1983-Vibration Studies and Tests of Liquid Storage Tanks

EARTHQUAKE ENGINEERING AND STRUCTURAL DYNAMICS, VOL. 11, 179-206 (1983)

VIBRATION STUDIES AND TESTS OF LIQUID STORAGE TANKS

MEDHAT A. HAROUN*

University of California, Irvine, California, U.S.A.

SUMMARY

Theoretical and experimental investigations of the dynamic behaviour of ground-supported, deformable, cylindrical liquid storage tanks were conducted. The study was carried out in three phases: (I) a detailed theoretical treatment of the coupled liquid-shell system for tanks rigidly anchored to their foundations; (11) an experimental investigation of the dynamic characteristics of full-scale tanks; and (111) a development of an improved seismic design procedure.

INTRODUCTION

The dynamic behaviour of ground-based liquid storage tanks has been the subject of an extensive study at Caltech during the past 5 years. Theoretical and experimental investigations have been conducted to seek possible improvements in the design of such tanks to resist earthquakes. This paper records the principal results obtained during this research programme and presents a comprehensive review of the subject which makes it more understandable and useful to both researchers and practising engineers.

Historical background Early developments of seismic response theories of liquid storage tanks considered the container to be

rigid and focused attention on the dynamic response of the contained liquid.I3. l 4 HousnerZ3 formulated an idealization for estimating liquid response in seismically excited rigid, rectangular and cylindrical tanks. The study presented values for equivalent masses and their locations that would duplicate the forces and moments exerted by the liquid on the tank. In 1964, the Alaska earthquake caused large scale damage to tanks of modern design’, l 9 and initiated many investigations into the dynamic characteristics of flexible containers. In addition, the evolution of both the digital computer and various associated numerical techniques has significantly enhanced solution capability. Several studies4* 6 9 2 1 were carried out to investigate the dynamic interaction between the deformable wall of the tank and the liquid by the finite element method. The tank was regarded as anchored to its foundation and restrained against cross-section distortions. A different approach to the solution of the problem was developed by Velet~os.’~ The tank was assumed to behave as a single degree of freedom system and to vibrate in a prescribed mode. Later, Veletsos and Yang” obtained the natural frequencies of the liquid-filled shell by the Rayleigh-Ritz method. In the past, experimental data were obtained by testing reduced scale models; however, most of these studies were concerned with dynamic problems associated with aerospace applications.’ It was not until recently that experimental investigations of the seismic response of aluminium tank models were carried out at the University of California, Berkeley.’, l 6

Outline of the present study (see Figure I ) A necessary first step in analysing seismic response of structures is to compute the natural frequencies of

vibration and the associated mode shapes. These are determined by means of a discretization scheme in

* Assistant Professor; formerly Graduate Student and Research Fellow at California Institute of Technology

0098-8847/83/020 179-28$02.80 @ 1983 by John Wiley & Sons, Ltd.

Received 15 June 1981 Revised 22 June 1982

180

MECHANICAL

=I PARAMETERS

1 OF MECH. - MODELS

M. A. HAROUN

VIBRATION STUDIES AND TESTS OF LIQUID STORAGE TANKS

y R I N G SHELL ELEMENT

I 2

3

4 LIQUID

Y REGION

I Theoret ical Study II. V i b r a t i o n Tes ts m. Seismic Design

( A ) Free Vlbral ion Analysis of Ful I - Scale Simpl i f ied Analysis ( 5 ) Eorthquahe Response Liquid Storage Tanks and Design Curves

Figure I . Outline of the present study

which the elastic shell is modelled by finite elements and the fluid region is treated as a continuum by the boundary solution technique. In this approach, the number of unknowns is substantially less than in those analyses where both tank wall and fluid are subdivided into finite elements.

Having established the basic approach to be used, the analysis is applied to investigate the influence of the static hoop stress on wall vibrations, the effect of the coupling between liquid sloshing modes and shell vibration modes, the effect of the flexibility of the foundation soil, and the influence of the rigidity of the roof. The remainder of the theoretical phase of the study was devoted to analysing tank response to earthquake excitation. Special attention was first given to the cos &type modes for which there is a single cosine wave of deflection in the circumferential direction. The significance of the cosne-type modes on the response of irregular tanks was then investigated.

The second phase of research involved vibration tests of full-scale tanks. The vibrations of three water storage tanks with different types of foundations were measured. Ambient as well as forced vibration measurements were made of the natural frequencies and mode shapes. Measurements were made at selected points along the shell height, at the roof circumference and around the tank bottom. Comparison with computed mode shapes and frequencies showed good agreement with the experimental results, thus confirming the reliability of the theoretical analysis.

The main objective of the final phase of research was to close the gap between academic studies and practical design considerations, and to provide practising engineers with a simple and sufficiently accurate tool for estimating seismic response of tanks. A mechanical model, which takes into account the deformability of the tank wall, was developed. The parameters of such a model can be found from charts and the maximum seismic loading can be predicted by means of a response spectrum characterizing the design earthquake. The outline of the overall study is presented diagramatically in Figure 1.

DYNAMIC BEHAVIOUR O F GROUND-BASED TANKS

Ground-based tanks can be classified in two categories depending on their support conditions: anchored and unanchored tanks. Because unanchored tanks are free to lift off their foundations in response to strong shaking, a non-linear analysis is required to estimate their seismic behaviour. For an anchored tank, vertical motion of the shell at the foundation level is prevented; therefore, its seismic behaviour can be analysed by evaluating the natural modes of vibration and superposing them properly. Such an approach is presented herein.

VIBRATION STUDIES OF LIQUID STORAGE TANKS 181

The free lateral vibration modes of an anchored circular cylindrical tank can be classified as the cost?- type modes for which there is a single cosine wave of deflection in the circumferential direction, and as the cos n8-type modes for which the deflection of the shell involves circumferential waves having n higher than 1. Figure 2 illustrates the circumferential and the vertical nodal patterns of these modes. It should be noted that earthquake motions tend to strongly excite the cos @type modes. For a tall tank, the cos &type modes can be denoted beam-type modes because the tank behaves like a vertical cantilever beam. This is not true for a broad tank because both the amplitude and the axial distribution of the radial displacement at 8 = 0 are different from those of the circumferential displacement at 8 = n/2.

In addition to the shell vibrational modes, there are the low-frequency sloshing modes of the contained liquid shown also in Figure 2.

m = l m.2 m = 3

Ie r t i ca l nodal p a t t e r n

I I I I I I I

I I I I I I I

r

I

I Circumferent ial ' I

nodal p a t t e r n

I ( a ) S h e l l M o d e s I

Fixed base

ANCHORED TANK UNANCHORED TANK

Figure 2. Seismic behaviour of ground-based tanks

TANK GEOMETRY AND CO-ORDINATE SYSTEM

The tank under consideration is shown in Figure 3. It is a ground-supported, circular cylindrical, thin-walled liquid container of radius R , length L and thickness h, with the wall connected to a rigid base. The tank is partly filled with liquid to a height H.

A cylindrical co-ordinate system is used with the centre of the base being the origin. The radial, circumferential and axial co-ordinates are denoted r, 8 and z , respectively, and the corresponding displacement components of a point on the shell middle surface are denoted by w, u and u, respectively. To describe the location of a point on the free surface during vibration, let 5 measure the superelevation of that point from the quiescent liquid free surface.

Lastly, let S , denote the quiescent liquid free surface, and S2 and S3 denote the wetted surfaces of the shell and the bottom plate, respectively.

