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Rocking response of tanks containing two liquids
Nuclear Engineering and Design 152 (1994) 103-115
ELSEVIER
Nuclear Enoineed.ng and Design
Rocking response of tanks containing two liquids Yu Tang Reactor Engineering Division, Argonne National Laboratory, Argonne, IL 60439, USA
(Received 8 March 1994)
Abstract
A study of the dynamic response of upright circular cylindrical liquid storage tanks containing two different liquids under a rocking base motion with an arbitrary temporal variation is presented. Only rigid tanks were studied. The response quantities examined include the hydrodynamic pressure, the sloshing wave height and associated frequencies, and the base shear and moments. Each of these response quantities is expressed as the sum of the so-called impulsive component and convective component. Unlike the case of tanks containing one liquid, in which the response is controlled by one parameter, the height-to-radius ratio, the response of tanks containing two different liquids is controlled by three parameters: the height-to-radius ratio and the mass density ratio and height ratio of the two liquids. The interrelationship of the responses of the tank-liquid system to rocking and lateral base excitations is established by examining numerical results extensively. It is found that some of the response quantities for a tank-liquid system under a rocking base motion can be determined from the corresponding response quantities for an identical tank under a horizontal base motion.
1. Introduction
Liquid storage tanks are important components of industrial facilities and, when located in earthquake-prone regions, should be designed to withstand the earthquakes to which they may be subjected. The dynamic response of liquid storage tanks subjected to earthquakes has been a subject of numerous studies in the past 30 years. For reviews o f the previous studies on responses of laterally, vertically and rocking-excited tank-liquid systems the reader is referred to the papers of Veletsos (1984), Veletsos and Tang (1986) and Veletsos and Tang (1987) respectively. Most of the previous studies were focused on the tank Elsevier ScienceS.A. SSDI 0029-5493(94)00699-Y
containing only one liquid. However, there are cases in which the density of the tank content is not uniform. For such cases the dynamic responses of tanks containing liquids with different densities must be studied. To respond to this need, Tang (1993a) presented a solution for the dynamic response of rigid tanks containing two liquids under a horizontal base motion. In that study, however, the effect of gravitation on the interface motion of the two liquids was neglected. It should be mentioned that this gravitational effect was later considered in the analysis and the findings are described in detail in an A N L / R E report (Tang, 1993b). The dynamic characteristics of flexible tanks containing two liquids were stud-
104
Y. Tang /Nuclear Engineering and Design 152 (1994) 103-115
ied by Tang (1994) and a paper by Veletsos and Shivakumar (1983) dealt with the sloshing response of layered liquids in rigid tanks. All these studies are aimed at the understanding of the dynamic behavior of tanks containing two liquids under lateral base motions. This paper describes the dynamic response of tanks containing two liquids of different densities under rocking base motions. It should be noted that rocking base motion can occur in a ground-supported tank or in an elevated tank under earthquake motions. Owing to the flexibility of either the supporting soil or the supporting tower, the tank base will experience a rocking component of motion, even for a purely translational free-field motion. The objectives of this paper are (1) to present the exact solution for the dynamic response of tanks containing two liquids under rocking base motions, (2) to elucidate the interrelationship of the responses of a tank to rocking and lateral base motions and (3) to provide a foundation for the study of the soil-structure interaction analysis of tanks containing two liquids. The response functions examined are the hydrodynamic pressure, the sloshing motion and associated natural frequencies and the base shear and moments. How to utilize these response functions in tank design can be found in Valetsos (1984). In this paper each of these response functions is expressed as the sum of the so-called impulsive and convective components of the response. This division is necessary because it is essential to the approach used in Veletsos and Yang (1977), Veletsos and Tang (1989) and Tang (1994) for the analysis of flexible tanks. The impulsive component of the response represents the effects of the part of the liquid that moves in unison with the tank, whereas the convective component represents the effects of the part of the liquid that associates with the sloshing motion. In this paper the interrelationship of the responses of a tank containing two liquids under rocking base motion and horizontal base motion is established by examining numerical data of the response functions under the two base motions extensively. It is found, similarly to the case of the responses of tanks containing one liquid, that for tanks containing two liquids some of the response quantities may be evaluated from the existing data
for the same tank-liquid system under lateral excitations. Similarly to the response of tanks containing two liquids under horizontal base motions presented in Tang and Chang (1993b), a tank containing two liquids under a rocking base motion has two natural frequencies associated with each sloshing mode of vibration. These natural frequencies are the same as those for tanks in lateral excitation.
2. System description The tank-liquid system investigated is shown in Fig. 1. It is a ground-supported upright circular cylindrical tank of radius R that is filled with two liquids to a total height H. The lower portion liquid, identified as liquid I, has heavier mass density, p], and the upper portion liquid, identified as Liquid II, has lighter mass density, P2. The heights of liquids I and II are H~ and H2 respectively. The tank wall is assumed to be of uniform thickness and clamped to a rigid base. Both liquids are considered to be incompressible and inviscid. The response of the liquid is assumed to be linear.
L__.J °.°°.°oo°
° °° ~°"
°-'"
Q Q ~ °--~--...~o.~o~°°o.~
f
.....
Z~
/
•
....... "~
i~b (t)
Fig. 1. System considered.
105
Y. Tang 1 Nuclear Engineering and Design 152 (1994) 103- I15
Let r, 19and z, denote the radial, circumferential and vertical axial coordinates of a point in liquid I and let r, 8 and z2 be the corresponding coordinates for a point in liquid II as shown in Fig. 1. The origins of the two coordinate systems are on the central axis of the cylindrical tank. The base motion experienced by the tank is an angular acceleration, denoted by &(t), acting in the direction along the 0 = 90” coordinate axis. The temporal variation in &(t) can be arbitrary.
The boundary conditions for liquid II are as follows. (a) The radial acceleration along the tank wall is given by
3. Governing equations and boundary conditions
where g is the gravitational acceleration. The boundary conditions at the interface of the two liquids are as follows. (a) Continuity of vertical accelerations, i.e.
Given the conditions that the liquids are incompressible and inviscid, the hydrodynamic pressures induced at liquids I and II, denoted by p, and p2 respectively, must satisfy the Laplace equations v’p,
= 0
(la)
in the region 0 < r < R, 0 < 13G 21r, 0 d z, < H,, and V’p, = 0
(lb)
in the region O
__-Lapl rl-
for points in
and
P2 an
for points in liquid II. The boundary conditions for liquid I are as follows. (a) The vertical acceleration of liquid I at the tank base must equal the acceleration of the baseplate, i.e.
ar
= - (z2 +
H, ) cos 0 &b(t)
(44
iR
(b) At the free surface the linearized boundary condition is
(44
--- 1 ap, PI aZl I,=H,
1 aP2 p2az2r2=0
(44
(b) Kinematic and pressure conditions. (bl) Kinematic condition. If q(r, 8, t) represents the height of a small disturbance at the interface above the still interface level, then q(r, 8, t) is related to p, by
1 ap,
at2= - ZaZ,
r,=H,
(b2) Pressure condition. If the gravitational effect is considered for the interface motion, there is a discontinuity of the hydrodynamic pressure to the extent of (p, - p2)gq at the interface, i.e. -P21q=o
= (P, -
Pzkrl
(6)
Eliminating q between Eqs. (5) and (6) and making use of Eq. (4e), one obtains the following equation for the interface boundary conditions in addition to Eq. (4e) :
Also,
--- 1 ap, pI
P2
Pllr,=H,
1 3P2
a,=---
--- 1 ap2
az,
=r
cos 8
l&(t)
p,
r,=O
and p2 are finite at r = 0
(Q)
(b) The radial acceleration of liquid I adjacent to the tank wall must equal the acceleration of the tank wall, i.e.
The solutions for p, and p2 are expressed as the sum of the impulsive component and convective component, i.e.
--- 1 33
p, =py
P,
ar
= r=R
-z,
cos 8
&(t)
(4b)
+py
Pz=pi?‘+PF
(7) U-9
Y. Tang / Nuclear Engineering and Design 152 0994) 103-115
106
where the superscript i represents the impulsive component, the superscript c represents the convective component and the superscript r is used as a reminder for rocking. The detailed derivations of these solutions are given in the Appendix. The solutions for these functions are summarized as
p~r(r, 0, zl, t) = f i r ( r , Zl)XT(t)plR COS 0
(9)
pier(r, O, z2, t) = Cm(r, z2)XT(t)pER cos 0
(10)
plcr (r,O, Zl,t) ~- ( ~ , ~,, C.k(r, Ir Z l ) A ~ ( t ) ) pl R COS0 \n=l k=l (11)
where the superscripts I and II denote liquids I and II respectively. The expressions for the dimensionless functions C~r(r,zl), C~Ir(r,z2), Ir C,a.(r, zO and ['~lIr[.. .~,u,~r, zz) are given in the Appendix; 2 T ( t ) = HOb(t) is the horizontal acceleration of the tank wall at the level of the still liquid surface and the functions A~,k(t), k = 1 and 2, are the pseudoacceleration functions for the n th sloshing mode of vibration, J~(2. r/R), and are defined by
2,g
(16a) (16b)
c = ( 1 - a) tanh fil. tanh f12.
(16c)
in which ill, =2,H1/R,
fiE, =2,H2/R and a =
PE/Pl. It can be shown that the discriminant of Eq. (15) D = b 2 - 4 a c > 0 for a > 0 ; therefore Eq. (15) has two real and unequal roots. Explicitly, these two roots are given by
b Anl --
-~-D 1/2
2a
(17)
2a
(18)
Obviously A,~ >An2 ; hence co,~ > co~2. The reasons for this reverse order numbering for the natural frequencies were explained in Tang and Chang (1993b). Note that Eq. (15) is identical with the characteristic equation presented in Tang and Chang (1993b) for tanks subjected to a lateral base motion, i.e. the natural frequencies for the sloshing motion in a tank undergoing rocking motion are the same as those of the sloshing motion in an identical tank subjected to a lateral excitation. The numerical results for values of co,a, are available in Tang and Chang (1993b).
(13)
in which co,a,, k = 1 and 2, are the natural frequencies associated with the n th sloshing mode of vibration. Note that for the nth sloshing mode of vibration there are two natural frequencies, co.~ and co,2. This phenomenon has been discussed in Tang and Chang (1993b). Introducing the notation A,a, for a non-dimensional coefficient that is related to co,a, by the equation
co2kR
fl2n
b = tan ill, + tanh 1/2,
A,2
(12)
A.k -
a = 1 + ~ tanh fl~, tanh
b - D 1/2
p~r(r, O, z2, t) -
A~(t) = co~k fot 2T(t) sin[co,k(t -- Z)] dz
where
(14)
4. Presentation of results The dynamic response of tank containing two liquids under rocking base excitations is controlled by three parameters, H/R, H2/H~ and a, which are the same control parameters as those for the dynamic response of a tank containing two liquids under lateral base excitations.
