Journal of Sound and Vibration (1988) 122(3), 491506
FREE VIBRATION ANALYSIS OF LIQUID STORAGE TANKS R. K. GUPTA AND G. L. HUTCHINSON
Department of Civil and Agricultural Engineering, University of Melbourne, Parkville, Victoria, Australia 3052 (Received 19 March 1987, and in revised form 14 July 1987) To obtain the response of structures subjected to dynamic loading a knowledge of the free vibration characteristics of the structure is essential. The enclosed fluid in the tank exerts inertial loading on the vessel, and part of the fluid may be coupled to the motion of the vessel. For partially filled tanks/shells the free surface motions of the fluid may be coupled with the tank/shell motion, further complicating the analysis. In this paper, variational principles have been used to obtain a functional describing coupled oscillations between a linear elastic body and a liquid of small wave heights. A complementary Rayleigh's quotient has then been introduced to obtain the natural frequencies of circular cylindrical tanks partially filled with liquid and oscillating in an axisymmetric manner.
1. INTRODUCTION To obtain the response of a structure subjected to earthquake excitation a knowledge of the free vibration characteristics is necessary. In a seismically excited tank containing fluid, the fluid exerts an inertial loading on the tank vessel and part of the fluid may be coupled to the motions o f the vessel. For partially filled tanks, the liquid free surface motions may be coupled to the shell motions of the vessel resulting in further analytical complications. The free vibration characteristics of liquid storage tanks have been studied by using various methods such as finite element methods, boundary integral techniques and variational methods. In typical finite element analyses one models the tank shell alone or both the t a n k shell and liquid [18]. The classical RayleighRitz method has been used [911] to obtain a set of equations describing the behaviour o f the vibrating shell, and the method of separation o f variables may be used [12] to obtain series form solutions for the coupled oscillations o f t h e liquid and structure for an axisymmetric cylindrical tank. In order to account for sloshing effects appropiately in this paper variational principles (Hamilton's and Toupin's) are used to obtain a functional describing coupled oscillations between a linear elastic body with infinitesimal deformations and a liquid with small wave heights. A complementary Rayleigh's quotient is then introduced to obtain the natural frequencies of a circular cylindrical tank partially filled with liquid and oscillating in an axisymmetrie manner. Apl~roximation~ are made in the analysis to obtain simplified formulae for natural coupled eigenvalues and eigenvectors in both sloshing and bulging modes. Appropriate comparisons with exactly calculated values are made wherever possible, and it is demonstrated that these simplified formulae may be used for the (preliminary) analysis o f liquidfilled tanks, thus avoiding the complicated exact analyses. Results have been obtained for two different depth o f water (d) to tank height ( H ) ratios. Throughout the approximate analyses the liquid is assumed to be incompressible and dissipative forces are neglected. 491 0022460x/88/090491 + 16 $03.00/0 O 1988 Academic Press Limited
492
R.K.
G U P T A A N D G. L. I I U T C H I N S O N
2. TANKLIQUID SYSTEM The tank is considered to be an elastic solid lying in the domain D, as shown in Figure 1. The Cartesian coordinate system x~ ( i = 1, 2, 3) is introduced as a reference frame and the displacement, u , is expressed as a function of x~ and time t. The elastic modulus tensor, E o u , and the density of the solid, p~, are independent of time. The displacement ~ is prescribed as a function of x~ and t on S~ in /3, (see Figure 1). The body force, f~, and the surface traction, ~ , are prescribed as functions of xs and t in D, and S,, ( = D ,  S , ) , respectively. Assuming u~ takes the prescribed value fi~ on S , and the variation 8ui vanishes at two arbitrary time instants t~ and t2 one has u~ = t~ on S,, and ~u~l,,.,~= O. Free
s~ sp
o~ x3
Su
sp s,
ol
F
X2
Xl
Figure I. Tank liquid system.
. 3. APPLICATION OF VARIATIONAL PRINCIPLES 3..
HAMILTON'S PRINCIPLE FOR A SOLID DOMAIN
Hamilton's principle for a solid domain for small displacements may be stated as 8
L, dt = 0,
where L, = T,  ~ .
(1, 2)
I
L , , T, and 1/, represent the Lagrangian for the solid domain, the kinetic energy and the potential energy of the system, respectively. Equation (1) may be expressed as
8 f " [" f P,~,C~,dD. f (89
dD,+f ~u, dS,.] dt=O.
