PARAMETRIC STUDY OF SEISMIC SOIL-TANK INTERACTION. II: VERTICAL EXCITATION

By Medhat A. Haroun,1 Member, ASCE, and Wajdi Abou-Izzeddine2

ABSTRACT: Analysis of the behavior of ground-based tanks under vertical seismic excitations is quite relevant to their seismic-resistant design as vertical acceleration is transmitted to a horizontal hydrodynamic loading on the tank wall. This effect results primarily in amplified hoop tensile stress and, coupled with the effect of horizontal excitations, may lead to inelastic buckling of the shell. However, to realistically consider the effect of vertical excitations in the design of tanks, one must consider the dynamic interaction of the tank and its content with foundation soil. A simplified analysis is used as the basis for a thorough parametric evaluation of the effects of numerous factors that influence the seismic response of an elastic tank supported on a rigid base (solid concrete slab) and subjected to a vertical excitation, taking into account shell-liquid-soil interaction. Results are presented in the form of design graphs that practicing engineers can use to estimate the seismic response of tanks under vertical excitations.

INTRODUCTION

Analysis of ground-motion records from past seismic events has revealed that the maximum amplitude of the vertical component of ground accel­eration can exceed the peak horizontal amplitude, especially near the origin of the earthquake. This is quite relevant when considering that, in a liquid-filled tank, vertical acceleration is transmitted to a horizontal hydrodynamic loading on the tank wall. This effect results primarily in amplified hoop tensile stress and, coupled with the effect of horizontal excitations, may lead to inelastic buckling of the shell. In 1979, a simplified study by Marchaj (1979) focused attention on the importance of vertical acceleration in the design of tanks and attributed the failure of metallic tanks during past earthquakes to lack of consideration of vertical acceleration in their design. A critical study of the axisymmetric seismic behavior of tanks was carried out by Kumar (1981), in which the radial motion of partly filled tanks was considered but the effect of axial deformations was neglected. Later, Ve-letsos and Kumar (1984) presented a simple design procedure for evaluating the effects of vertical earthquake shaking on cylindrical tanks. At the same time, a comprehensive study of the effects of vertical seismic excitations on the response of anchored tanks was conducted by Haroun and Tayel (1984, 1985a, 1985b, 1985c). However, in all these studies, the vertical ground motion was assumed to be specified at tank-base level with no allowance for modification of ground motion by the motion of the tank itself. Haroun and Abdel-Hafiz (1986) developed a simplified method to consider the interaction of an elastic circular cylindrical tank on a rigid base with the foundation soil in evaluating its dynamic response under a vertical excitation. In another study, Veletsos and Tang (1986) considered the tank-liquid sys­tem to respond as a single-degree-of-freedom system and applied Galerkin's

T>rof. and Chair, Dept. of Civ. Engrg., Univ. of California, Irvine, CA 92717. 2Grad. Res. Asst., Dept. of Civ. Engrg., Univ. of California, Irvine, CA. Note. Discussion open until August 1, 1992. Separate discussions should be sub­

mitted for the individual papers in this symposium. To extend the closing date one month, a written request must be filed with the ASCE Manager of lournals. The manuscript for this paper was submitted for review and possible publication on September 6, 1990. This paper is part of the Journal of Structural Engineering, Vol. 118, No. 3, March, 1992. ©ASCE, ISSN 0733-9445/92/0003-0798/$1.00 + $.15 per page. Paper No. 438.

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method to evaluate the response of the tank to a vertical component of ground shaking, considering also the flexibility of the supporting medium. The main purpose of the present investigation is to conduct a parametric study of numerous factors affecting the response of tanks under vertical excitations following a similar procedure as presented in the companion paper (Haroun and Abou-Izzeddine 1992) dealing with the response under horizontal excitations.

TANK GEOMETRY AND COORDINATE SYSTEM

The tank under consideration is a ground-supported, circular cylindrical, thin-walled liquid container of radius R, length L, and thickness hs, filled with liquid of mass density p, to height H, and subjected to a vertical, free-field displacement Gv(t). A cylindrical coordinate system (r,z) is used, with the origin situated at the center of the base, as shown in Fig. 1. The radial displacement of the shell is denoted by w{z,t), whereas the total vertical motion of base is denoted by v'(t).