182 M. A. HAROUN

\ J

CYLINDRICAL S H E L L

W E T T E D SURFACE OF SHELL ( S e )

W E T T E D SURFACE OF BOTTOM P L A T E ( S 3 1

1/

Figure 3. Tank geometry and co-ordinate system

PHASE (1)-THEORETICAL ANALYSIS

Equations governing liquid motion

Laplace equation For the irrotational flow of an incompressible inviscid liquid, the velocity potential, 4(r , 8, z, t ) , satisfies the

v 2 4 = 0 (1)

in the region occupied by the liquid (0 < r < R, 0 < 8 < 2n, 0 < z < H). Since the velocity vector of the liquid is the gradient of the velocity potential, the liquid-container boundary conditions can be expressed as follows:

1. At the rigid tank bottom, z = 0, the liquid velocity in the vertical direction is zero

84 az -(r, 8,0, t ) = 0

2. The liquid adjacent to the wall of the elastic shell, r = R, must move radially with the same velocity as the shell

where w(0, z , t ) is the shell radial displacement. At the liquid free surface, z = H+ l ( r , 8, t ) , two boundary conditions must be imposed. The kinematic

condition states that a fluid particle on the free surface at some time will always remain on the free surface. The other boundary condition specifies that the pressure on the free surface is zero. By considering only


small-amplitude waves, the free surface boundary conditions become

(5 ) 34

PI dt (r, e, H , t ) + PI g a r , 0, t) = 0

where p1 is the mass density of liquid and g is the acceleration of gravity.

a free surface is given by Following the variational formulation presented by Luke, the appropriate functional for a liquid having

J C ( 4 5 ) = I:'UA t )d t (6)

where Lc is the complementary Lagrangian functional given by

in which V indicates the volume occupied by the liquid and ( ' ) means differentiation with respect to the time, t . By requiring that the first variation of J c be identically zero, the Laplace equation and the linear boundary conditions [equations (2) , (3), (4) and (5 ) ] can be obtained.

Equations governing shell motion The strain energy of the shell,17 including the effect of both stretching and bending, can be written as

L 2 n

U( t ) = {crJT{E}Rd8dz 2 0 0

where {cr} is the generalized stress vector given by

{ o } ~ = [N,, No, fl, M,, Mo, A1

{ E } ~ = CE,, Eel E Z e , K, , Ke, KzoI

(9)

(10)

and { E } is the generalized strain vector defined by

In equation (9), N , and No are the membrane force resultants; and M , and M , are the bending moment resultants.The quantities fl and A? are referred to as the effective membrane shear force resultant and the effective twisting moment resultant, respectively.

The shell material is assumed to be homogeneous, isotropic and linearly elastic. Hence, the force and moment resultants can be expressed in terms of the normal and shear strains in the middle surface E,, E~ and E , ~ ; in terms of the midsurface changes in curvature K , and KO; and in terms of the midsurface twist K , , as follows:

(01 = CDl { E l ( 1 1)

where [D] is the constitutive matrix. The potential energy expression [equation (S)] can be rewritten in terms of shell displacements as

where { d } is the displacement vector defined by

{d}T = [u, v, w l (13) and [PI is a differential operator matrix relating the generalized strain vector to the shell displacement vector.

184 M. A. HAROUN

Lastly, the kinetic energy of the shell, neglecting rotary inertia, can be expressed as L 257

T(t) = j 1 (m(z) {d}T{d}) R dOdz 2 0 0

where m(z) is the mass of the shell per unit area.

FREE VIBRATION ANALYSIS

Basic approach The finite element method provides a convenient and reliable idealization of the liquid-shell system.

However, for the problem under consideration, it is advantageous to treat the liquid region as a continuum by the boundary solution technique and to model the elastic shell by finite elements. In essence, the boundary solution technique consists of choosing a set of trial functions which satisfies, a priori, the differential equations throughout the domain; and consequently, only the boundary conditions have to be satisfied in an average integral sense. Because the boundary solution technique involves only the boundary, the number of unknowns can be much less than those of a standard finite element analysis. At this point, we must remark that the boundary solution technique is limited to relatively simple, homogeneous and linear problems for which suitable trial functions can be identified.

I/ariutionaEformuiation. If one combines equations (7), (12) and (14), the variational functional for the free lateral vibration of the liquid-shell system can be expressed as

By selecting suitable trial functions which satisfy the Laplace equation identically, the volume integral in equation ( I 5) can be replaced by a surface integral using Green's theorem

where V is the volume occupied by the liquid and bounded by the surface S = S , + S, + S,; and v is the outward normal vector.

In the basic analysis, only the impulsive pressure of the liquid is considered; and, therefore, the shell vibrational motion becomes independent of the free surface motion. Given this new situation, the functional J takes the form

Expansion ofthe velocity potential function. Once a set of trial functions, f i i ( r , 8, z), which are solutions of the Laplace equation, are identified, then one can assume that

I

4 ( r , Q, z , t ) = c ~ ~ ( t ) . f i i ( r , 8, z ) (18) i = 1

where I is the number of trial functions. These functions can be obtained by the method of separation of variables. Those which have vanishing derivative with respect to z at z = 0 and satisfy that the time derivative of the potential function at z = H be zero for all time, are given by


where I , are the modified Bessel functions of the first kind of order n;

(2i- 1)n ; and i = 1,2, ..., I ff. = ~

’ 2H

Idealization of the shell. The shell is divided into an appropriate number of finite ring-shaped elements as shown in Figure 4. The displacement components u, u and w are first expanded in Fourier series as follows:

00

u(O,z, t ) = c un(z, t ) cos (no) n = I

00

o(0, z , t ) = C u,(z, t ) sin (ne) n = 1

m

~ ( e , Z, t ) = C w,(z, t ) cos (ne) n = 1

This displacement functions u,(z,t), u,(z,t) and w,(z,t) are then expressed in terms of the nodal displacements of the finite elements by means of an appropriate set of interpolation functions. The shape functions associated with the axial and tangential displacements are taken to be linear between the nodal points. However, those associated with the radial displacement are cubic Hermitian polynomials to assure slope continuity at the nodes.

R I N G ELEMENTS

Figure 4. Finite element definition diagram: (a) finite element idealization of the shell; (b) shell element

Matrix equation of motion. Employing the variational functional J , one can derive the matrix equation of motion of the liquid-shell system. With the aid of the shell displacement model, the scalar energy quantities can be expressed in terms of the assemblage nodal displacement vector of the shell, ( q } , as follows:

where [ K , ] and [ M , ] are the stiffness and mass matrices, respectively, of the shell. The velocity function expression [equation (18)] can be rewritten in a matrix form as

186 M. A. HAROUN

Inserting this expression into the third term of the functional J , and noting that the trial functions satisfy the conditions that 4 = 0 along S , and t?$/az = 0 along S, , one can write

where [C] is a diagonal matrix whose elements are given by

nRp, aiH 2

c.. = I,,(cqR)~'l,(cqR); i = 1,2, ..., 1

where 'I, is the derivative of 1, with respect t o Y. With the aid of the finite element model of the shell and the expression of the velocity potential function, the last term of J becomes

PI I, w4 ds = {4ITCc^l {A) (25)

The calculation of the matrix [el is straightforward' and will not be presented herein.

yields Inserting equations (21), (23) and (25) into the variational functional, and applying the variational operator

CMSI i41 +[&I {d+ c c^l{4 = (0 )

[ C l W - IIQTM = (0)

( A ) = cc1- 'CC IT{(i)

Since the matrix [C] is non-singular, one can write

and consequently, the first of equations (26) takes the form

CMsI{ci'> + CKS1{4) + [el CCl -1[c^lT{4} = ( 0 ) (28) The governing matrix equation of the free lateral vibration of the liquid-filled shell is therefore given by

(CMSl+ CDW) { 4 } + C K J ( 4 ) = (0) (29)

where [DM] is an added mass matrix due to the effect of the liquid. The matrix [ D M ] is symmetric and partially complete (i.e. not banded); the elements are well distributed over the matrix.

