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Engineering Structures 24 (2002) 909921

www.elsevier.com/locate/engstruct

Seismic response of liquid storage tanks isolated by slidingbearings

M.K. Shrimali, R.S. Jangid *

Department of Civil Engineering, Indian Institute of Technology Bombay, Powai, Mumbai 400 076, India

Received 23 July 2001; received in revised form 12 November 2001; accepted 13 December 2001

Abstract

The response of liquid storage tanks isolated by the sliding systems is investigated under two horizontal components of realearthquake ground motion. The continuous liquid mass is lumped as convective mass, impulsive mass and rigid mass. The corre-sponding stiffness associated with these lumped masses is calculated depending upon the properties of the tank wall and liquidmass. The governing equations of motion of the tank with a sliding system are derived and solved by Newmarks step-by-stepmethod with iterations. The frictional forces mobilized at the interface of the sliding system are assumed to be velocity dependentand their interaction in two horizontal directions is duly considered. A parametric study is also conducted to study the effects ofimportant system parameters on the effectiveness of seismic isolation of the liquid storage tanks. The various parameters consideredare (i) the period of isolation (ii) the damping of isolation bearings and (iii) the coefficient of friction of sliding bearings. It hasbeen found that the bi-directional interaction of frictional forces has noticeable effects and if these effects are ignored then thesliding base displacements will be underestimated which can be crucial from the design point of view. Further, the dependence ofthe friction coefficient on relative velocity of the sliding bearings has no significant effects on the peak response of the isolatedliquid storage tanks. 2002 Elsevier Science Ltd. All rights reserved.

Keywords: Liquid storage tank; Sliding isolation system; Bi-directional interaction; Seismic response; System parameters

1. Introduction

The integrity of a structure can be protected from theeffects of severe earthquakes either through the conceptof resistance or isolation. In designing a structure byresistance, it is assumed that the earthquake forces aredirectly transmitted to the structure and each member ofthe structure is required to resist the maximum possibleforces that may be induced by earthquakes based onvarious ductility criteria. In the category of earthquakeisolation, however, one is interested in reducing the peakresponse of the structure through implementation of cer-tain isolation devices between the base and foundationof the structure which prevents the transmission of earth-quake acceleration. The main concept in isolation is toincrease the fundamental time period of structuralvibration beyond the energy containing periods of earth-quake ground motion. The other purpose of an isolation

* Corresponding author.

0141-0296/02/$ - see front matter 2002 Elsevier Science Ltd. All rights reserved.

PII: S0 1 4 1 - 0 2 9 6 ( 0 2 ) 0 0 0 0 9 - 3

system is to provide an additional means of energy dissi-pation, thereby reducing the transmitted acceleration intothe superstructure. This innovative design approach aimsmainly at the isolation of a structure from the supportingground, generally in the horizontal direction, in orderto reduce the transmission of the earthquake motion tothe structure.

A variety of isolation devices including elastomericbearings (with and without lead core), frictional/slidingbearings and roller bearings have been developed andused practically for aseismic design of buildings duringthe last 20 yr [1,2]. A significant amount of recentresearch in the base isolation has focused on the use offrictional elements to concentrate flexibility of the struc-tural system and to add damping to the isolated structure.The most attractive feature of the frictional base isolationsystem is its effectiveness for a wide range of frequencyinput. The other advantage of a frictional type system isthat it ensures the maximum acceleration transmissibilityequal to the maximum limiting frictional force. The sim-plest sliding system device is a pure-friction (P-F) sys-

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910 M.K. Shrimali, R.S. Jangid / Engineering Structures 24 (2002) 909921

tem without any restoring force [3]. More advanced

devices involve P-F elements in combination with a

restoring force. The restoring force in the sliding system

reduces the base displacements and brings the system

back to its original position after an earthquake. Someof the commonly proposed sliding systems with restor-

ing force include the resilient-friction base isolator (R-FBI) system [4], the friction pendulum system (FPS) [5],

Electricitede France system (EDF) [6] and elliptical rol-

ling rods [7]. The sliding systems perform very wellunder a variety of severe earthquake loading and are very

effective in reducing the large levels of the superstruc-

tures acceleration without inducing large bearing dis-placements [4,8]. In addition, the sliding systems are

also less sensitive to the effects of torsional coupling in

asymmetric base-isolated buildings [9].There have been several studies investigating the

effectiveness of seismic isolation for buildings but very

few studies are reported for the seismic isolation of

liquid storage tanks which has a vital and strategic use.

