Seismic Response of Liquid-Filled Tank with Baffles

Mohammad Reza Shekari

Department of Civil Engineering, Yasouj University,Yasuj,Iran

Seismic Response of Liquid-Filled Tank with Baffles

Mohammad Reza Shekari＊

Department of Civil Engineering, Yasouj University,Yasuj,Iran

In thispaper, the effects of a rigid baffle on the seismic response of liquid in a rigid cylindrical tank are evaluated. A baffle is an annular plate which supplies a kind of passive control on the effects of ground excitation. The contained liquid is assumed incompressible, inviscid and has irrotational motion. To estimate the seismic response, the method of superposition of modes has been applied. To analyze the rigid tank response, Laplace′s equation is considered as the governing equation of the fluid domain, in both time and frequency domains. The boundary element method (BEM) is employed to evaluate the natural modes of liquid in a cylindrical tank. To gain this goal, the fluid domain is divided into two upper and lower parts partitioned by the baffle. Linearized kinematic and dynamic boundary conditions of the free surface of the contained liquid have been considered.

seismic response; liquid storage tank;liquid-filled tank;rigid baffle;boundary element method (BEM)

1 Introduction

Recently, the civil engineering community has been concerned with the development and performance of ingenious design concepts for seismic protection of structures, particularly for the control of earthquake effects on buildings.

To safeguard the liquid storage tanks against the excitation of severe earthquakes, they may regularly be strengthened which results in higher magnitude of seismic energy attraction. Seismic isolation, as another alternative approach for this purpose, has been introduced in recent years in some practical projects alongside a number of researches. Several experimental and numerical investigations on base isolated tanks have revealed that reduction in hydrodynamic pressures and increase in water surface displacement are the consequences of using isolation techniques (Choet al., 2004; Shrimali and Jangid, 2004; Shekariet al., 2009).

Seismic response reduction systems need not be installed between the base and bottom of the tank. Several attempts have been made to locate them in different parts of the structure, either in the form of additional response reduction masses, or dampers, friction devices, etc.

In liquid tanks, breaking of surface waves, while is highly dependent on ground excitation amplitude and frequency, isthe main mechanism of energy dissipation. Liquid dampers have been used in space satellites and marine vessels. The amount of additional damping can increase with low viscosity of the liquid, with a smooth bottom of the container, and with a sufficient space between the liquid and the roof of the container. Another approach to the response reduction systems is the coupling of special devices, one alongside another with different stiffness, and intermediate energy absorbing systems.Vondorn (1966) studied the damping effect of the bottom boundary layer on liquid motion. Miles (1958) also investigated the ring damping of free surface oscillations in a cylindrical tank.

Several obstacles like baffles can be supported around the tank periphery and positioned slightly below the liquid surface. Baffles are known as devices for preventing sloshing effects in moving liquid tanks and several analyses have been performed in this field. Fluid separation around baffles causes energy dissipation and reduction in sloshing magnitude and consequent hydrodynamic pressures. Several researches have been performed in this regard, particularly on their application in fuel tanks of space vehicles whose stability is very sensitive to uncontrolled excitations.

Results show that the ring baffles in cylindrical tanks significantlyaffect the frequency and damping ratio of sloshing mode. The comparisons in circular-cylindrical tanks shows that the resonant frequency can be up to 15% higher than the unbaffled tank value when a horizontal ring baffle intersects the liquid surface (Dodge, 2000; Garza and Abramson, 1963).

The effects of baffle on the free and forced vibration of liquid storage tanks were studied by Gedikli and Erguven (1999). Gedikli (1996) developed a variationally coupled BEM-FEM to analyze dynamic response, including free-surface sloshing motion of liquid in cylindrical storage tanks with/without baffles subjected to horizontal ground motion. Fluid-structure interaction analysis of 3D rectangular tanks by a variationally coupled BEM–FEM was solved by Kohet al.(1998).

The boundary element method can be applied to evaluate the natural frequencies and the natural modes of the shaking liquid. The technique of superposition of the modes has then been used for the seismic scrutinizes (Hunt, 1987). Lately, Askari and Daneshmand (2009) inspected the coupled vibration of a partially fluid-filled cylindrical tank with an internal body analytically and formulated the velocity potential in terms of eigenfunction expansions appropriate to two distinct fluid regions which can be matched across theircommon verticalboundary(an artificial vertical boundary). Mikelis and Journee (1984) performed some experiments on the cargo tanks used to transport liquid cargo in ships. Results of their research show that the integration of pressures around the tank walls yields to overall forces and moments that are transmitted by the liquids onto the tank structure and consequently on the ship. Nielsen (2003) studied a variety of depths and radiuses in an excited container including sloshing. The investigation, on sloshing dealtwith pressure differences with respect to depth. In the present research, to assess the effectiveness of baffle for liquid oscillations, the forces acting on the foundation of the baffe-tank system, caused by the hydrodynamic pressure of the fluid, is determined by employing the boundary element method.