Numerical examples. A digital computer program was written to compute the natural frequencies and mode shapes of vibration of the coupled liquid-shell system. The shell nodal displacements (eigenvectors) are a direct result of the solution, and these are then used to solve for the shell force and moment resultants, and for the hydrodynamic pressure acting on the wall of the tank. Because of the efficiency of the method of analysis, the program consists only of about 500 cards and the execution time on the Caltech digital computer (IBM 370/3032 system) is less than 10sec.

Throughout this investigation, attention is given to two particular tanks: a broad tank (B) and a tall tank (T). Tank (B) has the following dimensions: R = 18.3 m (60 ft), L = 12.2 m (40 ft) and h = 254 cm (1 in) while the dimensions of tank (T) are: R = 7.32m (24ft), L = 21.96m (72ft) and h = 2.54cm (1 in). Both tanks are assumed to be completely full of water of density p1 = lo3 kg/m3 (0.94 x Ib . sec2/in4). The tank wall is made of steel whose properties are: E = 20.67 x lo7 kPa (30 x lo6 Ib/in2), v = 0.3 and ps = 7,84 x lo3 kg/m3 (0.733 x 1 O - j Ib. sec2/in4). The analysis was applied to compute the natural frequencies and modes of vibrations of tank (B). The number of shell elements was taken to be 12; and, therefore, the number of expected modes is 4 x 12 = 48. The fundamental mode shape of the cos &type modes is shown in Figure 5(a). To investigate the influence of the aspect ratio (length to radius ratio), the dynamic characteristics of tank (T) were computed and the fundamental mode shape was displayed in Figure 5(b). Inspection of Figure 5 shows that the mode shapes of broad and tall tanks are indeed quite different. The hydrodynamic pressure


distributions for these two tanks and for similar rigid tanksz4 are shown in Figure 6 for comparison. It is important to note that, in the numerical examples under consideration, attention is given to the cos 8-type modes only; these modes are unaffected by the hydrostatic pressure of the liquid.

W

CIRCUMFERENTIRL WRVE NUMBER = I

NRTURRL FREQUENCY = 6.18 CPS

(a)

CIRCUMFERENTIAL WRVE NUMBER = 1 NRTURRL FREQUENCY = 5.31 CPS

(b)

Figure 5. Fundamental vertical modes of full tanks: (a) ‘broad‘ tank (B) (L/R = 0.67); (b) ‘tall’ tank (T) (L/R = 3.00)

TRNK

L / R = 0.67 L/R = 3.00

HYORODYNRMIC PRESSURE D I S T R I B U T I O N

( FUNDRMENTRL MODE )

Figure 6. Hydrodynamic pressure distribution on full flexible and rigid tanks

Eflect of the hydrostatic hoop stress In the preceding analysis, it was assumed that the only stresses present in the shell are those arising from

the vibratory motion. However, tank walls are subjected to hydrostatic pressures which cause hoop tensions. The presence of such stresses affects the vibrational characteristics of the shell, especially the cos &type modes.

188 M. A. HAROUN

To incorporate such an effect, it was necessary to modify the strain energy expression of the shell, and to generalize accordingly the equations of motion. The strain energy can be written conveniently as

U t ) = U,(t)+ U2(d (30)

where U,(t) is the strain energy employed in deriving the stiffness matrix of equation (21) and U2(t) is given by

where NL is the initial membrane force resultant in the circumferential direction, and cO is the midsurface strain which can be expressed as

& ‘ - R - y ae + .) +;{(;$$ +[ ;( $+ .)I’ + [ ;( v -g)T} The non-linear terms in equation (32) are given by Washizu.26 However, it should be mentioned that the linear terms of the strain-displacement relationships developed by Washizu are identical to those of Novozhilov theory” which has been used in U,(t).

With the aid of the finite element displacement model of the shell, the strain energy expression U2(t) can be rewritten as

where [KB] is an added stiffness matrix due to the presence of the initial hoop stress. The matrix equation of motion of the tank wall now takes the familiar form

[MI ( 4 ) + [Kl H4) = 0 (34)

where [ K ] = [ K , ] + [KL]. Numerical examples. The analysis was applied to compute the natural frequencies and mode shapes of tank

(B) and tank (T). As expected, the influence of the hoop stress on the coso-type modes of vibration is insignificant. The stiffening effect due to the hydrostatic pressure has a considerable influence upon the frequencies of the cosn0-type modes of tall tanks. Such an effect becomes more significant as the circumferential wave number n increases. On the other hand, the stiffening effect on the cos no-type modes of broad tanks is, for practical purposes, negligible.

To illustrate the effectiveness of the analysis under consideration, a comparison between the computed dynamic characteristics and those found experimentallyz2 is made. The physical model employed in the test was partly filled with water, and had the following dimensions and properties: R = 10.2cm (4in), L= 31.8cm (12.5in), H = 27.9cm (ll in), h = 0.013cm (0.005in), E = 506x lO’kPa ( 0 . 7 3 5 ~ 1061b/in2), ps = 1.42 x lo3 kg/m3 (0.133 x Ib. sec2/in4) and v = 0.3. As is seen from Figure 7, the computed characteristics are in good agreement with the experimental results. This confirms the accuracy of the analysis, and the significant role played by the initial hoop tension during the cos no-type vibrations of tall tanks.

Interaction between liquid sloshing modes and shell vibration modes The dynamic interaction between liquid sloshing waves and shell vibrations has not yet been investigated.

The coupling is usually neglected on the ground that the significant sloshing modes are of much lower natural frequencies than those of the vibrating shell. In the following analysis, emphasis is placed on the question of when the coupling effect can be significant; in other words, is it necessary to consider the liquid- shell-surface wave system, or only the two uncoupled cases: (i) the liquid-shell system plus (ii) the free surface gravity waves in a similar rigid tank?,,

VIBRATION STUDIES OF LIQUID STORAGE TANKS I89

i SHAKING D

W INITIAL STRESS INCLUDED -&--A- INITIAL STRESS EXCLUDED

SHRKING TABLE TESTS / /

.-.

I I 1 1 .J 5 6 7 8 9

CIRCUMFERENTIFlL WAVE NUMBER . n (b)

TESTS I n .6

w

(a) Figure 7. Comparison between calculated and measured characteristics of the cos no-type modes: (a) mode shape; (b) natural frequencies

As was mentioned earlier, a finite element discretization of the liquid region itself is not necessary; only its boundaries need to be discretized. The shell is modelled by a series of ring-shaped elements as before, and the quiescent liquid free surface is represented by concentric annular rings. The trial functions are chosen from the solutions of the Laplace equation which are non-singular at r = 0 and have a vanishing derivative with respect to z at z = 0. These can be written as

J,(kr) cosh (kz)

fi(r, O, z ) = cos (ne) r n ( n > 1) i I&) cos (kz)

where J , are Bessel functions of the first kind of order n.

