Green's function solution for axisymmetric vibration of flexible liquid-filled storage tanks

Green's function solution for axisymmetric vibration of flexible liquid-filled storage tanks A. N. Williams and W. I. Moubayed

Department of Civil and Environmental Engineering, Cullen College of Engineering, Univer- sity of Houston, Houston, TX 77204-4791, USA (Received April 1989; revised June 1989)

The dynamic response of a flexible, liquid-filled cylindrical storage tank subjected to high frequency vertical ground motions is investigated theoretically. The tank motion is idealized as that of a uniform, elastic, thin-walled cylindrical shell undergoing radial displacements; all axial deformations are neglected. The fluid is assumed to be inviscid and linearly compressible and to undergo small-amplitude, irrotational motion. Both the structural and fluid motions are expressed in terms of appropriate Green's functions. The axisymmetry of the response then leads to a pair of coupled line integral equations for the fluid velocity potential and its normal derivative on the walls of the tank. These equations are then solved numerically. Results are presented which illustrate the influence of the frequency of ground excitation and the various geometric and material parameters on the hydrodynamic pressure distribution and associated dynamic response of several example structures.

Keywords: Green's function, liquid storage tank, seismic response

In recent years, there has been considerable interest in the effect of axisymmetric vibrations, caused by the vertical component of ground motion, on the seismic response of liquid-filled storage tanks. It is now recognized that in a liquid-filled tank, vertical accelerations can be transmit- ted into horizontal hydrodynamic forces and so result in radial deformations of the tank walls.

The importance of the vertical component of ground acceleration in the design of liquid storage tanks was first highlighted in a simplified analysis by Marchaj t. Subse- quently, Kumar z investigated analytically the effect of vertical accelerations on the radial motion of flexible, liquid-filled storage tanks. A simple design procedure for evaluating the effects of axisymmetric vibrations due to seismic excitation was later presented by Veletsos and Kumar 3. Haroun and Tayel *-6 have studied the dynamic response of empty and partially filled cylindrical storage tanks to vertical ground motions using both numerical (finite element) and analytical techniques. Recently, Fischer and Seeber ~ presented a semi-analytical solution for the behaviour of a liquid-filled tank under vertical ground excitation which included foundation interaction effects.

In the present paper, a Green's function approach is utilized to calculate the dynamic response of a flexible, liquid-filled cylindrical storage tank to high frequency

0141-0296/90/010049- I I/$03.00 © 1990 Buttcrworth & Co (Publishcrs) Lid

vertical ground motions. The tank is idealized as a uniform, elastic, thin-walled, cylindrical shell. Only radial displacements are considered, all axial deformations are neglected. The fluid is assumed to be inviscid and linearly compressible and to undergo small-amplitude, irrotational motion. Expressing both the structural and fluid motions in terms of appropriate Green's functions, then the axisymmetry of the response leads to a pair of line in¢grai equations for the fluid velocity potential and its normal derivative on the walls of the tank. These two coupled equations are solved numerically by discretizing the contour of integration, thereby replacing the contin- uous problem by two discrete systems of algebraic equations. Numerical results are presented for the hydrodynamic pressure distributions and associated dynamic response of several example structures which illustrate the influence of the frequency of ground excitations and the various geometric and material properties on these quantities.

Formulation

The liquid-shell system under consideration is shown in Figure/. A vertical circular cylindrical tank of radius a, height H, and of constant wall thickness h is completely

Eng. Struct. 1990, Vol. 12, January 49

Axisymmetric vibration of flexible fiquid-filled storage tanks: A.N. Williams and W.I. Moubayed

h - - ~

Figure 1 tank

V

H

O

~ Uoe- i~ t

Definition sketch for liquid-filled cylindrical storage

filled with a homogeneous liquid. Cylindrical polar coor- dinates (r, 0, z) are used, with the origin of the coordinate system being located at the centre of the circular base. The rigid base of the tank is subjected to vertical, small- amplitude harmonic ground excitation of magnitude U o and frequency f~. The resulting structural motion is analysed by assuming that the tank behaves as a uniform, elastic, thin-walled cylindrical shell, in general, a cylindrical shell undergoingaxially symmetric v~ration is gov- erned by two differential equations, one each in the axial and radial directions s. However, for completely filled tanks, it has been shown that axial displacements can be neglected L6 and the motion is primarily in the radial direction. After adding an appropriate inertia term, Fliigge's differential equation of motion for a thin-walled cylindrical shell acted upon by an axisymmetric pressure distribution P(z, t) = Re[/~(z)e -~n'] may be written as