2 Fundamental equations

The cylindrical coordinate system (r,θ,z) for the baffle tank system is fixed as revealed in Fig. 1. A Cartesian coordinate system is fixed at the bottom of the tank.To make easy the definition of shear force and the overturning moment acting on the foundation are provided. The relation between these two co-ordinate systems is:

Fig.1 cylindrical liquid tank with a baffle and coordinate systems.

To analyze the behavior of the liquid velocity potential is used as in references (Aslamet al., 1979; Hunt and Priestley, 1978). A harmonic boundary value problem can be represented by using the velocity potential as follows:

in which,φis the velocity potential. In order to solve the Laplace equation, the following appropriate boundary conditions are used.

a)Linearized kinematic and dynamic boundary conditions of the free surface of the contained liquid are considered as follows:

b) Assuming the tank wall to be rigid, the boundary condition at the fluid-structure interface is given by

in which,φis the gravitational acceleration, andnthe outward normal vector at the tank wall. The hydrodynamic pressure acting on the tank wall may be obtained employing the linearized Bernoulli equation as

Furthermore, the hydrodynamic pressure at the liquid free surface is calculated by

whereηis the sloshing height, andρthe mass density of the liquid.

Superposition of modes

The velocity potential field of the liquid in the cylindrical tank can be written as

whereφndenotes the value of the velocity potential on the plane ofθ=0 andψknis the weighting factor.kandnare the numbers for the modes in the circumferential and radial directions, respectively. Casesk=0 andk=1 are related to axially symmetric and asymmetric mode shapes, respectively. In the present paper, for the sake of simplicity and sufficiency, only the termk=1 has been considered, so omitting indexkfrom equation (7) one has

For the modes of natural vibration, by takingψkn= sinωntanda(t)=0 then substituting Eq.(8) into Eq.(3), the following condition for the nth mode is obtained:

in which,ωnis the natural frequency of thenth mode.

The velocity potential field for the liquid under the effect of the recorded earthquake acceleration can be estimated by the use of mode shapes and natural frequencies of natural vibration. Substituting equations (8) and (9) into equation (3) and using partial integration of resulted equation, the following equation is obtained with respect to time

here

whereГfsis the intersection of liquid surface plane andθ=0 plane. Duhamel’s integral (11) has been numerically calculated by using the trapezoidal integration rule.

3. Boundary element method

The boundary integral equation form of Eq. (2) can bewritten for any mode shape of the velocity potential, upon omitting indexnas (Brebbia and Dominguez, 1992)

in whichG*is the free space Green function for the axially-asymmetric problem, i.e. cosθtype.αpis identified by the position of the source pointP(Brebbia and Dominguez, 1992).The boundary element method with the constant elements is applied for the solution of the initial value problem. By estimating the integrals in Eq.(12) over constant boundary elements, the following linear system of equations is acquired

whereis called the free space Green function which is the potential on the pointSdue to the unit source placed on pointP. Discretized form of boundary element equations of the liquid region may be written and separate into the following expressions, according to the free surface and tank nodes of the liquid region (Shekariet al., 2009):

in which, the subscriptstandfshow the boundary nodes on the tank wall and liquid free surface, respectively.

4 Illustrative numerical analyses

When the mode shapes and eigenfrequencies of Eq. (14) are determined, the shear force and the overturning moment at the bottom of the tank subjected to the horizontal ground motion can be determined. Duhamel′s integral (11) is numerically evaluated by using the trapezoidal integration rule.Only a few of the mode shapes corresponding to the smallest eigenfrequencies are needed. The most effective modes, in response to the ground excitation, correspond to the smallest eigenfrequencies, because of the fact that they always include most of the whole system energy. In the present study numerical examples were assessed by using only the first two mode shapes. It has been checked numerically that these two modes provide sufficient exactness.

4.1. Liquid in the cylindrical tank

To assess the validity of the present algorithm, some comparisons are made with the published data of Kim and Lee (2005). The dimensions of the slender tank are: radiusR=1 m, baffle thicknesst= 4 mm,Ri/R=0.7 andH=0.2R. Fig. 2 displays the sloshing frequency variation with various liquid levels for the baffled storage tank. It is observed that the discrepancy between the results is less than 3% for sloshing modes and the present algorithm performs well.

