SEISMIC RESPONSE OF SOIL-SUPPORTED UNANCHORED

LIQUID-STORAGE TANKS

By Praveen K. Malhotra,' Member, ASCE

ABSTRACT: A systematic study is made of the effects of base uplifting on the seismic response of cylindrical liquid-storage tanks that are supported directly on flexible soil foundations. First. a detailed investigation is made of the effects of system parameters on the uplifting resistance and energy dissipation capacity of the partially uplifted base plate. It is shown that: (1) the hydrodynamic base pressures reduce the uplifting resistance as well as the energy dissipation capacity of the base plate; (2) the uplifting resistance increases with increase in the thicknesses of the base plate and the tank: wall. and the stiffness of the foundation soil; and (3) the energy dissipation capacity increases with an increase in the base-plate thickness and the yield level of plate material, and reduces with increase in foundation stiffness and the thickness of the tank wall. Next. an efficient method is presented for the dynamic response analysis of flexibly supported unanchored tanks. It is shown that the flexibility of the foundation reduces the overturning base moment, and reduces significantly the axial compressive stresses in the tank: wall. but these reductions are accompanied by increased values of plastic rotations and (in some cases) base uplifting and hoop compressive stresses.

INTRODUCTION

Numerous studies have been conducted on the seismic re­sponse of ground supported liquid-storage tanks. Whereas, the initial studies by Jacobsen (1949) and Housner (1963) were concerned with the hydrodynamics of fl uid in a rigid tank. the later studies by Veletsos and Yang (1977), Haroun and Hous­ner (1981), Veletsos and Tang (1990), among others, explored the effects of fluid-structure and fluid-stfUcture-foundation in­teraction for fully anchored, flat-bottom tanks. In reality. how­ever, complete anchorage is not always warranted or feasible; as a result, a large number of tanks are either unanchored or only partially anchored at their base. During intense ground shaking these tanks experience partial base uplifting and re­spond in a nonlinear manner. Recent studies on the response of unanchored tanks supported on rigid concrete mat founda­tions have shown that base uplifting influences significantly the dynamic response of tanks and leads to axial compressive stresses in their walls that are substantially higher than those in similarly excited fixed-base systems (peek 1986; Natsiavas 1987; Haroun and Badawi 1988; Lau and Clough 1989; Mal­hotra and Veletsos 1994, 1995).

In practice, many unanchored tanks are supported directly on flexible soil foundations . When subjected to earthquake ground shaking. these tanks uplift on one side and penetrate their flexible foundation on the opposite side; the resulting response is therefore highly nonlinear. Such tanks have sus­tained damage in the fonn of: ( I) the failure of the piping connections to the wall, caused by large base uplifting: (2) rupture at the plate-shell junction, caused by excessive joint stresses; (3) buckling of the tank wall, caused by large axial compressive stresses; and (4) failure of the soils underneath, caused by excessive foundation penetrations (Hanson 1973; Smoots 1973; USDOC 1973; Gates 1980; Manos and Clough 1985 ; "Northridge" 1995).

Manos and Clough (1982), Cambra ( 1982), Sakai et al. (1988), and Akiyama and Yamaguchi (1988) conducted static­tilt and shaking table tests on scale models of flexibly sup­ported unanchored tanks. The flexibility of the foundation soils

lAdjunct Fac. , Dept. of Civ. Engrg., California State Univ., Sacra­mento, CA 95819-6029.

NOie. Assoc iate Editor: Nicholas P. Jones. Discussion open until Sep­tember I , 1997. To extend the closing date one month, a written request must be filed with the ASCE Manager of Journals. The manuscript for this paper was submitted for review and possible publication on April 24, 1995. This paper is part of the ]ourlUll of Structural Engineering, Vol. 123. No.4. April. 1997. O ASCE, ISSN 0733-9445/97/0004-0440-04501 $4.00 + $.50 per page. Paper No. 10577.

440 / JOURNAL OF STRUCTURAL ENGINEERING / APRIL 1997

was simulated by placing a 2.54 cm (I in.) thick rubber pad under the base plate. It was shown that the foundation flexi­bility reduces significantly the axial compressive stresses, but increases base uplifting, foundation penetration. and hoop compressive stresses in the tank wall.

The current study is an analytical approach to the dynamic analysis of flexibly supported unanchored tanks. An important step in this direction-the response analysis of partially up­lifted base plate under static loading-was the subject of an earlier paper (Malhotra 1995). This paper has a twofold ob­jective: (I) to present an in-depth study of the effects of system parameters on the uplifting resistance and energy dissipation capacity of the base plate under dynamic loading; and (2) to present an efficient method for the seismic response analysis of flexibly supported unanchored tanks.

SYSTEM CONSIDERED

Shown in Fig. I(a), the system considered is a cylindrical tank of radius R, filled to a height H with a liquid of mass density PI. Presumed resting on a Winkler foundation of sub­grade modulus K, the tank is excited by a unidirectional hor­izontal ground motion x,(t), the intensity of which is sufficient to induce rocking of its wall and partial uplifting of its flex ible base plate, as shown in Fig. I(b). The maximum width of the uplifted portion of the base plate is denoted by L. Points in the tank are specified with a cylindrical coordinate system (r,

(.J (bJ

+ ['-"l - L- i,(t)

FIG. 1. System Considered

~ b 1 n/2 - 1

T2 3

FIG. 2. Base Plate Model

<1>, z), the origin of which is taken at the center of the base plate.

UPLIFTING RESISTANCE AND DAMPING OF BASEPLATE

Problem Statement

In its "at rest" condition, the base plate is subjected to a unifonn hydrostatic pressure p = PlgH on its surface and a unifonn line load W/2'ITR along its boundary, where W is the weight of the tank wall. In an earthquake, the base plate is subjected also [Q a hydrodynamic pressure Pd on its surface and an overturning moment Mr transmitted to its boundary by the wall of the tank. As a result of these forces, the base of the tank rotates by an angle IjI and a portion of the base plate uplifts, as shown in Fig. I(b). Of interest herein is the Mr -tV relationship.

Model and Solution Method

The plate is represented by n unifonnly loaded, flexibly sup­ported, semiinfinite beams that are connected at their ends to the cylindrical wall of the tank, as shown in Fig. 2. For the sake of clarity, only a few beams are shown in this figure. It

(a)

8

4

IOOMrO W/R

-4

-8 -0.5 0 0 .5 -0. 5

is assumed that the plate boundary remains in one flat plane at all times. A detailed analysis of the accuracy of the model may be found elsewhere (Malhotra 1995; Malhotra and Yelet­sos 1995).