ASSUMPTIONS

The following assumptions are made:

1. Tank is completely filled (L = H) and anchored to a rigid base (solid concrete slab).

2. Tank wall is represented by a membrane shell offering resistance only to the radial deformation w(z, t) of prescribed form as

w(z,t) = w(f)cos — (1)

3. Liquid free-surface oscillations are neglected since experimental tests have shown that sloshing under arbitrary vertical excitations is minimal.

Foundation Soi

FIG. 1. Tank Geometry and Coordinate System

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4. Soil is represented by a spring of a stiffness K and a dashpot of a constant C; both are frequency-independent. An added mass may be used in the soil model.

EQUATION OF MOTION OF SHELL

A cylindrical shell undergoing axisymmetric vibrations is governed basi­cally by two differential equations: one of the second order governing the dynamic equilibrium in the axial direction, and one of the fourth order governing the dynamic equilibrium in radial direction (Novozhilov 1964). Neglecting bending rigidity and the effects of the shell axial displacement, the governing equations are reduced to only one equation of the form

d2w Eshsw p A ^ + ( T ^ ^ = ^ 2 ' r ) (2)

where ps, Es, and vs = mass density, modulus of elasticity, and Poisson ratio of shell material, respectively; and p(R,z,t) = the pressure exerted by liquid on a tank wall at any time f.

EQUATION OF MOTION OF TANK BASE

The equation of motion of the tank base can be expressed as

Mv' + Co + Kb = - J P(r,0,t) dA (3)

where v' = the total vertical acceleration of the base; v = the relative vertical displacement of the base to the free-field motion Gv(t); A = the area of tank base; M = the mass of base and shell; and C and K = damping and stiffness coefficients, respectively. The right-hand side of (3) accounts for both static and hydrodynamic pressures on the base.

LIQUID PRESSURE

The pressure distribution, p(r,z,t), can be determined from the Bernoulli equation and is given by

p{r,z,t) = -p,-ft + Plg(H - z) (4)

where p, = mass density of liquid; $ = the velocity potential function obtained from solving the Laplace equation subject to the appropriate boundary conditions; and g = the acceleration of gravity. The first term in (4) is the hydrodynamic pressure, pd, which, after substituting for 4>, be­comes (Abou-Izzeddine 1989)

Pd(r,z,t) = -p, A-i 4>(Ai-K)

(5)

where I0 = the modified Bessel function of first kind of order zero; and Xx = (ir/2H).

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MODELING OF SOIL

The structure and the soil on which it is founded form a combined dynamic response mechanism, and there may be significant feedback from the struc­ture to soil layers. The structure affects the earthquake motions, and the extent of this effect depends on the relative mass and stiffness properties of soil and structure. A quantitative determination of these properties adds to the complexity of the modeling process due to soil nonhomogeneity and strain dependency. In principle, a frequency-dependent foundation model could be used directly in a frequency-domain analysis of the soil-structure interaction problem. However, analysis of a complicated soil-structure ideal­ization involving a large number of degrees of freedom can be greatly sim­plified by modeling the soil with a lumped-parameter system with frequency-independent characteristics (Fig. 2).

Solutions are available for a rigid plate, either circular or rectangular, resting on an isotropic, homogeneous, linearly elastic half-space under steady-state vibration. Such solutions were obtained assuming that the distribution of the contact stresses is the same as under static loading, independent of the frequency of vibration. A comparison of the exact solution for a rigid circular plate on an elastic half-space with the solution based on the same distribution as under static conditions shows that the latter is satisfactory up to and somewhat beyond the resonant frequency (Das 1984; Newmark and Rosenblueth 1971; Wolf 1985). So it is advantageous to represent the elastic half-space foundation medium by frequency-independent compo­nents, with their properties chosen in a way to reproduce the true frequency-dependent behavior as well as possible.

