Water Tankunder Dynamic

Engineering Structures 31 (2009) 2426–2435

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Engineering Structures

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Generalized SDOF system for seismic analysis of concrete rectangular liquidstorage tanksJ.Z. Chen, M.R. Kianoush ∗Department of Civil Engineering, Ryerson University, Toronto, Ontario, Canada

a r t i c l e i n f o

Article history:Received 4 December 2008Received in revised form25 May 2009Accepted 26 May 2009Available online 17 June 2009

Keywords:Reinforced concreteLiquid containing rectangular tankSeismicDynamic analysisTank flexibilityImpulsive hydrodynamic pressure

a b s t r a c t

This paper presents a simplified method using the generalized single degree of freedom (SDOF) systemfor seismic analysis and design of concrete rectangular liquid storage tanks. In most of the current designcodes and standards for concrete liquid storage tanks, the response of liquid and tank structures isdetermined using rigid boundary conditions for the determination of hydrodynamic pressures. Also, thelumped mass approach is used for dynamic analysis. However, it has been shown that the flexibility of atank wall increases the hydrodynamic pressures as compared to the rigid wall assumption. On the otherhand, the consistent mass approach reduces the response of liquid containing structures as comparedto the lumped mass approach. In the proposed method, the consistent mass approach and the effectof flexibility of a tank wall on hydrodynamic pressures are considered. The prescribed vibration shapefunctions representing the first five mode shapes for the cantilever wall boundary condition are studied.The application of the proposed shape functions and their validity are examined using two different casestudies including a tall and a shallow tank. The results are then compared with those using the finiteelement method from a previous investigation and ACI 350.3 commonly used in current practice. Theresults indicate that the proposed method is fairly accurate which can be used in the structural designof liquid containing structures. It is also concluded that the effect of the second mode should also beconsidered in the dynamic analysis of liquid containing structures.

© 2009 Elsevier Ltd. All rights reserved.

1. Introduction

Liquid storage tanks as part of environmental engineeringfacilities are primarily used for the treatment of water and sewageas well as other industrial wastes. Normally, they are constructedfrom reinforced concrete in the form of rectangular or circularconfigurations. Early investigation of the dynamic response ofliquid storage tanks subjected to earthquakes was conducted byHousner [1]. An approximate method was proposed to includethe effect of hydrodynamic pressure for a two-fold-symmetric-fluid container subjected to horizontal acceleration as shownin Fig. 1. The hydrodynamic pressures induced by earthquakeswere separated into two parts, impulsive pressure and convectivepressure, and approximated by the lumped added masses. Theadded mass in terms of impulsive pressure is rigidly connectedwith the tank wall and the added mass in terms of convectivepressure is connected to the tank wall using springs. The boundarycondition in the calculation of hydrodynamic pressures was

∗ Corresponding address: Department of Civil Engineering, Ryerson University,350 Victoria Street, M5B 2K3, Toronto, Ontario, Canada.E-mail addresses: [email protected] (J.Z. Chen), [email protected]

(M.R. Kianoush).

0141-0296/$ – see front matter© 2009 Elsevier Ltd. All rights reserved.doi:10.1016/j.engstruct.2009.05.019

treated as rigid. Yang [2] and Veletsos [3] studied the effect ofwall flexibility on themagnitude and distribution of hydrodynamicpressures and associated tank forces. They used Flűggle’s shelltheory to analyze circular tanks and assumed that the tank–fluidsystem behaved like a single degree of freedom system. The baseshear andmoment were evaluated for several prescribedmodes ofvibration. It was found that for tanks with realistic flexibility, theimpulsive forces are considerably higher than those in rigid walls.Most of the research conducted on liquid storage tanks, as

mentioned above has been on circular configurations made ofstructural steel. For rectangular tanks, Haroun [4] presented a verydetailed method of analysis on the typical system of loadings. Thehydrodynamic pressures were calculated by a classical potentialflow approach. The formula of hydrodynamic pressures onlyconsidered the rigid wall condition. Park et al. [5] studied thedynamic behaviour of rectangular tanks using boundary elementmodeling for the fluid motion and finite element modeling forthe solid walls. The time history analysis was used to obtain thedynamic response of fluid storage tanks subjected to earthquakes.Subsequently, they presented an analytical method for thecalculation of hydrodynamic pressures based on a 3D analysisof tanks. They applied the Rayleigh–Ritz method using assumedvibration modes of a rectangular plate with boundary conditionsas admissible functions [6].

J.Z. Chen, M.R. Kianoush / Engineering Structures 31 (2009) 2426–2435 2427

List of symbols

Aa pseudo-accelerationc generalized damping of systemd displacementE modulus of elasticity of materialEc modulus of elasticity of concreteg acceleration due to gravityhi height above the base of thewall to the center of the

gravity of the impulsive lateral forcehc height above the base of thewall to the center of the

gravity of the convective lateral forcehw height from the base of the wall to the center of

gravity of tank wallHL height of fluidHw height of tank wallI moment of inertiak generalized stiffness of systemLx, Lz half inside length of a rectangular tank in the

directions of x and zm effective mass of systemmL effective added mass of hydrodynamic pressuremW effective inertial mass of tank wallm generalized mass of systemmL generalized added mass of hydrodynamic pressuremW generalized inertial mass of tank wallMB base momentMc convective mass of contained liquid per unit width

of a rectangular tank wallMi impulsivemass of contained liquid per unit width of

a rectangular tank wallbase moment due to impulsive hydrodynamicpressure

ML total effective added mass of hydrodynamic pres-sure per unit width of a rectangular tank wall

ML total generalized added mass of hydrodynamicpressure per unit width of a rectangular tank wall

Mw total effective inertial mass of tank wall per unitwidth of a rectangular tank wall

