Transient failure analysis of liquid-filled shells PART II: Applications

Nuclear Engineering and Design 117 (1989) 141-157 North-Holland, Amsterdam

141

TRANSIENT FAILURE ANALYSIS OF LIQUID-FILLED SHELLS PART II: APPLICATIONS * Wing Kam LIU * * and Rasim Aziz URAS * * * Northwestern University, Department of Mechanical Engineering Evanston, Illinois 60208, USA Received 16 May 1989

In this paper, applications of the theory developed in the companion paper by Liu and Uras [1] are presented. The effects of various types of ground motion on the dynamic stability of the fluid-structure system are analyzed. The stability criteria of liquid-filled shells subjected to horizontal and rocking excitation, shear loading, bending/shear combined loading, and vertically applied load are established. The resulting instability regions and stability charts are given in tables and in 0~- c plots. Under horizontal and rocking motion, modal coupling in the circumferential direction as well as in the axial direction is observed. The possible buckling modes for a tall tank can be identified as cos20, cos30 and cos40 under horizontal and rocking motion, and cos50 and cos60, under vertically applied load. For a broad tank two sets of instability modes are found: cos60 through cos90, and cos120 through cos140 under horizontal and rocking motion. When subjected to vertically applied load, the failure modes of a broad tank shift to cos100 through cos120, and cos140 through cos150. The effect of shear load on a broad tank appears to be important only if damping is relatively small. Under bending/shear combined loading, the bending forces dominate the stability of the fluid-filled shells.

1. Introduction

In a companion paper, Liu and Uras [1], the governing finite element equations for the dynamic buckling analysis of fluid-structure systems have been developed. In this paper, the stability characteristics of anchored liquid-filled shells subjected to horizontal and rocking, shear load, bending/shear combined load, and vertically applied load are studied to provide answers to the following questions: (1) Which of the loads (horizontal, vertical and rocking) that the tank sustains during seismic excitation is responsible for buckling damage? (2) Is the behavior of the enclosed fluid of importance in correctly identifying the buckling loads and modes? (3) For failure conditions in which the buckling modes are not exactly known from the limited shaking table experiments, can the significant bifurcation solutions be identified (dynamic instability)? (4) Is buckling predominantly influenced by the stresses resulting from the lowest response mode (i.e. beam type (cos0) mode)? If this is true, how can the cos(n0)-type modes, especially the cos(30) and cos(40) modes for a tall tank, be excited? Preliminaries

As frequently encountered in seismic analysis, the response of the fluid-structure system is dominated by only a few modes. With this assumption, eq. (7.32) of the Part I paper by Liu and Uras [1] can be simplified to ii + C h + A u + cGu cos tot = 0, * This research is supported by National Science Foundation Grant No. CES-8614957. * * Professor. * * * Graduate Student.

0029-5493/89/$03.50 © Elsevier Science Publishers B.V.

(1.1)

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W.K. Liu, R.A. Uras / Transient failure analysis of liquid-filled shells H

where o~e and { represent a typical dominant frequency and the normalized amplitude of the seismic excitation, respectively. A superposed dot denotes time differentiation; and u is the vector of modal displacement amplitudes. When mass proportional damping is adopted, the ith and nth components of the damping matrix, C, are given by 0a? ---'"

C,,=2~"

fori=l ..... I

omtn O~rnin

and

n = l . . . . . N;

(1.2)

where I and N are the total number of modes in the axial and circumferential directions, respectively; ~Omi. and ~min are the lowest frequency and the corresponding damping coefficient, respectively. A is a diagonal matrix of natural frequencies, 0a~,. G is the block-tri-diagonal matrix (G)I 1

(G)I 2

0

(a)=

0

0

0

0

o

o

o

o

0

0

0

(G)n,n+I

0

0 0

0

G=

0 0

0 0

(G)n,n-1 0

(G)nn (a).+a,.