The Mr - 4t relationship is established by considering small increments of base rotation .6.\jJ. For each increment, the ver­tical displacement at the end of each beam is obtained first, the uplifting force at the end of each beam computed next from the beam analysis (Malhotra 1995), and the moment re­sultant of the uplifting forces computed in the end to yield a value of Mr. Beams in this model are loaded unifonnly by the hydrostatic base pressure only. The effect of hydrodynamic base pressures is considered later.

Numerical Results

A representative plot of the Mr - \jJ relationship for three different load cycles of size IIjI I = 0.2°, 0.35°, and 0.5" is shown in Fig. 3(a). The moment Mr is expressed as a per­centage of W,R. where WI = -rrR2p is the total weight of the liquid. For the results shown, the base plate thickness h = RI2,OOO, the nonnalized yield stress rr,lp = \,800, and the nonnalized Young's modulus of elasticity Elp = 1.5 X 10'. The plate is constrained at its boundary by the tank wall of unifonn thickness h, = R/I ,OOO and weight W = 0.015W,. The slope of the Mr - IjI plot represents the uplifting stiffness of the base plate, whereas the area enclosed within a load cycle represents the hysteretic energy lost by plastic yielding at the plate boundary. It is desired to separate these two effects.

For increasing values of MT (loading), the plastic hinging at the plate boundary starts at <I> = 0° and 180° (not necessarily simultaneously) when the radial bending moment, per unit cir­cumferential length of the plate, becomes equal to the yield moment

" h' M =-' -, 4 ( I )

where uy = yield stress of the plate material. With further in­crease in MT , the plastic hinging spreads in the circumferential direction. The direction of the yield moment at 4> = 0° is shown in Fig. 4(a). The loading path after plastic hinging is along the upper S-shaped curve in Fig. 3(a). For decreasing values of MT (unloading), the plastic hinging begins once again at <P = 0° and 180°, but the direction of yield moment is re­versed in this case, as shown in Fig. 4(b). The unloading path is along the lower S-shaped curve in Fig. 3(a). Since the load­ing and unloading paths are nearly parallel to each other, they represent the same stiffness. The size of the hysteresis loop,

(b) + (c)

0 0.5 -0.5 0 0 .5

Base Rot ation , '" (d egrees)

FIG. 3. Plots of: (a) Moment-Rotation Diagram; (b) Skeleton Stiffness; (e) Hysteresis Loops

JOURNAL OF STRUCTURAL ENGINEERING I APRIL 1997 / 441

Ca) Loading Cb) Unloading

Shell

Base plate

M,

FIG. 4. Plastic Hinging at Plate Boundary at 01> = ()" [Fig. 1 (a))

TABLE 1. Values of Es , ED. and ~II for Three Load Cycles Shown in Fig 3(8 )

1 " I 10'Es 10sEo '" (degrees) W, R W, R (percent) (1 ) (2) (3) (4)

0.2 94 89 7.5 0.35 216 167 6.2 0.5 363 236 5.2

Note: Results are for Khlp = 2; hlR = 0.0005; h.lR = 0.001; a~/p = 1,800; £/p = 1.5 x 106

; and WIW/ = 0.015.

hence the amount of energy dissipated, is controlled by the distance between the two paths. The distance reduces as My reduces; in the limiting case when My = a the loading and unloading paths overlap and the hysteresis loop disappears completely, as shown in Fig. 3(b). The plot of Fig. 3(b) is, therefore, the skeleton stiffness of the base plate, which is obtained by assuming a pin condition (zero moment) at the plate-shell junction. The damping curve (hysteresis loop) is obtai ned by subtracting the skeleton stiffness from the plot of Fig. 3(a); it is shown in Fig. 3(c).

The area Es under the skeleton stiffness curve [Fig. 3(b)] denotes the elastic strain energy, while the area ED within the hysteresis loop [Fig. 3(c)J denotes the energy d issipated in a load cycle. Values of Es and ED for the three load cycles are shown in Table 1. Also shown in this table are the values of the quantity

{ _ 100Eo h - 41TEs (2)

where 'h = percentage effective viscous damping due to hys­teretic action (Chopra 1995; pages 94 - 100). If the tank wall were rigid, 'h would denote the contribution of the base plate to the overall system damping. For a real tank, the effect of this contribution is reduced because only a pan of the total deformation takes place in the base plate, while the remaining takes place in the tank wall. In Table I, both Es and ED are seen to increase with increase in the size of the load cycle; the latter, however, increases at a slower rate. The hysteretic damping 'h, therefore, reduces with increase in the size of the load cycle.

BASE UPLIFTING UNDER HYDRODYNAMIC LOADING

Hydrodynamic Pressures

In a seismically excited tank, the hydrodynamic pressures are generated by the impulsive action of the liquid movi ng rigidly with the tank wall, and the convecti ve action of the liquid moving in sloshing modes near the free surface. Of the two, the impUlsive action usually dominates the response, and

442 I JOURNAL OF STRUCTURAL ENGINEERING I APRIL 1997

in most cases the fundamentaJ impulsive mode alone provides satisfactory results. The hydrodynamic wall pressures under these assumptions may be expressed as (Veletsos and Tang 1990)

p,(z, 01>. t) = -a(z)p,RA(t)cos 01> (3)

where a(z) = dimensionless fu nction that defines the height­wise variation of pressures; and A(t) = instantaneous pseu­doacceleration of a single degree of freedom (SDOF) model of the tank-liquid system. The hydrodynamic wall pressures can be integrated to obtain the following expression for the overturning base moment MT •

MT(t) = miiA(t) (4)

where m = effective or modal mass of the liquid; and Ii = height of the resul tant of hydrodynamic wall pressures. The hydrodynamic base pressures may be expressed as

pir, 01>, t) - -a(r)p,RA(t)cos <I> (5 )

where a(r) = dimensionless function with defines the radial variation of pressures. Values of m, Ii, and a are reported in literature as functions of the properties of the tank and the contained liquid (Haroun and Housner 1981; Veletsos and Tang 1990); m and ii are usually expressed as fractions of the totaJ liquid mass ml and liquid height H, respectively.