MW total generalized inertial mass of tank wall per unitwidth of a rectangular tank wall

p hydrodynamic pressurep generalized force of systemp effective mass of liquid–tank systemP hydrodynamic forcePi hydrodynamic force due to impulsive componentq the factor of external load appliedt timetw thickness of tank wallTn period of vibration in the nth mode.u, u, u displacement, velocity and acceleration respectivelyug , ug , ug ground displacement, velocity and acceleration

respectivelyv Poisson’s ratiovx, vy velocity component in the directions of x, yVB base shear of tank wallx, y, z Cartesian coordinatesWI internal workWE external workφ velocity potential function for liquidψn shape function for nth modeλi,n wavelength for impulsive pressure for the nthmodeρl mass density of liquidρw mass density of tank wallωn natural frequency of the nth mode of vibrationζ damping ratio

Chen and Kianoush [7] used a procedure referred to as thesequential method for computing hydrodynamic pressures basedon a 2Dmodel for rectangular tanks inwhich the effect of flexibilityof a tankwall was taken into consideration. The sequential methodis a coupling technique in which the two fields of fluid andstructure are coupled by applying results from the first analysis asloads or boundary conditions for the second analysis. Comparedto Housner’s model, it was shown that the lumped mass approachoverestimates the base shear and base moment significantly. Also,Kianoush and Chen [8] investigated the response of concreterectangular liquid storage tanks subjected to vertical groundacceleration in which the importance of the vertical component ofground motion in the overall seismic behaviour of liquid storagetanks was evaluated. It was concluded that the response of tanksdue to vertical ground acceleration can be significant and shouldbe considered in the design.Kianoush et al. [9] and Ghaemian et al. [10] applied the stag-

gered method to solve the coupled liquid storage tank problemsin 3D space. The staggered method is a partitioned solution pro-cedure that can be organized in terms of sequential execution ofa single-field analyser. The scheme of the staggered method is tofind the displacement and hydrodynamic pressure at the end of thetime increment i + 1, given the displacement and hydrodynamicpressure at time i. The staggered method is general and applicableon any shapes of storage tanks and takes into account both convec-tive and impulsive components. Also, the seismic excitation can beapplied in any direction to the system. The results of these studiesalso showed that the lumped mass approach is too conservative interms of tank response. It is worth noting that both the sequen-tial method and the staggered method use the sequential analysisalgorithmwhichmeans that the two domains transfer the data be-tween each other through a sequential procedure.The previous methods developed by the authors as described

above are rather elaborate to be used for the design of liquid con-taining structures. There is a need to develop a simple designapproach since the current lumped mass approach is too conser-vative. As part of this ongoing research effort, in this paper a sim-plified method using the generalized SDOF system is proposed tostudy the dynamic response of liquid storage tanks. The predeter-mined shape functions are used to approximate the vibration ofa liquid containing system in which only one degree of freedomexists in dynamic analysis. The effect of impulsive hydrodynamicpressure is incorporated into the liquid containing system usingthe flexible wall boundary condition based on the consistent massapproach. The dynamic responses of a tall and a shallow rectan-gular tank are studied to illustrate the application of the proposedmethod. The effect of higher modes on the dynamic response ofliquid containing structures is also considered in this study.

2. Theories

Fig. 2(a) shows a 3D rectangular tank. It is assumed that theliquid storage tank is fixed to the rigid foundation. A Cartesiancoordinate system (x, y, z) is used with the origin located at thecenter of the tank base. It is assumed that the direction of groundmotion is in the ‘‘x’’ direction. Furthermore, it is assumed that thewidth of tank 2Lz is sufficiently large so that a unit width of tankcan represent the tank wall, and the corresponding 2D model isshown in Fig. 2(b).

2.1. Hydrodynamic pressure

The fluid filled in the rectangular tank as shown in Fig. 2 isof height, HL above the base. The fluid is considered to be ideal,which is incompressible, inviscid, and with a mass density ρl. Thehydrodynamic pressure is obtained using the velocity potential

2428 J.Z. Chen, M.R. Kianoush / Engineering Structures 31 (2009) 2426–2435

UndisturbedFluid Surface

OscillatingFluid Surface

Hw

LxLxtw tw

ML

HI

LxLx

hi

hc

tw tw

Mc

Mi

ImpulsivePressure Convective

Pressure

y

x

(a) Fluid motion in tank. (b) Dynamic model for rigid wall tank.

Fig. 1. Housner’s model.

LxLx

Lz

Lz

Hw

HL

x

z

y

LxLx

Hw

HL

tw tw

x

y

(a) 3D model of rectangular tank. (b) 2D model of rectangular tank.

Fig. 2. Schematic of rectangular tank.

method as presented in the previous study [7]. Only the impulsivecomponent is considered in this study. The hydrodynamic pressuredistribution on the flexiblewall related to the velocity potential canbe expressed as:

p =∞∑i=1

2 · ρl · tanh(λi,n · Lx)λ·i,nHL

· cos(λi,n · y)

×

∫ HL

0cos(λi,n · y) · u(y, t)dy (1)

where λi,n = (2i − 1)π/2HL. As the series in the above equationconverges very fast, only the first term of the series may be usedfor practical applications.For the rigid tank, Eq. (1) can be simplified further as:

prigid =∞∑i=1

2 · (−1)i · ρlλ2i,n · HL

tanh(λi,n · Lx) · cos(λi,n · y) · ug(t). (2)

The total lateral force P due to hydrodynamic pressure at thespecific time t is calculated by integrating the pressure distributionp along the height of the liquid as follows:

P =∫ HL

0

∞∑i=1

2 · ρlλi,n · HL

tanh(λi,nLx) cos(λi,ny)

×

∫ HL

0cos(λi,ny)u(y, t)dy · dy. (3)

The height at which the lateral force P is applied above the base is:

hi =

(∫ HL

0

∞∑i=1

2 · ρlλi,n · HL

tanh(λi,nLx) cos(λi,ny)

×

∫ HL

0cos(λi,ny) · u(y, t)dy · ydy

)/P. (4)

2.2. Generalized SDOF system

For a systemwith distributedmass and stiffness characteristics,the structure can exhibit an infinite number of degrees of freedomfor a flexural mode of deformations. If there are some predeter-mined shapes to approximate the vibration of the system, and onlythe amplitude of vibration varies with time, then themotion of thesystem can be described by a single variable, or a generalized co-ordinate in which only one DOF exists. The system idealized in thismanner is referred to as generalized SDOF systems [11]. The the-ories of generalized SDOF are well known and simple to be usedin practice. In this section, the generalized SDOF system is appliedto solve the dynamic response of liquid containing structures sub-jected to earthquakes.