Kim and Lee [10] experimentally investigated seismic

performance of liquid storage tanks isolated by elasto-

meric bearings and found that the isolation system is

effective in reducing the dynamic response. Malhotra

[11,12], Chalhoub and Kelly [13] and Shrimali and Jan-gid [14] studied the seismic response of isolated liquid

storage tanks and observed that isolation is quite effec-

tive in reducing the earthquake forces. It is to be noted

that in all the above studies elastomeric bearings were

used and there is a need to study the performance of

sliding systems for seismic isolation of liquid storage

tanks.In this paper, the response of liquid storage tanks iso-

lated by the sliding systems under two horizontal compo-

nents of earthquake ground motions is investigated. The

specific objectives of the present study can be summar-ized as (i) to present a method for earthquake analysis

of liquid storage tanks supported on sliding systems by

duly incorporating the effects of bi-directional interac-

tion and velocity dependence of frictional forces of the

isolation system, (ii) to investigate the effects of bi-

directional interaction of frictional forces under theearthquakes (by comparing the response of the system

with and without interaction), (iii) to study the effects

of velocity dependence of the friction coefficient of thesliding system on the peak response of the system and

(iv) to study the influence of important parameters onthe effectiveness of sliding systems for liquid storagetanks. The various important parameters considered are:

the period, damping, friction coefficient of the slidingsystem and the aspect ratio of the tank.

2. Model of liquid storage tank and the sliding

system

Fig. 1 shows a model of a liquid storage tank sup-

ported on the sliding system. The sliding system is

Fig. 1. Model of a liquid storage tank mounted on a sliding system.

installed between the base and the foundation of the

tank. The tank is modelled by a lumped mass model

suggested by Housner [15] and Rosenblueth and New-mark [16]. The contained continuous liquid mass is

lumped as convective, impulsive and rigid masses

referred as mc, mi and mr, respectively. The convectiveand impulsive masses are connected to the tank wall by

corresponding equivalent springs having stiffness kc andki, respectively. The damping constant of the convective

and impulsive masses are cc and ci, respectively. The

system has 6 degrees of freedom under bi-directional

earthquake ground motion, 2 degrees of freedom at eachlumped mass in two horizontalx- andy-directions. These

degrees of freedom are denoted by (ucx, ucy), (uix, uiy)

and (ubx

,uby

) which denote the absolute displacement of

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911M.K. Shrimali, R.S. Jangid / Engineering Structures 24 (2002) 909921

convective, impulsive and rigid masses in the x- and y-

directions, respectively. The tank model is assumed to

have a deformable cylindrical shell. The parameters of

the tanks considered are liquid height H, radius, R and

average thickness of tank wall, th. The effective massesare defined in terms of the liquid mass, m from the para-

meters expressed [17] as

Yc 1.013270.87578S 0.35708S2 (1)

0.06692S3 0.00439S4

Yi 0.15467 1.21716S0.62839S2 (2)

0.14434S30.0125S4

Yr 0.01599 0.86356S0.30941S2 (3)

0.04083S3

where S H/R is the aspect ratio (i.e. ratio the liquid

height to radius of the tank) and Yc, Yi, and Yr are the

mass ratios defined as

Yc mc

m (4)

Yi mi

m (5)

Yr mr

m (6)

m pR2Hrw (7)

The natural frequencies of sloshing mass, wc and

impulsive mass, wi are given by the followingexpressions

wi P

HE

rs(8)

wc 1.84gRtanh1.84H

R (9)

whereEand rsare the modulus of elasticity and densityof tank wall, respectively; g is the acceleration due to

gravity; andP is a dimensionless parameter expressed as

P 0.037085 0.084302S0.05088S2 (10)

0.012523S30.0012S4

The sliding system is considered as isotropic (i.e.

same coefficient of friction in two orthogonal directionsof the motion in the horizontal plane) and the restoring

force provided by the sliding systems is considered tobe linear (i.e. proportional to relative displacement). The

additional damping (other than friction) is assumed as

viscous damping. The frictional forces mobilized at slid-ing system are assumed to be coupled in two horizontal

directions and the friction coefficient is assumed to bedependent on the relative velocity. The limiting value of

the frictional force, Qs to which the sliding system can

be subjected in a particular direction is expressed as

Qs mMg (11)

where m is the friction coefficient of the sliding system;and M(i.e. mc+mi+mr) is the effective mass of the tank

(the mass of tank wall is neglected since it is very smallin comparison to the effective liquid mass).