The net pressure on the base plate, obtained by superim­posing the hydrodynamic pressure Pd on the hydrostatic pres­sure P = p,gH, is given by

p(r, <1>, t) = p,gH - a (r )p,RA(t)cos <I> (6)

The pseudoacceieration A(t) in (6) is eliminated by making use of (4), to give

MT(r) p(r, 01>, t) = p,gH - a (r )p,R --- cos <I> (7)

mh

Solution Method

The circumferential variation of base pressures is accounted for by assigning different values of pressure to different beams in the base plate model of Fig. 2. The pressure on the ith beam is

MT(t) [21T ] p ,(r, t) = p,gH - c«r)p,R -_- cos - (i - I) mh n

(8)

On account of the variation of a in the radial direction, the intensity of pressure varies along the length of each beam. This variation may, however, be neglected because the uplifted width of the plate is only a small fraction of the tank radius R. In this analysis, the value of pressure corresponding to T = R is used for the entire length of each beam. This leads to the following expression for the pressure on the ith beam .

p,(r) = { I M T(r) [27r ]} - ~ -- cos - (i - I) p,gH W,R n

(9)

in which

a (R)

~ = (mim,)(iilH )(HIR)' ( 10)

The solution method for dynamic uplifting is similar to that for static uplifting, except that the pressure on various beams needs to be recomputed at the end of each rotation increment Il.IjJ using (9) and (10).

Numerical Results

Results were obtained for different values of the hydrody­namic parameter r3 , subgrade modulus K, base plate th ickness

(a) (b) (c)

8 {! = 5 (! = 15

-4

-8 -0.5 a 0.5 -0.5 a 0.5 -0.5 a 0.5

Base Rotation, 'if; (degrees)

FIG. 5. Effect of Hydrodynamic Base Pressures on Skeleton Stiffness and Hysteresis Loop

(a) (b) (c)

8 K.h l p = 1 "hi p = 2 Khl p = 100

-8 -0.5 a 0.5 -0.5 a 0.5 -0.5 a 0.5

Base Rotation, 'if; (degrees)

FIG. 6. Effect of Foundation Stiffness on Skeleton Stiffness and Hysteresis Loop

(a) (b) (c)

8 hi R = 0.00025 hlR = 0.0005 hlR = 0.001

-8 -0.5 a 0.5 -0.5 a 0.5 -0.5 a 0.5

Base Rotat ion , 'if; (degrees)

FIG. 7. Effect of Base Plate Thickness on Skeleton Stiffness and Hysteresis Loop

h, shell thickness hs. and yield level of plate material crr Se­lected plots of skeleton stiffness and hysteresis loop are pre­sented in Figs. 5-7, and the values of E" ED, and {, in Table 2.

For a typical tank, a.(R) - 0.5 (Veletsos and Tang 1990). For steel tanks with height to radius ratio HIR between 0.5 and 3, mimi ranges from 0.3 to 0.7, and IiIH from 0.4 to 0.55. The value of 1>, determined from (10), therefore ranges from 0.14 for a very tall tank to 16 for a very broad tank. The plots

in Fig. 5 show that as f3 increases from 0 to 15 , the skeleton stiffness reduces. This is because the hydrodynamic base pres­sures assist uplifting by acting upward on the uplifted side of the base plate and downward on the contact side. The size of the hysteresis loop (ED in Table 2) reduces by 57%, as I> in­creases from 0 to 15; the corresponding reduction in the value of ~, is 46%.

In Fig. 6, an increase in K is shown to increase s ignificantly

JOURNAL OF STRUCTURAL ENGINEERING 1 APRIL 1997 / 443

TABLE 2. Effect of System Parameters on Values of E • • Eo. and., fo r Load Cyc le of S ize 1",1 = 0.5"

105E$ 105£ 0 .,

Parameter W,R W, R (percent) (1 ) (2) (3) (4)

~=O 363 236 5.2 ~=5 338 202 4.8 ~ = 15 292 101 2.8 Khlp = 1 286 202 6.0 Khlp = 2 338 202 4.8 Khlp = 100 551 257 3.7 hiR' = 0.0025 252 98 3.1 hiR' = 0.005 338 202 4.8 hiR' = 0.01 410 280 5.4 h,IR b = 0.005 288 173 4.8 h,IR b = 0.01 338 202 4.8 h,/R' = 0.02 418 209 4.0 u7 1p = 1,800 338 202 4.8 u7 1p = 2,700 338 247 5.8 u,lp = 3,600 338 259 6.1

Note: unless noted otherwise, 13 = 5; Khlp = 2; hlR = 0.0005; h,IR = 0.001; u,lp = 1,800; Elp = 1.5 X IO's; and WIW/ = 0.015.

-For these results. Khlp = I. 2, and 4. respectively. "For these results, WIW, = 0.0075, 0.015, and 0.03 , respectively.

the skeleton stiffness. Although the size of the hysteresis loop also increases with increase in K, the damping ratio 'h actually reduces as seen in Table 2. It should be pointed out that the values of Khlp = I and 100 imply that the base plate, when subjected to the hydrostatic pressure p, settles by h and hlI 00, respectively. These two values of Kh/p are expected to rep­resent a soft soil and a rigid concrete mat foundation, respec­tively.

In Fig. 7, an increase in the base plate thickness is shown to increase both the skeleton stiffness and the size of the hys­teresis loop. In Table 2, a fourfold increase in h is shown to increase ~, by 76%. Values in Table 2 show that the skeleton stiffness increases significantly with increase in the shell thick­ness h,. Although the size of the hysteresis loop (ED) increases, the value of 'h actually reduces. With increase in the yield level (j y the skeleton stiffness remains unchanged. but the size of the hysteresis loop increases. as seen in Table 2. A twofold increase in (j1 results in a 28% increase in ~h'

SEIS MIC R ESPONSE ANA LYS IS

System Model

The hydrodynamic pressures in a flat-bottom tank, respond­ing in its fundamental impulsive mode of vibration, are in­duced by the translational and rocking motion of the tank wall, as well as by the rocking motion of the tank base plate. For an uplifting tank, the latter contribution to the hydrodynamic pressures can be ignored because in this case, as seen later, only about 5-10% of the total area of the base plate actually participates in the rocking motion. Under these assumptions the system may be represented by the model shown in Fig. 8(a), in which the mass m represents the portion of the liquid associated with the fundamental impulsive mode, and Ii is the height of the resultant of the hydrodynamic wall pressures. The rotational spring at the base represents the rocking resis­tance of the base plate. The rotational damper accounts for (in an approximate manner) the effects of soil internal damping and the radiation damping.