2.2.1. Equation of motionFig. 3 shows a tank wall with the distributed mass m(y)

and stiffness EI(y) per unit height subjected to the earthquake

J.Z. Chen, M.R. Kianoush / Engineering Structures 31 (2009) 2426–2435 2429

Hw

HL

tw

y

p(y)

m(y)

EI(y)

u(y)

x

Fig. 3. Concrete rectangular tank in generalized SDF system.

ground motion ug(t). The distributed mass m(y) and stiffnessEI(y) can be either uniform or non-uniform. It is determined bythe configuration of the tank, such as the wall dimensions andthe material properties. As the tank wall possesses an infinitenumber of natural frequencies and corresponding natural modesin vibration, for an exact analysis, the wall must be treated as aninfinite degree freedom system.For simplicity, the distributed mass and stiffness system can be

treated as a generalized SDOF system. The equation of motion fora generalized SDOF system is that:

m · u+ c · u+ k · u = p(t) (5)

where m, c, k, p are defined as the generalized system of mass,damping, stiffness and force respectively. These generalized prop-erties are associated with the selected generalized displacementu(t) as discussed later.

2.2.2. Coupling analysisFor the coupling analysis between the structure and the

contained liquid, the direct couplingmethod is used in the analysis.This means that the responses of the liquid and structure can bedirectly solved through the equation of motion.Thework or energy principle is applied to obtain the equation of

dynamic equilibrium of the liquid–structure system. In this study,the principle of virtual displacement is used to deduce the equationof motion of a generalized SDOF system. The external virtual workdue to hydrodynamic pressure can be expressed in terms of thegeneralized coordinate and assumed shape function as:

δWE2 = −δu ·[∫ HL

0p(y) · ψ(y) · dy

](6)

where p(y) is the function of hydrodynamic pressure distributionalong the height of the wall as discussed in Section 2.1 andψ(y) isthe prescribed shape function discussed later in this paper. Sincethe external virtual work due to hydrodynamic pressure is also afunction of the acceleration of ground motion in an earthquake,after substituting Eq. (1) into Eq. (6), it results:

δWE2 = −δu · [u(t) · f1(y, ψ(y))+ ug(t) · f2(y, ψ(y))] (7)

where f1(y, ψ(y)) and f2(y, ψ(y)) are the two functions related tothe shape function and vertical coordinate along the height of thewall as follows:

f1(y, ψ(y))

=

∞∑i=1

2 · ρlλi,n · HL

tanh(λi,nLx)[∫ HL

0cos(λi,ny) · ψ(y)dy

]2(8)

f2(y, ψ(y))

=

∞∑i=1

2 · (−1)i+1ρlλ2i,n · HL

tanh(λi,nLx)∫ HL

0cos(λi,ny) · ψ(y)dy.

The generalized system of mass, stiffness and force in terms ofthe generalized coordinate and assumed shape function can beobtained as follows:

m =∫ HW

0m(y) · [ψ(y)]2 · dy+ f1(y, ψ(y)) (9)

k =∫ HW

0EI(y) · [ψ(y)]2 · dy (10)

p = ug(t) ·[∫ HW

0m(y) · ψ(y) · dy+ f2(y, ψ(y))

]. (11)

The generalized mass in Eq. (9) can be written as two separateparts, the generalized inertial mass of wall mw and the generalizedadded mass of the liquid due to impulsive hydrodynamic pressuremL as shown below:

mW =∫ HW

0m(y) · [ψ(y)]2 · dy (12)

mL = f1(y, ψ(y)). (13)

Also, from Eq. (11), the effective inertial mass of wall mw and theeffective added mass of the liquid due to impulsive hydrodynamicpressuremL can be defined based on the shape function as:

mW =∫ HW

0m(y) · ψ(y) · dy (14)

mL = f2(y, ψ(y)). (15)

Therefore the equation of motion for coupling the structure andthe contained liquid subjected to an earthquake is obtained bysubstituting Eqs. (9)–(11) into Eq. (5). Then by dividing both sidesof the equation by m, the following relationship is obtained:

u+ 2 · ζ · ωn · u+ ω2n · u = −q · ug(t) (16)

where ω2n = k/m represent the natural frequencies associatedwith its liquid containing system and q is the factor of external loadapplied, that is:

q = p/m =mW +mLmW + mL

(17)

where p is the total effective mass of the liquid–tank system asfollows:

p = mW +mL. (18)

If an estimated damping ratio ζ is assumed, then all the unknownparameters i.e. u, u, u can be determined by an assumed shapefunction. Therefore the infinite degrees of freedom of a liquidcontaining system can be simplified to a generalized SDOF system.It is worth noting that the generalized SDOF system used in this

study is not the same as the lumped SDOF system in Housner’smodel [1]. In Housner’s model, the entire inertial mass associatedwith the impulsive component of the liquid and the tank wall islumped at an effective height above the base of the tank wall. Inthis paper, a generalized coordinate system which is based on theconsistent mass approach is used to approximate the vibrationmode. As a result, the predefined shape function can reducethe infinite degrees of freedom system into a SDOF system. Theefficiency of the generalized SDOF system used for the dynamicresponse of liquid containing structures will be presented usingtwo different case studies.

