liquid finite element model.

M.S. Sobhan et al. Thin-Walled Structures 112 (2017) 173–183

174

natural frequencies and impulsive pressure distribution computed frommodal analysis are compared to those obtained from the analyticalrelation for the natural frequencies of the tank vibration modes and thenumerical results reported by Virella et al. [11] respectively. Also, thedynamic results are checked with the experimental results reported byManos and Clough [23].

4.1. Modal analysis

In this section, the shallow tank considered by Virella et al. [11,13](model A) is chosen to verify the results of the modal finite elementanalysis. The geometry specifications of the tank model designed in thisstudy are close to the tank (model A) used by Virella et al. [11]. Thetank model was first subjected to hydrostatic pressure and self-weightloads and then modal analysis is carried out. The analytical relation forthe natural period of the fundamental impulsive mode for ground-supported cylindrical tank suggested by API [21] is:

T C H ρDEt

=2000i i L

u (2)

where Ti is the fundamental impulsive period (seconds), HL is the liquidheight (m), tu is the equivalent uniform thickness of tank shell (mm)and is calculated as 10.2 mm, D is the tank diameter (m), E is the elasticmodulus of tank material (MPa) and the dimensionless coefficient Ci

that is dependent on the HL/D ratio, is determined as 6.9. Veletsos andAuyang [24] suggested the following relation for estimating the naturalfrequencies of the first and second sloshing (convective) modes for thecylindrical tank:

fπ

g λR

λ HR

= 12

tanhnn

nL⎛

⎝⎜⎞⎠⎟ (3)

In which fn is the natural frequency of the nth sloshing mode (Hz), Ris the radius of the tank (m), g is the acceleration of gravity and λn is thenth positive root of Bessel function of the first order J1(λ)=0. The firstand second positive roots of the first order Bessel function are

λ1=1.841 and λ2=5.331, respectively. Table 1 shows the naturalperiod of the fundamental impulsive mode and first and secondconvective modes computed by modal analysis for tank model A inthis study (FEA) and those obtained by Virella et al. [11], and analyticalsolutions. The natural periods of the tank computed by modal analysisare in close agreement with those obtained from analytical solutions aswell as the numerical results reported by Virella et al. [11].

The normalized impulsive pressure distribution computed usingmodal analysis for the tank model A at the meridian with maximumdisplacements and those reported by Virella et al. [25] are separatelyshown in Fig. 4. There is a good compatibility between the resultsobtained in this study and the one reported by Virella [25], with amaximum difference of 2.1%.

4.2. Dynamic analysis

To verify the accuracy of the prepared finite element model fordynamic analysis of the tank-liquid system, the results of the experi-mental investigations conducted by Manos and Clough [23] on seismicbehavior of the anchored cylindrical metal tank subjected to horizontalground motion is considered. The metal tank used in shaking table testhas a diameter of 3.66 m, a height of 1.83 m and an aspect ratio (H/D)of 0.5. The level of liquid in the tank was 1.53 m. The metal tank wasmade of aluminum with density of 2700 kg/m3, Young's modulus of71 GPa, Poisson's ratio of 0.3 and yield stress of 100 MPa. The thicknessof the bottom plate, the first shell course and the second shell course is2, 2 and 1.3 mm, respectively. The geometry and dimensions of the tankare shown in Fig. 5. The experimental tank model was subjected to thehorizontal component of El Centro 1940 record that is scaled up to PGAof 0.5 g. Because of similitude requirements, the input ground excita-tion was scaled in time by 1/ 3 .

The 3-D finite element model of experimental anchored tank isshown in Fig. 6. Since the geometry of structure, boundary conditionand loading is symmetric, only one half of the liquid-tank system wasmodeled. In the finite element model, both geometric and materialnonlinearities are considered. The tank model was first analyzed under

Fig. 3. The 3-D finite element mesh of the tank model TK040: (a) tank shell and (b) liquid.

Table 1The results of the free vibration analysis for the tank model A.

Vibration mode Natural period (s) Difference (%)

FEA Virella [11] API [21] Veletsos [24]

Fundamental impulsive 0.213 0.212 0.205 0.00a

3.90b

First convective 6.09 6.14 0.81Second convective 3.37 3.37 0.00

a The difference between the FEA with Virella's results.b The difference between the FEA with API's results.