4. I. Impulsive component of hydrodynamic pressure
it is shown in the Appendix that A,~, k = 1 and 2, are the roots of the characteristic equation given by
The impulsive pressure exerted on the tank wall is conveniently expressed in the form
aA~ -- bA, + c = 0
pit(O, z, t) = C~o(Z)gT(t)pl R COS 0
(15)
(19)
Y. Tang / Nuclear Engineering and Design 152 (1994) 103-115
CrO(Z)is given by
where
c~o(z) = C~or(r,z,)l,=R
for 0~
C~o(Z)=~CIJ~(r,z2)[==n
for
(20a)
H1<~z<<.H (20b)
The coordinate z used in Eq. (19) is related to z I and z2 by the equations z = zl
for 0 ~
z = z2 + H1
(21a)
for Hi ~
(21b)
It is clearly shown in Eq. (19) that the timewise variation in the impulsive pressure is the same as that of the base excitation. This indicates that the impulsive pressure is produced by a portion of liquid that moves in unison with the tank wall. The distributions of C[(z) for ~ =0.25, 0.5, 0.75 and 1 for a broad tank, H/R = 0.5, and a tall tank, H/R = 3, are shown in Fig. 2 for H2/H~ --0.5 and in Fig. 3 for H2/H~ = 2. Note that for o
a broad tank the impulsive pressure decreases monotonically from bottom to top, whereas for a tall tank the shape of the impulsive pressure distribution has a double curvature. The maximum value occurs at a point away from the bottom. These trends are in agreement with those presented in Veletsos and Tang (1987) for tanks containing one liquid undergoing rocking motion. The impulsive pressure decreases as the value of decreases. This reduction is more pronounced in all tanks and for larger values of Hz/H~.
4.2. Impulsive component of base shear The impulsive component of base shear, Qir(t), is given by
Z
o
r=R
+
0.75 0.5
.
.
.
i
.
.
.
.
i
0.5
Qir(t) =
c;
1.0
I
0.0
co'(z)
0.5
1.0
Co'(Z)
Fig. 2. Impulsive pressure exerted on tank wall for H2/H I --0.5. C3
Ca --
o t = l
0.75
a = l
.75
] ~10.5
I \ L k --°'5
_H= 3 . O.O
0.5
Co'(Z)
. . 1~.0
.
. 0.0
i 0.5
. . . .
r=n
R cos 0 dz2 dO
(22)
Substituting Eqs. (9) and (10) into Eq. (22) and performing the integration, one obtains the expression for Qir(t) as
o .
0.0
R COS 0 d z 1 dO
Qir(t) =
o
0.75 0.5 z ,~ 0.25
107
i t.O
co'(z)
Fig. 3. Impulsive pressure exerted on tank wall for H2/H I = 2.
SlorM,,.~T(t) + SXoIrM,22T(t)
(23)
in which Mn=plnR2Ht is the total mass of liquid I, 3'/12= p2nR2H2 is the total mass of liquid II and S~r and S[ I~ are dimensionless coefficients dependent on.the values of H/R, H2/HI and ct. In Eq. (23) the first term on the right-hand side is the base shear contributed from liquid I and the second term is that from liquid II. Since Eq. (22) involves only simple integrations of hyperbolic functions, the expressions for S[ r and S[ ~ are not given herein. The expression given by Eq. (23) has a physical meaning: it gives the volume ratio of each liquid that may be considered as added mass to the tank wall. However, to study the effect of two liquids on the total base shear, it is also desirable to have an expression that can be used for comparison with an identical tank that contains one liquid. Therefore Qir(t) is also expressed as
Oil(t) =
r~sMl 5~T(t)
(24)
Y. Tang / Nuclear Engineering and Design 152 (1994) 103-115
108
i n w h i c h M~ = nPl R 2 H is t h e t o t a l l i q u i d m a s s i f t h e t a n k is filled w i t h l i q u i d I a n d r ~ is a d i m e n s i o n l e s s c o e f f i c i e n t r e l a t e d t o S I r a n d S I ~r b y t h e equation
Table 2 Quantities in expressions for impulsive components of response for H2/HI = 2
H/R
sIrnl + ~sIIro2 r~ -
H
Table 1 Quantities in expressions for impulsive components of response for H2/H 1 = 0.5
S~or
S~r
CoMIr
C0Mllr
AC~M
(1)
(2)
(3)
(4)
(5)
0.396 0.380 0.346 0.324 0.312 0.306 0.303
0.200 0.223 0.286 0.358 0.428 0.489 0.540
0.149 0.147 0.142 0.143 0.147 0.153 0.159
0.470 0.526 0.679 0.851 1.020 1.169 1.292
4.561 1.575 0.341 0.130 0.063 0.035 0.021
0.469 0.447 0.398 0.368 0.352 0.342 0.337
0.193 0.212 0.264 0.326 0.389 0.445 0.493
0.187 0.182 0.172 0.171 0.174 0.179 0.184
0.455 0.500 0.628 0.778 0.930 1.068 1.185
5.037 1.725 0.366 0.137 0.065 0.036 0.021
0.539 0.507 0.441 0.403 0.381 0.369 0.360
0.188 0.202 0.246 0.302 0.360 0.413 0.459
0.223 0.214 0.197 0.192 0.194 0.198 0.202
0.442 0.478 0.587 0.722 0.863 0.994 1.108
5.508 1.864 0.387 0.143 0.067 0.036 0.022
0.605 0.562 0.476 0.430 0.404 0.389 0.378
0.183 0.194 0.232 0.283 0.337 0.388 0.434
0.257 0.243 0.217 0.209 0.209 0.212 0.215
0.431 0.459 0.553 0.678 0.811 0.937 1.049
5.945 1.994 0.404 0.147 0.069 0.037 0.022
= 0.25
=0.5 0.3 0.5 1.0 1.5 2.0 2.5 3.0 = 0.75 0.3 0.5 1.0 1.5 2.0 2.5 3.0 ~t = 1.0 0.3 0.5 1.0 1.5 2.0 2.5 3.0
Cov,'T
CItG
AC~oM
(2)
(3)
(4)
(5)
0.318 0.308 0.281 0.254 0.231 0.212 0.196
0.348 0.358 0.386 0.416 0.445 0.472 0.492
0.134 0.130 0.120 0.111 0.103 0.097 0.093
0.296 0.309 0.342 0.278 0.413 0.445 0.474
12.347 4.356 1.004 0.401 0.200 0.114 0.071
0.466 0.442 0.383 0.334 0.295 0.264 0.240
0.340 0.344 0.361 0.388 0.419 0.448 0.474
0.208 0.198 o. 174 0.155 0.139 0.128 0.120
0.289 0.297 0.322 0.356 0.393 0.427 0.458
16.280 5.654 1.247 0.479 0.232 0.129 0.078
0.606 0.564 0.467 0.295 0.342 0.302 0.272
0.333 0.332 0.341 0.366 0.399 0.431 0.459
0.279 0.260 0.218 0.188 0.166 0.151 o. 139
0.284 0.287 0.306 0.339 0.377 0.414 0.447
20.088 6.857 1.448 0.540 0.256 0.140 0.084
0.741 0.676 0.538 0.444 0.379 0.331 0.296
0.326 0.321 0.324 0.349 0.383 0.417 0.447
0.347 0.317 0.255 0.214 0.187 0.168 o. 154
0.278 0.278 0.292 0.325 0.365 0.405 0.439
23.778 7.974 1.617 0.588 0.275 0.148 0.088
= 0.25
T h e v a l u e s o f S Ir a n d S II~ f o r v a r i o u s c o n t r o l
0.3 0.5 1,0 1.5 2.0 2.5 3.0
S n"
(1) (25)
parameters H / R , Hz/HI and • are listed in c o l u m n s ( I ) a n d ( 2 ) o f T a b l e s 1 a n d 2, f r o m
H/R
Stor
0.3 0.5 1.0 1.5 2.0 2.5 3.0 =0.5 0.3 0.5 1.0 1.5 2.0 2.5 3.0 = 0.75 0.3 0.5 1.0 1.5 2.0 2.5 3.0 = 1.0 0.3 0.5 1.0 1.5 2.0 2.5 3.0
w h i c h o n e c a n see t h a t f o r t h e s a m e v a l u e s o f H / R a n d H2/H~, t h e v a l u e o f S I r i n c r e a s e s a s t h e v a l u e o f ~ i n c r e a s e s , w h e r e a s t h e v a l u e o f SIIr d e c r e a s e s w i t h i n c r e a s i n g v a l u e o f ~. E x a m i n a t i o n o f t h e n u m e r i c a l v a l u e o f r~s reveals that this quantity can be computed from the quantities in the expressions for the base moments of an identical tank-liquid system excited laterally. Specifically, f o r a n i d e n t i c a l s y s t e m e x c i t e d
Y. Tang / Nuclear Engineeringand Design 152 (1994) 103-115
109
laterally by an acceleration ~(t), the base moment M~(t) at a section immediately above the tank base can be expressed as Mi(t) = r OM M~ ns~(t)
(26)
and the base moment induced by the pressure exerted on the tank base, denoted by A M i ( t ) , is given by
0.5
o.o
1 .o
0.0
0.5 a
1.0
0.0 . . . .
0'.5 . . . . Gt
1.0
a E
AMi(t) = ArioMn~ HJ~(t)
(27)
Then it is found that the value of r~ is related to the values of rgM and ArgM by the equation ros
i r _ _--
i roM
q- Ar~M
4.3. Impulsive components of base moments
.
.
J.5 a
.
.
.
.
Fig. 4. Effect of mass densities on impulsive component of base moment. in which r~M is a dimensionless coefficient defined by • r[[M=
(y
nr H 2 OM~-~) + ~ COM ~-
(32)
(30)
R(z2+H1) cos 0 dz z dO
[r= R
and the result is expressed as Mir(t) = C~MS~T(t)M,,H, + CIo'~lXT(t)Ml2n2
.
AMir(t) = J0 J0 p ' =, =0 r2 COS0 dr dO
Jo P ' r = R R z l c ° s 0 d z ld0
pigr
0.0 .