Eh 3 d%i'

12(! - v z) d: ~ + ~ - ~:p.,h i = P ( z ) (l)

valid on r = a for0 ~< .-"~< 1"-I: In equation (1), if(z) is the amplitude of the radial displacement w ( z , t ) - ~ Re[i(z)e- ;a,], p, is the mass density of the shell, v is the Poisson's ratio, and E is the modulus of elasticity of the shell material. Two situations will be considered, corresponding to fixed (Case I) or pivoted (Case II) conditions

at z = 0. Thus, the structural boundary conditions at the base may be written

d i i = d--:-- = 0 at z = 0 (I ) (2a)

d ' i i = ~ = 0 at z = 0 (II) (2b)

together with

d ' i d 3 i dz ' = dz 3 at z = H (I&II) (3)

which implies that the top of the structure is radially free. The liquid is assumed to be homogeneous, inviscid,

and linearly compressible and to undergo small-amplitude, irrotationai motion. The governing equation for the axisymmetric liquid motion may be written in terms of the velocity potential 4~(r, z, t) = Re[@(r, z)e -~c"] as

0"@ 1 0 ¢ dzO Or" + -r ~ r + ~ z e + k ' O = 0 (4)

where k = ~ / c , and c is the acoustic speed in water. The boundary conditions on O(r, z) are

dO O--z = - i f l U ° on z = 0, 0 ~< r ~< a (5)

0O O-r- = - i f ~ i on r = a, 0 ~< z ~< H (6)

@ = 0 on z = H, O <~ r <~ a (7)

The last boundary condition, equation (7), is a modified form of the iinearized free-surface condition for high frequency oscillations where liquid sloshing effects are neglected. The dynamic pressure/~(z) appearing in equation (1) may be expressed in terms of the velocity potential on the structure through the iinearized Ber- noulli equation, namely

~(z ) = if~pO(a, z) for 0 ~< z ~< H (8)

where p is the liquid density.

Solution by Green ' s f u n c t i o n s

Equation (1) may be rewritten as

d 4 i dz" = ' i (z ) = iflO(a, z) on 0 ~< z ~< H (9)

where

~4 12(I - - V 2) Fp,~Sa 2 1 = i. ( 1 0 )

12(I -- v')~p P = Eh3 (1 l )

The structural Green's function V~(z, :o) is required to satisfy

d'V~ dz 4 ~4V G = 6(z - Zo) on 0 ~< z ~ H (12)

where 6( ) is the Dirac delta function, together with the

50 Eng. Struct. 1990, Vol. 12, January

Axisymmetric vibration of flexible liquid-filled storage tanks: A.N. Williams and W.I. Moubayed

following boundary conditions

V° = d z = 0 a t z = 0 ( I ) ( 1 3 a )

d~Vc V G = ~ = 0 at z = 0 (II) (13b)

and

d2Vo davG dz 2 = -~Fz3 = 0 at z = H (I&II) (14)

A suitable form for Vc(z; z o) for Case I (fixed supports) has been given by Liu and Cheng 9 for ~,4 > 0 as

1 {sinh =(z - Zo) - sin =(z - Zo)} v~(z; Zo) =

1

+ 4~,3(1 + cosh ,tH cos =H) {[C(z°Xsinh ,tH + sin ~H)

-- S(zoXcosh 0tH + cos aH)](cosh 0tz - cos r,,.-)

+ [S(zo)(sinh a H - sin a l l ) - C(zo)(cosh =H + cos =H)]

(sinh =z - sin a:)} for z > Zo (15)

in which

C(z o) = cosh ~,(H - Zo) + cos =(H - Zo) (16a)