M.S. Sobhan et al. Thin-Walled Structures 112 (2017) 173–183

175

constant initial loads and then subjected to horizontal ground accel-eration. Dynamic time history analysis of the tank–liquid system isconducted using the explicit time integration method.

The experimental result for the hydrodynamic pressure is used toevaluate the accuracy of the numerical results obtained in this study.Fig. 7 shows the time history of the hydrodynamic pressure measuredon tank wall at levels of 0.05 m and 0.45 m above the tank base, in thedirection of excitation, as well as the finite element results. The resultsof the finite element model for the hydrodynamic pressure are highlyconsistent with the experimental results in terms of shape, peakamplitude of pressure and timing of peak levels.

5. Static pushover analysis of steel tank

The SPO analysis is an efficient method for performance assessmentof new and existing structures while it requires less computation effortin comparison to the dynamic analysis. In this study, the bucklingbehavior of the steel tank is investigated using SPO analyses. Sinceoverall tank response is mainly affected by the impulsive mode relatedforces [17], the impulsive pressure distribution may be considered aslateral seismic load pattern for the tank-liquid system when under ahorizontal strong ground motion. The tank is first analyzed underhydrostatic pressure and self-weight loads and then subjected tomonotonically increasing horizontal loading. The SPO analyses of thesteel tank are carried out for the horizontal excitation alone and bothhorizontal and vertical excitations cases. The hydrodynamic pressuredistribution for the impulsive mode of a rigid tank-liquid system iscomputed as [10,26]:

P ς θ t C ς ρH A t θ( , , )= ( ) ( )cosih i L h (4)

where Pih is the rigid impulsive pressure due to horizontal excitation, ρis the density of a liquid, Ah(t) is the horizontal ground acceleration, HL

is the liquid height, ς z H= / L is the non-dimensional vertical coordinate,θ is circumferential position and C ς( )i is a function that describes theimpulsive pressure distribution along the tank height. The parameterC ς( )i can be calculated as [10,26]:

0

2

4

6

8

10

12

0 0.2 0.4 0.6 0.8 1

Hei

ght (

m)

Normalized pressure

FEA

Virella [25]

Fig. 4. Normalized pressure distribution at the meridian with maximum displacementsfor the fundamental impulsive mode of the tank model A.

Fig. 5. The experimental tank model considered by Manos and Clough [23].

Fig. 6. The 3-D finite element model of the experimental tank: (a) tank shell and (b) liquid.

(a)

)b(

-8

-4

0

4

8

0 1 2 3 4 5 6 7

Pres

sure

(kPa

)

Time (sec)

z = 0.05 mFEMExperimental

-8

-4

0

4

8

0 1 2 3 4 5 6 7

Pres

sure

(kPa

)

Time (sec)

z = 0.45 mFEMExperimental

Fig. 7. Experimental and finite element results for the hydrodynamic pressure timehistories of the anchored tank at different levels.

M.S. Sobhan et al. Thin-Walled Structures 112 (2017) 173–183

176

∑C ς I ν R HI ν R H ν

ν ς( )=2 (−1) ( / )′( / )

cos( )in

nn

n nn

=0

∞1

12 (5)

In which I x( )1 and I x′( )1 are the modified Bessel function of the firstorder and its first derivative, respectively and ν n π=(2 + 1) /2n . Thehydrodynamic pressure distribution for the fundamental impulsivemode of the flexible steel tank under horizontal excitation is obtainedusing modal finite element analysis. The normalized impulsive pressuredistributions along the tank wall for the rigid and flexible tanks underhorizontal excitation at the meridian and in the direction of excitationare presented in Fig. 8. The pressure is normalized with respect to itsmaximum value. The normalized impulsive pressure distribution for therigid tank is similar to that for the flexible tank. This agrees withprevious results obtained by Virella et al. [25]. The maximum impulsivepressure for the rigid and flexible tanks under horizontal groundexcitation occurred at tank's base and a height of 2.45 m from thetank's base, respectively. The hydrostatic and the impulsive pressuresapplied to the tank wall under the ground horizontal excitation, arecombined as

p z θ p z λp z θ( , )= ( ) + ( , )st ih (6)

where p is the total pressure, pst is the axisymmetric hydrostaticpressure, pih is the normalized impulsive pressure for the tank underhorizontal excitation and λ is the increased load factor. The impulsivepressure applied in the second step is multiplied by load factor λ. Theload factor λ, corresponds to the fraction of the impulsive pressurecausing tank buckling.