Method of Ana lysis

The equilibrium of forces on the mass in Fig. 8(a) requires that

444 1 JOURNAL OF STRUCTURAL ENGINEERING 1 APRIL 1997

h~~1

r m

I,· (-) L.... ;;,(t) (b) L.... ;;,(t )

FIG. 8. Modet of Tank-liquid System

milo + c(';o - Ii.j, ) + k(uo - IiljJ) = -mx, (t)

whereas the equilibrium of base moments requires that

c,

(lla;

(C(';o - Ii.j,) + k(uo - IiljJ)]1i = MT(IjJ) + c • .j, (lIb)

where Uo = overall horizontal displacement of the mass relative to the moving base (overdot denotes differentiation with re­spect to time t); C ::: damping coefficient for the tank in its fixed-base condition; c .. = soil damping coefficient; k ::: stiff­ness of the superstructure; and MT(~) the moment in the base spring, as a function of the time-dependent rotation 41.

Eqs. (lla) and (lIb) are solved incrementally, assuming linear relationship between each moment increment !l.MT and each rotation increment A~, i.e.

(1 2)

where k~ ::: instantaneous value of the rotational spring stiff­ness. With the prefix A used to represent a small increment for each remaini ng response as well, (Ila) and (lIb) may be written as

[m 0] {l:J.ilo } + [ c -c] {1:J.';0} o 0 lil:J.ijJ -c c + c.lli' lil:J..j,

[ k -k] {~uo} = _{mI:J.X,(I)} + -k k + k,lli' hl:J. IjJ 0 ( 13)

The preceding set of equations may be reduced to a single differential equation by making use of the following approx­imate relationship between Auo and !l.41, which is established from the second part of (l3) neglecting the effect of damping forces

( 14)

In (14), kr = the effective stiffness of the uplifting system; it is given by

I I Ft' - :::- + ­k. k k>JI

(IS)

On substituting (l4) into ( 13) and premultiplying the resulting expression by the transpose of the vector on the right side of (l4), one obtains

mAuo + c.Auo + kr 6.uo = -mAx,(t) (16)

which is the incremental equation of motion for the SDOF

model shown in Fig. 8(b). In ( 16), c, = the effective viscous damping of the uplifting system; it is given by

(k')' c, ( k')' c~ = C k + hi 1 - k (17)

Since the stiffness is inversely proportional to the square of the period, the stiffness ratio k/k may be expressed as

(18)

where T = period of the fixed-base system ; and t == effective period of the uplifting system. Substitution of (18) into ( 17) gives

c c, ( I )' c- - + - 1- -, - (tiT)' h' (tiT)'

(19)

Note that the effective damping c, depends on the period elon­gation t i T. For a system that derives its flexibility from the superstructu re only, t IT = 1; which gives Cr = c. For a system that derives its flexibility from the rotational spring only, f IT ;:: 00; which gives Ct ;:: c,/f? The percentage effective viscous damping {, is given by

100c,t { = --

r 4m'Tr (20)

It should be noted thaI the contribution of the hys teretic damp­ing in the base plate all) is not included in ~t.

Eq. ( 16) is solved for Il u. at uniform time steps II I = 0.0002 s by the linear acceleration method (Clough and Penzien 1993; pages 121-132). The results changed imperceptibly when Ilt was halved to 0.0001 s or doubled to 0.0004 s. The extremely small time step was needed to capture accurately the effects of sudden changes in the sti ffness of the sys tem due to up­lifting of the base plate and yielding at the plate boundary. The incremental base rotation a1.ll is computed next using the second part of (14), i.e.

(a)

120

60

0

=--60

6. -120

k,h IlIjI = - Ilu.

k. (21)

Corresponding to the new value of the base rotation 1.lI, new values of the overturning base moment M T , uplifting of plate boundary, radial separation between the base plate and foun­dation, plastic rotations at plate boundary, axial and hoop com­pressive stresses in tank wall, and shear force in tank wall are computed from the base plate analysis discussed earlier.

Numerical Results

Description of Tank, Conditions of Support, and Ground Motion

A 15.2 m (50 ft) radius steel tank, filled with water to a height of 12.2 m (40 ft) is selected for a detailed analysis. The shell of the tank has a maximum thickness h, = 1.37 cm (0.54 in.) at the base; and the base plate is of uniform thickness h = 1.04 cm (0.4 1 in .). The material properties for the lank are: Young 's modulus of elastic ity E = 200 GPa (29 X 10' ksi ), yield stress cry = 248 MPa (36 ksi), and Poisson's ratio v = 0.3. The unit weights of water and the tank material are: Pig = 9.81 kN/m' (62.4 pcf) and pg = 77 kN/m' (490 pef), re­spectively. The total weight of tank wall and roof W = 1.41 MN (31 6 kips). The system parameters for the first impulsive mode of vibration are obtai ned from Veletsos and Tang ( 1990) for an assumed value of the equivalent uniform thickness of the shell h, = 1.14 cm (0.45 in.). These parameters are: mg = 39.4 MN (8,855 kips), h = 4.9 m (16.2 ft), and T = 0.22 s. The damping factor c is such that the system exhibits 2% critical dampi ng in its fixed-base condition. The soil damping faclOr c. = 237 MN'm's (2.1 X 10' kip-in.-s), which upon using (19) and (20), together wi th an assumed period elon­gation of tiT = 2, gives an effective damping ratio {, = 5% critical.

The tank is examined for these three conditions of suppon: (I) fully anchored on a rigid foundation; (2) unanchored on a

(b)

6

3

0

-3

-6 :;;; 0 120 8 ~

60

3

0 ~ i!S

r Unanchored on rigid base,

-c: Q)

0 E 0

:::;; -60

Cl c: ·E -1 20

-3 ~ i!!

US -6 0;

-9 ~ :J t 120 Q)

<3 60

6

3 r Unanchored on flexible bas\

0 0

-60 -3

-120 0 2 3 4 5 6 7 0 2 3 4 5 6 7

-6

Time (sec) Time (sec)

FIG. 9. Histories of Overturning Base Moment and Axial Stress In Tank Wall for Three Different Cond itions of Support

JOURNAL OF STRUCTURAL ENGINEERING / APRIL 1997/ 445

rigid foundation; and (3) unanchored on a flexible foundation. The subgrade modulus for the flexible foundation is K = 54.3 N/cm) (200 pci), which is representative of a compacted gravel fill; a value I ()() times larger, K = 5.43 kN/cm' (20,000 pci), is assumed for the rigid foundation.