S(Zo) = sinh =(H - z o) + sin a(H - zo) (16b)

For Case II (pivoted supports) thc corrcsponding Green 's function, obtained by a Laplacc transform method (Appendix A), is

I {sinh ~,(z - Zo) - sin u(z - Zo)} v~(z; Zo) =

I

+ 4~,~(sinh aH cos ,tH - sin ,,H cosh a l l )

x {l'C(zo)(sinh ~H + sin ~H)

- S(zoXcosh :oH + cos ~H)]

x (sinh az + sin ~,z) + [S(zo)(cosh ~,H - cos ~,H)

- C(zo)(sinh aH -- sin aH'l

x (sinh ~,z - sin =z)} for z > z o ( 1 7 )

again for ~,t > 0. The Green's function for z < z o may be obtained in each case by interchanging z and z o on the right-hand sides of equations (15) and (17). Also, in situations where a 4 < 0, the above forms should be modified by replacing • by ae j~/'. Application of Green 's second identity to K(:) and V~(z; Zo) over [0, H'l leads to

~(:o) = i¢/ v~(:: Zo)ei,(a, z)dz (18)

for both Cases I and I!. Thus, the boundary condition on the tank walls may be rewritten as

c?O__ (a, z o) = ~ f l VG(-; Zo)O(a, z )dz (19) Or

The problem in the liquid domain may also be ap- proached through a Green 's function technique. How- ever, before application of the method, some preliminary

algebraic manipulation is necessary. The velocity potential is decomposed into two components

• (r, z) = ~P(r, z) + O'(r, z) (20)

where the term O'(r, z) is required to satisfy the inhomo- geneous condition on the bot tom of the tank, equation (5), together with equations (4) and (7). A suitable form for (b'(r, z) is

sin k(z - H ) O'(r, z) = - icU o (21)

cos k H

and the boundary-value problem for ~P(r, z) becomes

02~ 1 t ~ ~ 2 ~ ~r 2 + r - ~ r + ~ + k2~[J ---- 0 (22)

0W 0--z- = 0 o n z = 0, 0 ~< r ~< a ( 2 3 )

05' c3--r- = - if2~ on r = a, 0 ~< z ~< H (24)

= 0 on z = H, 0 6 r ~< a (25)

The axisymmetric liquid Green 's function G(r, z; r o, Zo) is required to satisfy

02G I OG 02G Or 2 + r ~r + ~Z z- + k2G

= - - 4 n ~(r - ro)3(z - Zo) (26)

r

in the liquid domain, together with the boundary conditions

~G ~ z = 0 on z = 0 ,0 ~ r ~< a (27)

G = 0 on z = H, 0 <~ rr <~ a (28)

A suitable axisymmetric Green 's function, developed on the basis of the eigenfunction expansion method outlined by Morse and Fesbach ~°, has been given by Williams tt a s

G(r, z; r o, Zo)

=

H . = t \Ho( '~ . . ro)Jo(2.r)] k , z cos knzo

4 ~ {Ko(,['nr)lo(A',ro) ~ , , C O S COS + ,=N+t k /iKo(~.,ro)lo().J)] k,z k,z o

(29)

In equation (29), Jo and Ho are the Bessel and Hankel functions of the first kind, respectively; Io and Ko are the modified Bessel functions of the first and second kinds, respectively; and

k. = (2n - l )n/2H n >/ I (30)

2 1/2 2. = (k e - k.) n ~< N (31a)

2~, = (k~ - k2) 1/2 n > N (31b)

where N is the largest integer satisfying the condition [k z - k, z] > 0. The upper terms in equation (29) are to be


Axisymmetric vibration of flexible liquid-filled storage tanks: A.N. Williams and W.L Moubayed

used for r I> re and the lower for r < re. It can be seen that for excitation frequencies fl > f~* = nc/2H, the solution for the Green's function is complex, arising from the inclusion of liquid compressibility in the formulation. At frequencies f~ < ~*, the Green's function is real and the solution is similar to that obtained by neglecting compressibility.