The coefficient of sliding friction,m at a resultant slid-ing velocity zb x2b y2b, may be approximated from[18] by the following equation

m mmax(m) exp (a|zb|) (12)

where xb and yb are the velocities of the sliding system

relative to the ground in the x- and y-directions, respect-

ively;mmax is the coefficient of friction at large velocityof sliding (after leveling off); m is the differencebetween the friction coefficient at large and zero velocity

of the system; and a is a constant which depends uponbearing pressure and condition of interface and its value

is taken as 20 s/m.

3. Governing equations of motion

The equations of motion of an isolated liquid storage

tank subjected to earthquake ground motion are

expressed in the matrix form as

[m]{z} [c]{z} [k]{z} {Q} (13)

[m][r]{ug}

where {z} {xc,xi,xb,yc,yi,yb}T and {Q}

{0, 0,Qx, 0, 0,Qy}T are the relative displacement and

frictional force vector, respectively; xc ucxubx andyc ucyuby are the displacements of the convective

mass relative to bearing displacements in the x- and y-

directions, respectively; xi uixubx and yi uiyubyare the displacements of the impulsive mass relative to

bearing displacements in thex- andy-directions, respect-

ively; xb ubxugx and yb ubyugy are the displace-ments of the bearings relative to the ground in the x- and

y-directions, respectively; [m], [c] and [k] are the mass,

damping and stiffness matrices, respectively; [r] is the

influence coefficient matrix; {ug} {ugx,ugy}T is theearthquake ground acceleration vector; (ugx,ugy) and (Qx,Qy) are the ground accelerations and the frictional forces

in thex- andy-directions of the system, respectively; and

Tdenotes the transpose.

4. Criteria for sliding and non-sliding phases

In a non-sliding phase (xb yb 0 and xb yb

0) the resultant of the frictional forces mobilized at the

sliding system interface is less than the limiting frictional

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force (i.e.Q2x Q2y Qs). The system starts sliding(xb yb 0 and xb yb 0) as soon as the resultant

of the frictional forces attains the limiting frictional

force. Thus, the sliding of the system takes place if

Q2x Q2y Q

2s (14)

Note that Eq. (14) indicates a circular interactionbetween the frictional forces mobilized at the interface

of the sliding system as shown in Fig. 2(a). The system

remains in the non-sliding phase inside the interactioncurve. Further, the governing equations of motion in two

orthogonal directions of the structures supported on the

sliding type of isolators are coupled during the sliding

phases due to interaction between the frictional forces.

However, this interaction effect is ignored if the struc-

tural system is modelled as a 2-D system. In such cases

Fig. 2. Interaction and incremental frictional forces in two orthogonal

directions of the sliding system.

the corresponding curve which separates the sliding and

non-sliding phases is a square as shown in Fig. 2(a) by

dashed lines.

Since the frictional forces oppose the motion of the

system, the direction of the sliding of the system withrespect to the x-direction is expressed as

q tan1 ybxb (15)

5. Solution of equations of motion

During the non-sliding phase (xb yb 0 and xb

yb 0), the rigid mass sticks to the foundation and the

system behaves as two single degrees of freedom (i.e.convective and impulsive mass) in two orthogonal hori-

zontal directions. These equations can be solved by an

exact or numerical integration technique until the fric-tional forces mobilized at the sliding surface are less than

the limiting value. As soon as the frictional force attains

the limiting value the sliding phase of motion begins in

which the additional degrees of freedom of the rigid

mass are included in the response analysis. Since the

frictional forces mobilized in the sliding system arecoupled and non-linear functions of the displacement

and velocity of the system are in two orthogonal direc-tions, as a result, the equations of motion are to be solved

in the incremental form during the sliding phase of

motion. Newmarks method has been chosen for the sol-ution of governing differential equations, assuming lin-

ear variation of acceleration over the small time interval,t. The incremental equations in terms of unknown

incremental displacements are expressed as

[keff]{z} {Peff} {Q} (16)

where [keff] is the effective stiffness matrix; {z}

{xc, xi, xb, yc, yi, yb}T is the incremental dis-

placement vector; {Peff} is the effective excitation vec-

tor; {Q} {0, 0, Qx, 0, 0, Qy}T is the incremental

frictional force vector; and Qx and Qy are the

incremental frictional forces in the x- and y-directions,

respectively.