The hydrostatic and impulsive pressures applied to the tank wallunder both horizontal and vertical ground excitations, are combined as

p z θ p z λ p z θ aa

p z( , )= ( )+ ( , )+ ( )st ihv

h iv

⎛⎝⎜

⎞⎠⎟ (7)

where piv is the normalized impulsive pressure for the tank undervertical excitation which is described below, ah and av are thehorizontal and vertical ground accelerations. The vertical to horizontalacceleration ratio (av/ah) is an important parameter that can affect thecapacity of the tank buckling. Therefore, the influence of the verticalseismic excitation on pushover analysis results may be investigated.

Pushover analysis of the tank subjected to both horizontal andvertical ground excitations is performed by considering the hydrody-namic pressure distributions calculated based on the Eqs. (4) and (8).The hydrodynamic pressure on the rigid tank wall due to verticalground acceleration is calculated as [10]

P ρA H ς= (1 − )iv v L (8)

where Piv is the rigid hydrodynamic pressure due to vertical excitationand Av is the vertical ground acceleration. The normalized hydrody-namic pressures along the rigid tank wall due to horizontal and verticalground accelerations are shown in Fig. 9. The pressures are normalizedwith respect to the density of the tank content, the radius of the tankand the unit acceleration.

5.1. Pushover analysis results

A step-by-step static nonlinear pushover analyses is performed forthe rigid and flexible pressure distributions due to horizontal excitationto capture the buckling of the tank wall. The maximum radialdisplacement of tank wall for the rigid and flexible pressure distribu-tions occurs at the meridian and in the symmetric plane at a level of2.6 m from the tank's base. The first yield of the tank wall occurred atthe first shell course at a level of about1.0 m from the tank's base. Thestatic base shear corresponding to the first yield of the tank wall for therigid and flexible pressure distributions are 14.39 and 15.14 kN,respectively. The maximum radial displacements related to the firstyield of the tank wall (without considering the displacements due to theinitial loads) for the rigid and flexible pressure distributions are0.0064 m and 0.0068 m, respectively. Therefore, the static base shearand the maximum radial displacement corresponding to the first yieldof the tank wall for the rigid pressure distribution are respectively 5.2%and 6.3% less than those for the flexible pressure distribution.

Fig. 10 shows the base shear versus the radial displacement curves(at the critical node where buckling occurs) obtained from the SPOanalysis for the rigid and flexible pressure distributions due tohorizontal excitation. The SPO curve for the rigid pressure is in verygood agreement with that for flexible pressure. Therefore, the bucklingcapacity curve of the steel tank under horizontal excitation may bedetermined with acceptable accuracy using SPO analysis for the rigidimpulsive pressure distribution.

The SPO curve in Fig. 10 indicates that at small radial displace-ments, the slope of SPO curve corresponds to the initial elastic stiffnessof the tank. At higher levels of base shear the slope of pushover curve isreduced indicating the increase in nonlinear behavior of tank wall.Upon increasing the lateral pressure, the structure approaches a criticalbase shear where the stiffness is substantially reduced. The SPO curveswere idealized as a bilinear curve to identify the critical static baseshear as illustrated in Fig. 11. The bilinear idealization technique adoptedhere for estimating the critical level of maximum ground acceleration (PGA)of steel tanks under seismic loadings was first introduced in Ref. [11] whilebeing employed in some recent studies [17–19]. Using the bilinearidealization of the SPO curve, one could determine the critical base

0

2

4

6

8

10

12

0 0.2 0.4 0.6 0.8 1

Hei

ght (

m)

Normalized Pressure

FEM (Flexible)

Eq. (4) (Rigid)

Fig. 8. Normalized impulsive pressure distribution for the rigid and flexible tanks underhorizontal excitation.

0

2

4

6

8

10

12

0 0.2 0.4 0.6 0.8

Hei

ght (

m)

Normalized Pressure, Pi/ R

HorizontalVertical

Fig. 9. Normalized pressure distribution for the rigid tank under horizontal and verticalexcitations.

M.S. Sobhan et al. Thin-Walled Structures 112 (2017) 173–183

177

shear force, i.e., the transition zone from the stable to an unstable state,as 18.51 MN. The maximum radial displacement associated to criticalbase shear is 0.0085 m. The ratio of post-buckling stiffness to the elasticone in the SPO curve is calculated as 4.9%.