2.2.3. Shape functionsIn order to obtain the approximate response of the generalized

SDOF systemwith distributedmass and stiffness, the deflections ofthe wall for liquid containing structures relative to the ground canbe assumed to be a single shape function ψ(y) that approximates

2430 J.Z. Chen, M.R. Kianoush / Engineering Structures 31 (2009) 2426–2435

the fundamental vibration mode in the form of:

ur(y, t) = ψ(y) · u(t) (19)

where u(t) is the defined time function related to a singlegeneralized displacement, andψ(y) is the assumed shape function.Therefore the total displacement can be expressed by:

u(y, t) = ur(y, t)+ ug(t) = ψ(y) · u(t)+ ug(t). (20)

Choosing a proper shape function is critical to accuratelyestimating the natural frequencies of liquid containing structureswhen using a generalized SDOF system. In principle, any shapefunction may be selected if it satisfies the displacement boundaryconditions at the supports. However, a shape function that satisfiesonly the geometric boundary conditions does not always ensure anaccurate result for the fundamental natural frequency.It is worth noting that the configuration of concrete rectangular

tanks in terms of boundary conditions could be different for asimple open top rectangular tank used in the study. The prescribedshape functions used in this study are based on the cantileverwall boundary condition. However, the generalized SDOF systemcan be applied to any configuration of concrete rectangular tanksprovided that the proper mode shape functions are used for theapproximation of vibration modes.In this study, the shape function SF1 representing the firstmode

shape based on the cantileverwall condition is selected for analysisas follows.

SF1(y) = ψ1(y) =32y2

H2W−12y3

H3W. (21)

It should be noted that for boundary conditions other than acantilever wall, different shape functions need to be defined.

2.2.4. Peak earthquake responseOnce the fundamental natural frequency of the generalized

SDOF system is known, the response of a liquid containing struc-ture can be easily calculated using the response spectrum methodfor the specific earthquake record or the design response spectrumwhich is specified in the design standards and codes.It is worth noting that the response spectra specified in the de-

sign standards and codes are normally based on the lumped massSDOF system. Since the equation of motion for the generalizedSDOF system for the nthmode indicated in Eq. (15) is similar to thatfor the lumpedmass SDOF system, the response spectrummethodis still applicable for the generalized SDOF system [11].The maximum displacement at the top of the tank wall can be

calculated using the formula:

umax =qω2n· Aa (22)

where Aa is the pseudo-acceleration which can be obtained fromthe response spectrum at period Tn = 2 · π/ωn for the dampingratio ζ .The base shear and base moment can be calculated using the

following relationships:

VB = p · q · Aa (23)

MB = p · q · Aa (24)

where q and p are defined in Eqs. (17) and (18) respectively, and pis defined as:

p =∫ HW

0m(y) · ψ(y) · ydy+

∫ HL

0f2(y, ψ(y)) · ydy

= mW · hW +mL · hi (25)

where hw and hi are the effective heights of inertialmass of the tank

wall and added mass of the liquid due to impulsive hydrodynamicpressure with respect to the tank base, respectively.The hydrodynamic force P and the effective height at which

the hydrodynamic pressure is applied, hi, at the peak earthquakeresponse can be obtained by substituting the acceleration functionEq. (26) into Eqs. (3) and (4) respectively.

u(t) = ψ(y) · q · Aa. (26)As a result, the dynamic response of a liquid containing structurecan be evaluated by the generalized SDOF system.

2.3. Contribution of higher modes

The effect of higher vibration modes on the dynamic responseof liquid storage tanks has generally been ignored in the pastbecause only the rigid wall condition is considered. However,when considering the flexibility of tank walls, the effect of highervibration modes must be included in dynamic analysis. In thissection, the contribution of higher modes to the dynamic responseof liquid storage tanks is studied.Since the generalized SDOF system is defined as the motion

of the system described by a single variable, or a generalizedcoordinate in which only one DOF exists, then, if the highermode vibration functions can be approximated by one generalizedcoordinate, the effect of high modes on the generalized SDOFsystem can be solved. It is worth noting that the generalized SDOFis not the single degree of freedom system based on one lumpedmass. It is the generalized coordinate to approximate the vibrationsystem.In a general condition, the beam vibrating function can be used

as an admissible function to approximate the vibration mode [12].The general form can be expressed as:ψn(y) = an sin(kny)+ bn cos(kny)

+ cn sinh(kny)+ dn cosh(kny) (27)where an, bn, cn, dn are constants and kn is the eigenvalue forthe nth mode. All these parameters are determined based on theboundary conditions.For the cantilever wall condition, the vibration function for the

nth mode is:ψn(y) = (cosh(kny)− cos(kny))− σn(sinh(kny)− sin(kny)) (28)where

σn =cos(knHW )+ cosh(knHW )sin(knHW )+ sinh(knHW )

. (29)

For the first mode, n = 1, kn = 1.875/HW and σn = 0.734. Thisshape function is defined as SF2 expressed as:

SF2 = ψn(y) =(cosh

(1.875

yHW

)− cos

(1.875

yHW

))− 0.734

(sinh

(1.875

yHW

)+ sin

(1.875

yHW

)). (30)

The natural frequencies and normalized modes of a cantileverwall for the first five modes of SF2 are summarized in Table 1.Although, the shape function SF1 is also for the first mode usingthe general vibration equation for the cantileverwall condition, theshape function SF2 for the first mode is more complex in terms ofmathematical expressions as compared to the shape function SF1.For practical application of SF2, the design tables and charts can bedeveloped to simplify the calculation.