The ground motion considered is the first 6.3 s of the N-S component of the 1940 EI Centro, Calif. , earthquake ground motion record, scaled to a peak value of O.4g.

Response Time Histories

The plots of the overturning moment Mr for the three dif­ferent conditions of support are shown in Fig. 9(a). A com­parison between the top and the middle plot shows that, for a rigidly supported tank, a change from fully anchored to unan­chored condition causes the response period to elongate from 0.22 s to about 0.4 s, and the peak base moment to reduce by nearly 25%. Higher damping in the uplifting system is partly responsible for this reduction. An estimate of damping in the uplifting system is given later. A comparison between the mid­dle and the bottom plot in Fig. 9(a) shows that a change from rigid to flexible fou ndation causes the response period to elon­gate further to about 0.55 s and the overturning base moment to reduce by an additional 10%.

Shown in Fig. 9(b) are the plots of the axial stress at the base of the tank wall computed at <p = 180° [Fig. I(a)]; neg­ative values imply compression. A comparison between the top and the middle plot shows that, for a rigidly supported tank, a change from fully anchored to unanchored condition causes the axial compressive stress to increase to nearly four times. This dramatic increase is due to considerably small con­tact between the wall and the foundation of a rigidly supported uplifting tank. A comparison between the middle and the bot­tom plot in Fig. 9(b) shows that, for an unanchored tank, a change from rigid to flexible foundation causes the axial com­pressive stress to reduce to less than one-third.

Shown in Fig. 100a) are the plots of base uplifting at q, = 180° for rigidity and flexibly supported unanchored tank. The maximum uplifting is nearly the same for both tanks, but the flexibly supported tank also experiences significant foundation penetration ( - ve uplift). Small nonzero values of penetrations for the rigid foundation are on account of the very high, yet finite, value of the subgrade modulus K for that foundation.

Shown in Fig. I O(b) are the plots of plastic rotation at the plate boundary at <p = 180°. A positive value of plastic rotation

(a)

signifies a tendency of the joint between the wall and the base plate to increase from 900 [Fig. 4(a)1. whereas a negative value signifies a tendency of the joint to reduce from 90° [Fig. 4(b)]. Several cycles of large plastic rotations at plate-shell junction may cause the joint to rupture. The plots in Fig. lO(b) show that the maximum positive plastic rotation is nearly the same for both tanks, but the flexibly supported tank also experiences significant negative plastic rotation, and is therefore more sus· ceptible to rupture at the plate-shell junction. Note that at the end of the shaking, some plastic rotation remains at the plate­shell junction.

Moment-Rotation Diagrams and Overall System Damping

The plots of the relationship between the overturning mo­ment and base rotation are shown in Fig. 11. The hysteresis loops for the flexibly supported tank are bigger, due 10 greater plastic action in the base plate of that tank.

An estimate of the overall system damping is obtained by adding the hysteretic damping 10 the effective viscous damping " . The hysteretic damping obtained from (2) is: " = 4% for the rigidly supported system (for !1jJ! = 0.25°) ; and ,, = 5.4% for the flexibly supported system (for !1jJ! = 0.3°). Since only a portion of the total defonnation takes place in the base plate, the net hysteretic damping for the overall system should be smaller than Sit. Further, since the deformations are inversely proportional to the stiffness, the net hysteretic damping s~ is given by

,; = " k,F.' (22) k.

which, upon making use of (1 5) and (I8), gives

,; = " [1 - (t/~)'] (23)

For the rigidly supported tank, f IT - 1.8; which gives St = 4% [from (J9) and (20)] and G = 2.8%. For the flexibly sup­ported tank, tiT - 2.5%; which gi ves " = 7.6% and ,; = 4.5%. The overall system damping (', + ,;) for the rigidly and flexibly supported tank is therefore 6.8 and 12.1 %, re­spectively.

Circumferential Distribution of Responses

The plots of circumferential distribution of base uplift, ra­dial separation, and plastic rotation are shown in Fig. 12, and

(b) 6

runanChored on rigid base 15

4 10 (i)

2 5 OJ ~

§. Ol 0 0 OJ

;E ~

g-2 -5 <= a OJ 6 r Unanchored on flexible base"""\ 15 ~ '" '" a: CD 4 10 ()

~ 2 5 '" 1i:

0 0

-2 0 2 3 4 5 6 7 0 1 2 3 4 5 6 7 -5

Time (sec) Time (sec)

FIG. 10. Histories of Base Uplifting and Plastic Rotation at Plate Boundary for Rigidly and Flexibly Supported Unanchored Tank

446 1 JOURNAL OF STRUCTURAL ENGINEERING 1 APRIL 1997

those of axial stress, hoop stress, and shear force are shown in Fig. 13. The plots are at an instant of time when the tank's response is maximum. The dashed line represents the footprint of the tank wal l and the solid line represents the magnitude of the response; positive values are plotted radially inward.

The rigidly supported tank experiences an uplift of 13.2 cm (S.2 in.) at 4> = 180° and practically no foundatio n penetration at 4> = 0°. The flexibly supported tank experiences an uplift of IS.5 cm (6.1 in.) and a foundation penetration of 4.1 em ( 1.6 in.). The maximum separation between the base plate and the foundation is 104 cm (4 1 in.) for the rigidly supported, and 94 cm (37 in.) for the flexibly supported tank. These val­ues are significantly small compared to the radius of the tank which is IS.2 m (SO ft). The net area of the uplifted portion of the base plate is about 5% of the total base area for each tank. The maximum positive plastic rotation, at <J> = 180°, is 12.3° for the rigidly supported, and 14.5° for the flexibly

(a)

80 Rigid foundation

:iF K = 20000 pci c. :;;: 0 40 0 0 :::. E Q)

E 0 0 ;:;; C> c ·c :; -40 1:: Q)

<3 -80

-0.4 -0.2 0 0.2

Base Rotation (degrees)

0.4

supported tank; the flexibly supported tank also experiences a negative plastic rotation of 7 .2° at 4> = 0°,

The axial force generated by the moment Mr and the weight of tank W is distributed over an arc of central angle equal to 40° for the ri gidly supported, and 1 \30 for the flexibly sup­ported tank (Fig. 13). The maximum axial compressive stress for the rigidly supported tank is S7 .2 MPa (8.3 ksi), which is more than three times the value of 17.9 MPa (2.6 ksi) for the flexibly supported tank.