A step-by-step SPO analyses were performed for the rigid pressuredistribution due to both horizontal and vertical ground excitations inaccordance with Eq. (7) to determine the buckling state of the tankshell. The influence of the vertical component of ground acceleration onthe buckling capacity curve of the tank is investigated by SPO analysiswith different vertical to horizontal acceleration ratios, av/ah, of 0.33,0.67 and 1.0. The SPO analyses results show that the maximum radialdisplacement of tank wall for av/ah of 0.33, 0.67 and 1.0 occurs at themeridian and in direction of horizontal excitation and the levels of2.6 m, 2.4 m and 2.4 m from the tank's base, respectively. Therefore, forlarger ratios of av/ah, the location of wall critical node with largestradial displacement shifts toward the tank's base in comparison to thecase of horizontal-only excitation. As presented in Fig. 14, the staticbase shear for the first yield of the tank wall for the ratios of av/ah of0.33, 0.67 and 1.0 are 10.74 MN, 8.59 MN and 7.30 MN, correspond-ingly. That indicates a reduction of 25.4%, 40.3% and 49.3% withrespect to the case of horizontal-only excitation, respectively.Therefore, vertical acceleration has significant effect on reducing thestatic base shear for the first yield of the tank wall. The maximum radialdisplacements corresponding to the first yield of the tank wall for theratios av/ah of 0.33, 0.67 and 1.0 are 0.0061 m, 0.0059 m and0.0060 m, respectively. These results indicate a reduction of 4.7%,7.8% and 6.3%, correspondingly, when compared to the case of

horizontal-only excitation. Therefore, vertical ground acceleration hasno significant effect on the maximum radial displacements correspond-ing to the first yield of the tank wall. Moreover, the static base shearcorresponding to the first yield of the tank wall as well as the slope ofthe elastic branch of idealized pushover curve has reduced withincreasing vertical acceleration. So, the radial displacement corre-sponds to the first yield of tank wall has remained almost constant.The SPO curves of the tank under the combined effect of vertical andhorizontal acceleration with the ratio av/ah of 0.33, 0.67 and 1.0plotted in Fig. 12 and are compared to static pushover curve for the caseof horizontal-only excitation. The deformed tank shapes at the end ofthe SPO analyses for the horizontal-only and combined horizontal andvertical excitations cases are illustrated in Fig. 13.

The bilinear idealization of the SPO curves for the tank undercombined horizontal and vertical excitations with the different av/ahratios are demonstrated in Fig. 14. The static critical base shear for thetank subjected to both horizontal and vertical excitations with av/ahratios of 0.33, 0.67 and 1.0 are determined as 15.70, 12.40 and9.81 MN, respectively. Those have been reduced by 15.2%, 33.0%and 47.0% compared to the case of the horizontal-only excitation case,correspondingly. The addition of vertical excitation reduces the slope ofthe elastic and the post-buckling branches of the idealized pushovercurves, where the reduction becomes larger for increasing verticalexcitation intensity. The ratio of post-buckling stiffness to the elasticstiffness for idealized SPO curves with the ratio av/ah of 0.33, 0.67 and1.0 are 2.7%, 2.6% and 3.0% respectively, which is less than thehorizontal- only excitation case.

6. Incremental dynamic analysis

The IDA is a parametric analysis procedure for estimating theseismic demand (damage measure) of structures at different levels ofseismic intensity (intensity measure), and is carried out using a numberof strong ground motions [27]. The IDA is carried out for the consideredtank model under uni-directional and bi-directional excitations, usingthe selected earthquake records scaled for a series of PGAs ranging from0.05 to 0.5 g. For the case of bi-directional (horizontal and vertical)excitations, the horizontal component with larger PGA together withthe vertical component are selected for time history analysis. Also, forbi-directional excitations both records are multiplied by the same scalefactor for each dynamic analysis. Prior to dynamic analysis of the tank-liquid system, the tank model is first subjected to hydrostatic pressureand self-weight loadings. Dynamic analysis of the structural system isperformed using the explicit time integration method [20]. This methodis based on explicit central difference integration rule and benefits fromthe lumped and diagonal mass matrix.