Two methodologies exist to select the soil parameters. In the first method, a simple spring-dashpot system is chosen, where the spring constant is se­lected to give the correct static displacement and the dashpot coefficient is chosen to provide the best possible agreement with the theoretical base-force amplitude and phase. The values of the spring constant, the damping coefficient, and the mass that appeared in (3) are given in Wolf (1985) as

' - ^ ( 6 )

c = rh/fcfR2 (y) M = 2irRHhsps + trR2hbPb (8)

where p/, vf, and Gf = the mass density, Poisson ratio, and shear modulus

EQUIVALENT SYSTEM

RIGID BLOCK HAVING EQUIVALENT MASS"/-

ACTUAL FOUN

VERTICAL | EXCITATION 1

mmmi

DATION

,V/1

EQUIVALENT | i | EQUIVALENT I I I S EQUIVALENT DAMPER U-|—' 5 SPRING

FIG. 2. Modeling of Foundation Soil

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of the foundation soil, respectively; c* = the shear-wave velocity of the foundation soil (cy = \^Gf/pf); and hb and pb = the thickness and mass density of the base, respectively.

The second approach introduces a virtual mass of soil so one has an additional parameter that permits a better adjustment over a limited range of frequencies. To compensate for the addition of the virtual mass, the damping coefficient should be readjusted. The damping coefficient is re­duced to 72% of that of (7), whereas the mass is increased by (1.5pfR

3).

GENERALIZED EQUATIONS OF MOTION

Since a displacement function for the shell has already been assumed, the generalized equation of motion is derived by using Hamilton's principle (Clough and Penzien 1975) expressed as

(9) 8(7 - V) dt 4- | hW dt = 0

where 7 and V = the kinetic and potential energies of the system, respec­tively; and bW = the virtual work done by the hydrodynamic pressure exerted on the tank wall through an arbitrary virtual displacement 8vv, i.e.

8W = 2TTR JO pd{R,z,i)hw dz (10)

Using (1) and (5), (10) becomes

W = -p, iRiptmm-^w) m^R)

8w (11)

Since the variation 8w is arbitrary, the equation of motion of the shell can be written as

/o(M) M*i*v

$p,RH2

v + irEshsH

R(l-v^ 8p,RH2

G„ (12)

where v' is replaced by its two components, v and Gv. Substitution for the hydrodynamic pressure into the equation of motion

of tank base yields

-2TTP, f Jo

/70(V) dr

V o ( M )

- ( M + TrR2Hp,)Gv

ti + (M + TtR2Hp,)v + Ci(t) + Kv{t)

(13)

Eqs. (12) and (13) are coupled differential equations that can be expressed in matrix form as

~MU

_M21

M12"

M22_ (f! W + "c„

0 0 "

C22_

f * l UJ

+ ~Kn

0 0 "

K22_ :}-{£}< (14)

where

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Mu = *9,Rh,H + 2pfiIP ^ | 1 ; M12 = - ( ^ - 2 ) (15)

M21 = M12; M22 = M + TrR2HPl (16)

C u = 2 £ V 7 Q ^ ; C22 = C (17)

irE H Ku= R(i - viy K22 = K ( 1 8 )

where £ = the damping ratio of the liquid-shell system without consideration of soil-structure interaction. Its value will be taken as 0.02 (steel tank) in this analysis.

The general equation of motion [(14)] is solved by a step-by-step inte­gration scheme using the trapezoidal rule.

PARAMETRIC ANALYSIS

The seismic response of a tank is greatly influenced by its geometric properties and the condition of the soil supporting it. Therefore, it is im­portant that the relationship among all these parameters and their influence on tank response be well understood. The variation of the tank response with changes in the height-to-radius ratio and the wall-thickness-to-radius ratio is studied. These two parameters are known to be important factors in influencing the seismic response of tanks. This study encompasses a range of shear-wave velocities representative of a variety of soil conditions. Throughout the study (Abou-Izzeddine 1989), the tank response as well as the different parameters under consideration are presented in nondimen-sional forms to facilitate the generalization of results. For example, the natural frequencies, co, will be reported in terms of nondimensional param­eters, u, such that

hs/R - 0.0010 hs/R - 0-0015 hs/R = 0.0020 hs/R = 0.0025 hs/R = 0.0030

-* hs/R = 0.0035

6

o

' 3 W Q

O Q

QJ

ram

et

0.06

£

ncy

0.0.