Note here that EokleklSeij = trOSeil = 89 o'08Ui J = ( % S U ~ ) j  o'ojSUi,
Ulj + Uj.i)  ~o'08Ujj t +~o'OSUj.I %8Ul. ~ = (O'oSUj).i  oO.,SU.
The first variation of the functional in equation (3) thus may be expressed as
(3)
FREE VIBRATION ANALYSIS OF LIQUID STORAGE TANKS
493
For 8u/l,=r,.12= 0, equation (5) may be expressed as
f," I
f
(  p , zT,+tr0.j+f/) 8u, d D , 
t
]
(trunk') ~) 6u, dSo dt =0.
(6)
If equation (4) is to be satisfied, then croj +/] = p,//, in D ,
and
tr0n) 'l = ~
on So,
(7, 8)
where n) ") represents the outward unit normal vector to the surface of the solid. 3.2. TOUPIN'S PRINCIPLE FOR A SOLID DOMAIN Toupin's complementary principle [13] introduces an impulse (v o) corresponding to the stress tro such that v# = ft o'# dt,
or
t~0 = o#.
(9)
Substituting equation (9) into equation (7) yields
~,=(1/p.)(o'ud+f~)=(1/p,)(f~,jd+f~),
~,=(1/ps)(o,jj+ f ~dt).
(10, 11)
For the condition 8ouJ,.,,,, , = 0, the complementary principle holds in the same form as equation (1): i.e., 9~"8 f f2 Lc, dt = 0,
where L . = V*, T..
(12, 13)
I
T . and V~./'epresent the kinetic energy and the complementary potential energy of the solid, respectively. Toupin's principle for the solid domain may be expressed by the functional i
+f
.
.
dr)
1
d,o.
.,,
where Cuu represents the flexibility coefficient tensor. Upon noting that ( l / p , ) v,~kS.v~jj= zi,Svuj = (~i,Sv0) J  ~iuSv0 = (~,Svu)j 89
+ i~j.,)Svo,
substituting into equation (14) and neglecting body forces J~, the first variation of equation (14) may be written as
494
R . K . GUPTA A N D O. L. H U T C t t l N S O N
Since Bv0b=,,.r2=0, equation (16) may be expressed as
f')[f {89
f (d,f~,)n,')SvodS.]dt=O.
(17)
For equation (17) to be satisfied
89
+ tij.,) = e0,
~(u/j + uz,) = E,j,
or
(18)
and t~i  t~i = 0,
or
ai = u~.
(19)
3.3. TOUPIN'S PRXNCIPLE FOR A LIQUID DOMAIN If nonlinear terms are omitted, the stress caused by liquid motion may be expressed in terms of a velocity potential r as
cro"= 60 = P~o = P~r
(20)
where p and pj represent the liquid pressure and the liquid mass density, respectively. and 8;i is Kronecker's delta. If 8~lt=,,.,,=0, then Toupin's principle for the liquid field may be obtained as a special case of that of the solid field: i.e., 8
fi ll Let d t = 0, I
where Let = V,~ Td.
(21, 22)
" .
Ta and V* represent, respectively, the kinetic energy and the complementary potential energy of the liquid domain. Toupin's principle for the liquid domain may be expressed by the functional
f p,g,:dS/89f
,5 f,i" I f p,~(bdSy+ f p,a,n~"Cl,dSp89
dD,] dr=0,
(23)
where ~" represents the component of the prescribed displacement on the fre e surface, Sy, inthe x3direction. The first variation of the functional in equation (23) may be written as
ft')[p,r Upon noting that
and
f',2ple~,'a'.,dD,=f,"p,C'.,n~d'cPdS,+f')p,9 CJ,n,(~
dSf f'PlSgb~." dD,,
and substituting these into equation (24) one obtains
f') [f (p,~p,c~.,n~") SC, dSz+ f (p,ti,n,t)pt~.,n~t)) ,clgdS,  f (g,+')p,d~ dSf + f p,~.,8~ dDt] =O.
(25)
FREE
VIBRATION
ANALYSIS
OF LIQUID
STORAGE
495
TANKS
For equation (25) to be satisfied
Sf, p,gr 9 (o  c~.in to and i on Sv, ~ = ~.,n~ 0
uin
on
(26, 27)
on SI, ~,,, = 0
i
on Di.
(28, 29)
Variational principles can now be applied to the coupled system. Referring to Figures l, one can assume that the solid is in contact with the fluid on Sp, which may be regarded as a portion of S~. The remainder of S~ is denoted as S* ( S,,  Sp). T* may be expressed as
= T*
on S*,
~pB0n)
~
on Sp.