In order to determine the incremental frictional forces,consider Fig. 2(b). At time t the frictional forces are at

point A on the interaction curve and move to point B attime t+t. Therefore, the incremental frictional forces

are expressed as:

Qx Qt+ts cos (q

t+t)Qtx (17)

Qy Qt+ts sin (q

t+t)Qty (18)

where Qt ts is the limiting frictional force at time t+t

which is obtained by Eq. (11) depending upon the coef-

ficient of friction, m expressed by Eq. (12); and thesuperscript in the above equations denotes the time.

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Since the frictional forces are opposite to the motion

of the system, therefore, the angle qt t is expressed interms of the relative velocities of the system at timet+tby

qt+t tan1

yt+tb

xt+t

b

(19)

Substituting for qt t from Eqs. (17) and (18), theincremental frictional forces [3] are expressed as

Qx Qt+ts

xt+tb

(xt+tb )2 (yt+tb )2Qtx (20)

Qy Qt+ts

yt+tb

(xt+tb )2 (yt+tb )2Qty (21)

In order to solve the incremental matrix Eq. (16), the

incremental frictional forces (Qx and Qy) should be

known at any time interval. The incremental frictional

forces involve the system velocities at time t+tby Eqs.

(20) and (21) which in turn depend on the incremental

displacements (xband yb) at the current time step. As

a result, an iterative procedure is required to obtain the

required incremental solution. The steps of the procedure

considered are as follows

1. Assume Qx Qy 0 for iteration, j 1 in Eqs.

(20) and (21) and solve Eq. (16) for xb and yb.2. Calculate the incremental velocity xband ybusing

the xb and yb.

3. Calculate the velocities at time t+tusing incremental

velocities (i.e. xt tb xtb xb and yt tb ytb yb) and compute the revised incremental frictional

forces Qxand Qyfrom Eqs. (20) and (21), respect-

ively.

4. Iterate further, until the following convergence cri-

teria are satisfied for both incremental frictionalforces i.e.

(Qx)j+1||(Qx)j||(Qx)

j| e (22)

|(Qy)j+1||(Qy)

j|

|(Qy)j|

e (23)

where e is a small threshold parameter. Thesuperscript to the incremental forces denotes the iter-

ation number.

When the convergence criteria is satisfied, the velocityof the sliding structure at time t+t is calculated using

incremental velocity. In order to avoid the unbalanced

forces, the acceleration of the system at time t+t is

evaluated directly from the equilibrium of the system inEq. (13). At the end of each time step the phase of the

motion of the system should be checked. The response

of the sliding structures is quite sensitive to the time

interval, t and initial conditions at the beginning of

sliding and non-sliding phases. The number of iterations

in each time step is taken as 10 to determine the

incremental frictional forces at the sliding support.

The base shear is a measure of the hydrodynamicforces generated in the tank which is directly pro-

portional to earthquake forces exerted in the tank. There-fore, the effectiveness of base isolation is measured in

terms of reduction of the base shear generated in the

tank during earthquake. The base shear is directly pro-portional to the axial compressive forces induced in the

cylindrical tank wall which causes the buckling [11].

The base shear generated in the x- and y-directions of

the tank are expressed by

Fbx mcucx miuix mrubx (24)

Fby mcucy miuiy mruby (25)

where Fbx and Fby are the base shear in the x- and y-

directions of the tank, respectively.

6. Numerical study

The seismic response of liquid storage slender and

broad tanks isolated by the sliding system is investi-gated. Three types of commonly used sliding base iso-

lation systems i.e. the pure-friction (P-F) system, the

friction pendulum system (FPS) and the resilient-friction

base isolator (R-FBI) are considered for the present

study. The properties of the sliding system (such as stiff-

ness, damping and friction) are kept the same in both

the x- and y-directions of the system. As a result, thesliding isolation system can be completely defined bythe three parameters namely the period of isolation, Tb(i.e. Tb 2p/wb; andwb kb/M), the damping ratios,xb(i.e.xb cb/ 2Mwb) and the coefficient of friction (i.e.mmax and m). However, other tank parameters such asdamping ratio of convective mass (xc) and the impulsivemass (xi) are taken as 0.5 and 2%, respectively. The tankwall considered is made of steel with a modulus of elas-

ticity of E 200 MPa and mass density, rs 7900kg/m3.