Applying Green's second identity to tP(r, z) and G(r, z; re, Zo) over the liquid domain yields

fs {~P(r, z) ~n (r, z; ro, Zo)

-- G(r, z; r o, Zo) ~-n (r, z) dS = 2n~P(ro, Zo) (32)

for (r o, Zo) on the surface S consisting of the side walls of the tank, S c, the free surface, S o, and the tank base, Sb. In equation (32), n is the unit normal to S at (r, 0, z), directed into the liquid domain. Applying the free-surface and base conditions and carrying out the 0-integration analytically reduces equation (32) to

f:{ z,0O a tP(a' ~ r (a, z; a, Zo)

- G(a , z; a, Zo) O--r (a, z ) d : = - ~ ( a , z o) (33)

where 0 ~ Zo ~ H. Equation (19) may now be rewritten a s

f: OtP-- (a, Zo) = ~// V~;(:.; Zo)tP(a, z)dz dr

icUo~ ~" ~ Jo Vc,(Z; Zo)sin k(,: - H)dz (34)

Equations (33) and (34) now constitute two simultaneous integral equations for ~P and ¢9~P/dr on So. These integral equations may be solved numerically by discretizing the contour of integration [0, HI into N small line segments, L., n = I, 2 . . . . . ~. The values of ~P and dW/c~r are then assumed to be constant within each segment, thus replacing the integral equations by two large finite systems of algebraic equations. These may be written in matrix form a s

AP + P' = B- (35a)

CP = DP' (35b)

where the matrix entries are given by

{a..} = -f~p I. v~(z; z.,)dz (36a) # / . m

icU of~p f n V~(z; zm)sin k(z - H)d'. (36b) {bin} = Jo

{c,,,} = 6,,, + a . d r (a, .'; a, z,)dz (36c)

{d~,,} = a I. G(a, z; a, zm)d2 (36d) # L M

{p,} = tP(a, z,) (36e)

d'P {p~,} = ~ (a, : ,) (36f)

To the same degree of approximation used in obtaining equations (35), the integrals in equations (36) may be approximated by the product of the integrand at the element node and the element length A: = H/~. How- ever, for m = n, the integrals in equations (36c) and (36d) exhibit singular behaviour at the node and the series for G and dG/dr are not convergent at this point. This difficulty may be overcome by the subtraction and analytical summation of the asymptotic forms, and the singular terms may be integrated analytically. A similar technique has been used by Williams tt, and in two dimensions by Williams and Mau ~2. Explicit expressions for the matrix coefficients {c,m} and {d,,m} in the present case are given in Appendix B.

Once the entries of the coefficient matrices have been computed, the matrix equations (35a) and (35b) may be solved to obtain the values of the velocity potential ~P and its derivative at the nodal points on the tank wall. The complex dynamic pressure distribution /~(z) may then be calculated from equation (8) and the complex radial displacement amplitude if(z) obtained from equation (24).

Numerical results and discussion

A computer program has been written to calculate the hydrodynamic pressure distribution and radial displacements along the walls of a liquid-filled cylindrical storage tank utilizing the above theory. Four example tanks were considered to demonstrate the application of the solution technique and to illustrate the effect of the various geometic parameters on the dynamic response. The chosen dimensions were: (a) H/a=0.5, h/a=O.O02; (b) H/a = 0.5. h/a = 0.004; (c) H/a = 1.0, h/a = 0.002; (d) H/a = 1.0, h/a = 0.004. The material properties taken were E = 2.107 x 10 l° kg/m ~, v = 0.3 and p, = 7846 kg/m 3. The liquid density p = 1000 kg/m 3, and acoustic speed c = 1430 m/s. The hydrodynamic pressure was normalized with respect to the static pressure at the base, pgH, and the dynamic displacement by the amplitude of the ground motion, Uo. The ground acceleration was set at a unit amplitude of 1 g, i.e., f12Uo = g. In obtaining the results presented herein the integration contour [0, HI was discretized into elements of length Az, according to Az/a = 1/40. Decreasing the element size did not change the computed values by more than I-2 %, indicating that numerical convergence had essentially been achieved.