(30,31)
Also /I ( I ) ~   n j t,)
on
Sp.
(32)
Summing Ls, from equation (2), and La, from equation (22), yields a combined functional, L, with ui, tp and ~" as its variational function. The first variation 8L is calculated for an interval t~ <~ t ~< t2 under the constraints to the variations 8u~,etc., as
: I 'f [ I (pSf~,Su, o'oBu,d+f ~Su,)dDs + f "* Su, dS* + I pin~"(fi,8~ + ~B~,) dSp  S ptg~8~dSf + f.p,(~SC,+ cPS~)dSf f
[email protected],~n~t,;8 @dSp
f
[email protected],tni{,, 8 c]gdSf
p~..8~ dDi dt =0, I P~fi,Su, dDs[::+ f p,n~'I~Bu,dSv ::+ I
(33) P'~B~'dSf]::
 f,:~ [f Pfii'SuidD" + I p'n~''*su' ds" +
,
o'on j
v
,
.
 ffi~uidD,f'*SuidS*  I Plnit')d
[email protected] d S.v + I plgr162dSzlP'~ScPdSz qf p,tP,
[email protected]+f p,cl'.,n~')
[email protected], tP.,
[email protected],]dtO. For 8u,
o,
(34)
0, and 8~'1,=,,.,,.0, equation (34) may be written as
fff [f (Pfiii+trqd+ f~) Su'dD'+ f (o'qn,')+ T*) Su, dS*~ + I p,C',,ScPdD, + f p,(~ cl'.,nl'I)SC' dSf f p,(gr + 4') 8~ dSf  f (crqnj'" +ptclgni " (" ) 8U i dSp + I pt(fti
[email protected]")n~')Scl'dS']
dt =
O.
(35)
496
R . K . GUPTA A N D G. L. [ I U T C H I N S O N
In equation (35) the boundaries aD, and ODI of D, and /91 are written as
OD, = S,, + S* + Sp,
OD,= Sf + Sp.
and
(36)
For equation (35) to be satisfied P, ffi+~oj+~ =0 9 ., = 0 g~'+q6=0
in D,,
in Di.
.
tron~")= T*
~,in~ ~ = ~
on S*,:
(37, 38)
on Sf.
o'unJ'~=pn~t~ on S~. a,n~I>= r ,n~~ on Sp.
(39, 40)
on SI,
(41,42,43)
4. FREE VIBRATION OF THE COUPLED SYSTEM A complementary Rayleigh quotient is now introduced to estimate the coupled natural frequency, to, of the oscillations of liquid storage tanks. In the analysis it is assumed that the tank material is linear elastic and the body forces, j~, and the traction forces, T*, are zero. The liquid pressure p (= plq~) has been considered as a known external force on
8p. The displacement ul may be obtained when equation (37) is integrated as follows to satisfy equations (38) and (42) and the given displacement fi on •u: P=ffl = o',jd = o./jn)S) = Pa wn,'z(o
on St, ,
(44)
(45) G,(P, Q; to:)p,n~')~(Q) dSo, Jsp Here GI(P, Q; to:) is the Green function (the induction or the influence function) for the linear elastic solid oscillating with radian frequency a~, P is the spatial point where the xrdirection of displacement is evaluated, and Q is the point where unit force is exerted in the xj direction. The Green function has a symmetric property, i.e., u,(P) =  f
G~(P, Q; to2) = GI(Q, P; to2)
(46)
and hence
l2
=

It
;
v
[pln~lJa,80dSe] dt.
(47)
!
From equation (39), 9 is integrated with integral constants undetermined and from equation (41) =
~']g.
(48)
Equation (40) may be transferred with the help of equation (48) as follows:
q~jn~l)+~/g = 0.
(49)
FREE VIBRATION ANALYSIS OF LIQUID STORAGE TANKS
497
Thus all the terms of equation (35), except those underlined, are satisfied. The underlined terms may be rearranged, by using equation (49): i.e.,
f"[_f
o,I[
r
+ )8 dSsf
,  a,)
q6\
d S , ] dt =0.
(50)
To derive the complementary Rayleigh quotient from equation (50), ~, ui and 6q~ are formed as follows: q~ = q3 cos tot,
ul = t~ito sin tot,
6q~ = eq~.