3. Dynamic response of tanks

To demonstrate the efficiency of the generalized SDOF systemfor dynamic analyses of liquid containing structures, a tall and ashallow tank that were studied previously [7,8] are used in this

J.Z. Chen, M.R. Kianoush / Engineering Structures 31 (2009) 2426–2435 2431

Table 1Natural frequencies and normal modes for cantilever wall.

Mode, n ψn(y) =(cosh(kny)− cos(kny))− σn(sinh(kny)− sin(kny))σn =

cos(knHW )+cosh(knHW )sin(knHW )+sinh(knHW )

, ωn = Cn√

EIm(y)L4

kn σn Cn

1 1.875/HW 0.734 3.5162 4.694/HW 1.018 22.0353 7.855/HW 0.999 61.6974 10.959/HW 1.000 120.0905 14.137/HW 1.000 199.860

study. Both empty and full tanks are considered. It is worth notingthat since the current dynamic analysis is based on the 2D model,the definition of tall and shallow tanks are the relative terms basedon the assumption of a sufficiently large width of tank 2Lz asdiscussed previously.

3.1. Tall tank

The dimensions and the properties of the tall tank studied inthis paper are as follows:

ρw = 2300 kg/m3 ρl = 1000 kg/m3

Ec = 2.0776× 104 MPa ν = 0.17 Lx = 9.8 mLz = 28 m Hw = 12.3 m HL = 11.2 m tw = 1.2 m.

In this study, the dynamic response of a tall tank obtained from thegeneralized SDOF system is compared with those using the finiteelement method (FEM) and ACI 350.3 as described subsequently.In the study by Chen and Kianoush [7], six models were presentedusing the finite element method (FEM). The mode superpositionmethod was used in Model 4 in which the distributed addedmass of liquid due to hydrodynamic pressure was considered.In Model 5, the time history analysis including the sequentialprocedure was used. The effect of flexibility of the tank wall onthe dynamic response for both the tank wall and hydrodynamicpressure was considered. As the distributed mass was consideredin bothmodels, they represented themore accurate analysis in thatstudy.ACI 350.3 [13] outlines the calculation procedure for dynamic

analyses of concrete rectangular liquid containing structures.Housner’s model is adopted and the lumped added mass approachassuming a rigid wall boundary condition for the calculation ofhydrodynamic pressure is considered. It is worth noting that theresponse modification factor R and the importance factor, I areassumed as 1.0. Therefore, the comparison between the proposedmodel and the ACI 350.3 procedure is on the basis of elasticanalysis.Table 2 shows the generalized mass of tank wall mW for the

first mode based on the shape function SF1. Compared to the totalgeneralizedmass of tankwall MW based on the rigidwall condition,themass percentage obtained from the shape function SF1 is 23.6%of the total generalized mass of the tank wall.For the effective mass of tank wall mW , the mass percentage

obtained from the shape function SF1 in terms ofMW is 37.6%.MWis the total effective mass of the tank wall based on the rigid wallcondition and equals MW for the shape function ψ(y) = 1.As expected, because there are infinite degrees of freedom for

the tankwall, the participation of generalized and effectivemassesof the tank wall for the first mode using the consistent mass is lessthan that using the lumped mass based on the rigid wall boundarycondition. It is worth noting that only the first mode is consideredin this section. The effect of highermodes on the dynamic responseof liquid containing structures is discussed later in this paper.For the full tank, the values of mL and mL, which are the gener-

alized and effective added masses of the liquid due to impulsive

Table 2Summary of dynamic response of tanks.

Parameters Tall tank Shallow tank

Empty tank mW (103 kg) 8.00 1.952% of MW 23.6 23.6mW (103 kg) 12.73 3.105% ofMW 37.6 37.6mWmW

1.591 1.591KW (103 kN/m) 4.823 6.610T1 (s) 0.256 0.108Aa (m/s2) 0.840 g 0.616 g

Full tank mL (103 kg) 4.320 1.137% of ML 7.2 7.0mL (103 kg) 13.46 3.648% ofML 22.5 22.4mW+mLmW+mL

2.126 2.187T1 (s) 0.318 0.136Aa (m/s2) 0.674 g 0.806 gPi (kN) 189.1 63.05Mi (kN m) 1086.2 172.8hi (m) 5.744 2.74hi/HL 0.513 0.498

hydrodynamic pressure, respectively, based on the first modeshape function are calculated as shown in Table 2. In addition, it isassumed that the generalized and effective addedmasses based onthe rigid wall boundary condition, ML and ML, represent the totalgeneralized and effective added masses of the liquid due to hydro-dynamic pressure for the liquid containing system. Since the shapefunction ψ(y) = 1 is applied to evaluate the rigid wall bound-ary condition, the total generalized and effective added masses ofthe liquid due to impulsive hydrodynamic pressure, ML andML areboth equal.It can be found that only part of generalized and effective added

masses of the liquid for the first mode, mL and mL, participate inthe dynamic analysis as compared to the total generalized andeffective added masses ML and ML. The same trend can be foundin the generalized and effective inertial masses of the tank wall forthe first mode shape function, mW and mW , as discussed above.For the shape function SF1, the mass participation factors for thegeneralized and effective added masses of the liquid are 7.2% and22.5% of the total generalized and effective added masses of theliquid due to impulsive hydrodynamic pressure, respectively.The generalized stiffness of the structure is calculated using

Eq. (10). Based on a unit load applied at the top of the wall, thewall stiffness can also be determined using the following simplerelationship:

k =Ec4·

(tWHW

)3. (31)

Based on the above equation, the stiffness of the tank wall is4823 kN/m. This agrees well with the results obtained from theshape functions SF1 and SF2 for the first mode.The fundamental natural period of the empty tank based on the

shape function SF1 is 0.256 s. The fundamental natural period ofthe first mode for the full tank is 0.318 s for shape function SF1.These values are similar to those obtained using the finite elementmethod (FEM) in the previous study.The maximum response of the structure can be obtained using

the pseudo-ground acceleration of the response spectrum. The ElCentro 1940 Earthquake used in the previous investigation is alsoused in this study. The response spectrum for such a record isbased on a 5% damping ratio. The pseudo-ground accelerations Aacorresponding to the periods are listed in Table 2. It should benoted that the actual response spectrum rather than the designresponse spectrum is used in this study. This is because theprevious study was based on a time history analysis using the ElCentro record which is used as the basis for comparison.