CD

LU O

o

/',-'-'" ///^ -

I-/' -

i//^ /•// / /

f §

1/ H — i a

_____ ..-——— "

-—*" --—* _ ,

--~~~~~~" . — — ~ ~ • " * " "

- - » hs/R - 0.0040 - - hs/R - 0.0045 - •* hs/R - 0.0050

- - * hs/R - 0.0060

—i —|

<-) Height-to-Radius ratio, H/R (b) Height-to-Radius ratio, H/R

FIG. 3. Variation of Frequency Parameter, &s, of Fixed-Base Shell

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c, = 600 ft/sec cr = 5000 ft/sec

hs/R - 0.001 hs/R - 0.002 hs/R - 0.003 hs/R - 0.004 hs/R - 0.005 hs/R - 0.006

° 0 (a) Height-to-Radius ratio, H/R (b)

hs/R - 0.001 hs/R •= 0.002 hs/R = 0.003 hs/R = 0.004 hs/R - 0.005 hs/R - 0.006

Height-to-Radius ratio, H/R

FIG. 4. Variation of Fundamental Frequency Parameter with Soil and Tank Properties

c, = 600 ft/sec c, = 5000 ft/sec

•a

ii

hs/R -0.001 hs/R - 0.002 hs/R - 0.003 hs/R - 0 .004 hs/R = 0.005 hs/R - 0.006

hs/R - 0.001 hs/R = 0.002 hs/R = 0.003 hs/R = 0.004 hs/R = 0.005 hs/R - 0.006

<") Height-to-Radius ratio, H/R (b) Height-to-Radius ratio, H/R

FIG. 5. Variation of Second Natural Frequency Parameter with Soil and Tank Properties.

w = 1 -Es

.Pitt - vf). (19)

where w = the frequency parameter. The study is carried out in three stages: first, the natural frequency of

fixed-base shell is investigated; second, the natural frequency of the tank system is obtained by solving the eigenvalue problem. Finally, the equations of motion of a two-degree-of-freedom system and that of an idealized one-

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7940 EL CENTRO EARTHQUAKE c, = 7000 ft/sec

Time (sec)

1971 SAN FERNANDO EARTHQUAKE c, = 7000 ft/sec

—« FLEXIBLE FOUNDATION - -B RIGID FOUNDATION

(b) Time (sec)

FIG. 6. Time History of Wall Displacement (Broad Tank, c, = 1,000 ft/sec)

degree-of-freedom system are solved by a step-by-step integration scheme. The variation of results with respect to the aforementioned parameters are thoroughly studied.

NATURAL FREQUENCY OF FIXED-BASE SHELL

At first, the natural frequency of fixed-base shell is computed under the assumption that interaction between shell and base and between base and soil is neglected.

The natural frequency of the shell is given by

(20)

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1940 EL CENTRO EARTHQUAKE c, = 2500 ft/sec

1940 EL CENTRO EARTHQUAKE c. = 5000 ft/sec

Time (sec)

FIG. 7. Variation of Wall Displacement Amplitude with Shear-Wave Velocity (Broad Tank)

where

= -nEshsH R(i - v?)

M = TrpsRhsH + 2p,RH2

X - — K~2H

/i(M0.