(51)
Here e is an arbitrary small parameter and ~ and ~i are functions of xj, independent of time. If @ and u~ of equation (51) are exact then the first variation of the functional vanishes in the interval ti <~ t <~ t2, and equation (50) may be solved with respect to to2, yielding a complementary Rayleigh quotient: to2=
lsts e ptn~Z)c~.,c~ds
(o,/g),b,b ds +I5,
ds}"
(52)
Equation (52) is a quadratic form of integral constants of q~ which is identified as an eigenvalue problem. This equation may be solved through a successive iteration method to obtain to and the integral constants of @ which make the first variation ofthe functional, 8L, vanish. 5. APPLICATION TO A CIRCULAR CYLINDRICAL TANK I. The dimensions of a circular cylindrical tank are shown in Figure 2. The shell thickness, h, is assumed to be constant and d is the fluid depth. In the present study, the free vibration frequencies are obtained for the cylindrical tank oscillating in an axisymmetric manner. At lower eigenvalues the motions are attributable mainly to liquid surface sloshing while at higher eigenvalues socalled "bulging modes" emerge and in this study sloshing and bulging eigenvalues and eigenvectors are dealt with separately. Equation (39), known as Laplace's equation, may be expressed in cylindrical coordinates (r, z) as C2~,,i =
V 2 t ~ = 02 t ~ / c g r 2 q  ( l / r )
rgtY15/Ortc~2CJS/az 2
(53)
which may be solved by separation of variables, with appropriate boundary conditions. Upon assuming
= ~T(t),
(54)
; h /2
H
F i g u r e 2. Profile o f a c y l i n d r i c a l tank.
a.K. GLIPTA AND G. L. HUTCHINSON
498
where T(t) is harmonic and utilizing the boundary conditions along the z axis, i.e., (a)
(~2~/
[email protected]/~z)]~.a=O
(b)
0q~/0z[~.~o= 0,
(free surface condition),
(55) (56)
the frequency equation may be obtained as
to2=  ( g / a ) h, tan (,X,d/a).
(57)
Equation (57) determines a set of eigenvalues at real values o f An (n = 1, 2, 3 , . . . ) . For n = 0 an imaginary value of i,Xo/a is assumed and equation (57) is modified to
to2 = (g/ a)Ao tanh ()rod/a).
(58)
A , l o ( A , r / a ) cos (,X.z/ a) c~ = AoJo( ,Xor/ a ) cosh ( Aoz / a ) ~ y~ , Jo(,Xo) cosh (Aod/a) ,~l Io(,X,)
(59)
The ftinction ~ is obtained as
where Ao and A, are integral constants, Jo is the Bessel function of the first kind, o f zero order, and Io represents the modified Bessel function of the first kind of zero order. 5 . 1 . ESTIMATION OF t~ In this study the radial displacements of the cylindrical Shell, tT, are determined by using membrane shell theory. If the edges of the shell are free, the internal pressure, p, as shown in Figure 3, produces only a hoop stress, t..
m =pa/h
(60)
and the increase in the radius of the shell in the radial direction is
u = atr,/E =pa2/Eh = pjd~a2/Eh,
(61)
where p (=  p ~ ) is the dynamic liquid pressure. Since u = tito sin tot and @ = t/3 cos tot, qb = tot~ sin tot,
(62)
substituting into equation (61) yields
u = (p~a~/Eh)to~ 2 sin tot.
(63)
r
': h
V
H
p
p 9
O
It
P
Figure 3. Pressure distributioncylindrical shell tank.
FREE VIBRATION ANALYSIS OF L I Q U I D STORAGE TANKS
499
Comparing equations (61) and (63) gives
[(p, a2/Eh)clj2,
0
u = 1.0,
(64)
Substituting equation (64) into equation (52) yields
dr+ fo"
d'
+Ido (pia21Eh)(cJ~J)l,:adz }.
(65,
6. DEVELOPMENT OF APPROXIMATE FORMULAE FOR EIGENVALUES AND EIGENVECTORS 6.1. S L O S H I N G #*lODE In sloshing modes (oscillation of the liquid free surface) the eigenvector, ~, is approximated as
= AoJo(knr/a) cosh (k~z/a)/Jo(kn)
cosh
(knd/a),
(66)
where Ao is replaced by k~ (a set of n values) and the etIect of the second term on the fighthand side of equation (59) is assumed to be small. Approximate formulae are developed for two cases, as follows. 6.1.1. Flexible tank ~i~ll When the tank wall is considered to be flexible and deforms under the liquid dynamic pressure, p, equation (65) may be simplified, as follows. From equation (66),
~0z= Ao(kJ a)Jo(knr/a) sinh (k,z/a)/Jo(k,) 0~
~Or= Ao(kJa)Jl(knr/a)
cosh
(knz/a)/Jo(kn)
cosh
(k,d/a),
cosh
(67)
(k~d/a).