2432 J.Z. Chen, M.R. Kianoush / Engineering Structures 31 (2009) 2426–2435

Top Displacement

Dis

plac

emen

t (m

m)

Base Shear

Bas

e S

hear

(kN

)

(a) Top displacement. (b) Base shear.

Bas

e M

omen

t (kN

m)

Base Moment

(c) Base moment.

Fig. 4. Dynamic response of tall tank.

It is worth noting that the pseudo-ground acceleration variesin the range of periods between 0.1 and 0.7 s for this specificsite response spectrum. However, if a standard design spectrum isused, this kind of deviation can be eliminated because the designspectrum is not intended to match the response spectrum forany particular ground motion but is constructed to represent theaverage characteristics of many ground accelerations.Fig. 4 shows the comparison of results obtained from the

generalized SDOF system with those using the FEM and the ACI350.3 in terms of the maximum top displacement, base shear andbase moment. The comparison shows that the response for boththe empty and the full tank using the shape function SF1 agreeswith the FEM results. However, the ACI 350.3 overestimates thebase shear and the base moment. For the full tank, the maximumdisplacement based on the previous study is 32.7 mm for Model4, but it is 26.7 mm for Model 5. This difference in results may beattributed to the response within the small range of periods in theresponse spectrum curve.The fundamental natural periods obtained using ACI 350.3

are 0.131 s and 0.225 s for the empty and full tank conditionsrespectively. The fundamental natural periods are about 49% and29% lower than those using the generalized SDOF system andthe FEM for the empty and full tanks respectively. Since thefundamental natural period is critical to determining the responseof a structure, the estimated fundamental natural periods basedon ACI 350.3 result in higher dynamic response of tall tanks ascompared with those obtained using the generalized SDOF systemand the FEM as shown in Fig. 4.

The hydrodynamic pressure is calculated by using Eq. (1). Thetotal hydrodynamic pressure Pi is calculated by integration of thehydrodynamic pressure along the depth of the liquid as shownin Table 2. The distribution of hydrodynamic pressure along theheight of the wall is demonstrated in Fig. 5. The overall responsefrom this study compares very well with that obtained usingModel 5 in which the effect of wall flexibility was considered inthe analysis. However, hydrodynamic pressure distribution in thelower portion of the tank wall obtained from this study is lessthan that of Model 5. This is due to the difference in magnitudeof acceleration along the height of the tank wall. The accelerationsfor the lower part of tank calculated using generalized SDOF is lessthan those from Model 5 in the previous study.The effective height atwhich hydrodynamic pressure is applied,

hi, and the ratio of the height at which hydrodynamic pressure isapplied to the depth of the stored liquid hi/HL are shown in Table 2.The value of hi is 5.7 m and the hi/HL ratio is 0.51 for the shapefunction SF1. As stated earlier, in current design standards andcodes, Housner’s model [1] is commonly used. The effective heightat which the hydrodynamic pressure is applied is calculated usingthe following equations.For tanks with 2LxHL < 1.333,

hiHL= 0.5− 0.09375

(2LxHL

). (32)

For tanks with 2LxHL ≥ 1.333,

J.Z. Chen, M.R. Kianoush / Engineering Structures 31 (2009) 2426–2435 2433

2

4

6

8

10

0

12H

eigh

t (m

)

10 15 205

Pressure (kPa)

250

Flexible (Model 5)RigidGeneralized SDOF

Fig. 5. Hydrodynamic pressure distribution along the height of the wall—tall tank.

hiHL= 0.375. (33)

Based on the above equations, the value of hi is 4.2 m and thehi/HL ratio is 0.375, by using the above equations in this case. Itcan be seen that the height at which the hydrodynamic pressure isapplied is higher than the one obtained using Housner’s formula.This difference is due to the effect of the flexibility of the tank wallwhich is not considered in Housner’s model.

3.2. Shallow tank

Another example of a shallow tank studied previously [7,8] isanalyzed herein to further verify the efficiency of generalized SDOFsystem on dynamic analysis of liquid containing structures. Thedimensions and properties of the shallow tank are as follows:

ρw = 2300 kg/m3 ρl = 1000 kg/m3 Ec = 2.644× 104 MPaν = 0.17 Lx = 15 m Lz = 30 m Hw = 6.0 mHl = 5.5 m tw = 0.6 m.

The results of the analysis are summarized in Table 2 in thesame forms that were presented for the case of the tall tank. Thehydrodynamic pressure distribution along the height of the wallis shown in Fig. 6. A similar trend in the behaviour to that of thetall tank is observed for the shallow tank. However, there are smalldifferences between the results of the prescribed shape function ascompared to the results of the previous study using the FEM. Thiscan be attributed to the difference in the periods of vibration forthe response spectrumanalysis. If similar periodswere used for thetwo cases, the difference of results would be very small as shownin Table 3.

3.3. Effect of high modes

To study the effect of higher modes, both the tall and shallowtanks studied previously are analyzed in this section. The designresponse spectrum based on ASCE 7-05 [14] is used to obtain thedynamic response of liquid storage tanks. The site is assumed to bein the west coast of US inWashington State and the parameters forthe design response spectrum are:

(1) Short period maximum spectral response acceleration: Ss =1.25;

1

2

3

4

5

0

6

Hei

ght (

m)

105 150Pressure (kPa)

Flexible (Model 5)RigidGeneralized SDOF

Fig. 6. Hydrodynamic pressure distribution along the height of the wall—shallowtank.