(29)
f2. f,,. j
,,10
q
The value of r ~ is presented graphically in Fig. 4 for various values of H2/HI as a function of 0t. Four different values of H / R are considered, namely 0.5, 1, 2 and 3. From Fig. 4 it is noted that the value of r~[M decreases with decreasing value of • and that the decrease is more rapid for large than for small values of H:/HI. The base moment induced by the impulsive pressure exerted on the tank base is denoted by AMir(t), which is given by
The base moment Mir(t) at a section immediately above the tank base is computed from
+
_E
(28)
The relation defined by Eq. (28) may be explained by a generalization of Betti's principle stated in Veletsos and Tang (1987). In that paper the same relation as that defined by Eq. (28) is found for a tank containing one liquid under rocking motion. Numerical values for rgM and Ar[M have been given by Tang (1993a) and Tang and Chang (1992); therefore the value of r~ may be computed by direct application of Eq. (28).
Mir(t)=Jo
1
~2n
~R
ir[
(33)
Again, since Eq. (29) involves only simple integrations of hyperbolic functions, the expressions for lr 'llr COM and ¢~0M are not given herein. The numerical values of COM ~r and '-'OM / ' I I r are listed in columns (3) and (4) of Tables 1 and 2. For the same values of H / R and H2/H~, the value of COM~ increases as the value of ~ increases, but the value of "-OMCm ' shows the opposite trend. Eq. (30) can also be rewritten as
in which AC~M is a dimensionless coefficient. The value of AC[~M is tabulated in column (5) of Tables 1 and 2. It is seen that the value of AC~M decreases rapidly as the value of H/R increases. For the reason identified earlier, Eq. (34) can also be expressed as
Mir(t) = r~MM~ H2T(t)
AMir( t) = Ar~MSCT(t)M I H
(31)
and may be expressed in the form AMir(t) = AC[~M5~T(t)MH H1
(34)
(35)
Y. Tang/ NuclearEngineeringand Design152 (1994)103-115
110
.,.r
H
.£.
-R o.o
I-I=1 R
=0.5
0.5
t.o
[%o . . . .
d)r.5. . . . ct
I.O
E
H~"I I = 0.5
that the system is experiencing, the heightwise distribution of the convective hydrodynamic wall pressure induced by the rocking motion will be identical with that induced by the lateral motion. If the heightwise distribution of the convective component of the hydrodynamic pressure under lateral excitation is given by the function C~k(Z), then, following from the above argument, the convective pressure p¢r(z, O,t) under rocking can be expressed as
HI/H~ = 0.5
/
'
pCr(g'O't)=( ~'n=lk=l~~)nkfnk(g)ark(t)) RIRcOSO (37) _H=a
H --=2
R
R
o.o . . . .
o1.5 . . . . et
I .o
o.o
0.5 GI
1.o
Fig. 5. Effect of mass densities on impulsive component of base moment.
to compare the result with that corresponding to = 1. The coefficient Ar~M is related to AC~M by the equation Ar~rM
= AC~M(~) 2
in which 7~ is a dimensionless proportionality factor that remains to be determined.
4.5. Convectivecomponent of base shear It is shown in Tang and Chang (1993b) that the convective base shear QC(t) for a tank-liquid system excited laterally is given by
(36)
The values of Ar~M are shown in Fig. 5. It can be seen that for a tall tank this moment is much smaller than for a broad tank. Note that in Fig. 5 the vertical scale for H/R = 1 is one-fifth of that for H/R = 0.5.
4.4. Convectivecomponent of hydrodynamic pressure The convective pressure exerted on the tank wall may be obtained by evaluating the pressures defined by Eqs. (11) and (12) at r=R. This approach is similar to that used in obtaining the impulsive pressure exerted on the tank wall presented above. However, in this paper a physically motivated approach is used to derive the convective pressure instead and the results obtained by the two approaches will be checked against each other. Since the natural modes of vibration and the natural frequencies of the sloshing motion in a tank-system are independent of the excitation
+
S~A.k(t) M,2
(38)
lk=l
or it may be expressed differently as Q¢(t)=(~,=l~=,~
rSkA"k(t))M~
(39,
in which A~(t) is the instantaneous pseudoacceleration induced by a base acceleration J~(t) in an SDF system of circular natural frequency o~.k. A,k(t) is obtained from Eq. (13) by replacing 5/T(t) by J/(t). By application of the same analogy used in obtaining the convective pressure, Eq. (37), for the system in rocking motion from that for the system in lateral motion, one obtains the convective base shear for the system in rocking motion from Eqs. (38) as M I r, 1 1 L,,a,S~A,k(t))
Q=(t) = ( ~ n=l
k=l
r +(~, ~,?,gS,kAnk(t)) n=l
k=l
(40)
Y. Tang / NuclearEngineeringand Design 152 (1994) 103-115
the convective components of the base moments for the system in rocking motion as
or from Eq. (39) one obtains
Q~( t) =
\n=l k=l
y,u,r,k A ~,( t)
(41)
Note that Eqs. (40) and (41) can also be obtained from Eq. (22) by replacing the function pi~(t) by the function p=(t) of Eq. (37) and performing the integrations.
4.6. Convective components of base moments It is shown in Tang and Chang the convective base moment M~(t) immediately above the tank base liquid system excited laterally can as Me(t) =
\n=l k=l
+
(1993b) that at a section for a tankbe expressed
)
C~IIA,k(t) M~2H2
AC~A,a,(t) )M,1 H,
~'~ ~,a,C,a, MIA,a,(t) r )roll Hi
n=lk=l
+
lk=l
r,a,C,~IIA~(t) MIEHz
)
AMCr(t)=( ~n=lk=l~"~"kACnkA~k(t) M r MI1HI
(46)
(47)
M~r(t) =
n=lk=l
~kr,,kA~(t) M1H
AMCr(t)=(n=~l~=l~nkAr,~A~(t))M~H
(48) (49)
4. 7. Surface and interface sloshing displacements (42)
and the base moment induced by the pressure exerted on the tank base, denoted by AM¢(t), is given by
AM~(t) = (,=~ ~
MC~(t) = ( ~
or from Eqs. (44) and (45) one obtains
C,~IA~(t)MnHl
k=l
111
(43)
Eqs. (42) and (43) can be expressed differently as Me(t) : (,=,~' k=l ~ r,~A,,k(t))M~H
(44)
AMe(t)=(,=~I~__IAr~A,a,(t))M~H
(45)
Eqs. (42) and (43) are useful for providing information about the contribution of each liquid and Eqs. (44) and (45) are useful for comparing with the results of an identical tank filled with one liquid to assess the effect of two-liquid interaction. Applying the same analogy as that used in obtaining the convective components of the hydrodynamic pressure and base shear for the system in rocking motion from the corresponding results for the system excited laterally, one obtains from Eqs. (42) and (43) the expressions of
The surface sloshing wave height dr(r, O, t) of an arbitrary point on the liquid surface may be determined from
p~rlz:=n 2 = p2gd(r, O, t)
(50)
and the sloshing wave height at the inteface of the two liquids, r/r(r, 0, t), may be determined from
1
q(r, 0, t) --(Pl-pu)g
(p~rlz'=n* --p~rlz2=°)
(51)
Substituting p~r and p~r in Eqs. (50) and (51) by Eqs. (11) and (12), one may obtain the expressions for d(r, 0, t) and r/(r, 0, t). The expressions for the maximum values of these two functions are obtained by evaluating d(r, 0, t) and ~/(r, 0, t) at r = R and 0 = 0. Alternatively, the expressions for these maximum values may be obtained as follows. If the expressions for computing these maximum values for the twoliquid-tank system excited laterally are given by
Z
d(R, o, t) = Z E
A~(t)
R
(52)
~/(R, 0, t) = ~ ~ rl.kA.~(t) R n=l k=l g
(53)
n=lk=l
g
112
Y. Tang / Nuclear Engineeringand Design 152 (1994) 103-115
their counterparts for the same system in rocking motion may be expressed as 2 zr nk(t) dr(R, O, t) = ~ ~ ?..d.. R (54) n=~ k=~ g
?jr(R, 0, t) = ~ ~ 7..~/.. Ark(t) R nffilk=l
(55)
g
Once the proportionality factor 7.. is known, the maximum values of the surface and interface sloshing wave heights may be computed from the corresponding solutions for the liquid-tank system excited laterally by making use of Eqs. (54) and (55). Now the generalized Betti principle stated in Veletsos and Tang (1993b) is invoked herein to determine the proportionality factor ?"*. By noting that for a tank subjected to lateral and rocking base motions of the same timewise variations and for each mode of vibration of the sloshing motion in the tank the work done by the base shear for the tank in rocking motion through the displacement of the laterally excited tank is equal to the work done by the foundation moment for the laterally excited tank acting through the rotation of the tank in rocking motion, from Eqs. (41), (44) and (45) one obtains ?"*rS"*
=
r,~, + Arm
r ~ + Ar~ s Ink
5. Conclusions
The complete solutions for tanks containing two liquids under rocking excitation have been presented. Each dynamic response quantity is expressed as the sum of the impulsive and convective components so that the solutions presented may provide a rational basis for evaluating the effects of tank flexibility and soil-structure interaction. It is found that many response quantities for a tank in rocking motion may be evaluated from the corresponding response quantities for an identical tank excited laterally; in particular, all the convective components of the response quantities can be evaluated in such a way.
(56)
Therefore ~nk - -
results obtained by the two approaches confirm the accuracy. Extensive numerical data for a laterally excited tank containing two liquids are available in Tang and Chang (1993b). Also, because all the convective components of the response quantities for a tank in rocking motion can be computed from their counterparts for the laterally excited tank, no numerical results for the convective components of the response are presented herein.
Acknowledgment
(57)
With ~"* determined, all the convective components of the response quantities for the tank in rocking motion can be calculated. Note that the determination of ~"* from Eq. (57) requires only the results for the laterally excited tank; therefore there is no need to solve the governing equations and the boundary conditions to obtain the convective components of the response for the tank in rocking motion. However, in this paper, for the purpose of confirming the accuracy, the convective components of the response quantities are also computed from the convective pressures defined by Eqs. (11) and (12) which are obtained by directly solving the governing equations and the boundary conditions. The identical numerical
This work was performed in the Engineering Mechanics Program of the Reactor Engineering Division of Argonne National Laboratory under the auspices of the US Department of Energy, Contract W-31-109-Eng-38.