All dynamic quantities are presented as functions of a dimensionless frequency f l /~ where ~ represents the

Table 1 First two natural frequencies of liquid-filled cylindrical tank for fixed and pivoted base conditions

H/a h/a f l 1 I f l fZz/fl

Fixed 0.5 0.002 0.21 0.38 0.5 0.004 0.31 0.51 1.0 0.002 0.12 0.26 1.0 0.004 0.19 0.37

Pivoted 0.5 0.002 0.21 0.36 0.5 0.004 0.28 0.49 1.0 0.002 0.12 0.25 1.0 0.004 0.17 0.36


~J

O.

f,,,. e,

x

a

"0

O.

C <u

_m O.

"0

E ::) E x

Axisymmetric vibration of flexible liquid-filled storage tanks: A.N. Williams and W.I, Moubayed

_ , , i i

t

3 ~ I 3 eL ~ _ ~ ~ ~

" ~ ' i

1 ~ .~ 1 ' '

. = ~ . ' , 0 I I ~ I 0

0.00 I , I ~ I

0.10 0.20 0.30 Dimensionless frequency

q I

3

2- -

~ i---~ I ~ ~'~ O_ 0.0o 0 . 1 o

0.q0

U I. ~ I ~ I

0.20 0.30 0.~0 Dimensionless frequency b

Figure 2 Variation of (a) maximum dynamic pressure and (b) displacement amplitudes with dimensionless frequency ~ / ~ for H/a ,= 0.5, h/a = 0,002 (O, fixed; Z~, pivoted)

q - = : :

3 , ,,

E tb

J . ' . ~

0 , I , I , I , I , I ,

0 . 0 0 . 1 0 . 2 0 . 3 0 . 4 o . s a Dimensionless frequency

~ - q i l " o -

L : ' '

- t

u~ 2 j / ._~ 1

:E 0 - , I ~ I ~ I ,

0.0 0.1 0.2 0.3 0.~I O.S

b Dimensionless frequency

Figure 3 Variation of (a) maximum dynamic pressure and (b) displacement amplitudes with dimensionless frequency Q/~ for H/a = 0.5, h/a = 0,004 (O. fixed; A , pivoted)

0.0 0.1 0.2 0.3 a Dimensionless frequency

5

i ',

° i i _

2 - "o .

E ._E 1 x

0 ! J I

b 0.0 0.1 0.2 0.3 Dimensionless frequency

Figure4 Variation of (a) maximum dynamic pressure and (b) displacement amplitudes with dimensionless frequency I~/1~ for H/a = 1.0, h/a = 0.002 (O. fixed; A , pivoted)

t • "- 4 cL

3 '~ "

a. 2 , i E .E x 1 J

0 , I , I , i , 0.0 0.1 0.2 0.3 0.q

a Dimensionless frequency S t Jl i i

-

i f

3 " " :' ,

41

Ut ~ I I d

:5 2

0 , I , I i 0.0 0.1 0.2 0.3 0.4

b Dimensionless frequency

Figure5 Variation of (a) maximum dynamic pressure and (b) displacement amplitudes with dimensioness frequency Q/~ for H/a = 1.0, h/a = 0.004 (O, fixed; L~, pivoted)



fundamental breathing frequency of a circular ring with the same cross-sectional dimensions as the tank, and is given by

= a p , (37)

The two lowest natural frequencies of the four example tanks filled with liquid are presented in Table 1. From the

table it is noted that, for constant h/a, the natural frequencies of the liquid-tank system decrease as H/a increases. Also, for constant H/a, it can be seen that these natural frequencies increase with increasing h/a. For a tank with pivoted base conditions the natural frequencies of the system are always smaller (but relatively close to) those corresponding to a tank of the same geometric dimensions with fixed base conditions.