(35)

With the aid of the shell displacement model, the free surface displacement modc. and the triz functions of the velocity potential, the variational functional J [equation (1 5)] leads to the matrix equation of motion of the overall system

CM1{4 + CKI {x) = (0) (36) where the overall mass and stiffness matrices are written in the following partitioned forms:

and {x} is the nodal displacement vector for the entire assemblage which can be written as

where (4) is the assemblage nodal displacement vector of the free surface. Numerical examples. The analysis was applied to the tall tank (T). The quiescent liquid free surface is

divided into 12 elements, and the elastic shell is modelled by 12 elements; therefore, the number of expected modes is 60. The lowest natural frequencies of the coupled system are in good agreement with the sloshing frequencies in a similar rigid tank. Furthermore, the 13th, 14th, ..., etc. ascending frequencies are, for practical purposes, the same as those computed for the liquid-filled shell. Therefore, it may be concluded that the coupling effect is negligible. This is further substantiated by the mode shapes. Figure 8 displays the

190 M. A. HAROUN

fundamental mode shape of the coupled system; it is clear that this mode has predominantly free surface motion.

.I

CIRCUMFERENTIRL WRVE NUMBER = 1

NRTURRL FREQUENCY = 0.25 CPS

Figure 8. Fundamental mode shape of the coupled liquid-shell-surface wave system (shell displacements are magnified 500 times)

Dynamic interaction between shell and foundation Many studies have dealt with the dynamic interaction between different types of structures and the

supporting soil during earthquakes. However, no attempt has been made so far to extend such analysis to the soil-tank system. A complete analysis of the soil-tank system by the finite element method is relatively expensive and complicated; however, a simplified model of the soil can be employed with a finite element model of the shell to exhibit the fundamental dynamic characteristics of the overall system and to assess the significance of the interaction on the seismic response of tanks. Since the cos no-type deformations of the shell have no lateral force or moment, only the influence upon the cose-type modes should be investigated. Furthermore, rocking motion is most pronounced for tanks having aspect ratios (height to radius ratio) 2 1. Thus, the soil-tank interaction problem is governed by a beam-type behaviour rather than by a shell-type response. The system is therefore modelled by a vertical cantilever beam (Figure 9) supported by a spring- dashpot model to represent the flexibility of, and the damping in, the foundation soil.

The procedure for analysing the liquid-tank interaction differs from the method presented in the preceding sections. A series representation of the liquid-velocity potential function is obtained by proper specification of the velocities at and normal to the liquid boundaries.

These conditions are

(39) 84 -(R, e, z, t ) = {n(t) + z q t ) + a(z, t ) } cos (0) 8r

and

(40) 84 - (r, 8,0, t ) = - rdr(t) cos (8) 8Z

where x(t) is the horizontal translation of the tank base; u(t) is the angle of rotation about a transverse axis through the base; and w(z, t ) is the deflection of the beam axis. The finite element analysis of the liquid-shell interaction problem revealed that the fundamental mode shape of completely full, moderately tall tanks can


H

Figure 9. Hydrodynamic forces on tank-foundation system

be approximated by a sine curve; therefore, one can assume that

w(z, t ) = y ( t ) . sin - (E) By neglecting liquid sloshing modes, one can express the velocity potential function in terms of the time derivatives of x(t), ~ ( t ) and y(t). The hydrodynamic pressure is related to the velocity potential function by

(42) 84 at

Pd(r, e, z, t , = -pl-(r, e, z, t ,

and the lateral force exerted on the tank at any elevation z above the base can be computed from

j ( z , t ) = jO2'pd(l, e, z , t ) R cos (e) de (43)

The variation of the hydrodynamic pressure acting on the base produces an overturning moment M , which can be calculated from

M, = JRJ2'..(..8,O, t)r2cos(0)dedr 0 0

(44)

Having obtained the hydrodynamic forces, the base shear force, Q(t), and the overturning moment, M(t), can be computed. These are related to the base motion by"

192 M. A. HAROUN

where K , and K , are the real parts of the foundation impedance functions, and C, and C , are the imaginary parts. Taking y(t), x ( t ) and $(t) = Ru(t) as the generalized degrees of freedom, one can write the matrix equation which governs the free vibration of the soil-flexible tank system as

Numerical examples. A water storage tank is anchored to a 61 cm (2ft) thick R.C. slab on deep alluvium. The tank, which was tested experimentally, is 7.32 m (24ft) in radius, 21.64m (71 ft) in height, and consists of a thin steel shell of varying thickness; the maximum thickness at the bottom is 1.75 cm (0.69 in) and the minimum thickness at the top is 0.64cm (0.25 in). The fundamental natural frequency of the soil-tank system is computed for different values of shear wave velocity of the soil, and the results are shown in Table I. Inspection of this table indicates that the deformability of the foundation soil reduces the natural frequency of the tank; this was observed experimentally in phase (11). As the shear wave velocity increases, the fundamental natural frequency approaches the value computed assuming a rigid foundation.

Table I. Fundamental natural frequency of the soil-tank system ~

Shear wave velocity 304.8 365.8 426.7 487.7 609.6 731.5 1097.3 m/sec (ftjsec) (1000) (1200) (1400) (1600) (2000) (2400) (3600)

Natural frequency (Hz) 2.79 3.03 3.21 3.34 3.49 3.59 3.12

Effect of’the roof on shell vibration A complete analysis of the effect of the fixed-type roof on the dynamic characteristics of tanks requires

consideration of the equations of motion of the roof simultaneously with the equations of motion of the shell, and enforcement of the conditions of continuity of the generalized forces and displacements at the junction. Such analysis was carried out in Reference 4 where the dynamic problem of a tank covered by a dome was treated.

In this section, a simple roof model, commonly used in civil engineering tanks, is considered. It consists essentially of a thin steel plate supported by steel trusses. The plate has a considerable stiffness in its own plane; therefore, i t restrains the tangential and radial displacements of the shell at their mutual boundaries. It affects the cosStype modes by restricting the motion of the tank top to be a rigid body translation; i.e.

w(0, L, t ) = - u -, L, t (n = 1) (; 1 (47)

In addition, i t restrains the cos no-type modes against cross-sectional deformations at the tank top; i.e.

w(0, L, t ) = u(o, L, t ) = 0 (n 3 2) (48)

Furthermore, by virtue of its thinness, the plate has very little stiffness in the z-direction transverse to its plane; consequently, it will generate negligible moment M , and membrane force N , at the shell top as the tank vibrates. Although the foregoing boundary conditions are highly simplified, the computed frequencies and mode shapes of real full-scale tanks are in good agreement with those measured by vibration tests.

Numerical exumples. The effect of the roof in-plane rigidity on the cos &type modes is generally negligible. A slight reduction in the values of the natural frequencies is observed due to the additional mass of the roof. On the other hand, the roof has a significant effect on the cosno-type modes of the shell as illustrated in Figure 10.


+K-+K- ROOF EFFECT INCLUDED -&----A- RO8F EFFECT EXCLUDED

TANK ( T )

I I I I 1 2 3 rl 5 G 7

CIRCUMFERENTIAL WRVE NUMBER , n

Figure 10. Effect of the roof rigidity upon the natural frequencies of the cosns-type modes of a tall tank

EARTHQUAKE RESPONSE ANALYSIS

The only special feature of the earthquake response problem, compared with any other form of dynamic loading, is that the excitation is applied in the form of support motions rather than by external loads; thus the essential subject of the present discussion is the method of defining for the tank wall the effective external load history resulting from a given form of support motion.