(68)
Considering each term of equation (65) in turn,
(~ z~)
r dr (kJa,Ag sinh (k~d/a) cosh (k~d/a)(a2/2)[Jg(kn) +J~(k.)] Jo2(k~) cosh 2 (k~d/a) ' Io~ ~ "1 knA~Jo(kn)Jl(k,)[{asinh(2k~d/a)/4k~}+d/2] (qo.,qo) ,=~a dz jo2(k,,) cosh 2 (k~d/a) ,
9
rdz(a2~176
Io' 1 ( q ~ ) I g
z~
~ P~h " " J
(t/l~) ,=,,adz 
"
(P'a3/Eh)agj2(k~)[{asinh(2k~d/a)/4k~}+d/2] jo2(k,) cosh 2 (k,d/a) "
[k,,tanh(k~ d ) fJ21(kn) 1} Jl(k~) {tanh ( k d ) + d
EfJ kn'
(70) (71)
J~(k~) cosh ~ ( k ~ d / a )
and substituting in equation (65) Yields
~02
(69)
zd
a
(72)
500
R.K.
G U P T A A N D G . L. H U T C H I N S O N
When kn~0, Jo(kn)~ 1 and Jl(kn)~0. Hence to2 = k, tanh (knd/a)/[(a/g) + (p~a3/ Ehkn){tanh (knd/a) + (d/ a)kn sech 2 (knd/a)}], or
to2 = an tanh 0 J [ 1 + (fl/an) {tanh On+ 0n sech 2 0,}],
(74)
where an = k~g/a, fl = g2pla/Eh and 0n = k J d / a . Equations (66) and (74) may be used to determine the approximate sloshing eigenvectors and eigenvalues, respectively, for a cylindrical tank partially filled with liquid. 6.1.2. Rigid tank wall If the tank wall is considered to be rigid and does not deform under tile liquid dynamic pressure, p, in the radial direction, then
~fi/Or[,=a =0.
(75)
OfilOrlr=a = Ao(kJ a)Jl(kn) cosh ( knz/ a) = 0,
(76)
kn=3"8317, 7.0156, 10"1735, 133237, 164706 . . . . ,
(77)
From equation (66) which gives
corresponding to n = 1, 2, 3,14, 5, .. These values of kn are known as the zeros of the Bessel function of the first kind. The approximate forumlae for natural eigenvalues may be derived by substituting tT= 0 in equation (65): i.e., to2= an tanh 0n,
(78)
which corresponds to results derived by using other methods. 6.2. B U L G I N G M O D E For bulging modes (where the tank walls oscillate with the liquid) the eigenvector, fib, may be approximated as
fi = AnIo(lx,,r/a) cos (I.tmZ/a)/Io(tZ,,),
(79)
where An is replaced by/tin and the effect of the first term on the right side of equation (59) is assumed to be small. It is reasonable to anticipate that in the bulging modes the values of to in equation (57) are large. For large to
Ixmd/a=mrr/2,
tzm'(m~)rra/d,
m= 1,2,3,....
(80,81)
For these values of/.tin, fi vanishes at z = d and equation (65) reduces to
to=fo(fi.',fi)lr=oadz/f (afi)l,=oadz. o
(82)
Substituting for fi and t~ from equations (79) and (64) yields !
.
toz =(Eh/pla3)~,nll(~tm)/io(ttm).
(83)
Equations (79) and (83) are approximate expressions for the bulging eigenvectors and eigenvalues for a cylindrical tank partially filled with liquid. : : It should be noted that the assumption of uniform internal pressure, p, in the estimation of u viol/ates the boundary condition at t h e bottom of the tank and the discontinuity condition at the depth d in partially filled tanks. The significance of these factors will be examined by comparing the results of typical tanks found by using the approximate methods outlined above with results found by Using more accurate analyses [12].