Three real earthquake ground motions are used to

study the response of isolated tanks. The details of these

motions are given in Table 1. The components S90W,

N90E and N90E of Kobe, Loma Prieta and Imperial Val-

ley earthquake ground motions, respectively are applied

in the x-direction of the tank. The other orthogonalcomponents are applied in the y-direction. The displace-

ment and acceleration spectra of the above earthquake

ground motions are shown in Fig. 3. The seismic

response of the isolated tanks is compared with the cor-

responding response of non-isolated tanks in order tomeasure the effectiveness of the sliding systems. The

response quantities of interest in both thex- andy-direc-

tions of the tank are: base shear (Fbx,Fby), displacements

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Table 1

Properties of earthquake ground motions

Earthquake x-direction y-direction

Component PGA (g) Component PGA (g)

Imperial Valley, 1940 (El-Centro) S90W 0.2144 S00E 0.3486

Loma Prieta, 1989 (Los Gatos Presentation Center) N90E 0.6079 N00E 0.5704Kobe, 1995 (JMA) N90E 0.6297 N00E 0.8345

Fig. 3. Displacement and acceleration spectra of three earthquake ground motions (a) x-direction components and (b) y-direction components.

of convective mass (xc, yc), impulsive mass (xi, yi) and

displacements of sliding system (xb,yb). For comparative

and detailed parametric study two different types of

tanks, namely the broad and slender tanks are con-

sidered. The properties of these tanks are: (i) aspect ratio

(S) for slender and broad tanks is 1.85 and 0.6, respect-ively; (ii) the height, H, of water filled in the slenderand broad tanks is 11.3 and 14.6 m, respectively; (iii)

the natural frequencies of convective mass and impulsive

mass for the broad and slender tank are 0.123, 3.944 Hz

and 0.273, 5.963 Hz and (iv) the ratio of tank wall thick-

ness to its radius (th/R) is taken as 0.004 for both thetanks. Note that the same value of th/R 0.004 is used

in deriving Eqs. (13) and (10). The base shear of thetank is normalized by the effective weight of the tank,

W (i.e. W Mg).

The time variation of base shear and relative displace-ments of the convective mass, impulsive mass and rigid

mass isolated by the FPS system for the slender tank

is shown in Figs. 4 and 5 for the x- and y-directions,

respectively under Imperial Valley, 1940 earthquake

ground motion. The isolation parameters considered areTb 2 s, mmax 0.05 and m 0. It is observed thatthere is significant reduction in the base shear and impul-sive displacement of the tank implying that the sliding

system is quite effective in reducing the earthquakeresponse of the tanks. On the other hand, the sloshing

displacement of the tank remains the same for both iso-

lated and non-isolated conditions. This is due to the fact

that the period of sloshing mass is 3.66 s which is well

separated from the period of the isolation systems hence

isolation has no significant effect on sloshing displace-ment. The peak bearing displacements are 6.19 and 5.12

cm in the x- and y-directions of the tank, respectively

which are considerably less in magnitude to accommo-

date the sliding system.

The peak response of non-isolated and isolated tanksunder three earthquake ground motions are shown in

Tables 2, 3 and 4 for P-F, FPS and R-FBI systems,

respectively. It is observed that due to isolation, peak

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Fig. 4. Time variation of response of slender tank in the x-direction

isolated by the FPS system under Imperial Valley, 1940 earthquake

(Tb 2 s, mmax 0.05 and m 0).