(21)

(22)

(23)

Note that iS(Xi-R) = I^^R). The two terms in (22) represent the contri-

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1940 EL CENTRO EARTHQUAKE c, = 1000 ft/sec

FLEXIBLE FOUNDATION RIGID FOUNDATION

Time (sec)

1971 SAN FERNANDO EARTHQUAKE c, = 1000 ft/sec

FLEXIBLE FOUNDATION RIGID FOUNDATION

FIG. 8. Time History of Wall Displacement (Tall Tank, cf = 1,000 ft/sec)

bution of shell mass and the contribution of liquid mass, respectively. For practical purposes, shell contribution is relatively small compared with liquid-mass contribution; and it could be neglected. Therefore, the frequency can be represented in the form of nondimensional parameters as

(24)

Figs. 3(a) and 3(b) present the values of the nondimensional frequency parameter, ws, given by (24) for different values of HIR and hJR. The frequency parameter increases monotonically.to a value of HIR ~ 2.0. Thereafter, &s shows slight change with changes of HIR. If the ratio of HIR is held fixed, it is observed that as hJR increases, so does the frequency

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H/fl - 0.4 -* H/fl = 0.6 - H/fl - 0.8 - H/fl =1.0 -> H/fl = 1.6 ->•• H/R = 2.0 -« H/fl = 3.0

0 1000 2000 3000 4000 B000

(a) Shear Wave Velocity (ft/sec) W 1000 2000 3000 4000 5000

Shear Wave Velocity (ft/sec)

FIG. 9. Variation of Ratio of Maximum Wall Displacement (Flexible/Rigid) for Broad Tanks: (a) hJR = 0.0014; (b) hJR = 0.003

<") W00 2000 3000 4000 5000

Shear Wave Velocity (ft/sec) <b)

H/R -0.4 H/R -0.6 H/R - 0.8 H/fl =1.0 H/fl =1.5 H/fl =2.0 H/fl -3.0

1000 2000 3000 4000 5000

Sh&ar Wave Velocity (ft/sec)

FIG. 10. Variation of Ratio of Maximum Wall Displacement (Flexible/Rigid) for Tall Tanks: (a) hJR = 0.003; (b) hJR = 0.006

parameter value. The increase is larger for values of HIR > 1.0, whereas for values of H/R ^ 1 . 0 , the difference is small and gets even smaller as H/R approaches zero.

NATURAL FREQUENCY OF TANK-SOIL SYSTEM

The evaluation of natural frequencies of a tank-soil system is somewhat inaccurate because of the large values of damping exhibited by the soil; nonetheless, these calculations show the trend in frequency shifting. Ac­curate calculation of response is obviously achieved by using a step-by-step

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0 WOO 2000 3000 4000 5000 Shear Wave Velocity (ft/sec)

FIG. 11. Variation of Ratio of Maximum Wall Displacement (Flexible/Rigid) for Tall Tank [hJR = 0.003) under San Fernando Record

integration technique. The natural frequencies of the system under consid­eration are calculated by solving the following eigenvalue problem

(K - co2M)i|< = 0 (25)

where stiffness and mass matrices are those given in (14). The analysis is carried out under the following assumptions based on

current design considerations: p, = p,„ (the liquid is water); pb = 2.5p,„ (concrete base); pf = 2pH, (typical soil density); p̂ = 7.8p>v (shell material is steel); and hbIH = 1/20.

Fig. 4 shows the variation of the fundamental frequency parameter, do, obtained from the eigenvalue problem with changes in geometric properties of the tank, as well as changes in soil stiffness. Similarly, Fig. 5 shows the variation of the second natural-frequency parameter.

Effect of HlR and Soil Stiffness on Fundamental Frequency It is observed that the fundamental-frequency parameter increases mon-

otonically as HlR increases. If HlR is kept constant and the shear-wave velocity is increased, the value of do increases. However, it is noted from Fig. 4 as well as similar plots with shear-wave velocities of 1,000 ft/sec (305 m/s) and 2,500 ft/sec (762 m/s), that the percentage of increase is smaller for higher values of shear-wave velocities. It is also worth noting that at a shear-wave velocity of 5,000 ft/sec, the values of OJ are very close to those obtained in fixed-base shell analysis.

Effect of hJR and Soil Stiffness on Fundamental Frequency As hJR increases, the frequency parameter values are higher for every

given value of HlR. This effect is pronounced at high values of shear-wave velocity (stiffer soils), and it decays as the shear-wave velocity is decreased (softer soils).