FREE VIBRATION ANALYSIS OF LIQUID STORAGE TANKS
501
7. COMPARISON OF PROPOSED APPROXIMATION WITH MORE ACCURATE ANALYSES [12] By using equations (74), (78) and (83), eigenvalues corresponding to the sloshing and bulging modes have been calculated for a typical cylindrical steel tank, partially filled with water. Results can be compared with values obtained in reference [12]. Data for the tank and liquid are as follows: tank diameter, D = 50 m; tank height, H   3 0 m; wall thickness, h  0.03 m; water depth, d = 216 m; elastic modulus o f steel, E = 206 GPa; Poisson ratio of steel, v = 0.3; density o f steel, p~ =7850 kg/m3; density of liquid, pt = 1000 kg/m 3. Table 1 shows a comparison of the first five natural frequencies for the sloshing and bulging modes as obtained by using both methods. Very close agreement can be observed for the sloshing modes and good agreement for the bulging modes. For practical purposes any differences are sufficiently small to be ignored. The corresponding first and second mode shapes for sloshing and bulging are given in Figure 4. TABLE 1
Comparison of natural frequencies (tad/s) Sloshing mode
Bulging mode
^
^
Mode No.
Approximate values
From reference [12]
Approximate values
1 2 3 4 5
12244 1"6591 19979 2.2864 2"5422
12238 1.6582 1.9969 22852 2.5409
22"3494 44"1699 58.2442 69"5125 79.1894
From reference [12] 22.0956 437619 56.8292 " 66.8874 75.3469
Note.The results taken from reference [ 12] are based on an exact integration of the equations Of motion for the tank wall (with the axial inertia force neglected) rather than solving for deformations by expandifig in a trigonometric series. Further, reference [12] shows that the axial component of inertia force has a negligible influence on the values of to. To study the effect of tank geometry and liquid depth on natural frequencies, parametric studies for the cases HID = 0 . 6 and H/D 12 (corresponding to a 6 0 m high tank with all Other parameters identical) and with the ratio d/a varied from 01 tO 1.0 were carried out. Such values represent realistic field cases and the results of the hnalyses are listed in Table 2. In the case of the sloshing modes, natural frequencies for the cases of the tank walls treated as flexible and rigid are presented. Insignificant differences occur between the computed values. For the case of the bulging modes the natural frequencies have been calculated with the assumption that the tank wall was flexible. Figures 5 and 6 show the variation o f the first five sloshing natural frequencies with d/a for HID equal to 0.6 and 12 respectively. For the first frequency with H / D = 0 . 6 (Figure 5) once the water depth reaches approximately 50% of the height (d/a 2 0 . 5 ) the frequency remains almost constant regardless of any increase in the depth of the water column. Therefore, only this upper fraction o f the liquid column takes part in sloshing, the remainder of the column being essentially steady. For the second and higher frequencies, the upper fraction of the liquid column participating in sloshing gradually declines. Similarly, for HID = 1.2, for corresponding frequencies, the upper fraction of the liquid column participating in sloshing m a y b e deduced from Figure 6. For all frequencies the increase i n H / D to 1.2 results in a significantly lowerproportion of the. water column participating in sloshing. :. :
502
R, K. G U P T A A N D G . L. H U T C t l l N S O N
Ii
i
I I First
(o)
I
•
Seomd
/
T
i i i
) /
First
Second
(b)
Figure 4. Sloshing and bulging modes. (a) Sloshing modes; (b) bulging modes. With reference to equations (74) and (78), as d/a is increased and other parameters are unchanged oJ~0.9, 0.492 and 0.21 for n = 1, 2 and 5 respectively. For H/D equal to 0.6 and 1.2 respectively, Figures 7(a) and (b) show the variation of the first five bulging natural frequencies with d/a. For both cases, as the water depth increases, the natural frequencies gradually reduce indicating that the full height o f the liquid column is participating. Further, the corresponding frequencies for HID = 1.2 are substantially lower than for the case HID = 0.6. Moreover, for both cases, for lower d/a values, the difference between consecutive frequencies is larger than for higher d/a values. This.occurs because, at lower water depths the radial displacement, if, of the cylindrical shell, is lower. 8. CONCLUSIONS (1) The approximate formulae developed may be used to estimate the coupled natural frequencies for a cylindrical tank oscillating in an axisymmetric manner. For the sloshing modes the example considered here shows excellent agreement between natural frequencies calculated by using the approximate'formulae and those calculated by using reference [12]. Similarly, forthe bulging modes good agreement is observed. : (2) The calculation of the radial displacement (t~), based on membrane shell theory, is a good approximation, so that one can avoid solving fourth order differential equations. (3) Only a certain upper fraction of the liquid column participates in sloshing and the remainder of the liquid column is essentially steady. (4) For bulging frequencies all of the liquid column participates. For higher liquid 9depth to tank height ratios the frequencies gradually decrease and, for a given liquid depth, corresponding frequencies decrease as the tank height to diameter ratio increases.