(2) At a period of 1 s, the maximum spectral response; accelera-tion: S1 = 0.60

(3) Site class B.

The results of the analysis are summarized in Tables 4 and 5. For thefirstmode, the results obtained from shape function SF2 are similarto those obtained from the shape function SF1. For the empty tank,the fundamental periods obtained from SF2 are 0.260 s and 0.110 sfor the tall and shallow tanks, respectively. This is consistent withthe values of 0.256 s for the tall tank and 0.108 s for the shallowtank based on the shape function SF1.For the full tank, the fundamental periods obtained fromSF2 are

0.326 s and 0.139 s for the tall and shallow tanks, respectively. Thisis also consistent with the values of 0.318 s for the tall tank and0.136 s for the shallow tank based on the shape function SF1. Thedynamic response of liquid storage tanks is also calculated basedon the design response spectrum as shown in Tables 4 and 5. Itshould be noted that these results are different form those shownin Fig. 4 since the El Centro earthquake response spectrum wasused in producing the results in that figure. The results obtainedfromshape function SF1 agreewellwith those obtained fromshapefunction SF2 for the first mode.The results obtained from the second to the fifth mode shapes

are also presented in Tables 4 and 5. For the highermodes, a param-eter referred to as the modal participation factor is calculated us-ing Eq. (17). This factor is used to evaluate the degree to which thenth mode participates in the response. Compared to other build-ing structures, the participation factor in the dynamic analysis ofliquid storage tanks includes not only the structure but also theadded mass of the liquid due to hydrodynamic pressure. Table 4shows that the modal participation factor decreases significantlyfor the empty tank from 1.566 to 0.868 compared to the full tankfrom 2.074 to 1.649 corresponding to the first mode and the sec-ond mode, respectively. This indicates that the added mass of theliquid due to impulsive hydrodynamic pressure affects the modalparticipation factor significantly and the second mode shape mustbe considered in the dynamic response of liquid storage tanks.Tables 6 and 7 show the combination of dynamic responses

of liquid storage tanks for the first two modes for the tall andshallow tanks, respectively. The Square Root of Sum of Square(SRSS) method is used for the combination. It can be seen thatfor the empty tank the contribution from the second mode is not

2434 J.Z. Chen, M.R. Kianoush / Engineering Structures 31 (2009) 2426–2435

Table 3Comparisons of results of analysis—shallow tank.

Cases SF1 Model 4a Model 5a ACI 350.3

Empty tank T1 (s) 0.109 0.109 – 0.055Aa (m/s2) 0.604 g – – 0.473 gdmax (mm) 2.84 3.03 2.87 –VB (kN) 29.3 33.0 33.4 38.4MB (kN m) 128.7 143.5 138.5 115.2

Full tank T1 (s) 0.148 0.148 – 0.097Aa (m/s2) 0.657 – – 0.665 gdmax (mm) 7.82 5.91 4.50 –VB (kN) 95.1 78.9 78.7 167.9MB (kN m) 333.2 300.0 240.2 396.9Pi (kN) 51.4 – – –Mi (kN m) 140.8 – – –

a [7].

Table 4Summary of dynamic response of tall tank for higher modes.

Mode n = 1 (SF1) n = 1 n = 2 n = 3 n = 4 n = 5

Empty tank mW /mW 1.591 1.566 0.868 0.509 0.382 0.283KW (103 kN/m) 4.823 4.969 195.144 1530 5815 16 050Tn (s) 0.256 0.260 0.041 0.015 7.553× 10−3 4.568×10−3

Aa (m/s2) 0.833 g 0.833 g 0.547 g 0.411 g 0.373 g 0.357 gdmax (mm) 21.56 21.85 0.202 0.011 2.021× 10−3 5.237×10−4VB (kN) 165.45 170.02 34.29 8.857 4.492 2.379MB (kN m) 1492 1519 88.22 13.88 6.83 2.07

Full tank mL (103 kg) 4.32 4.876 10.02 6.375 2.523 1.629% of ML 7.2 8.1 16.7 10.6 4.2 2.7mL (103 kg) 13.46 14.43 23.16 10.13 7.594 7.103% ofML 22.5 24.1 38.7 16.9 12.7 11.9mW+mLmW+mL

2.126 2.074 1.649 0.972 0.989 0.939Tn (s) 0.318 0.326 0.061 0.020 8.613× 10−3 4.988×10−3

Aa (m/s2) 0.833 g 0.833 g 0.651 g 0.438 g 0.378 g 0.359 gdmax (mm) 44.362 45.574 0.999 0.041 6.888× 10−3 2.084×10−3VB (kN) 454.832 469.739 321.389 60.34 39.614 31.434MB (kN m) 3337 3409 1352 102.23 93.283 88.67Pi (kN) 233.767 244.523 243.842 42.306 27.838 23.491Mi (kN m) 1343 1396 1152 73.967 75.373 81.759hi (m) 5.744 5.710 4.726 1.748 2.708 3.480hi/HL 0.513 0.510 0.422 0.156 0.242 0.311

Table 5Summary of dynamic response of shallow tank for higher modes.