The matrix equation which governs the earthquake response of the liquid-shell system for a particular value of n can be written as

CMI {g} + CCI (4) + CKI ( 4 ) = { P e f f } (49) where [MI and [ K ] are the mass and stiffness matrices defined in equation (29); [ C ] is the damping matrix; and (Perf} is the effective earthquake load vector resulting from a given ground motion G(t). For a perfect circular cylindrical shell, the effective earthquake load vector takes the form

and consequently, the earthquake response can be obtained by superposition of the vertical modes corresponding to n = 1 only.

Efjiective force vector The total displacement vector of the shell can be considered as the sum of two components: the relative displacement vector { d } defined by equation (13), and the displacement vector { d g } , associated with the ground displacement G(t), which can be expressed as (Figure 11)

0

194 M. A. HAROUN

The external forces acting on the shell due to ground motion G(t) include: (i) the distributed inertia force of the shell which is given by

V g } = -Psh{J,j (52)

and (ii) the hydrodynamic pressure on the wall of a similar rigid tank. This pressure can be expressed as

The work done by these external loads during arbitrary virtual displacements

dU , cOS (e) (54)

can be expressed as

6 W = lLj 2z({Fg)T(6d}) R d8 dz + ]oH]02z(pg(R, 8, Z, t ) 6w, cos (8)) R dB dz ( 5 5 ) 0 0

With the aid of the finite element model of the shell, this expression becomes

6W = - & ) { 6 q } T { F } (56)

{Peff} = - if-) (57)

and, therefore, the effective earthquake load vector is given by

Numerical examples The analysis was applied to estimate the earthquake response of the tall and broad tanks (T and B) presented in the preceding sections. Their free lateral vibrational modes were superimposed to evaluate the response to the N-S component of the 1940 El Centro earthquake. Although a modal damping ratio of about 2 per cent seems appropriate for the vibration of the tank wall, the maximum values of the seismic response were computed for different values of damping ratio. Table I1 presents the impulsive response of the tall tank and displays that of a similarly excited rigid tank for comparison. It is found that the flexibility of the tank wall has a significant effect on the response of both broad and tall tanks. This is due to the fact that the impulsive loads arise through acceleration of the shell. If the shell is flexible, two acceleration components must be considered: (i) the acceleration of the undeformed shell, i.e. the ground acceleration, and (ii) the relative acceleration due to shell deformations. In a rigid tank, only the acceleration of the undeformed shell is considered which introduces the noticeable difference in the magnitude of shell stresses. These results were further substantiated by comparing the computed responses of aluminium tank models with those measured. 5 * This comparison also indicated that the computed fundamental frequency is higher than the measured frequency.


Table 11. Impulsive earthquake response of tank (T) ~

Damping

2%* 5%t lO%t

Maximum radial component of shell displacement

cm (in) W(O,72, t )

Maximum axial force resultant

Maximum tangential force resultant

N/mm (Ib/in) NB(O, 6, t )

Maximum base shear

1.13 (0.445)

1,467 (8,375)

379 (2,166)

2.27 x 107 (5.1 1 x lo6)

0.87 0.75 (0.344) (0296)

1,134 975 (6,473) (5,5641

1.75 x lo7 1.51 x lo7 (3.95 x 106) (3.39 x 106)

Mechanical Rigid tank modell

- 1.1 7 (0.462)

603 1,655 (3,444) (9,454)

1.21 107 2.30 107 (2.72 x lo6) (5.17 x lo6)

*Computed by time integration. t Computed by response spectrum. $Refer to phase (111).

Effect of cos &-type modes As was mentioned earlier, such modes cannot be excited in a perfect circular tank; however, fabrication

tolerances in civil engineering tanks permit a departure from a nominal circular cross-section and this tends t o excite the cos n8-type modes. An analysis of the effect of irregularity of flexible tanks was conducted as a part of the present study. The fact remains that the magnitude and distribution of fabrication error cannot be predicted and, therefore, only a hypothetical analysis can be made. It is of interest to note that a recent experimental study22 showed that buckling of plastic models, which are almost full of water, depends largely on the stresses associated with the cos 0-type modes as shown in Figure 12. These tests were carried out by

I .4

1.2 2

I-

W _I W 0 u

0 a LK 1.c

a 0.E

z 0 ox m n N 0.4 a

u A Y

3

W

A

z P 0.2

0

I I I I 1

EXPERIMENT. FREE TO!

R / h H/L

0 833 0.93 0

00 0 1250 0.89

a v 093

H / L = 0.92 o\

0.2 0.4 0.6 0.8 I .O 1.2

NORMALIZED FREQUENCY, w / w , Figure 12. Buckling tests of plastic models

196 M. A. HAROUN

fixing the frequency of excitation and increasing the amplitude of the shaking table motion until buckling occurred. Theoretically, the buckling was assumed to occur when the axial membrane stress at the base, computed from the present analysis, reached the classical value.

PHASE (11)-FULL-SCALE VIBRATION TESTS

Adequate understanding of the dynamic behaviour of complex structures is dependent upon the use of both theoretical and experimental techniques to support each other. Therefore, a series of ambient and forced vibration tests of three, full-scale, water storage tanks was conducted to determine the natural frequencies and, if possible, the mode shapes of vibrations, and to select two tanks on which permanent instruments would be installed to record future earthquakes. Figure 13 shows schematic sections of these tanks and their foundations. Tank no. (1) is anchored to a 61 cm (2 ft) thick R.C. slab on deep alluvium, and tank no. (2) rests on a 3.66 m (12 ft) deep concrete ring wall without anchor bolts. The third tank is anchored to a 153 m (5 ft) thick R.C. foundation slab supported by 100 R.C. caissons.

NO. ( 1 ) NO. ( 2 )

I r --

/R.C. SLAB

I I I I

18.29 'rn (60'1

/RING /COMPACTED WALL GRAVEL

NO. ( 3 )

1 I onchor ' b o l l s I

L! R. C. CAISSONS 1 Figure 13. Schematic sections of the tanks and their foundations

Experimental arrangements and procedures Measurements of ambient and forced vibrations were made at selected points along the shell height, at the roof circumference and around the tank bottom. The first series of tests was conducted to measure the vertical pattern of vibrational modes. Six Ranger seismometers were mounted along the tank height to measure the radial motion of the shell. In addition, two seismometers were placed on the foundation slab oriented to detect vertical motion and thus to obtain a measurement of the amount of rocking of the base of the tank. The objective of the second series of tests was to monitor the motion around the circumference. However, it was impractical in this preliminary investigation to mount the transducers around the tank at arbitrarily selected elevations and, therefore, it was decided to depend on measurements made along the circumference of the roof to identify the number of circumferential waves, n. Three Rangers were placed on an aluminium plate in such a way that three orthogonal components of the motion at a point could be measured. This package of transducers was moved from point to point and the motion was recorded at ten different locations around the perimeter. One vibration generator was used in the forced vibration test. It was anchored to a concrete slab resting on the ground adjacent to the tank. The horizontal sinusoidal force exerted by the vibration generator was transmitted through the ground and produced small amplitude


vibrations of the tank. Figure 14 is a schematical diagram showing the experimental set-up and the instrumentation used in testing tank no. (1). Slightly different arrangements were made for the other two tanks. During the tests, the tanks were maintained full whenever possible.

detail ( a I

6 i

oscillograph 1 r recorder

Figure 14. Experimental set-up for tank no. (1)

Presentation and discussion of test results The data recorded in the test program are far too much for a detailed presentation herein. Only selected

data which provide a qualitative indication of the general nature of the dynamic behaviour as well as the quantitative evidence for verification of the theoretical analysis are presented.