Appendix. Solution for rigid tanks
A.I. Impulsive component of solution
The impulsive component of the hydrodynamic pressures p~r and piEr must satisfy Vp~r = 0
(Ala)
Vpi2r = 0
(hlb)
113
Y. Tang/ NuclearEngineeringand Design152 (1994)103-115
in which cIr(r, zl) and c~Xr(r,z2) are dimensionless coefficients given by
and the boundary conditions
1 ~pi(
= rOb(t) COS0
Pl Ozl Izl=o 1 ap~r
(A2a)
I
I
Pl Or ~=R
= --Z I Ob(t) cos 0
r Zl R°~G~ I cIr(r'zl)='Rn-]-nn~=l Z A.
(A2b) ,
f. 2"1'~q Jl(2.r/R)
+ B. slntz. ~ ) j 1 Opizr • = R = - (z2 + H1 )0b(t) cos 0 P2 ~r
(A2d)
10p~ r
=
__
P2
ir
(AZf)
p~r and p~ are finite at r = 0
Jlt/'n) J
where 2 A~ = ~ (sinh fil. cosh fiE" + ct cosh fiE" sinh fiE") + (~ - 1)//lr cosh fiE" _ H
(A3) x sinh ill. cosh fiE"
n=l
}
..
os0 (A4)
In Eqs. (A3) and (A4), ~T(0 = ttO'b(t), Jl is the Bessel function of the first kind of order one, ~. is the nth zero of J~ (2), the first derivative of J1, and A., B., C~ and D. are constants of integration that may be determined from the boundary conditions defined by Eqs. (A2a), (A2d), (A2e) and (A2f). After evaluating these constants of integration and substituting them back into Eqs. (A3) and (A4), the results are cast into the forms
p~r(r, O, Zl, t) = Clor(r,zl)p,R~T(t ) COS0 pigr(r, 0, z2, t) = CIol~(r,Zz)p2RS~T(t) COS0
(A9c)
2 cosh fi2. q- (o~ - I) -~HI sinh fil.cosh fiz~ 2.
D. -
R ~ [ C~ cosh ( 2. 7,2) ~r
1
(A9a)
2 H H1 C. = ~ sinh fiE" - ~ cosh fil. + (1 - a) -~-
and similarly, on satisfying Eqs. (Alb), (A2c) and (A2g), piEr takes the form
Z2 Jl (2.r/R)
A8)
(A9b)
+ B~sinh()t.~)]Jl(.2-~.r,[----ff)}plRSiT(t)cosO
•
cosh
B~ = -- ~ (cosh fii. cosh fiE"+0t sinh filn sinh fiE.)
{~ Z1H+ HR.~, ...°c [ An cosh ( Z2.R )
+
R
(A2g)
The method of separation of variables is employed to solve Eqs. (Ala) and (Alb) and the integration constants are determined from the boundary conditions• On satisfying Eqs. (Ala), (A2b) and (A2g), the function p~r takes the form
pi2r={RZ2"Jf-Hl ---H---
(A7)
. L zA1Jl(&dR) +O. smt . ;/
~g2 Iz2=0
=-, --P2 I,z=0
p~=
cllr (r, Z2)
(AEe)
1 ~pi~
0zl ~ =HI
~
(A2c)
ir =n2 - 0 P21~2 Pl
(el)
cosh 2 . ~
-
H ~ a sinh ill.
2
G. = 2 2.-1
(A9d) (A9e)
A. = cosh fil. cosh fi2n -4- 0~sinh filn sinh fi2n (A9f)
A.2. Convective component of solution The convective component of the hydrodynamic pressures p]r and p~r must satisfy Laplace's equations
(A5)
V2p~r = 0
(A10a)
(A6)
VEp~r = 0
(A10b)
Y. Tang / Nuclear Engineering and Design 152 (1994) 103-115
114
along with the boundary conditions ~p~r
=0
and (A12b) into Eq. (A1 ld), one obtains a differential equation (A1 la) sinh flln sinh fl2nJ~n +" g2n ~ sinh flln cosh flz.E.
~ZI zl=0 1 0p~ r
=0
(A1 lb)
Pl 0r r=R
g2. + cosh f12,,i~. + ~ sinh fl2,,F~
1 0p~ r
Pz ~r .=R = 0
_
(Allc)
(~2cr Op~,[ o-pF + _Op~r~ _ Ot2 g Oz2/I~:=H2 = --g~-2-z2 ~2=H2 1 0 p ~ r z,=/~, = Pl ~Zl
1 ~p~ ~2 P2 822 =0
(A1 ld) (Alle)
g2. R G. (A.
R H A. \ - ~ + C. sinh flz.
+ D. cosh flZn )XT(/)
and substituting Eqs. (A6) and (A12b) into Eq. (A1 lf), one obtains another differential equation g2n cosh fll~/~. + ( 1 - a) ~ sinh fll,,E. - aP,,
(OP~r ~P~r l OP~r~ Zl "~i- + g ~zl + g "~zl ) =~I,
_ ap~~-t-' g "-~z2 ~pivr'~ = {ap~" ~-~T + g -~z2 ) 2= 0 p~r and p~r are finite at r = 0
p~r(r, O, zl, t)=[n~=lDn(t)cosh(J,n~-~) J'(2nr/R) _]
(A12a)
and p~r takes the form
P~r(r,O, zz, t)={.=~[E.(t)cosh(2. R ) + Dn(t) sinh Bin sinh 2~ ~ ~
~
; P 2 " cos0
Cnl(r,z,)= (A12b)
in which E,(t) and D~(t) are integration functions that can be determined by satisfying Eqs. (A1 ld) and (A1 If). Substituting Eqs. (A5), (A6), (A12a)
(AI4)
Then Eqs. (A13) and (A14) can be solved for E.(t) and F.(t). Taking the Laplace transformation on both sides of Eqs. (A13) and (A14) and assuming homogeneous initial conditions for E.(t) and F.(t), one obtains two algebraic equations for determination of the Laplace transforms of E.(t) and F.(t). The required solutions for E.(t) and F.(t) are then obtained by finding the inverse Laplace transforms, Then, replacing E.(t) and F.(t) in Eqs. (A12a) and (A12b) with the results obtained, one can cast the final expressions into the form presented in Eqs. (11) and (12). The derivation is not difficult but is tedious; hence it is omitted herein. However, the details of the derivation are available in Tang and Chang (1993b). The expressions for C,e.(r, ~r zl) and i'~IIr/ "-~.k~r, z2) for k = 1 and 2 are given as lr
J~(2,r[R)]
A,,
R HA. \(1-a)~-]+A~sinhfll~
+ B. cosh ill. - aD. )£T(t)
(Allg)
× p~R cos 0
g2,,RG. (
(Allf)
Again the method of separation of variables is employed to solve Eqs. (A10a) and (A10b) and the integration constants are determined from the boundary conditions. On satisfying Eqs. (A10a), (A10b), (Alla), (Allb), (Allc), (Alle) and (Allg), the function p ~ takes the form
x
(A13)
( zl']Jl(2.r/R ) d.A.~X, cosh 2.-~/
G~ 1
(A15a)
,r C.2(r, zl) =
G. 1 (z,)J,(2.r/R) A.A.~2 yl cosh 2. ~ Jl(2.) (A15b)
115
Y. Tang /Nuclear Engineering and Design 152 (1994) 103-115
.l ~', z2) =
An A.l x2 cosh 2. + xl sinh ill. sin 2.
Jl().n) (A16a)
An
n2 ~,J', Z 2 ) ~-
2
Y2 c o s h
(Al6b) where X 1 --
An1 ql - q2 A,1 -- An2
(A17a)
Yl-
q2 -- A n 2 q l A . I - - An2
(A17b)
Am q3 -- q4
(A17c)
A n l -- An2
and q4 -- A n 2 q 3 Y2 --
Am
(A17d)
_ An 2
in which ql =
sl cosh f12~ + xs2 An Sl sinh fiE.
q2 --
s2 cosh ill. - sl sinh fl]. sinh f12. q3 = q 4 ~-
(A18a) (A18b)
An
An
(A18c)
s2(1 - ~) sinh ill. - Sl sinh ill. cosh fiE. An (A18d)
with An Sl = ( 1 - a) ~ + A. sinh ill. + Bn cosh flln - aDn (A19a) A. s2 = ~ + C. sinh f12. + D. cosh 132.
g2;
2J-sin i l l . sinh flEn cos
+ y] sinh fl,. sin( 2"zER/j~]J'(2"r/R)J, (2.)
X2 --
The characteristic equation for the natural frequencies can be obtained from the associated eigenvalue problem of Eqs. (A13) and (A14). Letting 5~x(t) = O, E.( t) = exp(kot) and F.(t) = F. exp(kot) in Eqs. (A11) and (A12), one obtains two homogeneous equations which are expressed by
(A19b)
cosh t l. [s inh ill. cosh f12. sin 0 x L ( 1 - a) sinh ill.
AtFo + A[F.)=0
(A20)
It is easy to show that the determinant of Eq. (A20) is Eq. (15), the characteristic equation.
References Y. Tang, Dynamic response of tank containing two liquids, J. Eng. Mech. Div., ASCE 119(3) (1993a) 531-548. Y. Tang and Free vibration analysis of a tank containing two liquids, J. Eng. Mech. Div., ASCE in press. Y. Tang and Y.W. Chang, Dynamic response of a tank containing two liquids, Argonne National Laboratory Rep. ANL/RE 92/1, 1992. Y. Tang and Y.W. Chang, The exact solutions to the dynamic response of tanks containing two liquids, Argonne National Laboratory Rep. ANL/RE 93/2, 1993b. A.S. Veletsos, Seismic response and design of liquid storage tanks, in Guidelines for the Seismic Design of Oil and Gas Pipeline Systems, ASCE, New York, 1984, pp. 255-370 and 443-461. A.S. Veletsos and P. Shivakumar, Sloshing response of layered liquids in rigid tanks, J. Earth. Eng. Struct. Dyn. 22 (1983) 801-821. A.S. Veletsos and Y. Tang, Dynamics of vertically exicted liquid storage tanks, J. Struct. Div., ASCE 112(10) (1986) 1228-1246. A.S. Veletsos and Y. Tang, Rocking response of liquid storage tanks, J. Eng. Mech. Div., ASCE 113(11) (1987) 17741792.
A.S. Veletsos and Y. Tang, Soil-structure interaction effects for laterally excited liquid storage tanks, J. Earth. Eng. Struct. Dyn. 19 (1989)473-496. A.S. Veletsos and J.Y. Yang, Earthquake response of liquid storage tanks, Advances in Civil Engineering Through Engineering Mechanics, Proc. ASCE/EMD Spec. Conf., Raleigh, NC, 1977, pp. 1-24.