base shear and peak impulsive displacement are signifi-

cantly reduced. Moreover, the peak impulsive displace-ment is quite small in comparison to sloshing and basedisplacement which is further reduced due to isolation

of the tank. The percentage reductions of peak base shear

with interaction effect in slender and broad tanks isolated

by P-F, FPS and R-FBI due to Imperial Valley are 79.84,

79.84, 88.39 and 70, 76.47, 84.41, respectively. Simi-

larly due to the Loma Prieta earthquake the percentage

reductions of peak base shear are 90.63, 58.33, 83.99

and 83.37, 45.81, 80.45, respectively. The percentage

reductions due to the Kobe earthquake are 88.79, 58.89,90.6 and 87.21, 55.17, 89.18 respectively. The above

reduction in base shear indicates that the sliding systems

are relatively more effective for slender tanks in com-

parison to broad tanks. Further, there is no significantchange in the sloshing displacement of both slender and

broad tanks due to Imperial Valley, 1940 earthquake butthe other earthquake ground motions have some influ-ence on it. It is to be noted that there is significant slosh-ing displacement in the slender tank in the y-direction

for both isolated and non-isolated conditions under

Loma Prieta, 1989 earthquake motion. This is due to thefact that at the sloshing period of slender tank (i.e. 3.66

s) this earthquake has a maximum displacement spectra

ordinate (see Fig. 3), as a result, the sloshing displace-

Fig. 5. Time variation of response of slender tank in the y-direction

isolated by the FPS system under Imperial Valley, 1940 earthquake

(Tb 2 s, mmax 0.05 and m 0).

ment is amplified. The maximum bearing displacement

is found to of the order of 50 cm for a slender tankisolated by the R-FBI system under Loma Prieta, 1989earthquake motion.

In Tables 24, the response of isolated tanks is alsocompared with the corresponding response without con-

sidering the interaction of friction forces of the sliding

systems (i.e. earthquake response of tanks in two hori-

zontal directions obtained as a 2-D idealization). It is

observed that the base shear for both slender and broad

tanks is not significantly influenced by the interactionof frictional forces for all sliding systems. Further, theconvective and impulsive displacements due to the inter-

action effect are slightly decreased in the slender tank

while in the broad tank it is increased. The bearing dis-

placement is found to be increased due to the interaction

of the friction forces of the sliding system. This is due

to the fact that when the interaction is taken into con-sideration the system starts sliding at a relatively lower

value of the frictional forces mobilized in the sliding

system (refer to the sliding Eq. (14)), as a result, the

sliding displacements in the isolated system are

increased. Thus, if the interaction of the frictional forcesof the sliding system is ignored then the sliding displace-

ments will be underestimated which can be crucial from

the design point of view of the isolation system.

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Table 2

Peak seismic response of liquid storage tanks isolated by P-F system (Tb , xb 0 and mmax 0.1 and m 0)

Earthquake/tank condition Response quantity

Fbx/W xc (cm) xi (cm) xb (cm) Fby/W yc (cm) yi (cm) yb (cm)

Slender tank

Imperial Valley, Non-isolated 0.319 41.54 0.39 - 0.491 23.71 0.74 -1940

Isolated (with interaction) 0.100 39.00 0.22 3.80 0.100 22.73 0.22 4.02

Isolated (no interaction) 0.100 39.84 0.26 2.33 0.100 24.87 0.29 3.21

Loma Prieta, 1989 Non-isolated 0.771 36.14 0.96 - 1.056 142.8 1.53 -



Kobe, 1995 Non-isolated 0.594 52.05 0.66 - 0.883 37.85 0.91 -



Broad tank

Imperial Valley, Non-isolated 0.258 53.08 1.28 - 0.340 42.29 1.95 -

1940



Loma Prieta, 1989 Non-isolated 0.440 17.50 2.37 - 0.609 55.31 2.95 -Isolated (with interaction) 0.100 21.63 0.68 9.17 0.100 55.35 0.93 13.68





Table 3

Peak seismic response of liquid storage tanks isolated by FPS (Tb 2 s, xb 0, mmax 0.05 and m 0)



Slender tank


1940






Kobe, 1995 Non-isolated 0.594 52.05 0.66 - 0.883 37.85 0.91 -Isolated (with interaction) 0.195 73.05 0.17 14.69 0.363 64.27 0.31 31.12


Broad tank


1940









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Table 4

Peak seismic response of liquid storage tanks isolated by R-FBI system ( Tb 4 s, xb 0.1, mmax 0.04 and m 0)



Slender tank

Imperial Valley, Non-isolated 0.319 41.54 0.39 - 0.491 23.71 0.74 -1940









Broad tank


1940



Loma Prieta, 1989 Non-isolated 0.440 17.50 2.37 - 0.609 55.31 2.95 -Isolated (with interaction) 0.074 24.79 0.34 11.46 0.119 65.41 0.58 30.85