Fig. 3 shows the natural frequencies of the liquid in the cylindrical tank for different tank geometry aspect ratios (height to radius) and for different radial wave numbersn.

From the Figure, it is considered that the natural frequency corresponding to the first mode reaches its limit value at aboutH/R＞0.9, while the analogous value for the second mode reaches its limit value at aboutH/R＞0.3. Likewise, it is obvious that the limits in relation to the highest modes will occur at smaller ratiosH/R.

Fig. 4 reveals the results of a parametric study on the effects of baffles with different dimensions on the natural frequencies of the liquid in cylindrical tank with the aspect ratio of 1.2.It is considered from the figure that, for the baffled cylindrical tank withRi/R=0, assumingn=1, the natural frequency is 0.485 rad/sec, while the analogous value for the cylindrical tank without baffle is 0.492 rad/sec when the aspect ratio is 0.25. The above result indicates that, when the internal radius of the baffle vanishes, the baffle separates the liquid domain into two domains. In this case, the liquid under the baffle behaves like a rigid body and has no vibration, because it has no moving surfaces. Liquid in the upper domain has natural vibrations, because it has a free surface. Similar situations are true for all of the points with the same numbers in the figures, because they indicate equal situations in physical meaning.

Fig. 2 Variation of the sloshing frequencies with various liquid levels

Fig. 5 shows the effect of location on the natural frequencies of the liquid in cylindrical tanks with different aspect ratios.

As a result, Figs. 4 and 5 show that as the depth of the baffle decreases, the effect of the baffle on the frequency is more considerable.

Fig. 3 natural frequencies for different aspect ratios

Fig. 4 Variation of natural frequencies versusRi/RH/R=1.2;h/H=0.25

Fig. 5 Variation of natural frequencies versusRi/R;H/R=1.2;h/H=0.1

4.2. Analyses in the time domain

A time-history analysis for different aspect ratios is performed using the 1994 Newhall earthquake as the input ground motion. In order to achieve the dynamic response of the structure in the time domain, the inverse Fourier transformation is used.

4.2.1 Unbaffled tank seismic response

The peak shear forces and the overturning moments at the bottom of the tank without a baffle, for different aspect ratios, are given in Fig.6. Some of the liquid, near to the base of the tall tank, has a rigid nature like a solid. The liquid surface has a behavior like a spring-mass system. A spring-mass system is the mechanical model (Shrimali and Jangid, 2004; Shekariet al., 2009).

Fig. 6 Foundation forces at the bottom of the cylindrical tank considering various aspect ratios

4.2.2. Baffled tank seismic response

Any of the foundation forces can then be depicted by two components that are caused by the rigid lumped mass and the sloshing part. The liquid below the baffle behaves as a rigid part. The rigid part leads to larger shear force than the sloshing part. Results indicate, as the ratioh/Hdecreases, the peak base shear at the bottom of the tank will increase (see Fig.7)and, in reality, the overturning moment will decrease (see Fig.8).

If the inner radius of the baffle vanishes, the liquid inside the volume surrounded by the rigid surfaces behaves, of course, like a solid.

When the baffle is located as near as possible to the free surface of the liquid, it slightly affects the base shear and overturning moment at the bottom of the tank (see Figs. 7 and 8).

Table 1Maximum seismic base shears and overturning moments, for different radial modes

As mentioned above (Section 4), numerical results show that considering two modes provide sufficient exactness for the whole system response because of the fact that the first two modes always include most of the whole system energy.

Table 1 shows the peak base shears and overturning moments for different radial modes. It is distinguished by investigating the table that moderately up to 95% of the whole system response is gained considering the first two modes.

5. Conclusions

The success of a baffle for damping liquid oscillations has been examined in an effort to develop more efficient baffle configurations for seismic analysis of the tank. The baffles typically consist of rigid annular rings or plates which are fitted around the internal periphery of the tank. For a useful passive control system, configurations can be designed by freely suspending baffles between limits along the tank wall and by locating them slightly below the liquid surface. If stiffeners are required in the tank design for structural integrity, the baffles and support rings may provide the dual purpose of slosh damper and stiffener.

A baffle can be effectively used to reduce the whole system response. For an effective baffled liquid tank, the inner radius shown to be greater than a half of the outer radius and that the baffle should be located as near as possible to the liquid free surface. In this paper it has been supposed that the baffle is always surrounded by the liquid.