As the base of the tank uplifts, the tendency of the tank wall to move radially inward induces hoop stress cr~4> in the wall (Malhotra 1995). The hoop compressive stress is maximum on the uplifted side of the tank and nearly zero on the contact side; values of 14S MPa (2 \.1 ksi) for the rigidly supported, and 190 MPa (27.6 ksi ) for the flexibly supported tank, are quite significant. Nonunifonn distribution of the axial com­pressive stress at the base of the tank wall induces shear force

(b)

80 Flexible foundation " = 200 pci

40

0

-40

-80

-0.4 -0.2 0 0.2 0.4

Base Rotation (degrees)

FIG. 11. Base Moment-Rotation Diagrams

-1.6

(a) Rigidly Supported

5.~·:-0,' ,.--..... . . , " ................. , ,'

, , ,

Base Uplift (in.)

(b) Flexibly Supported

",--,',' -.. " ,.

"' ........ '"

• , • ,

6.1 :--

,"

, , , ..-

Base Uplift (in.)

, . , . , ,

G'···---·····

41 t-, , . , . , . ,

.. ............. ..

Radial Separation (in.)

.. .......... .. .... ,,\

, , , l-, , , ,

Radial Separation (in.)

-7.2

.............. """"

• , • , , .-­, , , ,

" .. ........

,

Plastic Rotation (deg)

Plastic Rotation (deg)

FIG. 12. Circumferential Plots of Responses at Base of Rigidly and Flexibly Supported Unanchored Tank

JOURNAL OF STRUCTURAL ENGINEERING 1 APRIL 1997 1447

(a) Rigidly Supported

, ,

.. .. ........

-8 .3 - : 40·

, , , , , , , , , ,

-2.6

, ,

, " ...... " .... _-- _ .. ...

Axial Stress (ksi)

(b) Flexibly Supported

...... _ .. ...... ' , ,

, • • • ,

Axial Stress (ksi)

Hoop Stress (ksi)

Hoop Stress (ksi)

Shear Force (kips)

\ ----.. .. .. ... .. , " , , , , , ,

• •

Shear Force (kips)

FIG. 13. Circumferential Plots of Responses at Base 01 Rigid ly and Flexib ly Supported Unanchored Tank

TABLE 3. Effects of Ground Motion Intensity on Maximum Responses

M, w, 9, Ii, 103 kN ' m ~ L "u " .. v., (g) (10' kip-ft) (degrees) em (in.,. em (in.)b em (in.) (degrees)· (degrees)!> MPa (ksi) MPa (ksi) MN (kips) (1 ) (2) (3) (4) (5) (6) (7) (8) (9) (10) (1 1)

0.2 70.2 (51.8) 0 .10 3.3 (1.3) -2.0 (-0.8) 56 (22) 2.6 -1.2 -10.3 (-1.5) -11.0 (- 1.6) 1.63 (367) 0.3 82.7 (61.0) 0 .22 8.6 (3.4) -3.0 (-1.2) 76 (30) 8.1 -3.9 -13.8 (-2.0) -67.6 (-9.8) 1.96 (440) 0.4 95.8 (70.7) 0 .37 15.5 (6.1) -4.1 (- 1.6) 94 (37) 14.5 -7.2 -17.9 (-2.6) -190.3 (-27.6) 2.30 (518) 0.5 114.1 (84.2) 0.58 25 .2 (9.9) -5.6 ( -2.2) 119 (47) 21.8 -11.4 - 23.4 (-3.4) -4 13.2 (-59.9)' 2.75 (618)

Note: Results are for H :: 12.2 m (40 ft); R = 15.2 m (50 ft) ; w"", 1.41 MN (316 kips); h.:z: 1.37 em (0.54 m.); h = 1.04 em (0.4 1 tn.): and ~ = 200 pel. -Maxi mum + lU' vaJue bMaximum -ue value. cStress eltceeds yield value.

V"'t on vertical sections of the wall (Malhotra 1995). Due to symmetry. the shear force is zero on vertical sections of the tank wall passing through 4> = 0° and 180°; on a section in­between it altains a maximum value of 2.81 MN (631 kips) for the rigidly supported and 2.30 MN (518 kips) for the flex­ibly supported tank.

Effects of Ground Motion Intensity

The sensitivity of the responses of the flexibly supported unanchored tank to the intensity of ground shaking was in­vestigated by considering scaled versions of the EI Centro ground mOlion record. The effects of varying the frequency content of the ground motion is not examined here. Table 3 lists the maximum tank responses for different values of the peak ground acceleration x" ~ The responses most sensitive to change in intensity are the base rotation tV. base uplifling w 1( + ve), foundation penetration WI( - ve) , plastic ro tations 9, and the hoop compressive stress CT<tllb; these responses in­crease at a rate faster than the peak ground acceleration X" The responses that are relatively less sensitive to changes in intensity are the overturning base moment Mr. radial separa­tion L, axial compressive stress CTw and the shear force V<t> x;

these responses increase at a rate slower than the peak ground acceleration Xg •

In Table 3, the hoop stress CT<t>41 exceeds the yield value of 248 MPa (36 ksi) for x, = O.5g. This is on accounl of the assumption made in the analysis that the tank wall behaves in a linear-elastic manner, which is not true for this particular

448 / JOURNAL OF STRUCTURAL ENGINEERING / APRIL 1997

case. The results are st ill shown to identify trends in the var­ious responses.

Effec ts of Subgrade Modulus

The results for progressively increasing flexibility of the foundation (reducing values of subgrade modulus K) are pre­sented in Table 4. An increase in foundation flexibility reduces the upli ft ing resistance of the base plate (increases the effective period of the system), and allows greater contact between the wall and the foundation after uplifting. As a result, the over­turning base moment Mr reduces, the base rotation $ increases, and the axial compressive stress in the tank wall CT~: reduces. In addition, the values of the base uplifting, foundation pen­etration, plastic rotations, and hoop compressive stresses in­crease, and the values of the radial separation L. and the shear force V4l: reduce. Responses most sensitive to changes in foun ­dation flexibility are the axial compressive stresses in the wall and the negative plastic rotations in the base plate.