ELSEVIER
Nuclear Enoineed.ng and Design
Rocking response of tanks containing two liquids Yu Tang Reactor Engineering Division, Argonne National Laboratory, Argonne, IL 60439, USA
(Received 8 March 1994)
Abstract
A study of the dynamic response of upright circular cylindrical liquid storage tanks containing two different liquids under a rocking base motion with an arbitrary temporal variation is presented. Only rigid tanks were studied. The response quantities examined include the hydrodynamic pressure, the sloshing wave height and associated frequencies, and the base shear and moments. Each of these response quantities is expressed as the sum of the so-called impulsive component and convective component. Unlike the case of tanks containing one liquid, in which the response is controlled by one parameter, the height-to-radius ratio, the response of tanks containing two different liquids is controlled by three parameters: the height-to-radius ratio and the mass density ratio and height ratio of the two liquids. The interrelationship of the responses of the tank-liquid system to rocking and lateral base excitations is established by examining numerical results extensively. It is found that some of the response quantities for a tank-liquid system under a rocking base motion can be determined from the corresponding response quantities for an identical tank under a horizontal base motion.
1. Introduction
Liquid storage tanks are important components of industrial facilities and, when located in earthquake-prone regions, should be designed to withstand the earthquakes to which they may be subjected. The dynamic response of liquid storage tanks subjected to earthquakes has been a subject of numerous studies in the past 30 years. For reviews o f the previous studies on responses of laterally, vertically and rocking-excited tank-liquid systems the reader is referred to the papers of Veletsos (1984), Veletsos and Tang (1986) and Veletsos and Tang (1987) respectively. Most of the previous studies were focused on the tank Elsevier ScienceS.A. SSDI 0029-5493(94)00699-Y
containing only one liquid. However, there are cases in which the density of the tank content is not uniform. For such cases the dynamic responses of tanks containing liquids with different densities must be studied. To respond to this need, Tang (1993a) presented a solution for the dynamic response of rigid tanks containing two liquids under a horizontal base motion. In that study, however, the effect of gravitation on the interface motion of the two liquids was neglected. It should be mentioned that this gravitational effect was later considered in the analysis and the findings are described in detail in an A N L / R E report (Tang, 1993b). The dynamic characteristics of flexible tanks containing two liquids were stud-
104
Y. Tang /Nuclear Engineering and Design 152 (1994) 103-115
ied by Tang (1994) and a paper by Veletsos and Shivakumar (1983) dealt with the sloshing response of layered liquids in rigid tanks. All these studies are aimed at the understanding of the dynamic behavior of tanks containing two liquids under lateral base motions. This paper describes the dynamic response of tanks containing two liquids of different densities under rocking base motions. It should be noted that rocking base motion can occur in a ground-supported tank or in an elevated tank under earthquake motions. Owing to the flexibility of either the supporting soil or the supporting tower, the tank base will experience a rocking component of motion, even for a purely translational free-field motion. The objectives of this paper are (1) to present the exact solution for the dynamic response of tanks containing two liquids under rocking base motions, (2) to elucidate the interrelationship of the responses of a tank to rocking and lateral base motions and (3) to provide a foundation for the study of the soil-structure interaction analysis of tanks containing two liquids. The response functions examined are the hydrodynamic pressure, the sloshing motion and associated natural frequencies and the base shear and moments. How to utilize these response functions in tank design can be found in Valetsos (1984). In this paper each of these response functions is expressed as the sum of the so-called impulsive and convective components of the response. This division is necessary because it is essential to the approach used in Veletsos and Yang (1977), Veletsos and Tang (1989) and Tang (1994) for the analysis of flexible tanks. The impulsive component of the response represents the effects of the part of the liquid that moves in unison with the tank, whereas the convective component represents the effects of the part of the liquid that associates with the sloshing motion. In this paper the interrelationship of the responses of a tank containing two liquids under rocking base motion and horizontal base motion is established by examining numerical data of the response functions under the two base motions extensively. It is found, similarly to the case of the responses of tanks containing one liquid, that for tanks containing two liquids some of the response quantities may be evaluated from the existing data
for the same tank-liquid system under lateral excitations. Similarly to the response of tanks containing two liquids under horizontal base motions presented in Tang and Chang (1993b), a tank containing two liquids under a rocking base motion has two natural frequencies associated with each sloshing mode of vibration. These natural frequencies are the same as those for tanks in lateral excitation.
2. System description The tank-liquid system investigated is shown in Fig. 1. It is a ground-supported upright circular cylindrical tank of radius R that is filled with two liquids to a total height H. The lower portion liquid, identified as liquid I, has heavier mass density, p], and the upper portion liquid, identified as Liquid II, has lighter mass density, P2. The heights of liquids I and II are H~ and H2 respectively. The tank wall is assumed to be of uniform thickness and clamped to a rigid base. Both liquids are considered to be incompressible and inviscid. The response of the liquid is assumed to be linear.
L__.J °.°°.°oo°
° °° ~°"
°-'"
Q Q ~ °--~--...~o.~o~°°o.~
f
.....
Z~
/
•
....... "~
i~b (t)
Fig. 1. System considered.
105
Y. Tang 1 Nuclear Engineering and Design 152 (1994) 103- I15
Let r, 19and z, denote the radial, circumferential and vertical axial coordinates of a point in liquid I and let r, 8 and z2 be the corresponding coordinates for a point in liquid II as shown in Fig. 1. The origins of the two coordinate systems are on the central axis of the cylindrical tank. The base motion experienced by the tank is an angular acceleration, denoted by &(t), acting in the direction along the 0 = 90” coordinate axis. The temporal variation in &(t) can be arbitrary.
The boundary conditions for liquid II are as follows. (a) The radial acceleration along the tank wall is given by
3. Governing equations and boundary conditions
where g is the gravitational acceleration. The boundary conditions at the interface of the two liquids are as follows. (a) Continuity of vertical accelerations, i.e.
Given the conditions that the liquids are incompressible and inviscid, the hydrodynamic pressures induced at liquids I and II, denoted by p, and p2 respectively, must satisfy the Laplace equations v’p,
= 0
(la)
in the region 0 < r < R, 0 < 13G 21r, 0 d z, < H,, and V’p, = 0
(lb)
in the region O
__-Lapl rl-
for points in
and
P2 an
for points in liquid II. The boundary conditions for liquid I are as follows. (a) The vertical acceleration of liquid I at the tank base must equal the acceleration of the baseplate, i.e.
ar
= - (z2 +
H, ) cos 0 &b(t)
(44
iR
(b) At the free surface the linearized boundary condition is
(44
--- 1 ap, PI aZl I,=H,
1 aP2 p2az2r2=0
(44
(b) Kinematic and pressure conditions. (bl) Kinematic condition. If q(r, 8, t) represents the height of a small disturbance at the interface above the still interface level, then q(r, 8, t) is related to p, by
1 ap,
at2= - ZaZ,
r,=H,
(b2) Pressure condition. If the gravitational effect is considered for the interface motion, there is a discontinuity of the hydrodynamic pressure to the extent of (p, - p2)gq at the interface, i.e. -P21q=o
= (P, -
Pzkrl
(6)
Eliminating q between Eqs. (5) and (6) and making use of Eq. (4e), one obtains the following equation for the interface boundary conditions in addition to Eq. (4e) :
Also,
--- 1 ap, pI
P2
Pllr,=H,
1 3P2
a,=---
--- 1 ap2
az,
=r
cos 8
l&(t)
p,
r,=O
and p2 are finite at r = 0
(Q)
(b) The radial acceleration of liquid I adjacent to the tank wall must equal the acceleration of the tank wall, i.e.
The solutions for p, and p2 are expressed as the sum of the impulsive component and convective component, i.e.
--- 1 33
p, =py
P,
ar
= r=R
-z,
cos 8
&(t)
(4b)
+py
Pz=pi?‘+PF
(7) U-9
Y. Tang / Nuclear Engineering and Design 152 0994) 103-115
106
where the superscript i represents the impulsive component, the superscript c represents the convective component and the superscript r is used as a reminder for rocking. The detailed derivations of these solutions are given in the Appendix. The solutions for these functions are summarized as
p~r(r, 0, zl, t) = f i r ( r , Zl)XT(t)plR COS 0
(9)
pier(r, O, z2, t) = Cm(r, z2)XT(t)pER cos 0
(10)
plcr (r,O, Zl,t) ~- ( ~ , ~,, C.k(r, Ir Z l ) A ~ ( t ) ) pl R COS0 \n=l k=l (11)
where the superscripts I and II denote liquids I and II respectively. The expressions for the dimensionless functions C~r(r,zl), C~Ir(r,z2), Ir C,a.(r, zO and ['~lIr[.. .~,u,~r, zz) are given in the Appendix; 2 T ( t ) = HOb(t) is the horizontal acceleration of the tank wall at the level of the still liquid surface and the functions A~,k(t), k = 1 and 2, are the pseudoacceleration functions for the n th sloshing mode of vibration, J~(2. r/R), and are defined by
2,g
(16a) (16b)
c = ( 1 - a) tanh fil. tanh f12.
(16c)
in which ill, =2,H1/R,
fiE, =2,H2/R and a =
PE/Pl. It can be shown that the discriminant of Eq. (15) D = b 2 - 4 a c > 0 for a > 0 ; therefore Eq. (15) has two real and unequal roots. Explicitly, these two roots are given by
b Anl --
-~-D 1/2
2a
(17)
2a
(18)
Obviously A,~ >An2 ; hence co,~ > co~2. The reasons for this reverse order numbering for the natural frequencies were explained in Tang and Chang (1993b). Note that Eq. (15) is identical with the characteristic equation presented in Tang and Chang (1993b) for tanks subjected to a lateral base motion, i.e. the natural frequencies for the sloshing motion in a tank undergoing rocking motion are the same as those of the sloshing motion in an identical tank subjected to a lateral excitation. The numerical results for values of co,a, are available in Tang and Chang (1993b).
(13)
in which co,a,, k = 1 and 2, are the natural frequencies associated with the n th sloshing mode of vibration. Note that for the nth sloshing mode of vibration there are two natural frequencies, co.~ and co,2. This phenomenon has been discussed in Tang and Chang (1993b). Introducing the notation A,a, for a non-dimensional coefficient that is related to co,a, by the equation
co2kR
fl2n
b = tan ill, + tanh 1/2,
A,2
(12)
A.k -
a = 1 + ~ tanh fl~, tanh
b - D 1/2
p~r(r, O, z2, t) -
A~(t) = co~k fot 2T(t) sin[co,k(t -- Z)] dz
where
(14)
4. Presentation of results The dynamic response of tank containing two liquids under rocking base excitations is controlled by three parameters, H/R, H2/H~ and a, which are the same control parameters as those for the dynamic response of a tank containing two liquids under lateral base excitations.