~= 0.5

I .0 I .0

== 0.5

1 2 Pressure amplitude

0.0 0.0 0 3 0

_• I

t 2

Pressure amplitude 1 . 0 1.0 /

! -... 0.5 ~ 0.5

0.0 0.0 0

Displacement amplitude Displacement amplitude 1 . 0 1 , 0

::1: :z: .~ o.s .~ o.s -

m

o.o , 1 , I , I , o .o 1 ' t 2 3 4

a Phase (Rad) Phase (Rad)

1 . 0 ' ' ! . 0 i O0 O0

• 0.0 0.5 1.0 0.0 O . q 0 . 8

Pressure amplitude Pressure amplitude 1 . 0 , 1 , 0 I

I -~. 0 . 5 - 0.5

0 0 0,0 t l 1 2 3 1 2 3 4 5

Displacement amplitude Displacement amplitude I .0 1.0

~ o.s ~2 o.s ~

0.0 ~ 0.0 , t l = - 0 1 2 3 - ' 0 1 2 3

C Phase (Rad) d Phase (Rad)

Figure 6 Dimensionless dynamic pressure and displacement profiles at selected frequencies: (a) fZ/Q = 0.17, (b) D./~ = 0.25, (c) Q / ~ = 0,34, (d) £1/(~ = 0.40. Hla = 0.5, hla = 0.002 (O , f ixed: O, pivoted)

5 4 E n g . S t r u c t . 1 9 9 0 , V o l . 1 2 , J a n u a r y

Axisymmetric vibration of flexible l iquid.f i l led storage tanks: A.N. Williams and W.I. Moubayed

Figures 2-5 present the variation of the maximum hydrodynamic pressure and maximum dynamic displacement with dimensionless frequency for the four example structures. From the figures it can be seen that, as expected, the dynamic response quantities become infin- ite as the frequency of ground excitation approaches a natural frequency of the liquid-tank system.

The hyrdodynamic pressure distribution, radial dis-

1.0

0.5

0.0 1 2 3

placement profile and their (common) phase are shown at two pairs of selected frequencies for each of the example structures in Figures 6-9. These selected frequencies are chosen close to, and either side of, the first and second natural frequencies of the liquid-tank system. It can be seen from these figures that when the excitation frequency is less than the fundamental frequency fZ t of the system the maximum dynamic pressure occurs at the

1.0

0.5

0.0 0 1 2 Pressure amplitude Pressure amplitude

.~ o.s ~ o.5

0.0 0.0 2 3 0 1 2 3 q S

1.0 Displacement amplitude 1.0 Displacement amplitude

.~ o.s .~ o.s

0.0 J [ t I J I I 0.0 I 1 I 2 3 - 0


1.0 ~ ~ . 1.0 ~ - - - ~ ~ ~ ]

~: 0,5 L - , ~ ' < ~ ° ~ 4 r '~B 0.5

O 0 O 0 • 0.0 0.3 0.6 0.9 .2 0.0 0.1 0.2 0.3 0.4 O.

Pressure amplitude Pressure amplitude 1.0 1.0 [

I ~. O.S O.S

0.0 0 0 t I 1 2 " 0 1 2 3 q

-~ 0.5

Displacement amplitude 1.0

0.0 ~ - 0 1 2 3

C Phase (Rad ] d

1.0

-~ 0.5

0.0 , I -1 4

Displacement amplitude

I I , I t I , I 0 1 2 3

Phase (Rad]

Figure 7 Dimensionless dynamic pressure and d isp lacement profi les at selected frequencies: (a) ~ / ~ = 0.25, (b) ~ / Q = 0.34, (c) Q /~ = 0.47, (d) ~ / ~ = 0.56. H/a = 0.5, h/a = 0.004 ( 0 , f ixed; O, pivoted)



base of the tank; however, for Q > t3 t the maximum dynamic pressure occurs at some point in the interval [0, H]. At excitation frequencies slightly below the first two natural frequencies of the system, the pressure and displacement amplitudes obtained for fixed base conditions are smaller than those for a corresponding tank with pivoted base conditions. However, at excitation frequen-

cies slightly higher than these natural frequencies the opposite is true. The dynamic displacements presented in Figures 6-9 predict a region of extreme curvature near the tank base, indicating that large bending stresses may exist there. Thus, the vertical component of ground excitation may be a significant factor in the failure of the liquid-tank system during earthquakes, It is also noted