Effects of Base Plate and Wall Thicknesses

The responses of the flexibly supported tank were computed for several different values of the base plate thickness h. Re­sults showed that an increase in base plate thickness is asso­ciated wi th: (1) reduced values of base rotation, base uplifting, foundation penetration, plastic relations, and hoop compres­sive stress; and (2) increased values of the overturni ng base moment MT , radial separation L, axial compressive stress CTw and shear force V4It • Most significant effect of an increase in

TABLE 4 Effects of Subgrade Modulus K on Maximum Responses

M, w, e, K 103 kN'm ~ L a" a" V"

(pci) (10' kip-ft) (degrees) em (in.)- em (in .)!> em (in.) (degreest (degrees)!> MPa (ksi) MPa (ksi) MN (kips) (1 ) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11 )

20,000 104.2 (76.9) 0.26 13.2 (5.2) 0.5 ( 0.2) 104 (41) 12.3 0.1 57.2 ( 8.3) 145.4 ( 21.1 ) 2.81 (631) 1,000 102.5 (75.6) 0.32 15.2 (6.0) -1.8 (-0.7) 102 (40) 14.2 -4.0 -27.5 (-4.0) -185.4 (-26.9) 2.59 (582)

500 100.7 (74.3) 0.34 15.5 (6.1) -2.8 (-1.1 ) 99 (39) 14.5 -5.4 -23.5 ( -3.4) -193.0 (-28.0) 2.50 (561) 200 95 .8 (70.7) 0.37 15.5 (6.1) -4.1 ( -1.6) 94 (37) 14.5 -7.2 -17.9 (-2.6) -190.3 (-27.6) 2.30 (518) 100 94.2 (69.5) 0.43 17.0 (6.7) - 6.1 (-2.4) 91 (36) 15 .7 -9.7 -15.8 (-2.3) -219.2 (-31.8) 2.20 (495)

Note: Resuils are for H = 12.2 m (40 ft); R = 15.2 m (50 ft); W = 1.41 MN (316 kips); h, = 1.37 cm (0.54 In .); h = 1.04 cm (0.4 1 In.); and x, - 0.4 g. "Maximum + tie value. ~Maximum - tit' value.

base plate thickness is to reduce the negative plastic rotation at the plate boundary,

An increase in the thickness of the tank wall hs has a three­fold effect: ( I) it reduces the fixed-base period of the system; (2) it increases the weight of the tank wall. hence the resistance to uplifting; and (3) it further increases the resistance to up­lifting by increasing the resistance against inward movement at the plate boundary. The results obtained for several different values of the shell thickness showed that an increase in shell thickness is associated with reduced values of base rotation, base uplifting, positive plastic rotation, axial compressive stress, and hoop compressive stress. and an increased value of the negati ve plastic rotation. Responses not particularly sen­sitive to changes in shell thickness are the overturning base moment Mr , foundation penetration w!( - ve), radial separation L, and shear force V.,.

CONCLUSIONS

Seismic response of unanchored liquid-storage tanks, sup­ported directly on flexible soil foundations. has been examined using an analytical approach. First, a detailed insight has been provided into the uplifting stiffness and damping of the base plate under hydrodynamic loading. Next, an efficient method has been presented for the dynamic response analysis of tanks under seismic loading. The trends observed in the numerical results are in agreement with those observed in past experi­mental studies. The following conclusions are based on the results presented in this paper.

For unanchored tanks, the hysteretic dampi ng due to plastic yielding in the base plate may range from 2.5 to 5%. The value of the damping reduces with increase in the size of the load cycle.

The hydrodynamic base pressures reduce the uplifting stiff­ness and energy dissipation capacity of the base plate. The effect of hydrodynamic base pressures is significant for broad tanks only.

Increase in the foundation stiffness increases the uplifting stiffness of the base plate, but reduces its energy dissipation capacity. An increase in the thickness of the base plate or the tank wall increases the uplifting stiffness of the base plate, but only an increase in base plate thickness increases its energy dissipation capaci ty. Increase in the yield level of the base plate material increases the energy dissipation capacity of the base plate. but does not affect its uplifting stiffness.

Base uplifting significantly reduces the magnitude of the hydrodynamic forces generated by ground shaking. The flex­ibility of the supporting foundation further reduces the hydro­dynamic forces.

Unlike rigidly supported tanks, base upl ifting for flexibly supported tanks is not associated with significant increase in the axial compressive stress in the lank wall. Unanchored tanks on flexible soil foundations are, therefore, less likely to ex­perience "elephant-foot" type buckling of their walls as com­pared to similar tanks on rigid concrete .mat foundations.

The flexibility of the supporting soils may allow significant foundation penetration and may lead to greater values of base uplifting and hoop compressive stresses in the tank walL Due to the nonlinear behavior of soils (not considered in this study), soil supported tanks may experience uneven and per­manent senlement around the perimeter.

The foundation flexibility leads to greater plastic rotations at the plate boundary and, therefore, greater dissipation of en­ergy due to hysteretic action. For the same reason, the likeli­hood of rupture at the plate-shell junction is greater for a flex­ibly supported tank, The radial separation between the base plate and the foundation is not significantly affected by changes in foundation flexibility. The shear force in the tank wall reduces as the foundation flexibility increases.

Responses most sensitive to changes in intensity of ground shaking are the base rotation, base uplifting, foundation pen­etration, plastic rotations at plate boundary, and hoop com­pressive stresses in tank walL Responses relatively less sen­sitive to changes in intensity are the overturning base moment, radial separation between base plate and foundation, axial compressive stresses in tank wall. and shear force in tank wall.

An increase in either the base plate thickness or the tank wall thickness reduces the base rotation and the uplifting of plate boundary. In addition, an increase in plate thickness re­duces the foundation penetration and plastic rotations, and an increase in wall thickness reduces the axial and hoop com­pressive stresses in tank wall.

APPENDIX I, REFERENCES

Akiyama, N., and Yamaguchi, H. (1988). "Experimental study on lift­off behavior of flexible cylindrical tank." Proc., 9th World Con! on Earthquake Engrg., Vol. 6, Tokyo-Kyoto. Japan, 655-660.

Cambra. F. (1982). "Earthquake response considerations of broad liquid storage tanks." Rep. EERC 82-25. Earthquake Engrg. Res. Clf., Univ. of California. Berkeley, Calif.

Chopra, A. K. (1 995). Dynamics of structures: theory and applications to earthquake engineering. Prentice-Hall, Inc., Englewood Cliffs, N.J.

Clough, R. W., and Penzien , J. ( 1993). Dynamics of structures, 2nd Ed., McGraw-Hill Book Co., Inc .. New York. N.Y.