4. I. Impulsive component of hydrodynamic pressure
it is shown in the Appendix that A,~, k = 1 and 2, are the roots of the characteristic equation given by
The impulsive pressure exerted on the tank wall is conveniently expressed in the form
aA~ -- bA, + c = 0
pit(O, z, t) = C~o(Z)gT(t)pl R COS 0
(15)
(19)
Y. Tang / Nuclear Engineering and Design 152 (1994) 103-115
CrO(Z)is given by
where
c~o(z) = C~or(r,z,)l,=R
for 0~
C~o(Z)=~CIJ~(r,z2)[==n
for
(20a)
H1<~z<<.H (20b)
The coordinate z used in Eq. (19) is related to z I and z2 by the equations z = zl
for 0 ~
z = z2 + H1
(21a)
for Hi ~
(21b)
It is clearly shown in Eq. (19) that the timewise variation in the impulsive pressure is the same as that of the base excitation. This indicates that the impulsive pressure is produced by a portion of liquid that moves in unison with the tank wall. The distributions of C[(z) for ~ =0.25, 0.5, 0.75 and 1 for a broad tank, H/R = 0.5, and a tall tank, H/R = 3, are shown in Fig. 2 for H2/H~ --0.5 and in Fig. 3 for H2/H~ = 2. Note that for o
a broad tank the impulsive pressure decreases monotonically from bottom to top, whereas for a tall tank the shape of the impulsive pressure distribution has a double curvature. The maximum value occurs at a point away from the bottom. These trends are in agreement with those presented in Veletsos and Tang (1987) for tanks containing one liquid undergoing rocking motion. The impulsive pressure decreases as the value of decreases. This reduction is more pronounced in all tanks and for larger values of Hz/H~.
4.2. Impulsive component of base shear The impulsive component of base shear, Qir(t), is given by
Z
o
r=R
+
0.75 0.5
.
.
.
i
.
.
.
.
i
0.5
Qir(t) =
c;
1.0
I
0.0
co'(z)
0.5
1.0
Co'(Z)
Fig. 2. Impulsive pressure exerted on tank wall for H2/H I --0.5. C3
Ca --
o t = l
0.75
a = l
.75
] ~10.5
I \ L k --°'5
_H= 3 . O.O
0.5
Co'(Z)
. . 1~.0
.
. 0.0
i 0.5
. . . .
r=n
R cos 0 dz2 dO
(22)
Substituting Eqs. (9) and (10) into Eq. (22) and performing the integration, one obtains the expression for Qir(t) as
o .
0.0
R COS 0 d z 1 dO
Qir(t) =
o
0.75 0.5 z ,~ 0.25
107
i t.O
co'(z)
Fig. 3. Impulsive pressure exerted on tank wall for H2/H I = 2.
SlorM,,.~T(t) + SXoIrM,22T(t)
(23)
in which Mn=plnR2Ht is the total mass of liquid I, 3'/12= p2nR2H2 is the total mass of liquid II and S~r and S[ I~ are dimensionless coefficients dependent on.the values of H/R, H2/HI and ct. In Eq. (23) the first term on the right-hand side is the base shear contributed from liquid I and the second term is that from liquid II. Since Eq. (22) involves only simple integrations of hyperbolic functions, the expressions for S[ r and S[ ~ are not given herein. The expression given by Eq. (23) has a physical meaning: it gives the volume ratio of each liquid that may be considered as added mass to the tank wall. However, to study the effect of two liquids on the total base shear, it is also desirable to have an expression that can be used for comparison with an identical tank that contains one liquid. Therefore Qir(t) is also expressed as
Oil(t) =
r~sMl 5~T(t)
(24)
Y. Tang / Nuclear Engineering and Design 152 (1994) 103-115
108
i n w h i c h M~ = nPl R 2 H is t h e t o t a l l i q u i d m a s s i f t h e t a n k is filled w i t h l i q u i d I a n d r ~ is a d i m e n s i o n l e s s c o e f f i c i e n t r e l a t e d t o S I r a n d S I ~r b y t h e equation
Table 2 Quantities in expressions for impulsive components of response for H2/HI = 2
H/R
sIrnl + ~sIIro2 r~ -
H
Table 1 Quantities in expressions for impulsive components of response for H2/H 1 = 0.5
S~or
S~r
CoMIr
C0Mllr
AC~M
(1)
(2)
(3)
(4)
(5)
0.396 0.380 0.346 0.324 0.312 0.306 0.303
0.200 0.223 0.286 0.358 0.428 0.489 0.540
0.149 0.147 0.142 0.143 0.147 0.153 0.159
0.470 0.526 0.679 0.851 1.020 1.169 1.292
4.561 1.575 0.341 0.130 0.063 0.035 0.021
0.469 0.447 0.398 0.368 0.352 0.342 0.337
0.193 0.212 0.264 0.326 0.389 0.445 0.493
0.187 0.182 0.172 0.171 0.174 0.179 0.184
0.455 0.500 0.628 0.778 0.930 1.068 1.185
5.037 1.725 0.366 0.137 0.065 0.036 0.021
0.539 0.507 0.441 0.403 0.381 0.369 0.360
0.188 0.202 0.246 0.302 0.360 0.413 0.459
0.223 0.214 0.197 0.192 0.194 0.198 0.202
0.442 0.478 0.587 0.722 0.863 0.994 1.108
5.508 1.864 0.387 0.143 0.067 0.036 0.022
0.605 0.562 0.476 0.430 0.404 0.389 0.378
0.183 0.194 0.232 0.283 0.337 0.388 0.434
0.257 0.243 0.217 0.209 0.209 0.212 0.215
0.431 0.459 0.553 0.678 0.811 0.937 1.049
5.945 1.994 0.404 0.147 0.069 0.037 0.022
= 0.25
=0.5 0.3 0.5 1.0 1.5 2.0 2.5 3.0 = 0.75 0.3 0.5 1.0 1.5 2.0 2.5 3.0 ~t = 1.0 0.3 0.5 1.0 1.5 2.0 2.5 3.0
Cov,'T
CItG
AC~oM
(2)
(3)
(4)
(5)
0.318 0.308 0.281 0.254 0.231 0.212 0.196
0.348 0.358 0.386 0.416 0.445 0.472 0.492
0.134 0.130 0.120 0.111 0.103 0.097 0.093
0.296 0.309 0.342 0.278 0.413 0.445 0.474
12.347 4.356 1.004 0.401 0.200 0.114 0.071
0.466 0.442 0.383 0.334 0.295 0.264 0.240
0.340 0.344 0.361 0.388 0.419 0.448 0.474
0.208 0.198 o. 174 0.155 0.139 0.128 0.120
0.289 0.297 0.322 0.356 0.393 0.427 0.458
16.280 5.654 1.247 0.479 0.232 0.129 0.078
0.606 0.564 0.467 0.295 0.342 0.302 0.272
0.333 0.332 0.341 0.366 0.399 0.431 0.459
0.279 0.260 0.218 0.188 0.166 0.151 o. 139
0.284 0.287 0.306 0.339 0.377 0.414 0.447
20.088 6.857 1.448 0.540 0.256 0.140 0.084
0.741 0.676 0.538 0.444 0.379 0.331 0.296
0.326 0.321 0.324 0.349 0.383 0.417 0.447
0.347 0.317 0.255 0.214 0.187 0.168 o. 154
0.278 0.278 0.292 0.325 0.365 0.405 0.439
23.778 7.974 1.617 0.588 0.275 0.148 0.088
= 0.25
T h e v a l u e s o f S Ir a n d S II~ f o r v a r i o u s c o n t r o l
0.3 0.5 1,0 1.5 2.0 2.5 3.0
S n"
(1) (25)
parameters H / R , Hz/HI and • are listed in c o l u m n s ( I ) a n d ( 2 ) o f T a b l e s 1 a n d 2, f r o m
H/R
Stor
0.3 0.5 1.0 1.5 2.0 2.5 3.0 =0.5 0.3 0.5 1.0 1.5 2.0 2.5 3.0 = 0.75 0.3 0.5 1.0 1.5 2.0 2.5 3.0 = 1.0 0.3 0.5 1.0 1.5 2.0 2.5 3.0
w h i c h o n e c a n see t h a t f o r t h e s a m e v a l u e s o f H / R a n d H2/H~, t h e v a l u e o f S I r i n c r e a s e s a s t h e v a l u e o f ~ i n c r e a s e s , w h e r e a s t h e v a l u e o f SIIr d e c r e a s e s w i t h i n c r e a s i n g v a l u e o f ~. E x a m i n a t i o n o f t h e n u m e r i c a l v a l u e o f r~s reveals that this quantity can be computed from the quantities in the expressions for the base moments of an identical tank-liquid system excited laterally. Specifically, f o r a n i d e n t i c a l s y s t e m e x c i t e d
Y. Tang / Nuclear Engineeringand Design 152 (1994) 103-115
109
laterally by an acceleration ~(t), the base moment M~(t) at a section immediately above the tank base can be expressed as Mi(t) = r OM M~ ns~(t)
(26)
and the base moment induced by the pressure exerted on the tank base, denoted by A M i ( t ) , is given by
0.5
o.o
1 .o
0.0
0.5 a
1.0
0.0 . . . .
0'.5 . . . . Gt
1.0
a E
AMi(t) = ArioMn~ HJ~(t)
(27)
Then it is found that the value of r~ is related to the values of rgM and ArgM by the equation ros
i r _ _--
i roM
q- Ar~M
4.3. Impulsive components of base moments
.
.
J.5 a
.
.
.
.
Fig. 4. Effect of mass densities on impulsive component of base moment. in which r~M is a dimensionless coefficient defined by • r[[M=
(y
nr H 2 OM~-~) + ~ COM ~-
(32)
(30)
R(z2+H1) cos 0 dz z dO
[r= R
and the result is expressed as Mir(t) = C~MS~T(t)M,,H, + CIo'~lXT(t)Ml2n2
.
AMir(t) = J0 J0 p ' =, =0 r2 COS0 dr dO
Jo P ' r = R R z l c ° s 0 d z ld0
pigr
0.0 .