0

5

10

15

20

25

30

0 0.02 0.04 0.06 0.08

Base

shea

r (M

N)

Radial displacement (m)

FlexibleRigid

First yield

Fig. 10. The SPO curves for the rigid and flexible pressure distribution due to thehorizontal excitation.

0

5

10

15

20

25

30

0 0.02 0.04 0.06 0.08

Base

shea

r (M

N)

Radial displacement (m)

FEMBilinear curve

First yield

Critical base shear = 18.51 MN

Fig. 11. Bilinear idealization of the SPO curve for the tank under horizontal excitation.

0

5

10

15

20

25

30

0 0.02 0.04 0.06 0.08

Base

shea

r (M

N)

Radial displacement (m)

. . . .

av/ah=0av/ah=1/3av/ah=2/3av/ah=1

Fig. 12. The SPO curves of the tank under both horizontal and vertical ground excitationswith the ratio av/ah equal to 0.33, 0.67 and 1.0.

M.S. Sobhan et al. Thin-Walled Structures 112 (2017) 173–183

178

6.1. Ground motion selection

As IDA study is based on the time history dynamic analysis, theresults depend on the seismic ground motion to a great extent. Sevenpairs of far-field earthquake records listed in Table 2 are selected fromthe new NGA-West 2 PEER ground motion database [28] to investigatethe seismic performance and buckling behavior of the considered steelflexible tank. All selected ground motions were recorded on dense soilor rock at less than 20 km epicentral distances. According to the NEHRPsite classification scheme [29], the shear wave velocity of the top 30 msoil, Vs30, for very dense soils or a soft rocks is in the range of 360–760 m/s which is the case for the selected records in this study.

6.2. Dynamic buckling criterion

A number of measures have been addressed in the literature foridentifying the buckling of tank shell under the earthquake excitationthat is known as a dynamic buckling. The Budiansky and Roth [30]criterion which is the most used stability criterion for estimating thedynamic buckling load of structural systems is employed here. Based onthis criterion, the dynamic analyses of structure are carried out byvarying the intensity of the applied dynamic load. The time-displace-ment curve is plotted for several intensities of the applied earthquakerecords. According to this criterion, a specific value of the dynamic loadcausing a significant jump in displacement response for small loadincrement represents the dynamic buckling load of the system.

Babcock et al. [8] experimentally investigated the dynamic bucklingof a steel containment shell subjected to horizontal base excitation.Measuring the shell displacements and using the Budiansky-Roth

criterion, they determined the dynamic buckling mode of the steeltanks. Virella et al. [11] and Maheri and Abdollahi [18] employed theBudiansky-Roth criterion to estimate the critical PGA which initiatesthe elastic buckling of tank shell. Djermane et al. [19] evaluated theseismic code provisions relating to the dynamic buckling of steel tanksunder horizontal ground acceleration using dynamic finite elementanalysis. They employed the Budiansky-Roth and the phase planecriterions [19] to determine the critical PGA that initiates the dynamicbuckling of tank wall.

6.3. The results of IDA study

The hydrodynamic pressure induced due to seismic excitation inaddition to the hydrostatic pressure causes the tank shell to bedeformed. Here, the radial displacement of tank walls to the outsideis assumed to be positive. The mean, standard deviation and mean plusstandard deviation of maximum radial displacement of the tank walldue to horizontal-only and combined horizontal and vertical excitationare given in Table 3, and Table 4, respectively, for various PGA levels.The distributions of the mean maximum radial displacement of the tankwall along its height due to the horizontal-only and combinedhorizontal and vertical ground motions in the direction of the hor-izontal excitation are shown in Fig. 15.

From the IDA results provided in Table 3 and Table 4 it can beconcluded that the effect of vertical ground acceleration may eitherincrease or decrease the maximum radial displacement of the tank wall.Also, the results indicate that the mean maximum radial displacementof the tank wall due to bi-directional excitations is more significant thanthe uni-directional one for various PGA levels. The increase in the mean

Fig. 13. The deformed shapes of the tank due to the pushover analyses (a) av/ah=0, (b) av/ah=0.33, (c) av/ah =0.67, (d) av/ah=1.0 (Deformation scale factor is 15).