The summary of the maximum seismic responses of the baffled tanks is presented in Table 2. It is observed that the baffle causes an increase in the value of the base shear, soasmaller overturning moment is considered.

As observed from Table 2, the decrease in the ratio of overturning moments is strictly larger than the increase in the ratio of base shears (see Table 2).

This makes the usage of the baffle efficient. As an example, for the location of the baffle such ash/H=0.3 and inner radiusRi/Rd=0.75, base shear is increased 103% by the baffle. However, the overturning moment is decreased 92% by the baffle. The overturning moment can cause an uplift problem in the liquid storage tanks under the ground motion. Therefore, a baffle can be employed to avoid this problem.

Table 2 The effects of the baffle on thefoundationforces

Askari E, Daneshmand F (2009). Coupled vibration of a partially fluid-filled cylindrical container with a cylindrical internal body.Journal of Fluids and Structures, 25, 389–405.

Aslam M, Godden WG, Scalise DT (1979). Earthquake sloshing in annular and cylindrical tanks.Journal of the Engineering Mechanics Division,ASCE, 105(3), 371-389.

Brebbia CA, Dominguez J (1992).Boundary Elements, An Introductory Course. CMP & McGraw-Hill, New York.

Cho KH, Kim MK, Lim YM, Cho SY (2004). Seismic response of base-isolated liquid storage tanks considering fluid–structure–soil interaction in time domain.Soil Dynamics and Earthquake Engineering, 24, 839–852.

Fig. 7 Maximum base shear versusRi/R,H/R=1.2

Fig. 8 Maximum overturning moment versusRi/R,H/R=1.2

Dodge FT (2000). The new dynamic behavior of liquids in moving containers, San Antonio (TX). Southwest Research Institute.

Garza LR, Abramson HN (1963). Measurements of liquid damping provided by ring baffles in cylindrical tanks. Southwest Research Institute, Technical report prepared for NASA.

Gedikli A (1996). Fluid-structure interaction using variational BE–FE methods in cylindrical tanks. PhD Thesis, Istanbul Technical University, Istanbul, Turkey.

Gedikli A, Erguven ME (1999).Seismic analysis of a liquid storage tank with a baffle.Journal of Sound and Vibration, 223, 141–155.

Hunt B (1987). Seismic-generated water waves in axisymmetric tanks.Journal of Engineering Mechanics,ASCE, 113, 653-670.

Hunt B, Priestley N (1978).Seismic water waves in a storage tank.Bulletin of the Seismological Society of America, 68, 487-499.

Kim Young-Wann, Lee Young-Shin (2005). Coupled vibration analysis of liquid-filled rigid cylindrical storage tank with an annular plate cover.Journal of Sound and Vibration, 279, 217–235.

Koh HM, Kim JK, Park JH (1998). Fluid-structure interaction analysis of 3D rectangular tanks by a variationally coupled BEM–FEM and comparison with test results.Earthquake Engng Struct Dyn, 27, 109–124.

Miles JW (1958). Ring damping of free surface oscillations in a cylindrical tank.Journal of Applied Mechanics, 25, 274-276.

Mikelis NE, Journee JMJ (1984). Experimental and numerical simulations of sloshing behavior in liquid cargo tanks and its effect on ship motions.National Conference on Numerical Methods for Transient and Coupled Problems, Venice, Italy.

Nielsen BN (2003). Numerical prediction of green water loads on ships. Ph.D Thesis, Department of Mechanical Engineering, Technical University of Denmark.

Shekari MR, Khaji N, Ahmadi MT (2009). A coupled BE–FE study for evaluation of seismically isolated cylindrical liquid storage tanks considering fluid–structure interaction.Journal of Fluids and Structures, 25, 567–585.

Shrimali MK, Jangid RS (2004).Seismic analysis of base-isolated liquid storage tanks.Journal of Sound and Vibration,275, 59–75.

Vondorn WG (1966). Boundary dissipation of oscillatory waves.Journal of Fluid Mechanics, 24, 769-779.

Author biographies

Mohammad Reza Shekariis an Assistant Professor at Yasouj University, Iran. His current research interests include assessment the seismic behavior of marine structures employing numerical methods, and investigation of berm breakwater reshaping.

1671-9433(2014)03-0299-06

Received date:2014-01-25.

Accepted date:2014-07-03.

＊Corresponding author Email:m.shekari@yu.ac.ir

? Harbin Engineering University and Springer-Verlag Berlin Heidelberg 2014