The fundamental frequency of the cos8-type modes of tank no. (1) can be easily identified from Figure 15(a) in which the Fourier amplitude spectrum of the radial component of shell velocity at the tank top is displayed. The roof restrains the shell top against cos &type deformations and only the cos 0-type modes are observed. The computed fundamental frequency, assuming the foundation soil to be rigid, is higher than the measured one of 3.01 cps. It is believed that the reduction in the natural frequency is due to the rocking motion of the tank which was clearly observed by measuring the vertical motion of the foundation slab at the ends of the principal diameter. Figure 15(b) shows sample traces from the Brush recorder made simultaneously at stations (7) and (8) during the forced vibration test. These records show that the two vertical seismometers have the same amplitude and are 180" out of phase. This rocking motion occurs at 3.01 cps and is clearly seen in the Fourier amplitude spectrum shown also in Figure 15(b). The interaction of the cos nd-type deformation with the foundation was found to be insignificant. This was expected because a distributed radial force varying as cos n8 with n 2 2 has no lateral resultant force. It is of interest to note that the computed natural frequencies of the second and the third axial modes of the cos @type deformations are 10.38 and 15.11 cps, respectively; these are in reasonable agreement with those measured experimentally (96 and 14.3 cps, respectively).

No attempt was made in the test program to measure sloshing frequencies of the liquid; these can be reasonably estimated by testing small-scale rigid tanks. However, Figure 15(a) indicates a peak at a frequency corresponding to the computed sloshing frequency of the liquid, and this was attributed to the low-frequency sloshing waves.

198 M. A. HAROUN

I 1

0

8 2 I s $ 0

B

1.m-

0.75 -

0 Fundamental Natural Fnq. (n-1 3

r ig id foundation: 3.81 cps 3 flexible foundation: 3.01 cps !2

FREWENCY - n z . (a)

Forced Vibration lest

FREMNCY - HE.

(b) Figure 15. Ambient and sinusoidal tests of tank no. (1): (a) Fourier amplitude spectrum of the radial component of shell velocity at the tank top; (b) Fourier amplitude spectrum of the vertical velocity at station (8), and sample traces made simultaneously during forced

vibration test

VIBRATION STUDIES OF LIQUID STORAGE TANKS I99

One phenomenon, clearly observed in the motion of the shell away from the roof, was that cos no-type vibrations of the tank wall were developed. These modes were anticipated only in the ambient tests because of the nature of the excitation which tends to excite many modes. However, in the forced vibration test of tank no. (l), cosno-type modes were also excited by rigid base motion, presumably because of the initial irregularity of the shell. Figure 16 shows the axial pattern of the cos %-mode based on ambient and forced vibration measurements. It is clear that the roof does restrain the tank top against radial deformations. The computed natural frequency is 2.46 cps, which is in close agreement with the measured one of 2-42 cps. The computed mode shape is also presented in the same figure for comparison.

Tank no. (2), which is not anchored to the foundation, exhibited behaviour slightly different from tank no. (1). However, it is believed that it would behave much differently with a high level of excitation. This is a non- linear problem for which the relationship between small amplitude vibration characteristics and strong motion seismic response is not well defined.

F U L L - S C R L E V IBRRT ION TEST

FORCED VIBRATION

0 AMBIENT VIBRATION

C I R C U M F E R E N T I A L WAVE NUMBER = 5

TRNK NUMBER !

Figure 16. Fundamental mode shape of cos %-type modes

Figure 17 displays the Fourier amplitude spectrum of the radial component of shell velocity recorded at station no. (4) on tank no. (3). The circumferential modes with n up to 5 were identified from the ambient measurements. The availability of the computed frequencies and the good correlation between the measured and the computed frequencies helped in identifying the mode number with n>6. It should be mentioned that the low-pass filter of the signal conditioner was set, by mistake, to 4 cps in testing the third tank and therefore the peaks in the range 4 to 5 cps do not appear in their respective magnitude. Also, the high peak at 3.45 cps is attributed to environmental noise which was also observed in the calibration test. Figure 18 shows a comparison between the computed and measured frequencies. It clearly emphasizes the significant role played by the roof and the initial hoop stress in estimating the natural frequencies of the cos &type modes. It is also evident that the roof effect is more pronounced for small n, while the initial stress influence is more significant for large n.

PHASE (111)-SEISMIC DESIGN ANALYSIS

The two most common standards and codes currently used for the design of tanks are the API 6502 and the AWWA Seismic loads in these standards are based on the mechanical model derived by HousnerZ3 for rigid tanks. The design procedure considers two response modes of the contained liquid: (1) the ‘impulsive’

200 M. A. HAROUN

m II c -=-

I I i t 4

d II c

Not to scale

I " - -.--A 1 .m 2.m 3.m u.m 5 .m 0.03 1 o m

FAEQUENCY - H2.

Figure 17. Fourier amplitude spectrum at station no. (4)

-X--X- NO RUOF NO INITIHL STRESS

-&--A- ROOF -- NU IhlTIRL STRESS

itSt ROOF - INITIQL STRESS

W FULL - SCALE V I B R A T I O N TEST

/A /

U

-L___ - I-_-- -1- - i - - - i _ ~ - J 4 5 6 I 8 9

ZiRCUMrERENTIRL kRVE NUMBFR . n

Figure 18 Comparison between calculated dnd measured natural frequencles of the cos no-type modes

response caused by the portion of the liquid accelerating with the tank, and ( 2 ) the 'convective' response caused by the portion of the liquid sloshing in the tank. In a recent investigation of the effect of wall flexibility on the seismic response of anchored tanks, V e l e t ~ o s ~ ~ proposed an approximate method for the computation of the hydrodynamic forces associated with tank deformations. However, the method did not provide explicit values for the fundamental natural frequency of the system which is needed for the determination of the

VIBRATION STUDIES OF LIQUID STORAGE TANKS 20 1

spectral acceleration from a response spectrum. Later, Veletsos and Yang25 provided a diagram between the fundamental natural frequency of the liquid-shell system and the height/radius ratio; however, such a relation is applicable only to a shell thickness/radius ratio of 0.001. Recognizing the importance of wall flexibility, recent codes have adopted an increase in the acceleration coefficient to an ‘ad hoe’ value representing the short period amplified acceleration due to shell deformation. It should be noted that such acceleration is specified independent of the tank dimensions and of the support condition.

The principal aim of the final phase of research was to devise a practical approach which would allow, from the engineering point of view, a simple, fast and sufficiently accurate estimate of the seismic response of storage tanks. A mechanical model, shown in Figure 19, was developed. The equivalent masses m,, rnf and m, correspond to the forces associated with ground motion, wall deformation relative to the ground, and liquid sloshing, respectively.