Mode n = 1 (SF1) n = 1 n = 2 n = 3 n = 4 n = 5

Empty tank mW /mW 1.591 1.566 0.868 0.509 0.382 0.283KW (103 kN/m) 6.610 6.810 267.44 2097 7970 22 000Tn (s) 0.108 0.110 0.017 6.234× 10−3 3.187× 10−3 1.927×10−3

Aa (m/s2) 0.833 g 0.833 g 0.422 g 0.366 g 0.350 g 0.343 gdmax (mm) 3.84 3.89 0.028 1.803× 10−3 3.374× 10−4 8.954×10−5VB (kN) 40.35 41.47 6.45 1.92 1.03 0.56MB (kN m) 177.55 180.75 8.10 1.47 0.76 0.24

Full tank mL (103 kg) 1.137 1.284 2.667 1.609 0.650 0.410% of ML 7.0 7.9 16.4 9.9 4.0 2.5mL (103 kg) 3.648 3.91 6.266 2.826 2.124 1.922% ofML 22.4 24.0 38.4 17.3 13.0 11.8mW+mLmW+mL

2.187 2.132 1.702 1.055 1.077 1.011Tn (s) 0.136 0.139 0.026 8.323× 10−3 3.658× 10−3 2.110×10−3

Aa (m/s2) 0.833 g 0.833 g 0.469 g 0.377 g 0.352 g 0.344 gdmax (mm) 8.35 8.58 0.139 6.840× 10−3 1.259× 10−3 3.845×10−4VB (kN) 120.6 124.6 63.1 15.125 10.806 8.533MB (kN m) 422.5 431.6 129.2 14.232 13.605 12.160Pi (kN) 65.2 68.1 49.0 11.02 7.894 6.555Mi (kN m) 178.5 185.5 111.6 11.09 11.445 11.312hi (m) 2.74 2.723 2.275 1.007 1.450 1.726hi/HL 0.498 0.495 0.414 0.183 0.264 0.314

significant. The base shear by including the second mode onlyincreases by 2% and 1.2% compared to those obtained from the firstmode shape for the tall and shallow tanks, respectively. However,it increases by 21.2% and 12.1% for the tall and shallow tanks,respectively in the full tank condition. Also, the contribution of

the second mode shape for the tall tank is more than that forthe shallow tank. To investigate the effects of higher modes, theresults of analysis including the combination of the first threemodes using the SRSS method are presented in Tables 6 and 7. Itcan be observed that there is no significant increase in response

J.Z. Chen, M.R. Kianoush / Engineering Structures 31 (2009) 2426–2435 2435

Table 6Combination of response of higher modes—tall tank.

Items Response n = 1 n = 2 n = 3 2 modes 3 modesCombination Increase % Combination Increase %

Empty tank dmax (mm) 21.850 0.202 0.011 21.851 0.00 21.851 0.00VB (kN) 170.02 34.29 8.86 173.44 2.01 173.67 2.15MB (kN m) 1519 88.22 13.88 1521.56 0.17 1521.62 0.17

Full tank dmax (mm) 45.574 0.999 0.041 45.585 0.02 45.585 0.02VB (kN) 469.74 321.39 60.34 569.16 21.17 572.35 21.84MB (kN m) 3409 1352 102.2 3667.3 7.58 3668.7 7.62

Table 7Combination of response of higher modes—shallow tank.

Items Response n = 1 n = 2 n = 3 2 modes 3 modesCombination Increase % Combination Increase %

Empty tank dmax (mm) 3.89 0.028 0.002 3.89 0.00 3.89 0.00VB (kN) 41.47 6.45 1.92 41.97 1.20 42.01 1.31MB (kN m) 180.75 8.10 1.47 180.93 0.10 180.94 0.10

Full tank dmax (mm) 8.58 0.139 6.840× 10−3 8.58 0.01 8.58 0.01VB (kN) 124.6 63.1 15.1 139.7 12.09 140.5 12.75MB (kN m) 431.6 129.2 14.2 450.5 4.38 450.7 4.44

of the tall and shallow tanks as compared to those including thecombination of the first twomodes. Therefore, it can be concludedthat the secondmode should be considered in the dynamic analysisof liquid storage tanks, especially for tall tanks and the effect ofthird and higher modes can be ignored.

4. Conclusions

A simplified method using the generalized SDOF systemis presented to determine the dynamic response of concreterectangular liquid storage tanks. Only impulsive hydrodynamicdynamic pressure is considered in this study. The theoriesare based on the well-known principles of dynamic analysisin engineering practice. The consistent mass and the effectof flexibility of a tank wall on hydrodynamic pressures areconsidered. The advantage of the proposed method over othermethods such as the finite element method is its simplicity whichcan be used in the design practice. The proposed method iscomparable to the lumped mass approach currently being used interms of its simplicity.The prescribed shape functions representing the first five mode

shapes for the cantilever boundary condition are used for analysis.They represent the typical open top rectangular tanks commonlyused in water and waste water treatment plants. The proposedmethod can be easily extended to cover other types of rectangulartanks having different boundary conditions.A tall and a shallow liquid storage tank studied previously

are analyzed to demonstrate the efficiency of the generalizedSDOF system applied for the dynamic analysis of liquid storagetanks. Comparing the results obtained using the generalizedSDOF system as proposed in this study with those obtainedusing the finite element method from the previous investigationshows that the proposed method can provide sufficiently accurateresults. However, the results obtained using ACI 350.3 are notconsistent with the results of this study. The differences arein the fundamental natural frequencies for both empty andfull tank conditions. Also, the current practice overestimatesthe base shear and base moment, which is mainly due to theassumptionof lumpedmass approximation. It is concluded that theproposed shape functions SF1 and SF2 are appropriate functions to

approximate the response of liquid storage tanks for the cantileverwall boundary condition.The effect of higher modes on the dynamic response of liquid

containing structures is studied. It is concluded that the inclusionof the first two modes can provide sufficiently accurate results.This study also recommends that the effect of the flexibility of atank wall should be considered in the calculation of hydrodynamicpressures for concrete rectangular tanks. It is also recommendedto use the design response spectrum method when using thegeneralized SDOF system for a dynamic analysis of liquid storagetanks.

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