~: O.S

:Z: '-~ O.S

1.0 1.0

:~ O.S

0.0 0.0

1.0 1.0

1 2 3 Pressure amplitude

1 2 Displacement amplitude

-~ 0,5

0.0 0,0 3

1.0 1.0

1 Pressure amplitude

1 2 Displacement amplitude

.~ o.s .~ o.s

0.0 * I I I , I , I 0.0 I I 1 2 3 -1


1.0 1.0

0 .5

0.0 O 0 Op2 r : L 8 0 . . . . . . 5

essur pl i tude o

=: 0 . 5 ~ -~ 0.5

O 0 i O0 0

Displacement amplitude Displacement amplitude 1.0 1.0

~ o.s , , , ] "2 o.s [

0.0 ~ O.C ~ ~ I s I , I t

- 0 1 2 3 -1 0 1 2 3

C Phase (Rad) d Phase (Rad)

Figure 8 Dimensionless dynamic pressure and displacement profi les at selected frequencies: (a) Q/~ = 0.11. (b) ~ /~ = 0.15. (c) Q/~ = 0,23. (d) f~/~ = 0.29. H/a = 1.0, h /a = 0.002 (O, fixed; O, pivoted)



that for a constant H/a, an increase in h/a results in a slight upward shift in the locations of both the maximum and (where applicable) the null pressures and displacements, and that the opposite trend is true for a constant h/a and varying H/a, i.e., increasing H/a leads to a slight downward shift in the locations of the maximum and null pressures and displacements.

Conclusions

The dynamic response of a flexible, liquid-filled cylindrical storage tank subjected to high frequency vertical ground motion has been investigated utilizing a Green's function technique. Numerical results have been presented for several example structures which illustrate the

1.0

0.5

0.0

1.0

0.5

0.0

1.0

1 2 Pressure amplitude

1 2


1 . 0

:= O.S

0.0

1.0

0.5

0.0

1.0

0 1 2 Pressure amplitude

1 2 3


0.0 J I = I = I 0.0 I £ I 2 3

a 1 . o ~ ~ L ~ P h a s e (Rad) i i . i ~ P h a s e (Rad)

0.5

0.0 0.0 0.3 0.G 0.9 1.2 0.0 0.1 0.2 0.3 0.4 0.5

Pressure amplitude Pressure amplitude 1.0 1.0 ~ ~ ¢ ~ 1 ~ ~ ]

~. 0.5 0.5

0.0 0.0 0 I 2 3 q 5 0 I 2 3

Displacement amplitude Displacement amplitude 1.0 1.0

I • ,~ 0.5

7 I ~ I , I J I I I 0 1 2 3

Phase (Rad)

0.5

0.0 I 0.0 ' , I , I ~ I ~ I -1 - 0 I 2 3

c d Phase (.ad)

Figure 9 Dimensionless dynamic pressure and displacement profiles at selected frequencies: (a) Q/~ -- 0.15, (b) Q/Q -- 0.20, (c) JQ/Q = 0.33, (d) Q/~ = 0.39. H/a = 1.0, h/a = 0.004 (O, fixed; @, pivoted)



influence of the frequency of ground excitation and the various material and geometric properties on the hydrodynamic pressure distribution and associated dynamic response of the structure.

Acknowledgement

The authors would like to express their gratitude to S. T. Mau for a number of stimulating discussions on this topic.

R e f e r e n c e s

1 Marchaj, TJ. 'Importance of vertical acceleration in the design of liquid-containing tanks', Proc. 2nd US Nat. Conf. on Earthquake Engng, Standford, California, 1979

2 Kumar, A. 'Studies of dynamic and static response of cylindrical liquid storage tanks', Ph.D. Dissertation, Rice University, Hous- ton, TX, 1981

3 Veletsos, A.S. and Kumar, A. 'Numerical response of vertically excited liquid storage tanks', Proc. 8th World Conf. on Earthquake Engng, San Francisco, 1984, VII, 453~460

4 Haroun, M.A. and Tayel, M.A. 'Response of tanks to vertical seismic excitations', Earthquake Ending and $truct. Dyn. 1985. 13. 583-595