Figure 19. Mechanical model of a flexible tank

Convective component I t has been shown that the coupling between liquid sloshing modes and shell vibrational modes is weak;

and, consequently, the convective dynamic pressure can be evaluated with reasonable accuracy by considering the tank wall to be rigid. The maximum convective pressure due to the fundamental sloshing mode only is given by

where S,, is the spectral value of the pseudo-acceleration corresponding to the fundamental sloshing frequency which is given by

w,” = - 1 . 8 4 ~ tanh (y) R

The equivalent mass m, can be evaluated from the hydrodynamic pressure [equation (%)I by

m, = ( ~ o H ~ o z ’ p s ( R , 0, z ) . R cos (0) d0 dz

(59)

= 0.455npI R 3 tanh - (I*:H)

202 M . A. HAROUN

#

0

and its centre of gravity is at a distance H , from the base which is given by

I ' I I I 1 I I I 1 I I I I I I

Figure 20 displays the ratios (m,/m) and ( H J H ) for different values of (HIR), where m is the total mass of the liquid.

I \ I I \ I

I I 1

Impulsive and short period components

vibration of the deformable liquid-filled shell. The base shear force can now be expressed as To evaluate the equivalent masses m, and m,, one can consider only the fundamental natural mode of

Q(t) = m, Xf(t) + m, i;'ct)

Xf + 21, Of if + 0: X, = - G(t)

(62)

(63)

where x, is the solution of the differential equation

Since the first term in equation (62) is proportional to the relative acceleration of the shell, one must rearrange equation (62) before estimating the maximum seismic loads by means of a response spectrum; i.e. rewrite equation (62) as

Q(t) = rnf(Xf(t) + G(t)) + (m, - m,) %(t) (64)

One recognizes that the maximum value of I X,(t)+ G(t) I is the spectral acceleration Saf corresponding to the natural frequency 0,. The overturning moment, due to seismic forces, applied to the bottom of the shell can


be expressed as

M ( t ) = mf Hf x,(t) + rn, H , %(t)

It is of interest to note that the fundamental mode shape is normalized in such a way that the maximum amplitude of the radial component of shell displacement is 1.0 and, therefore, one can also estimate the maximum radial component of shell displacement by

I wmax I = P I Sdf (66)

where PI is the modal participation factor of the fundamental mode of vibration and S d f is the spectral displacement corresponding to the fundamental natural frequency of. Figure 2 l(a) displays the non- dimensional parameter (of H J ( p , / E ) for different values of ( H / R ) and (h/R), where p , and E are the mass density and Young’s modulus, respectively, of the shell material. These frequencies are for tanks completely filled with water. The remaining parameters PI, (m,/m), (H,/H), (m,/rn) and ( H J H ) are displayed in Figures 21(b)-(f), respectively. It should be noted that the effect of shell mass on the fundamental frequency of full tanks is negligible; consequently, one can estimate the natural frequency c5, of a tank filled with liquid of density p , by

;f = o f J(Pw/PJ (67)

where of is the natural frequency of the same tank when filled with water and pw is the mass density of water.

Numerical examples Consider the tall tank (T) discussed in the preceding sections. The analysis of the tank involves the

following steps: (a) Parameters of the mechanical model. The fundamental natural frequency of sloshing, o, = 1.57 rad/sec,

is obtained from equation (59); and consequently, T, = 2n/o, = 4sec. The convective mass m, and its elevation H , are given by (Figure 20) rn, = 5,585N. sec2/cm (3,1881b. sec2/in) and H , = 0.82H = 18m (59.04 ft). The fundamental natural frequency of vibration of the liquid-filled shell can be determined from Figure 21(a) for a value of (H/R) = 3.0 and a value of (h/R) = 0900347; hence, of = 33.25 rad/sec. The remaining parameters can be found from Figure (21): m, = 0.735 m = 27,251 N . sec2/cm (15,5551b.sec2/in); H , = 0.5548 = 12.17m (39.88ft); m, = 0.892rn = 33,073 N.sec*/cm (18,8771b.sec2/in); and H , = 0.442H = 9.71 m (31.82ft).

(b) Spectral values. The maximum ground acceleration of the N-S component of the 1940 El Centro earthquake is 0.3489. The spectral acceleration S,, corresponding to the sloshing frequency o, and for a damping ratio [, of 0.5 per cent can be found from the response spectrum; it is given by S,, = 0.063g. Lastly, the spectral displacement and acceleration corresponding to the frequency of and for a damping ratio of 2 per cent are Sdf = 0.757 cm (0.298 in) and S,, = 0.8569.

(c) Seismic response. The maximum free surface wave height is estimated from

Sas &,,ax = 0837R- = 38.6cm (15.19in)

9

In addition, the maximum radial component of shell displacement is given by [equation (66)]

IwrnaxI = PISdf = (155)(0.757) = 1.17cm(0.462in)

Because the maximum values of the convective, impulsive and short-period components of the base shear do not occur, in general, at the same time, one can estimate the maximum value of base shear from

I QT Imax = J((m, S a J 2 + (mf Saf12 + C(m,-mr) %max12)

= 2-3 x lo7 N (5.17 x lo6 lb)

204 M. A. HAROUN

3 1 I

i , , , , , , , 1 : I I

R I , , I 1 1 1 I 1 I l l ! , 1 1

I

o.uoo.so 0.80 >.on 1.x I.VO 1.60 1.m 2.00 2 . m 2 . ~ 0 2.60 2.80 3.00 3 2 0 3 . ~ 0 HEIMT-RADIUS R R T I O ( H / R )

(4

- I

- 1 ? j , , , l , l i l l l l l l , I

o.uOO.60 0.80 I.00 1.20 L V O 1 6 0 1.80 2.00 2 . 2 0 2 . C 2 . 6 0 2.80 3.00 3.20 3 . K HFIGHI-RADIUS R R T l O ( H I R )

(e) Figure 21. Parameters of

I I ZLLi1 I 1 I , , L - . . 1 I

0 . U 0.60 0.80 I 0 0 I 2 0 , !.UO 1.60 1.60 2.00 2 . 2 0 2.UO 2 . 6 0 2 60 3.0C 3 . 2 0 3 . W HEILHT-RRDIUS m r i o (HIR)

0.vOO.m 0.m 1.03 1.20 I . V O 1 . 1 1.80 2.00 2 . 2 0 2.W 2.60 2.BO H E I M T - R A D I U S RATIO ( H / R )

(4

--- 011 3.20 3.W

i ? I , I / , , , , , , , , , , - '

O . V O O . 6 0 0.80 >.DO 1.20 I.UO 1.60 1.60 2W 2.20 2.W 2 . 6 0 2.80 3.00 3.20 3.W HEIGHT-RADIUS RRTIO ( H / R )

(0 the mechanical model


Similarly, the maximum overturning moment applied to the bottom of the shell is given by

I M , lmax J((ms ~s S a J 2 +(mf Hf Sa,12 + ~ ( m r Hr-mf H,) Grnax12)

= 278.6 x lo6 N . m (205.3 x lo6 lb. ft)

and, consequently, the maximum axial force resultant at the bottom of the shell is given by

IMTlrnax 278.6 x lo6 1 .- - ~(7.32)’ 1000 Nz lmax = ~ - ZR2

= 1,655 N/mm (9,4541b/in)

Table I1 shows a comparison between the maximum response values obtained by the simplified mechanical model and those found by time integration of the modal equations of motion.

Design charts were also used to estimate the response of the broad tank (B) to the N-S component of the 1940 El Centro earthquake; the results are shown in Table 111.