Table 5

Effects ofm on the peak seismic response of liquid storage tanks isolated by FPS system ( Tb 2 s, xb 0 and mmax 0.05)

Earthquake Fbx/W xc (cm) xi (cm) xb (cm) Fby/W yc (cm) yi (cm) yb (cm)m

mmax(%)

Slender tank

Imperial Valley, 1940 0.0 0.103 47.60 0.11 6.20 0.098 29.41 0.11 5.12

10 0.103 47.90 0.11 6.16 0.096 29.58 0.11 5.06

20 0.102 48.12 0.09 6.14 0.095 29.74 0.10 5.01

30 0.102 48.33 0.08 6.12 0.095 29.87 0.10 5.04

Loma Prieta, 1989 0.0 0.159 52.67 0.15 14.20 0.440 207.90 0.40 40.46

10 0.159 52.65 0.15 14.20 0.440 207.89 0.40 40.44

20 0.159 52.65 0.15 14.20 0.440 207.88 0.40 40.42

30 0.159 52.64 0.15 14.19 0.439 207.87 0.40 40.40

Kobe, 1995 0.0 0.195 73.05 0.17 14.69 0.363 64.27 0.31 31.12

10 0.194 72.98 0.16 14.68 0.359 64.23 0.31 31.13

20 0.193 72.95 0.16 14.68 0.357 64.19 0.31 31.09

30 0.191 72.91 0.16 14.64 0.356 64.14 0.31 31.04Broad tank

Imperial Valley, 1940 0.0 0.076 52.27 0.39 3.04 0.078 43.76 0.42 3.39

10 0.076 52.28 0.38 3.09 0.076 43.92 0.38 3.44

20 0.076 52.29 0.36 3.15 0.076 44.06 0.35 3.50

30 0.077 52.33 0.34 3.23 0.077 44.11 0.32 3.55

Loma Prieta, 1989 0.0 0.182 24.95 0.56 13.74 0.330 64.13 0.98 30.42

10 0.182 24.99 0.56 13.75 0.331 64.18 0.99 30.45

20 0.182 25.02 0.56 13.75 0.331 64.23 0.99 30.48

30 0.182 25.03 0.56 13.76 0.331 64.27 0.99 30.50

Kobe, 1995 0.0 0.177 25.59 0.61 15.07 0.347 42.10 1.12 30.97

10 0.178 25.58 0.62 15.09 0.347 42.12 1.12 30.99

20 0.178 25.55 0.62 15.11 0.346 42.14 1.12 31.01

30 0.178 25.51 0.63 15.12 0.346 42.13 1.12 31.02

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6.1. Effects of velocity dependent friction coefficient

The friction coefficient of various sliding isolationsystems is typically dependent on the relative velocity

at the sliding interface and it will be interesting to studythese effects on the peak response of isolated liquid stor-

age tanks (refer to Eq. (12) for the expression of velocitydependent friction coefficient). In Table 5, the peak seis-mic response of slender and broad tanks isolated with

the FPS system is shown for different values ofm (i.e.0, 0.1mmax, 0.2mmax and 0.3mmax). Note that m 0denotes that the friction coefficient of the sliding iso-lation system is independent of the velocity at the sliding

interface (i.e. Coulomb-friction idealization). It is

observed from Table 5 that the dependence of the fric-

tion coefficient on the relative sliding velocity has nonoticeable effects on the peak response of liquid storage

tanks isolated by sliding systems for all the earthquake

ground motions. A similar trend in the results was also

observed for the tanks isolated by the P-F and R-FBI

systems. Thus, the effects of dependence of friction coef-

ficient on the sliding velocity may be ignored fordetermining the peak response of isolated tanks. These

effects are similar to that observed by Fan and Ahmadi

[8] for buildings isolated by the sliding systems.

Fig. 6. Effects of isolation period on the resultant seismic response

of liquid storage tanks isolated by the FPS system (mmax 0.05 andm 0).