FREE VIBRATION ANALYSIS OF LIQUID STORAGE TANKS
503
TABLE 2
Variation of natural frequencies with tank parameters Ratio
Sloshing mode
H/D
^
(tank geometry)
Ratio
Flexible tank wall (rad/s)
Rigid tank wall (tad/s)
Bulging mode (rad/s)
0.6
1"0
1"2255 1.6591 1.9979 2"2864 2.5422
1"2256 1"6592 1.9980 2"2865 2.5423
19.9822 406985 53"8761 64"3961 73"4191
09
1.2248 1.6591 1.9979 2.2864 2"5422
1"2250 1"6592 !'9980 2.2865 2"5423
21.6810 43"1783 56.9944 68"0478 77"5369
0"8
1.2234 1"6591 1.9979 2.2864 2"5422
12235 1"6592 19980 2"2865 2"5423
23"6275 46.0885 60.6662 72"3529 823948
0.7
12203 1.6590 1.9979 2.2864 2"5422
1.2205 1"6591 1"9980 2.2865 2"5423
25"9086 49.5773 65"0822 77"5369 88"2482
0"6
1.2137 1"6587 1.9979 2.2864 2.5422
1"2139 1.6588 1"9980 2"2865 2"5423
28"6598 53"8761 70.5401 83.9516 95.4955
0.5
11997 1"6576 19978 2.2864 2.5422
1.1999 1"6577 1"9979 2"2865 2"5423
32.1066 59.3707 77"5369 92.1844 104"8025
0"4
11701 1.6530 19973 2.2864 2.5422
1"1703 1"6531 1.9974 22865 2"5423
36"6603 66"7666 86"9815 103.3097 117"3869
0.3
1"1086 1.6346 1.9935 2.2857 2.5420
1"1087 1"6347 1.9936 22858 2"5421
43"1783 77"5369 100.7723 119.5721 135.7928
0"2
0"9845 1.5616 1.9641 22754 2.5387
0"9846 1"5618 1"9642 2"2755 2"5388
m m
07412 1.2908
0"7413 12909
d/a
0"1

504
R. K. GUPTA AND G. L. rtUTCHINSON TABLE 2
(continued)
Sloshing mode
Ratio
HID Flexible tank wall (rad/s)
Rigid tank wall (rad/s)
Bulging mode (rad/s)
1.7518 2.1324 24495
1.7519 2.1325 2.4496

1.0
1.2260 1.6591 1.9979 22864 2.5422
1.2262 1.6592 1.9980 2"2865 25423
8"9228 267647 36"6603 44.3655 50.9053
09
1.2260 1.6591 1.9479 22864 2.5422
1.2262 1"6592 1"9980 22865 2"5423
107112 28"6598 38"9564 47.0181 53"8761
0"8
1.2260 1.6591 1.9979 2.2864 2.5422
1"2282 1"6592 1"9980 2.2865 2.5423
12.6122 30"8605 41.6460 50"1350 573722
0"7
1.2261 1.6591 1.9979 22864 2.5422
1.2262 1"6592 1"9980 2.2865 2"5423
14.7084 334725 44"8646 53"8761 61"5748
0.6
12259 1.6591 1.9979 2.2864 2.5422
1.2261 1.6592 1.9980 2.2865 2.5423
17"1088 36"6603 48.8237 58.4909 66.7666
0.5
1.2255 16591 19979 2"2864 2.5422
1.2256 16592 1.9980 12865 2.5423
19.9822 40"6985 53.8761 64.3961 73"4191
0.4
1.2234 1.6591 1.9979 2.2864 2.5422
12235 1.6592 1.9980 2"2865 2.5423
23"6275 40.0884 60"6662 72"3529 82"3948
0"3
i.2137 1.6587 1.9979 2.2864 2.5422
1.2139 1.6588 1"9980 2"2865 2.5423
286598 53"8761 70"5401 83"9516 95.4955
0.2
1.1701 1"6530 19973 2.2864
1.1703 1"6531 1.9974 2.2865
(tank geometry)
Ratio
0"6
01
12
d/a
505
FREE V I B R A T I O N A N A L Y S I S O F L I Q U I D S T O R A G E T A N K S
(COtllinlted)
2
TABLE
Sloshing mode
Ratio
HID (tank geometry)
Ratio
d/a
Flexible tank wall (rad/s)
Rigid tank wall (rad/s)
12
0.2
25422
2,5423
01
0"9845 1.5616 1.9641 2.2754 2.5387
0.9846 1"5618 19642 2.2755 2.5388
1 1.0
1
2
i
*=
0.6
T T
* * *
I
.9
i
0.4
~..,"
02
0 0.7
,
f
,
,
1"0
~
J
I
i
I h, I
+
+
I
" 4
I I
J
x
,
5
i
/ x
4
&
T
0"8
Bulging mode (rad/s)
I
,
,,..~ T
I
i
i
4// i
I
1'5 2'0" Sloshing frequencies (racl/sl
f
25
Figure 5. Variation of first five sloshing frequencies; tt/D = 0.6.