Table 111. Seismic response of tank (B)

-4 r, (sec) (sec) N . sec2/cm (Ib . sec2/in)

~~~~ _-_

73,980 52,140 53,428 6.77 4.89 4.94 Parameters 6.89 0.1 62 (42,227) (29,761) (30,496) (22.2) (16.1) ( 16.2)

.. _ _ _ _ Spectral values S,, = 0.0289 S,, = 0.8289 S,, = 0544cm

(0.214 in)

Seismic t,,, = (0.837)( 1,829) (0.028) = 42.86 cm (16.88 in) w,,, = (1.21)(0.544) = 0.658 cm (0.259 in)

N,I,,, = 198N/mm (1,1281b/in)

response Q,,, = 42.4 x lo6 N (9.53 x lo6 lb)

Exact solution 0.620 cm (0.244 in) 39.7 x lo6 N (8.92 x lo6 lb) 190 N/mm (1,085 lb/in)

CONCLUSION

A simple and computationally effective method for computing the dynamic characteristics of anchored, ground-supported, cylindrical liquid storage tanks is presented. The elastic shell was modelled by finite elements and the liquid region was treated by the boundary solution technique.

It was concluded that the initial hoop stress due to the hydrostatic pressure and the in-plane rigidity of the roof system may affect considerably the cos &type modes of the tank wall. In addition, the flexibility of the foundation soil can reduce measurably the fundamental natural frequency of the cos 8-type modes due to the rocking motion. It was confirmed that the coupling between liquid sloshing modes and shell vibrational modes is weak; and, consequently, the ‘convective’ pressure can be evaluated with reasonable accuracy by considering the tank wall to be rigid, and the ‘impulsive’ pressure can be determined by analysing the liquid-shell system and neglecting the sloshing motion. The validity of the method of analysis was confirmed by both scale model testing and field measurements of the vibrational characteristics of full-scale tanks, Earthquake response of deformable storage tanks was obtained by superposing the free lateral vibrational modes; the dynamic stresses were much greater than those computed assuming rigid walls. Finally, a mechanical model which takes into account the deformability of the tank wall was developed and its parameters were displayed in charts. Comparison with the exact solution of the problem confirmed the validity of the model.

206 M. A. HAROUN

ACKNOWLEDGEMENTS

The author wishes to acknowledge the encouragement and advice of Professor G. W. Housner of the California Institute of Technology. The co-operation of the Metropolitan Water District of Southern California in making available its facilities for conducting tests is gratefully acknowledged. This research was supported by grants from the National Science Foundation and the Earthquake Research Affiliates program at the California Institute of Technology.

REFERENCES 1. H. N. Abramson (Ed.), ‘The dynamic behaviour of liquids in moving containers’, NASA SP-106, National Aeronautics and Space

2. API Standard 650, ‘Welded steel tanks for oil storage’, 6th ed., American Petroleum Institute, Washington, D.C. 3. AWWA D100, ‘AWWA Standard for welded steel tanks for water storage’, American Water Works Association, Denver, Colorado. 4. T. Balendra, and W. A. Nash, ‘Earthquake analysis of a cylindrical liquid storage tank with a dome by finite element method’,

5. D. P. Clough, ‘Experimental evaluation of seismic design methods for broad cylindrical tanks’, Report No. EERC 77-I0,

6. N. W. Edwards, ‘A procedure for dynamic analysis of thin walled cylindrical liquid storage tanks subjected to lateral ground

7. R. D. Hanson, ‘Behaviour of liquid storage tanks’, The Great Alaska Earthquake of 1964, Engineering, National Academy of

8. M. A. Haroun, ‘Dynamic analyses of liquid storage tanks’, Report No . EERL 80-4, Earthquake Engineering Research Laboratory,

9. M. A. Haroun and G. W. Housner, ‘A procedure for seismic design of liquid storage tanks’, Report No . EERL 80-5, Earthquake

Administration, Washington, D.C., 1966.

University of Massachusetts, Amherst, Mass., 1978.

Earthquake Engineering Research Center, University of California, Berkeley, CA, 1977.

motions’, PhD Thesis, University of Michigan, Ann Arbor, Michigan, 1969.

Sciences, Washington, D.C., 1973, pp. 331-339.

California Institute of Technology, Pasadena, CA, 1980.

Engineering Research Laboratory, California Institute of Technology, Pasadena, CA, 1982. 10. M. A. Haroun and G. W. Housner, ‘Seismic design of liquid storage tanks’, J . tech. councils ASCE, 107, 191-207 (1981). 1 I . M. A. Haroun and G. W. Housner, ‘Earthquake response of deformable liquid storage tanks’, J . appl. mech. ASME, 48, 411418

12. M. A. Haroun and G. W. Housner, ‘Dynamic interaction of liquid storage tanks and foundation soil’, Proc. 2nd ASCE/EMD

13. L. S. Jacobsen, ‘Impulsive hydrodynamics of fluid inside a cylindrical tank and of a fluid surrounding a cylindrical pier’, Bull. seism.

14. L. S . Jacobsen and R. S. Ayre, ‘Hydrodynamic experiments with rigid cylindrical tanks subjected to transient motions’, Bull. seism.

15. J. C. Luke, ‘A variational principle for a liquid with free surface’, J . Juid mech. 27, 395-397 (1967). 16. A. Niwa, ‘Seismic behaviour of tall liquid storage tanks’, Report No. EERC 7 8 4 4 , Earthquake Engineering Research Center,

17. V. V. Novozhilov, Thin Shell Theory, P. Noordhoff, Groningen, The Netherlands, 1964. 18. F. E. Richart Jr., J. R. Hall and R. D. Woods, Vibrations ofsoi ls and Foundations, Prentice-Hall, Englewood Cliffs, N.J., 1970. 19. J. E. Rinne, ‘Oil storage tanks’, The Prince William Sound, Alaska, Earthquake of 1964, and Aftershocks, Vol. 11, Part A, 1J.S. Coast

20. F. Sakai, M. Nishimura and H. Sakoda, ‘Studies on earthquake-resistance of liquified natural gas storage tanks’, 6th int. conJ

21. S. H. Shaaban and W. A. Nash, ‘Finite element analysis of a seismically excited cylindrical storage tank, ground supported and

22. C. Shih and C. D. Babcock, ‘Scale model buckling tests of a fluid filled tank under harmonic excitation’, 1980 press. vessels piping

23. U.S. Atomic Energy Commission, ‘Nuclear reactors and earthquakes’, TID-7024, Washington, D.C., 1963, pp. 367-390. 24. A. S . Veletsos, ‘Seismic effects in flexible liquid storage tanks’, Proc. 5th world con$ earthquake eng., Rome, Italy, 1, 63G639 (1974). 25. A. S. Veletsos and J. Y. Yang, ‘Earthquake response of liquid storage tanks’, Advances in Civil Engineering through Engineering

26. K. Washizu, Variational Methods in Elasticity and Plasticity, Pergamon Press, New York, 1975. 27. R. S. Wozniak and W. W. Mitchell, ‘Basis of seismic design provisions for welded steel oil storage tanks’, presented at the 43rd

(1981).

specialty con$ dynamic response struct., Atlanta, Georgia (1981).

SOC. Am. 39, 189-204 (1949).

soc. Am. 41, 313-346 (1951).

University of California, Berkeley, CA, 1978.

and Geodetic Survey, Washington, D.C., 1967, pp. 245-252.

exhibit. LNG, Kyoto, Japan (1980).

partially filled with liquid’, University of Massachusetts, Amherst, Mass., 1975.

techn. conj, ASME, San Francisco, California (1980) (Preprint 8GC2/PVP-66).

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Midyear Meeting, API, Toronto, Ontario, Canada (1978).

Documents

1983-Vibration Studies and Tests of Liquid Storage Tanks