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Effect of Tank Properties and Soil Parameters on Second Frequency In this case, the effects of shear-wave velocity are reversed. For higher

shear-wave velocities, the effect of hJR is minimal, and co increases steadily as HIR increases. On the other hand, at lower values of shear-wave velocity, the effect of hJR is apparent, being more pronounced at very low values of shear-wave velocities. The values of OJ generally decrease as the soil gets softer.

RESPONSE OF TANK-SOIL SYSTEM

The response of the tank-soil system is achieved by solving (14) with a step-by-step integration scheme. The time history of displacement response of two typical tanks was evaluated. Tanks considered have the following dimensions:

1. Tall tank: R = 24 ft (7.32 m), H = 72 ft (21.95 m). 2. Broad tank: R = 60 ft (18.29 m), H = 40 ft (12.19 m).

Both tanks have a uniform wall thickness of hs = 1.0 in. (2.54 cm). The mass densities of shell, base, and liquid are pb = 2.5p;; ps = 7.8p,; and p, = 1.949 lb sec2/ft4 (1,000 kg/m3), respectively. The foundation soil has the following properties: pf = 2p,; vf = 0.35; and the shear-wave velocities range from 1,000 ft/sec (305 m/s) to 5,000 ft/sec (1,524 m/s). Both tanks were subjected to the vertical component of the 1940 El Centro earthquake and the vertical component of the 1971 San Fernando earthquake, recorded at Pacoima Dam.

It is observed that displacement response of a tank-soil system modeled as a two-degree-of-freedom system (flexible foundation) is lower than that obtained from a single-degree-of-freedom idealization (rigid foundation), as seen from Figs. 6(a), 6(b), 8(a), and 8(b). However, as soil stiffness is increased by increasing the shear-wave velocity, the response of the flexible tank foundation system approaches that of a rigid foundation system, as shown in Figs. 7(a) and 1(b). Another interesting observation is that the response of the broad tank is higher than that of the tall tank for this particular analysis, as seen from Figs. 6 and 8.

Several graphs were generated to generalize the effects of geometric tank properties and soil properties on tank response. These graphs show the variation of the ratio of maximum displacement amplitude (flexible/rigid) for a range of shear-wave velocities and several ratios of HIR and hJR.

Figs. 9(a) and 9(b) depict results obtained for a tank of H = 40 ft (12.19 m), and Figs. 10(a), 10(b), and 11 present results obtained for a tank of H = 60 ft (18.29 m). After a close inspection, it is seen that at high values of shear-wave velocities, the ratios of maximum displacement amplitudes are very close to 1.0 in most curves. It is also noted that for lower values of shear-wave velocity, the ratio is in the range of 0.2 to 0.4, indicating a large reduction in response from rigid to flexible foundation. Although the response of tanks followed the same trend under both earthquakes, the response values were substantially higher when tanks were subjected to the San Fernando earthquake, partly due to the larger amplitude of acceleration of the excitation.

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CONCLUSIONS

The present analysis has demonstrated the relative importance of a num­ber of parameters on the seismic response of ground-based tanks subjected to vertical excitations. It is clearly seen that interaction of the tank and the foundation soil reduces the tank response, and that the magnitude of this reduction is a function of the soil shear-wave velocity as well as tank geo­metric properties, more specifically the ratios HIR and hs/R. This conclusion is quite relevant. Other recent studies of the effects of vertical earthquake motions have shown that, when soil-tank interaction is neglected, these effects are very significant and must be taken into account when designing or analyzing ground-based tanks. It is recommended herein that soil-tank interaction be included in such analyses to achieve a more realistic assess­ment of the effects of vertical excitations.

APPENDIX I. REFERENCES

Abou-Izzeddine, W. (1989). "Parametric study of seismic interaction of liquid storage tanks with foundation soil," thesis presented to the Univ. of California, at Irvine, Calif., in partial fulfillment of the requirements for the degree of Master of Science.

Clough, R. W., and Penzien, J. (1975). Dynamic of structures. McGraw Hill, New York, N.Y.