6.2. Effects of isolation period

So far the effectiveness of a sliding system for isolat-

ing the liquid storage tanks is investigated for the fixedparameters of the sliding systems. However, it will beinteresting to study the influence of isolator parameters

(such as period, damping and friction coefficient) on thebeahviour of isolated tanks. The variation of resultant

base shear, Fbz (i.e.F2bx F2by), sloshing displacement,zc and bearing displacement, zb for both slender and

broad tanks is shown against the period of isolation, Tbin Figs. 6 and 7 for the FPS and R-FBI systems, respect-

ively. The figures indicate that the base shear decreaseswith the increase offlexibility of isolation systems. Thisis due to the fact that with an increase of isolation period

the system becomes more flexible and, as a result, trans-mits less earthquake acceleration into the tanks leading

to a reduction in the base shear. However, the effect of

isolation period is not found to be significant on thesloshing displacement in both tanks. The base displace-

ment increases with the increase of isolation period for

Imperial Valley, 1940 and Loma Prieta, 1989 earth-

quakes whereas it decreases for Kobe, 1995 earthquake

ground motion. This trend is similar to the displacement

spectra of these ground motions as shown in Fig. 3.

Fig. 7. Effects of isolation period on the resultant seismic response

of liquid storage tanks isolated by the R-FBI system (xb 0.1,mmax 0.04 and m 0).

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Thus, the effectiveness of sliding systems for tanks

increases with the increased flexibility.

6.3. Effects of coefficient of friction

The effects of coefficient of friction, mmax on the

resultant response of isolated tanks are shown in Figs.8 and 9 for the FPS and R-FBI systems, respectively. It

is observed that the peak resultant base shear decreases

initially and after that it increases with the increase of

friction coefficient. This indicates that there exists anoptimum value of coefficient of friction for which thebase shear in the tank attains the minimum value. Onthe other hand, the bearing displacement for both tanks

decreases with the increase of coefficient of friction.This is expected as the friction coefficient of the slidingsystem increases it transmit higher earthquake acceler-

ation (due to increase of limiting frictional force) into the

tanks resulting in higher base shear. Since the limitingfrictional force is increased with the increase of friction

coefficient, as a result, less sliding takes place in theisolation system leading to reduction in the bearing dis-

placement. Further, the sloshing displacement mildly

decreases with the increase of friction coefficient of thesliding system.

Fig. 8. Effects of friction coefficient on the peak resultant response

of liquid storage tanks isolated by the FPS system (Tb 2 s and

m 0).

Fig. 9. Effects of friction coefficient on the peak resultant response

of liquid storage tanks isolated by the R-FBI system (Tb 4 s, xb 0.1 and m 0).

6.4. Effects of isolation damping

The peak resultant seismic response of slender andbroad tanks isolated by the R-FBI system is plotted

against the isolation damping, xb in Fig. 10. The figureindicates that the base shear initially decreases with

increases of damping and attains the minimum value

thereafter, it increases with increase of isolation damp-

ing. This implies that there exists an optimum value of

isolation damping at which there is a minimum value of

base shear. On the other hand, the sloshing displacement

and the base displacement decreases with the increaseof isolation damping. Thus, increase in the isolation

damping can reduce the displacement response of the

tank but under certain conditions the high damping may

produce more earthquake forces into the system.

The effects of friction coefficient and damping on thepeak base shear of the tank are similar to the well-knownphenomenon that high damping in isolation systems can

reduce the displacements but it may transmit higher

earthquake forces [19,20]. It happens due to the fact that

the high bearing damping transmits more acceleration

into the system for the earthquake energy at the higherfrequencies. Thus, one should select the optimum value

of coefficient of friction and damping in the sliding sys-tems for effective design of isolated liquid storage tanks.

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liquid storage tanks. Proceedings of the National Symposium on

Advances in Structural Dynamics and Design, VETOMAC-1,

Chennai, 2001:45966.

[15] Housner GW. Dynamic behaviour of water tanks. Bulletin of

Seismological Society of America 1963;53:3817.

[16] Rosenblueth E, Newmark NM. Fundamentals of earthquake

engineering. Englewood Cliffs, NJ: Prentice Hall, 1971.

[17] Haroun MA. Vibration studies and test of liquid storage tanks.Earthquake Engineering and Structural Dynamics 1983;11:179

206.

[18] Constantinou MC, Mokha AS, Reinhorn AM. Teflon bearing in

base isolation II: modeling. Journal of Structural Engineering,

ASCE 1990;116(2):45574.

[19] Inaudi J, Kelly JM. Optimum damping in linear isolation systems.

Earthquake Engineering and Structural Dynamics 1993;22:583

98.

[20] Jangid RS. Optimum damping in a non-linear base isolation sys-

tem. Journal of Sound and Vibration 1996;189(4):47787.

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