5 Haroun. M.A. and Tayel, M. A. "Axisymmetrical vibrations of tanks--numerical', J. Enflng Mech. Div., ASCE 1985, I I I. 329 345

6 Haroun, M.A. and Tayel, M. A. 'Axisymmetrical vibrations of tanks--analytical', J. Ending "Mech. Div.. ASCE 1985. I I 1,346-358

7 Fischer, F.D. and Sccbcr, R. 'Dynamic response of vertically excited liquid storage tanks containing liquid-soil interaction', Earthquake Engna and Struct. Dyn. 1988, 16, 329-342

8 Flfigge, W. Stresses in Shells, 2nd Edition, Springer-Verlage, P.crlin, 1973

9 Liu, P.L.-F. and Cheng, A.H.-D. 'Boundary solutions for fluid- structure interactions', J. Ilydraulics Dit,., ASCE 1984, 110, 51 64

10 Morse, P. and Fcsbach, H. Methc~ts of Theoretical Physics, McGraw-Hill, New York, 1953

I I Williams, A.N. 'Earthquake response of submerged circular cy- linder', d. Ocean Engng 1986, 13, 569-585

12 Williams, A. N. and Mau, S. T. 'Earthquake response of submerged circular arch', J. Waterway, Port, Coastal and Ocean Div., ASCE 1988, 114, 405-422

13 Abramowitz, M. and Stegun, l. A. Handbook of Mathematical Functions, Dover Publications, New York, 1974

Appendix A: s t ructural G r e e n ' s functions

The boundary-value problem for the structural Green's function Vc(z; zo) is given by equations (12), (13) and (14). Taking the Laplace transform of equation (12)

~o ° - ' - -,.- . tTds : , d ' [ v ~ ] = v~(...o)e d . = Zo) (38)

and applying the boundary conditions at z = 0 gives

sA + B + e-':° 17~(s; Zo) = s 4 _ ~4 (I) (39a)

s2C + D + e - ' ' ° 17G(s; :o) = s'* - ~'* (II) (39b)

in which A = V~(0) and B = V~(0) for Case I, and C = V~(0) and D = V~(0) for Case II, are constants to be determined. The inverse transforms of equation (39)

may be readily found t3. For Case I

A Vo(z; Zo) = 2-~ (cosh ,,z - cos ~z)

B + ~ (sinh az - sin az)

1 + ~ {sinh ~(z - Zo) - sin a(z - zo)}

for z > Zo (40)

while for Case II

C V~(z; zo) = ~ (sinh ~z + sin az)

D + ~ (sinh az - sin ~:)

1 + ~ {sinh ~(z - Zo) - sin ~(z - Zo) }

for z > z o (41)

Applying the boundary conditions at z = H allows the constants A, B, C, and D to be determined and results in the final forms of the Green's functions given in equations (16) and (17).

Appendix B: diagonal elements of C and D matr ices

This appendix contains explicit expressions for the matrix elements {c..} and {d=.) corresponding to the integral of the Green's function and its normal derivative ovcr the line scgmcnt containing the singularity,

{c,..} = i + aAz ~ iz, H'o(iz, a)Jo(P,a ) Lp = 1

4 Ko(ppa)io(IJpa) cos kez . H

+ -~ I~p K'o(ltpa)io(ppa) p=: |

aH 2 4ha

+ sin fl~, 2 2 2 c~s 2ft. + ~ (~ - 2ft.)

+2~-~a z In ~ - 1 f o r l ~ < m ~ < g

{d,m} = aA: ~ Ho(ppa)Jo(g~a ) !

4 Ko(l~pa)lcos 2 kpzm H

4 ~ [Ko(it, a)lo(l~,a) - H ] + ~ ~ cos 2 k~z. p = l

(42)



1 2ha [cos /~ . In(2 - 2 cos 2/~.) + sin/~.(2/~. - n)]

for 1 ~< m ~< ~7

where

H (44)

#p = ~-~ (43) 1.2p

l<~p<~N p > N (45)


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