2
1.0
r I i x I
0.8
•
3
06
4
5
Z~
+
A
+
x
I, i
0"4
/ 0"2
eJe
0"9 l'O
I j
1"5 2"0 Slosh{ng frequencies(rad/sl
2"5
Figure 6. Variation of first five sloshing frequencies; H/D = 1.2.
i
506
R. K. GUPTA AND G. L. ltUTCHINSON
1
2
0.4
4
(al
5
1
2
\
\ \
(b)
345
"~
x\\\
0.8
06
5
\ \\\
1.o
\\\
\
\.
\
\ \\o\, \ \ \o\.
\
02 o lO
I
[
.St3
I
I
I
I
I
100
I
I
I
t
~
0
A
I
t
I
i
50
I
I
t
I
100
Bulging frequency (rad/s)
Figure 7. Variation of first five bulging frequencies. (a) tt/D =0.6; (b) H/D = I2. (5) F o r the s l o s h i n g m o d e s there is no significant difference b e t w e e n the values o f n a t u r a l f r e q u e n c i e s c a l c u l a t e d w h e n a s s u m i n g the t a n k wall to b e either rigid o r flexible.
REFERENCES 1. A. M. J. ALNAJAFI and G. B. WARBURTON 1970 Journal of Sound and Vibration 13, 925. Free vibrations of ringstiffened cylindrical shells. 2. A. A. LAKlS and M. P. PAIDOUSSIS 1971 Journal of Sound and Vibration 19, 115. Free vibration of cylindrical shell partially filled with liquid. 3. L. KIEFLING and G. C. FENG 1976 American Institute of Aeronautics and Astronautics Journal 14, 199203. Fluid structure finite element vibration analysis. 4. M. A. HAROUN 1980 Earthquake Engineering Research Laboratory Report EERL 8004, California Institute of Technology, Pasadena. Dynamic analysis of liquid storage tanks. 5. M. A. HAROUN and G. W. HOUSNER 1982 Proceedings of the Engineering Mechanics Division American Society of Civil Engineers 108, 801818. Complications in free vibration analysis of tanks. 6. M. A. HAROUN and G. W. HOUSNER 1982 Journal of the Engineering Mechanics Division American Society of Civil Engineers 108,783800. Dynamic characteristics of liquid storage tanks. 7. M. A. HAROUN and G. W. HOUSNER 1982 Journal of the Engineering Mechanics Division American Society of Civil Engineers 108, 346360. Dynamic investigation of liquid storage tanks. 8. M. A. HAROUN 1983 Journal of Earthquake Engineering and Structural Dynamics 11,179206. Vibration studies and tests of liquid storage tanks. 9. W. E. STILLMAN 1973 Journal of Sound and Vibration 30, 509524. Free vibration of cylinders containing liquid. 10. S. TAN I, Y. TANAKA and N. HORI 1977 Proceedings of the Sixth World Conference of Earthquake Engineering, India, 12291234. Dynamic analysis of cylindrical shells containing liquid. 11. A. S. VELETSOS and J. Y. YANG 1976 Proceedings of U.S.Japan Seminar on Earthquake Engineering Research, 317341. Dynamics of fixedbased liquid storage tanks. 12. H. KONDO 1981 Bulletin of the Japan Society of Mechanical Engineers (JSME) 24, 215221. Axisymmetdc vibration analysis of a circular cylindrical tank. 13. R. A. TOUPIN 1952 Journal of Applied Mechanics American Society of Mechanical Engineers 19, 151152. A variational principle for the meshtype analysis of a mechanical system.