Das, B. M. (1984). Fundamental of Soil Dynamics. Elsevier Science Publishing Co., New York, N.Y.

Haroun, M. A., and Abdel-Hafiz, E. A. (1986). "A simplified seismic analysis of rigid base liquid storage tanks under vertical excitation with soil-structure inter­action." Soil Dyn. Earthquake Engrg., 5(4), 217-225.

Haroun, M. A., and Tayel, M. A. (1984). "Dynamic behavior of cylindrical liquid storage tanks under vertical earthquake excitation." Proc. Eighth World Conf. on Earthquake Engrg., Vol. VII, 421-428.

Haroun, M. A., and Tayel, M. A. (1985a). "Axisymmetrical vibrations of tanks— Analytical." /. Engrg. Mech., ASCE, 111(3), 346-358.

Haroun, M. A., and Tayel, M. A. (1985b). "Axisymmetrical vibrations of tanks— Numerical."/. Engrg. Mech., ASCE, 111(3), 329-345.

Haroun, M. A., and Tayel, M. A. (1985c). "Response of tanks to vertical seismic excitations." /. Earthquake Engrg. Struct. Dyn., 13(5), 583-596.

Haroun, M. A., and Abou-Izzeddine, W. (1992). "Parametric study of seismic soil-tank interaction. I: Horizontal excitation." J. Struct. Engrg., ASCE, 118(3), 783-797.

Kumar, A. (1981). "Studies of dynamic and static response of cylindrical liquid-storage tanks," thesis presented to Rice Univ., at Houston, Tex., in partial ful­fillment of the requirements for the degree of Doctor of Philosophy,

Marchaj, T. J. (1979). "Importance of vertical acceleration in the design of liquid containing tanks." Proc. of Second U.S. Nat. Conf. on Earthquake Engineering.

Newmark, N. M., and Rosenblueth, E. (1971). Fundamentals of earthquake engi­neering. Prentice-Hall, Inc., Englewood Cliffs, N.J.

Novozhilov, V. V. (1964). Thin shell theory. P. Noordhoof Ltd., Groningen, the Netherlands.

Veletsos, A. S., and Kumar, A. (1984). "Dynamic response of vertically excited liquid storage tanks." Proc. of Eighth World Conf. on Earthquake Engineering, Vol. VII, 453-460.

Veletsos, A. S., and Tang, Y. (1986). "Dynamic of vertically excited liquid storage tanks." J. Struct. Engrg., ASCE, 112(6), 1229-1246.

Wolf, J. P. (1985). Dynamic soil-structure interaction. Prentice-Hall, Englewood Cliffs, N.J.

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APPENDIX II. NOTATION

The following symbols are used in this paper:

A = area of tank base; C,y = coefficients of damping matrix; C = damping matrix; cf = shear-wave velocity;

.. Es = Young's modulus of shell material; Gv(t) = vertical ground acceleration;

Gf = shear modulus of foundation soil; g = acceleration of gravity;

H = liquid depth; hb,hs = thickness of base and shell;

I0 = modified Bessel function of first kind of order zero; , Kjj = coefficients of stiffness matrix; K = stiffness matrix; L = shell length; M = mass of tank base and shell;

Mjj = coefficients of mass matrix; M = mass matrix

p(R,z,t) = axisymmetric pressure exerted by liquid on tank wall; pd = hydrodynamic pressure; R = radius of tank; r = radial coordinate; T = kinetic energy; t = time; u = axial displacement of shell; V = potential energy;

v'(t) = total vertical displacement of base; v = vertical displacement of base relative to free-field motion;

W = work done by external loads; w{z,i) = radial displacement of shell;

w(f) = time-dependent displacement amplitude; z = axial coordinate; 8 = variational operator;

Xi = parameter equal to (ir/2H); Vf,vs = Poisson ratio of foundation soil and shell;

£ = damping ratio; Pb.Py = mass density of base and foundation soil; p;,ps = mass density of liquid and shell;

§(r,z,t) = velocity potential function; to = natural frequency of vibration; and o> = frequency parameter.

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