(29)
f2. f,,. j
,,10
q
The value of r ~ is presented graphically in Fig. 4 for various values of H2/HI as a function of 0t. Four different values of H / R are considered, namely 0.5, 1, 2 and 3. From Fig. 4 it is noted that the value of r~[M decreases with decreasing value of • and that the decrease is more rapid for large than for small values of H:/HI. The base moment induced by the impulsive pressure exerted on the tank base is denoted by AMir(t), which is given by
The base moment Mir(t) at a section immediately above the tank base is computed from
+
_E
(28)
The relation defined by Eq. (28) may be explained by a generalization of Betti's principle stated in Veletsos and Tang (1987). In that paper the same relation as that defined by Eq. (28) is found for a tank containing one liquid under rocking motion. Numerical values for rgM and Ar[M have been given by Tang (1993a) and Tang and Chang (1992); therefore the value of r~ may be computed by direct application of Eq. (28).
Mir(t)=Jo
1
~2n
~R
ir[
(33)
Again, since Eq. (29) involves only simple integrations of hyperbolic functions, the expressions for lr 'llr COM and ¢~0M are not given herein. The numerical values of COM ~r and '-'OM / ' I I r are listed in columns (3) and (4) of Tables 1 and 2. For the same values of H / R and H2/H~, the value of COM~ increases as the value of ~ increases, but the value of "-OMCm ' shows the opposite trend. Eq. (30) can also be rewritten as
in which AC~M is a dimensionless coefficient. The value of AC[~M is tabulated in column (5) of Tables 1 and 2. It is seen that the value of AC~M decreases rapidly as the value of H/R increases. For the reason identified earlier, Eq. (34) can also be expressed as
Mir(t) = r~MM~ H2T(t)
AMir( t) = Ar~MSCT(t)M I H
(31)
and may be expressed in the form AMir(t) = AC[~M5~T(t)MH H1
(34)
(35)
Y. Tang/ NuclearEngineeringand Design152 (1994)103-115
110
.,.r
H
.£.
-R o.o
I-I=1 R
=0.5
0.5
t.o
[%o . . . .
d)r.5. . . . ct
I.O
E
H~"I I = 0.5
that the system is experiencing, the heightwise distribution of the convective hydrodynamic wall pressure induced by the rocking motion will be identical with that induced by the lateral motion. If the heightwise distribution of the convective component of the hydrodynamic pressure under lateral excitation is given by the function C~k(Z), then, following from the above argument, the convective pressure p¢r(z, O,t) under rocking can be expressed as
HI/H~ = 0.5
/
'
pCr(g'O't)=( ~'n=lk=l~~)nkfnk(g)ark(t)) RIRcOSO (37) _H=a
H --=2
R
R
o.o . . . .
o1.5 . . . . et
I .o
o.o
0.5 GI
1.o
Fig. 5. Effect of mass densities on impulsive component of base moment.
to compare the result with that corresponding to = 1. The coefficient Ar~M is related to AC~M by the equation Ar~rM
= AC~M(~) 2
in which 7~ is a dimensionless proportionality factor that remains to be determined.
4.5. Convectivecomponent of base shear It is shown in Tang and Chang (1993b) that the convective base shear QC(t) for a tank-liquid system excited laterally is given by
(36)
The values of Ar~M are shown in Fig. 5. It can be seen that for a tall tank this moment is much smaller than for a broad tank. Note that in Fig. 5 the vertical scale for H/R = 1 is one-fifth of that for H/R = 0.5.
4.4. Convectivecomponent of hydrodynamic pressure The convective pressure exerted on the tank wall may be obtained by evaluating the pressures defined by Eqs. (11) and (12) at r=R. This approach is similar to that used in obtaining the impulsive pressure exerted on the tank wall presented above. However, in this paper a physically motivated approach is used to derive the convective pressure instead and the results obtained by the two approaches will be checked against each other. Since the natural modes of vibration and the natural frequencies of the sloshing motion in a tank-system are independent of the excitation
+
S~A.k(t) M,2
(38)
lk=l
or it may be expressed differently as Q¢(t)=(~,=l~=,~
rSkA"k(t))M~
(39,
in which A~(t) is the instantaneous pseudoacceleration induced by a base acceleration J~(t) in an SDF system of circular natural frequency o~.k. A,k(t) is obtained from Eq. (13) by replacing 5/T(t) by J/(t). By application of the same analogy used in obtaining the convective pressure, Eq. (37), for the system in rocking motion from that for the system in lateral motion, one obtains the convective base shear for the system in rocking motion from Eqs. (38) as M I r, 1 1 L,,a,S~A,k(t))
Q=(t) = ( ~ n=l
k=l
r +(~, ~,?,gS,kAnk(t)) n=l
k=l
(40)
Y. Tang / NuclearEngineeringand Design 152 (1994) 103-115
the convective components of the base moments for the system in rocking motion as
or from Eq. (39) one obtains
Q~( t) =
\n=l k=l
y,u,r,k A ~,( t)
(41)
Note that Eqs. (40) and (41) can also be obtained from Eq. (22) by replacing the function pi~(t) by the function p=(t) of Eq. (37) and performing the integrations.
4.6. Convective components of base moments It is shown in Tang and Chang the convective base moment M~(t) immediately above the tank base liquid system excited laterally can as Me(t) =
\n=l k=l
+
(1993b) that at a section for a tankbe expressed
)
C~IIA,k(t) M~2H2
AC~A,a,(t) )M,1 H,
~'~ ~,a,C,a, MIA,a,(t) r )roll Hi
n=lk=l
+
lk=l
r,a,C,~IIA~(t) MIEHz
)
AMCr(t)=( ~n=lk=l~"~"kACnkA~k(t) M r MI1HI
(46)
(47)
M~r(t) =
n=lk=l
~kr,,kA~(t) M1H
AMCr(t)=(n=~l~=l~nkAr,~A~(t))M~H
(48) (49)
4. 7. Surface and interface sloshing displacements (42)
and the base moment induced by the pressure exerted on the tank base, denoted by AM¢(t), is given by
AM~(t) = (,=~ ~
MC~(t) = ( ~
or from Eqs. (44) and (45) one obtains
C,~IA~(t)MnHl
k=l
111
(43)
Eqs. (42) and (43) can be expressed differently as Me(t) : (,=,~' k=l ~ r,~A,,k(t))M~H
(44)
AMe(t)=(,=~I~__IAr~A,a,(t))M~H
(45)
Eqs. (42) and (43) are useful for providing information about the contribution of each liquid and Eqs. (44) and (45) are useful for comparing with the results of an identical tank filled with one liquid to assess the effect of two-liquid interaction. Applying the same analogy as that used in obtaining the convective components of the hydrodynamic pressure and base shear for the system in rocking motion from the corresponding results for the system excited laterally, one obtains from Eqs. (42) and (43) the expressions of
The surface sloshing wave height dr(r, O, t) of an arbitrary point on the liquid surface may be determined from
p~rlz:=n 2 = p2gd(r, O, t)
(50)
and the sloshing wave height at the inteface of the two liquids, r/r(r, 0, t), may be determined from
1
q(r, 0, t) --(Pl-pu)g
(p~rlz'=n* --p~rlz2=°)
(51)
Substituting p~r and p~r in Eqs. (50) and (51) by Eqs. (11) and (12), one may obtain the expressions for d(r, 0, t) and r/(r, 0, t). The expressions for the maximum values of these two functions are obtained by evaluating d(r, 0, t) and ~/(r, 0, t) at r = R and 0 = 0. Alternatively, the expressions for these maximum values may be obtained as follows. If the expressions for computing these maximum values for the twoliquid-tank system excited laterally are given by
Z
d(R, o, t) = Z E
A~(t)
R
(52)
~/(R, 0, t) = ~ ~ rl.kA.~(t) R n=l k=l g
(53)
n=lk=l
g
112
Y. Tang / Nuclear Engineeringand Design 152 (1994) 103-115
their counterparts for the same system in rocking motion may be expressed as 2 zr nk(t) dr(R, O, t) = ~ ~ ?..d.. R (54) n=~ k=~ g
?jr(R, 0, t) = ~ ~ 7..~/.. Ark(t) R nffilk=l
(55)
g
Once the proportionality factor 7.. is known, the maximum values of the surface and interface sloshing wave heights may be computed from the corresponding solutions for the liquid-tank system excited laterally by making use of Eqs. (54) and (55). Now the generalized Betti principle stated in Veletsos and Tang (1993b) is invoked herein to determine the proportionality factor ?"*. By noting that for a tank subjected to lateral and rocking base motions of the same timewise variations and for each mode of vibration of the sloshing motion in the tank the work done by the base shear for the tank in rocking motion through the displacement of the laterally excited tank is equal to the work done by the foundation moment for the laterally excited tank acting through the rotation of the tank in rocking motion, from Eqs. (41), (44) and (45) one obtains ?"*rS"*
=
r,~, + Arm
r ~ + Ar~ s Ink
5. Conclusions
The complete solutions for tanks containing two liquids under rocking excitation have been presented. Each dynamic response quantity is expressed as the sum of the impulsive and convective components so that the solutions presented may provide a rational basis for evaluating the effects of tank flexibility and soil-structure interaction. It is found that many response quantities for a tank in rocking motion may be evaluated from the corresponding response quantities for an identical tank excited laterally; in particular, all the convective components of the response quantities can be evaluated in such a way.
(56)
Therefore ~nk - -
results obtained by the two approaches confirm the accuracy. Extensive numerical data for a laterally excited tank containing two liquids are available in Tang and Chang (1993b). Also, because all the convective components of the response quantities for a tank in rocking motion can be computed from their counterparts for the laterally excited tank, no numerical results for the convective components of the response are presented herein.
Acknowledgment
(57)
With ~"* determined, all the convective components of the response quantities for the tank in rocking motion can be calculated. Note that the determination of ~"* from Eq. (57) requires only the results for the laterally excited tank; therefore there is no need to solve the governing equations and the boundary conditions to obtain the convective components of the response for the tank in rocking motion. However, in this paper, for the purpose of confirming the accuracy, the convective components of the response quantities are also computed from the convective pressures defined by Eqs. (11) and (12) which are obtained by directly solving the governing equations and the boundary conditions. The identical numerical
This work was performed in the Engineering Mechanics Program of the Reactor Engineering Division of Argonne National Laboratory under the auspices of the US Department of Energy, Contract W-31-109-Eng-38.