M.S. Sobhan et al. Thin-Walled Structures 112 (2017) 173–183

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0

5

10

15

20

25

30

0 0.02 0.04 0.06 0.08

Base

shea

r (M

N)

Radial displacement (m)

FEMBilinear curve

First yield

Critical base shear = 15.70 MN

(av/ah=1/3)

0

5

10

15

20

25

30

0 0.02 0.04 0.06 0.08

Base

shea

r (M

N)

Radial displacement (m)

FEMBilinear curve

First yield

Critical base shear = 12.40 MN

(av/ah=2/3)

0

5

10

15

20

25

30

0 0.02 0.04 0.06 0.08

Base

shea

r (M

N)

Radial displacement (m)

FEMBilinear curve

First yield

Critical base shear = 9.81 MN

(av/ah=1)

Fig. 14. The bilinear idealized pushover curves for the tank under both horizontal and vertical ground excitations with (a) av/ah=0.33, (b) av/ah=0.67, (c) av/ah=1.0.

Table 2Characteristics of the earthquake records used for dynamic analysis.

ID Earthquake Year Station Magnitude (Mw) Vs30 (m/s) EpiD (km) PHAa (g) PVAb (g) PVA/PHA

EQ1 San Fernando 1971 Lake Hughes #4 6.61 600 24.18 0.198 0.167 0.84EQ2 Friuli 1976 Tolmezzo 6.50 505 20.24 0.357 0.277 0.78EQ3 Irpinia 1980 Brienza 6.90 561 46.16 0.220 0.203 0.92EQ4 Morgan Hill 1984 Gilroy - Gavilan Coll. 6.19 729 38.73 0.115 0.11 0.96EQ5 Nahanni 1985 Site 3 6.76 605 22.36 0.182 0.144 0.79EQ6 Northridge 1994 Big Tujunga, Angeles Nat F 6.69 550 31.55 0.253 0.179 0.71EQ7 Kocaeli 1999 Gebze 7.51 792 47.03 0.261 0.194 0.74

a Peak horizontal ground acceleration.b Peak vertical ground acceleration.

Table 3The maximum radial displacement of the tank wall due to the horizontal-only excitation.

PGA Ground motions Mean (m) St. Dev. (m) Mean+St. Dev. (m)

(g) EQ1 EQ2 EQ3 EQ4 EQ5 EQ6 EQ7

0.05 0.0019 0.0025 0.0067 0.0022 0.0012 0.0024 0.0021 0.0027 0.0017 0.00440.10 0.0033 0.0046 0.0324 0.0044 0.0021 0.0051 0.0035 0.0079 0.0100 0.01790.15 0.0048 0.0067 0.0470 0.0066 0.0030 0.0077 0.0053 0.0116 0.0145 0.02610.20 0.0063 0.0123 0.0584 0.0088 0.0041 0.0160 0.0068 0.0161 0.0177 0.03380.30 0.0163 0.0374 0.1067 0.0239 0.0061 0.0389 0.0162 0.0351 0.0312 0.06630.40 0.0293 0.0634 0.1320 0.0366 0.0082 0.0557 0.0273 0.0503 0.0374 0.08780.50 0.0400 0.1673 0.1860 0.0468 0.0156 0.1014 0.0407 0.0854 0.0627 0.1481

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maximum radial displacement of the tank wall under the bothhorizontal and vertical ground motions is at least 13.9% for aPGA=0.10g and 84.5% for a PGA=0.20g with respect to the onlyhorizontal case. The mean maximum radial displacement increasedlargely (57.6%) for PGAs above or equal to 0.20g in comparison to thePGAs below 0.20g (25.9%).

The IDA results in Fig. 15 illustrates that the mean of the maximumradial displacement of the tank wall for the case of the uni-directionalas well as the bi-directional excitations occurred at a height of 2.8 mabove the tank base. This maximum displacement is formed at almostmid-height of the second shell course which is thinner than the firstshell course. A jump in the displacement of the tank wall is observeddue to horizontal excitation in Fig. 15(a) for PGA above 0.30 g whilethe displacement jump occurred due to both horizontal and vertical

excitation in Fig. 15(b) for PGA above 0.20 g.Fig. 16 shows the maximum base shear versus the maximum radial

displacement curves (at the critical node where buckling occurs)obtained from IDA for the case of horizontal-only (H), and thecombined horizontal and vertical (HV) excitations. It is concluded thatthe effect of vertical ground acceleration may either increase ordecrease the buckling capacity of the steel tank depending on thefrequency content of the earthquake records. The results at the non-linear region of the IDA curves (for larger PGAs) due to bi-directionalexcitation exhibits more dispersion than the case for horizontal-onlyexcitation.