Gates. W. E. (1980). "Elevated and ground-supported steel storage tanks. reconnaissance report. Imperial County. California Earthquake of Oc­tober IS , 1979." Earthquake Engrg. Res. Inst.. Oakland, Calif.. 65-73.

Hanson. R. D. (1973). "Behavior of liquid-storage tanks. the Great Alaska earthquake of 1964." Nat. Academy of Sci., Washington. D.C.. Vol. 7, 331-339.

Haroun, M. A., and Badawi, H. S. (1988). "Seismic behavior of unan­chored ground-based cylindrical tanks." Proc., 9th World Con! on Earthquak.e Engrg., Vol. 5, Tokyo-Kyoto, Japan, 643 -648.

Haroun, M. A., and Housner, G. W. (1981). "Dynamic interaction of liquid storage tanks and foundation soil." Proc., 2nd ASCEIEMD Spec. Coni on Dyn. Response of Struct., ASCE. New York, N.Y.. 346-360.

Housner. G. W. (1963). "The dynamic behavior of water tanks." Bull. Seismological Soc. of Am .• 53(2). 381-387.

Jacobsen, L. S. (1949). " Impulsive hydrodynamics of fluid inside a cy­lindrical tank and of fluid surrounding a cylindrical pier." Bull. Seis­mological Soc. of Am., 39(3), 189-203.

Lau. D. T.. and Clough, R. W. (1989). "Static tilt behavior of unanchored

JOURNAL OF STRUCTURAL ENGINEERING / APRIL 1997 / 449

cylindrical tanks." Rep. EERC 89- 11 . Earthquake Engrg. Res. etc., Univ. of California. Berkeley, Calif.

Malhotra. P. K. (1995). "Base uplifting analysis of flexibly supported liquid-storage tanks." J. Earthquake Engrg. Sirucr. Dyn. , 24(12), 1591-1607.

Malhotra. P. K., and Veletsos. A. S. ( 1994). "Uplifting response of unan­chored liquid-storage tanks." J. Siruer. Engrg., ASCE, 120(12),3525-3547.

Malhotra. P. K., and Veletsos, A. S. (1995). "Seismic response of unan­chored and partially anchored liquid-storage tanks." Rep. TR·J05809, Electric Power Res . lnst. . Palo Alto, Calif.

Manos, G. C., and Clough, R. W. (1982) .• 'Further study of the earthquake response of a broad cylindrical liquid-storage tank model." Rep. EERC 82·07. Earthquake Engrg. Res. Ctr. , Univ. of California. Berkeley, Calif.

Manos, G. c., and Clough. R. w. (1985). "Tank damage during the May 1983 Coalinga Earthquake. " J. Earthquake Engrg. Siruct. Dyn., 13(4), 449-466.

Natsiavas, S. ( 1987). " Response and failure of fluid-filled tanks under base excitation," PhD thesis. California Inst. of Techno!.. Pasadena, Calif.

"Northridge earthquake of January 17, 1994, reconnaissance report." (1 995). 95-01, Earthquake Engrg. Res. Inst., J. F. Hall . ed., Oakland, Calif., Vol. I, 162-168.

Peek. R. (1986). " Analysis of unanchored liquid storage tanks." Rep. EERL 86·0 1, California Inst. of Techno!., Pasadena, Calif.

Sakai, F., Isoe. A. . Hirakawa. H. , and Mentani, Y. ( 1988). "Experimental study on uplifting behavior of flat-based liquid storage tanks without anchors." Proc., 9th World Con/. on Earthquake Engrg., Va!. 6, Tokyo­Kyoto, Japan, 649-654.

Smoots, Y. A. ( 1973). "Observed effects on foundations of structures, San Fernando Earthquake of February 9, 1971." U.S. Dept. of Commerce, Nat. Oceanic and Atmospheric Admin" Washington, D.C., Vol. I, 805 - 807.

U.S. Department of Commerce (DOC) (1973). "Earthquake damage to water and sewage facili ties, San Fernando earthquake of February 9, 1971." Nat. Oceanic and Atmospheric Admin ., Washington, D.C., Vo!' 2, 135-138.

Veletsos. A. S., and Tang. Y. (1990). "Soil-structu re interaction effects for laterally excited liquid-storage tanks." J. Earthquake Engrg. Struct. Dyn., 19(4), 473-496.

Ve letsos. A. S .. and Yang. J. Y. (1977). "Earthquake response of liquid storage tanks." Proc.. Ad\}. in Civ. Engrg. through Engrg. Mech.. Engrg. Meeh. Div. Spec. Con! , ASCE, New York, N.Y.. 1-24.

APPENDIX II, NOTATION

The following symbols are used in chis paper.'

A (c) = pseudoacceleration of SOOF model of tank· liquid system;

450 I JOURNAL OF STRUCTURAL ENGINEERING I APRIL 1997

c = viscous damping of fixed-base system; Ce effective viscous damping of uplifting system defined by

Eq. (17); c... soil damping coefficient; E Young's modulus of elasticity;

ED ; dissipated energy in a load cycle [Fig. 3(c)}; Es = strain energy associated with base plate uplifting [Fig.

3(b)}; g = acceleration due to gravity; H ; height of liquid in tank; h = thickness of base plate; h = height of the resultant of hydrodynamic wall pressures; hs = thickness of shell at the base of tank; k = stiffness of fixed-base system;

ke = effective stiffness of uplifting system; k. instantaneous stiffness of rotational spring representing

the base plate; L radial separation between base plate and foundation [Fig.

I(b)}; Mr overturning base moment due to hydrodynamic wall pres-

M, sures; plastic moment capacity per unit circumferential length of base plate;

m = fundamental impulsive mass o f system; mass of liquid in tank; m,

P R

PlgH = hydrostatic base pressure; radi us of tank:

T t ;

period of fixed-base system; effective period of uplifting system;

v~: W w,

overall horizontal displacement of mass with respect to moving base [Fig. S(a)}; shear force on vertical section of tank wall;

= weight of tank wall; total liquid weight;

w, ; uplift at plate boundary [Fig. I(b)}; x, peak horizontal ground acceleration; z = vertical distance from center of base plate;

13 hydrodynamiC base pressure parameter defined by Eq. ( 10);

~ hysteretic damping due to yielding at plate boundary; al plastic rotation at plate boundary;

subgrade modulus of foundati on; yield stress of tank. material ;

K

:; axial stress in tank wall; = hoop stress in tank wall;

circumferential coordi nate [Fig. l (a)]; and ; rOlalion of tank base [Fig. I (b)}.