Appendix. Solution for rigid tanks
A.I. Impulsive component of solution
The impulsive component of the hydrodynamic pressures p~r and piEr must satisfy Vp~r = 0
(Ala)
Vpi2r = 0
(hlb)
113
Y. Tang/ NuclearEngineeringand Design152 (1994)103-115
in which cIr(r, zl) and c~Xr(r,z2) are dimensionless coefficients given by
and the boundary conditions
1 ~pi(
= rOb(t) COS0
Pl Ozl Izl=o 1 ap~r
(A2a)
I
I
Pl Or ~=R
= --Z I Ob(t) cos 0
r Zl R°~G~ I cIr(r'zl)='Rn-]-nn~=l Z A.
(A2b) ,
f. 2"1'~q Jl(2.r/R)
+ B. slntz. ~ ) j 1 Opizr • = R = - (z2 + H1 )0b(t) cos 0 P2 ~r
(A2d)
10p~ r
=
__
P2
ir
(AZf)
p~r and p~ are finite at r = 0
Jlt/'n) J
where 2 A~ = ~ (sinh fil. cosh fiE" + ct cosh fiE" sinh fiE") + (~ - 1)//lr cosh fiE" _ H
(A3) x sinh ill. cosh fiE"
n=l
}
..
os0 (A4)
In Eqs. (A3) and (A4), ~T(0 = ttO'b(t), Jl is the Bessel function of the first kind of order one, ~. is the nth zero of J~ (2), the first derivative of J1, and A., B., C~ and D. are constants of integration that may be determined from the boundary conditions defined by Eqs. (A2a), (A2d), (A2e) and (A2f). After evaluating these constants of integration and substituting them back into Eqs. (A3) and (A4), the results are cast into the forms
p~r(r, O, Zl, t) = Clor(r,zl)p,R~T(t ) COS0 pigr(r, 0, z2, t) = CIol~(r,Zz)p2RS~T(t) COS0
(A9c)
2 cosh fi2. q- (o~ - I) -~HI sinh fil.cosh fiz~ 2.
D. -
R ~ [ C~ cosh ( 2. 7,2) ~r
1
(A9a)
2 H H1 C. = ~ sinh fiE" - ~ cosh fil. + (1 - a) -~-
and similarly, on satisfying Eqs. (Alb), (A2c) and (A2g), piEr takes the form
Z2 Jl (2.r/R)
A8)
(A9b)
+ B~sinh()t.~)]Jl(.2-~.r,[----ff)}plRSiT(t)cosO
•
cosh
B~ = -- ~ (cosh fii. cosh fiE"+0t sinh filn sinh fiE.)
{~ Z1H+ HR.~, ...°c [ An cosh ( Z2.R )
+
R
(A2g)
The method of separation of variables is employed to solve Eqs. (Ala) and (Alb) and the integration constants are determined from the boundary conditions• On satisfying Eqs. (Ala), (A2b) and (A2g), the function p~r takes the form
pi2r={RZ2"Jf-Hl ---H---
(A7)
. L zA1Jl(&dR) +O. smt . ;/
~g2 Iz2=0
=-, --P2 I,z=0
p~=
cllr (r, Z2)
(AEe)
1 ~pi~
0zl ~ =HI
~
(A2c)
ir =n2 - 0 P21~2 Pl
(el)
cosh 2 . ~
-
H ~ a sinh ill.
2
G. = 2 2.-1
(A9d) (A9e)
A. = cosh fil. cosh fi2n -4- 0~sinh filn sinh fi2n (A9f)
A.2. Convective component of solution The convective component of the hydrodynamic pressures p]r and p~r must satisfy Laplace's equations
(A5)
V2p~r = 0
(A10a)
(A6)
VEp~r = 0
(A10b)
Y. Tang / Nuclear Engineering and Design 152 (1994) 103-115
114
along with the boundary conditions ~p~r
=0
and (A12b) into Eq. (A1 ld), one obtains a differential equation (A1 la) sinh flln sinh fl2nJ~n +" g2n ~ sinh flln cosh flz.E.
~ZI zl=0 1 0p~ r
=0
(A1 lb)
Pl 0r r=R
g2. + cosh f12,,i~. + ~ sinh fl2,,F~
1 0p~ r
Pz ~r .=R = 0
_
(Allc)
(~2cr Op~,[ o-pF + _Op~r~ _ Ot2 g Oz2/I~:=H2 = --g~-2-z2 ~2=H2 1 0 p ~ r z,=/~, = Pl ~Zl
1 ~p~ ~2 P2 822 =0
(A1 ld) (Alle)
g2. R G. (A.
R H A. \ - ~ + C. sinh flz.
+ D. cosh flZn )XT(/)
and substituting Eqs. (A6) and (A12b) into Eq. (A1 lf), one obtains another differential equation g2n cosh fll~/~. + ( 1 - a) ~ sinh fll,,E. - aP,,
(OP~r ~P~r l OP~r~ Zl "~i- + g ~zl + g "~zl ) =~I,
_ ap~~-t-' g "-~z2 ~pivr'~ = {ap~" ~-~T + g -~z2 ) 2= 0 p~r and p~r are finite at r = 0
p~r(r, O, zl, t)=[n~=lDn(t)cosh(J,n~-~) J'(2nr/R) _]
(A12a)
and p~r takes the form
P~r(r,O, zz, t)={.=~[E.(t)cosh(2. R ) + Dn(t) sinh Bin sinh 2~ ~ ~
~
; P 2 " cos0
Cnl(r,z,)= (A12b)
in which E,(t) and D~(t) are integration functions that can be determined by satisfying Eqs. (A1 ld) and (A1 If). Substituting Eqs. (A5), (A6), (A12a)
(AI4)
Then Eqs. (A13) and (A14) can be solved for E.(t) and F.(t). Taking the Laplace transformation on both sides of Eqs. (A13) and (A14) and assuming homogeneous initial conditions for E.(t) and F.(t), one obtains two algebraic equations for determination of the Laplace transforms of E.(t) and F.(t). The required solutions for E.(t) and F.(t) are then obtained by finding the inverse Laplace transforms, Then, replacing E.(t) and F.(t) in Eqs. (A12a) and (A12b) with the results obtained, one can cast the final expressions into the form presented in Eqs. (11) and (12). The derivation is not difficult but is tedious; hence it is omitted herein. However, the details of the derivation are available in Tang and Chang (1993b). The expressions for C,e.(r, ~r zl) and i'~IIr/ "-~.k~r, z2) for k = 1 and 2 are given as lr
J~(2,r[R)]
A,,
R HA. \(1-a)~-]+A~sinhfll~
+ B. cosh ill. - aD. )£T(t)
(Allg)
× p~R cos 0
g2,,RG. (
(Allf)
Again the method of separation of variables is employed to solve Eqs. (A10a) and (A10b) and the integration constants are determined from the boundary conditions. On satisfying Eqs. (A10a), (A10b), (Alla), (Allb), (Allc), (Alle) and (Allg), the function p ~ takes the form
x
(A13)
( zl']Jl(2.r/R ) d.A.~X, cosh 2.-~/
G~ 1
(A15a)
,r C.2(r, zl) =
G. 1 (z,)J,(2.r/R) A.A.~2 yl cosh 2. ~ Jl(2.) (A15b)
115
Y. Tang /Nuclear Engineering and Design 152 (1994) 103-115
.l ~', z2) =
An A.l x2 cosh 2. + xl sinh ill. sin 2.
Jl().n) (A16a)
An
n2 ~,J', Z 2 ) ~-
2
Y2 c o s h
(Al6b) where X 1 --
An1 ql - q2 A,1 -- An2
(A17a)
Yl-
q2 -- A n 2 q l A . I - - An2
(A17b)
Am q3 -- q4
(A17c)
A n l -- An2
and q4 -- A n 2 q 3 Y2 --
Am
(A17d)
_ An 2
in which ql =
sl cosh f12~ + xs2 An Sl sinh fiE.
q2 --
s2 cosh ill. - sl sinh fl]. sinh f12. q3 = q 4 ~-
(A18a) (A18b)
An
An
(A18c)
s2(1 - ~) sinh ill. - Sl sinh ill. cosh fiE. An (A18d)
with An Sl = ( 1 - a) ~ + A. sinh ill. + Bn cosh flln - aDn (A19a) A. s2 = ~ + C. sinh f12. + D. cosh 132.
g2;
2J-sin i l l . sinh flEn cos
+ y] sinh fl,. sin( 2"zER/j~]J'(2"r/R)J, (2.)
X2 --
The characteristic equation for the natural frequencies can be obtained from the associated eigenvalue problem of Eqs. (A13) and (A14). Letting 5~x(t) = O, E.( t) = exp(kot) and F.(t) = F. exp(kot) in Eqs. (A11) and (A12), one obtains two homogeneous equations which are expressed by
(A19b)
cosh t l. [s inh ill. cosh f12. sin 0 x L ( 1 - a) sinh ill.
AtFo + A[F.)=0
(A20)
It is easy to show that the determinant of Eq. (A20) is Eq. (15), the characteristic equation.
References Y. Tang, Dynamic response of tank containing two liquids, J. Eng. Mech. Div., ASCE 119(3) (1993a) 531-548. Y. Tang and Free vibration analysis of a tank containing two liquids, J. Eng. Mech. Div., ASCE in press. Y. Tang and Y.W. Chang, Dynamic response of a tank containing two liquids, Argonne National Laboratory Rep. ANL/RE 92/1, 1992. Y. Tang and Y.W. Chang, The exact solutions to the dynamic response of tanks containing two liquids, Argonne National Laboratory Rep. ANL/RE 93/2, 1993b. A.S. Veletsos, Seismic response and design of liquid storage tanks, in Guidelines for the Seismic Design of Oil and Gas Pipeline Systems, ASCE, New York, 1984, pp. 255-370 and 443-461. A.S. Veletsos and P. Shivakumar, Sloshing response of layered liquids in rigid tanks, J. Earth. Eng. Struct. Dyn. 22 (1983) 801-821. A.S. Veletsos and Y. Tang, Dynamics of vertically exicted liquid storage tanks, J. Struct. Div., ASCE 112(10) (1986) 1228-1246. A.S. Veletsos and Y. Tang, Rocking response of liquid storage tanks, J. Eng. Mech. Div., ASCE 113(11) (1987) 17741792.
A.S. Veletsos and Y. Tang, Soil-structure interaction effects for laterally excited liquid storage tanks, J. Earth. Eng. Struct. Dyn. 19 (1989)473-496. A.S. Veletsos and J.Y. Yang, Earthquake response of liquid storage tanks, Advances in Civil Engineering Through Engineering Mechanics, Proc. ASCE/EMD Spec. Conf., Raleigh, NC, 1977, pp. 1-24.