The median IDA curves for the steel tank under the horizontal-only(H) and both horizontal and vertical (HV) excitations are plotted inFig. 17. The idealized bi-linear equivalent of the IDA curves shown in

Table 4The maximum radial displacement of the tank wall due to the combined horizontal and vertical ground excitations.

PGA Ground motions Mean (m) St. Dev. (m) Mean +St. Dev. (m)

(g) EQ1 EQ2 EQ3 EQ4 EQ5 EQ6 EQ7

0.05 0.0039 0.0030 0.0062 0.0023 0.0024 0.0025 0.0034 0.0034 0.0013 0.00470.10 0.0078 0.0060 0.0297 0.0043 0.0044 0.0048 0.0061 0.0090 0.0085 0.01760.15 0.0039 0.0176 0.0377 0.0063 0.0064 0.0070 0.0206 0.0142 0.0112 0.02550.20 0.0430 0.0329 0.0548 0.0076 0.0131 0.0118 0.0446 0.0297 0.0174 0.04710.30 0.0931 0.0605 0.1073 0.0087 0.0305 0.0211 0.0721 0.0562 0.0346 0.09080.40 0.1464 0.0976 0.1454 0.0123 0.0497 0.0484 0.1057 0.0865 0.0476 0.13410.50 0.2161 0.1658 0.2246 0.0169 0.0670 0.1135 0.1384 0.1346 0.0703 0.2049

0

2

4

6

8

10

12

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14

Hei

ght (

m)

Radial Displacement (m)

PGA=0.05gPGA=0.10gPGA=0.15gPGA=0.20gPGA=0.30gPGA=0.40gPGA=0.50g

0

2

4

6

8

10

12

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14

Hei

ght (

m)

Radial Displacement (m)

PGA=0.05gPGA=0.10gPGA=0.15gPGA=0.20gPGA=0.30gPGA=0.40gPGA=0.50g

Fig. 15. Distribution of the mean maximum radial displacement of the tank wall: (a)Horizontal excitation, (b) Horizontal-Vertical excitations.

0

5

10

15

20

25

30

35

0 0.02 0.04 0.06 0.08

Base

She

ar (M

N)

Radial Displacement (m)

IDA (Horizontal)

EQ1EQ2EQ3EQ4EQ5EQ6EQ7

0

5

10

15

20

25

30

35

0 0.02 0.04 0.06 0.08

Base

She

ar (M

N)

Radial Displacement (m)

IDA (Horizontal and Vertical )

EQ1EQ2EQ3EQ4EQ5EQ6EQ7

Fig. 16. Buckling capacity curves obtained from IDA for the steel tank: (a) Horizontal excitation, (b) Horizontal and vertical excitations.

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Fig. 18, defines the critical dynamic base shear force beyond which thestiffness of the steel tank is drastically reduced. Also, as Fig. 18illustrates, the critical dynamic base shear force of 14.68 MN due tocombined horizontal and vertical excitation (HV) is 23.5% smaller thanthe critical dynamic base shear force of 19.19 MN obtained for the caseof horizontal-only excitation (H).

6.4. Critical PGA for dynamic buckling

Fig. 19 depicts the mean maximum radial displacement of the tankwall at the critical node versus the corresponding PGA for the cases ofuni-directional (H) and bi-directional (HV) excitations. Such bilinearpresentation of the dynamic performance of steel tanks was firstintroduced by Budiansky and Roth [30]. Again, for a PGA higher thanthe critical PGA, the behavior of the steel tank could change to anunstable state with appreciably reduced lateral stiffness. Consideringthe PGA as a damage measure, the dynamic buckling of the steel tankdue to combined horizontal and vertical excitation (HV) occurred for acritical PGA=0.179 g is 31.9% smaller than the critical PGA=0.263 gobtained for the case of horizontal-only excitation (H).

7. Comparison of the results of SPO analyses and IDA

The mean IDA curve due to horizontal excitation (H) is comparedwith the corresponding SPO curve in Fig. 20. The elastic region of SPOcurve matches well with the mean IDA curve. The buckling capacityobtained from SPO analysis in the nonlinear region is less than the

result obtained from the IDA. The critical dynamic base shear estimatedfrom IDA study of the steel tank subjected to horizontal-only excitationis 19.19 MN which is 3.7% larger than the critical static base shear of18.51 MN determined by SPO analysis. The mean dynamic base shearforce corresponding to the first yield of the tank wall obtained from IDAstudy for the case of uni-directional excitation is 15.71 MN that is 7.8%less than the first yield static base shear force of 17.03 determined in

0

5

10

15

20

25

30

0 0.02 0.04 0.06 0.08

Base

shea

r (M

N)

Radial displacement (m)

Mean IDA (H)

Mean IDA (HV)

Fig. 17. The mean IDA curves for the steel tank under horizontal (H), and horizontal-vertical (HV) excitations.

0

5

10

15

20

25

30

0 0.02 0.04 0.06

Base

shea

r (M

N)

Radial displacement (m)

Mean IDA (H)Bilinear curve

Critical base shear = 19.19 MN

0

5

10

15

20

25

30

0 0.02 0.04 0.06

Base

shea

r (M

N)

Radial displacement (m)

Mean IDA (HV)

Bilinear curve

Critical base shear = 14.68 MN

Fig. 18. The idealized mean IDA curves for the steel tank subjected to (a) Horizontal excitation (H), (b) Horizontal and vertical excitation (HV).

0

0.1

0.2

0.3

0.4

0.5

0 0.05 0.1 0.15

PGA

(g)

Radial Displacement (m)

HHV

Critical PGA = 0.263g

Critical PGA = 0.179g

Fig. 19. The PGA- mean maximum radial displacement idealized curves and the criticalPGAs of the tank under uni-directional (H), and bi-directional (HV) excitations.

0

5

10

15

20

25

30

35

0 0.02 0.04 0.06 0.08

Base

She

ar (M

N)

Radial Displacement (m)

Mean IDA (H)

Pushover (H)

Fig. 20. Mean IDA and SPO curves for the steel tank subjected horizontal-only excitation(H).

M.S. Sobhan et al. Thin-Walled Structures 112 (2017) 173–183

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pushover analysis.The mean IDA curve due to both horizontal and vertical excitations

(HV) is compared with the corresponding SPO curves for various ratiosof av/ah in Fig. 21. The elastic region of the SPO curves approximatelymatches well with the mean IDA curve. The critical dynamic base shearforce estimated from IDA study of the steel tank subjected to bothhorizontal and vertical excitations is 14.68 MN which is close to thecritical static base shear force of 18.51 MN for ratio av/ah of 0.33computed by static pushover analysis.

8. Conclusion

This paper investigated the static and dynamic buckling of ananchored, shallow, steel cylindrical tank under the horizontal-only andboth horizontal and vertical ground excitations. The steel tank con-sidered in this study is designed based on the API seismic coderequirements. The SPO analyses and the IDA method are utilized forevaluation of the static and the dynamic buckling of the steel tank,respectively. The appropriate load patterns due to horizontal andvertical excitations are considered for SPO analyses. The results ofstatic pushover analyses were compared to those obtained from IDAstudy to assess the accuracy of the results. Static and dynamic bucklinganalyses indicate that the initiating buckling point of tank shell occursat a region near the base of the tank. The resulting critical base shearforce due to horizontal excitation obtained from SPO analysis was closeto those determined from the dynamic analyses. Therefore, the SPOmethod is able to provide conservative estimates of the critical baseshear for the steel tanks considered in this study. It is found that theeffect of vertical ground acceleration can either increase or decrease theresponse of maximum radial displacement of the tank wall as comparedwith that of horizontal-only excitation depending on the frequencycontent of the earthquake records. The mean critical PGA due to bothhorizontal and vertical excitations found to be 31.9% less than thatunder the horizontal excitation. The critical base shear force due to bi-directional excitation obtained from the SPO analysis for ratio av/ah=0.33 is similar to that obtained dynamic buckling analyses.

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0

5

10

15

20

25

30

35

0 0.02 0.04 0.06 0.08

Base

She

ar (M

N)

Radial Displacement (m)

Mean IDA (HV)Pushover (av/ah=0.33)Pushover (av/ah=0.67)

Pushover (av/ah=1.0)

Fig. 21. Mean IDA and SPO curves for the steel tank subjected both horizontal andvertical excitations (HV).

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