0

0

0

0

0

0

0

0

(G)N_I,

0

(G)N,N-1

(1.3) N

(G)NN

The components of G are given as (see eqs. (7.26) through (7.28), Liu and Uras [1]) z z 0 0 {Gij,m=rr[%azgijnmNil + {oaogij,mN;1]

for m = n,

(1.4a)

{Gijnm = 2 [%bzgijnmNh + {°bog~"'N6 - %°bzogijnmN;1 l

for m = n - 1,

(1.4b)

{Gijnm =

for m = n + 1,

(1.4c)

z

-~

z

zO

zO

{zbzgiSnmNiZl+ {obeg~,,,N°l + {zobzogijnmN~l

where N~ are the Fourier coefficients of the time-dependent membrane force amplitudes, N~(t), a = z, 0, z0. They are given in eqns. (7.15) of Liu and Uras [1]:

N~°(z, O, t) = N~(t)F~(z)[a~ + b~ cos 0],

(1.5a)

N°(z, O, t) = N°(t)Fo(z)[ao + be cos 0],

(1.5b)

Nz°(z, O, t ) = NZa(t)Fzo(z)b~o sin O,

(1.5c)

where a~, a o, b~, bo and b~o, which represent the effect of vertical, horizontal and rocking excitation, are the membrane force coefficients. The dynamic stability analysis is performed by considering eq. (1.1), and applying the criteria established by Hsu [2]. Accordingly, instability is possible when

{2 1<

2 {cr

0.2

(1.6)

Equation (1.6) is the equation for a hyperbola in which the location and shape are governed by (~, %r)

W.K. Lit~ R.A. Uras / Transient failure analysis of liquid-filled shells H

143

and o. In terms of the circumferential mode numbers n and m and axial mode numbers i and j, they are given by ~=(.OindrO~m

rn=n

or n + l ,

2 %r =

m = n

orn+l,

m

orn+l.

(1.7)

16~2i"t°~"t°3m

,.'L.C,.,.Gj,m. ~mi'(¢02n+t02m)

n

and

(1.8) (1.9)

tOmin

Generally, the resulting instability regions are depicted on to~¢ plots which are commonly referred to as stability charts. For seismic analysis, the frequency spectrum of interest is between 0.2 and 30 Hz. Consequently, for each circumferential mode n only two axial modes are required to establish the stability charts. In this case, the axial block matrices become (G).m--LG21

G2 2 ,,~

form--n-I,

n, n + l .

(1.10)

Hence, the location and the shape of the instability regions can be obtained by employing eqs. (1.7) through (1.9) for i, j = 1, 2. In the next section, the dynamic stability characteristics of liquid-filled shells subjected to horizontal and rocking ground motion are described. It is found t h a t dynamic buckling is caused by the beam bending mode. Dynamic buckling due to shear is treated in Section 3. The combined effect of bending and shear forces on the dynamic buckling of liquid-fiUed shells is discussed in Section 4. In Section 5, results are presented for the case of vertically applied load. Finally, a concluding remark is given in Section 6. The geometrical data for the broad tank and the tall storage tank used in this paper are: R = 2.6 m(102.0 in) = radius of the shell, h -- 0.0005 R = thickness of the shell, L = 4.292 R (tall tank) and 1.667 R (broad tank) -- length of the shell, H = 0.9 L = depth of the fluid, E -- 207.0 GPa(30.0 x 106 psi) - Young's modulus, ~,-- 0.3 -- Poisson's ratio, p = 7.86 x 103 kg/m3(0.735 x 10 -3 lb-s2/in 4) = structural mass density, PF = 0.127p = fluid mass density. The numbers of axial modes, I, and circumferential modes, N, taken are 7, and 20, respectively. Difficulties are observed when more than 7 modes are considered in the axial direction. The matrix equations of the fluid-structure system fail to yield reasonable eigenvalues and eigenvectors, probably due to the ill-conditioning of the resulting matrices. A combination of finite element and Galerkin functions can eliminate this numerical problem. In computing the added mass, the modified Bessel functions of the second kind of order n, denoted by I,, ranging from 0 to N are needed. A recursive relation can be used to obtain higher orders starting with n = 0 and n = 1. However, in computer implementations erroneous results are obtained due to round-off error accumulation. To avoid this difficulty, reverse recursion is implemented, Fr~Sberg [3].

144

W.K. Liu, R.A. Uras / Transient failure analysis of liquid-filled shells II

The axial integrals are evaluated using a 8-point Gauss quadrature scheme. It yields very accurate results except for the case when the tank is full since orthogonality conditions are not met for this case. In this study, ~ , is chosen accordingly to obtain approximately the same lowest values of %~, in an - - c chart for horizontal and rocking motion, shear and b e n d i n g / s h e a r load. To compare the effects of the breathing (n = 0) mode versus the bending mode, i.e. translational (n = 1) mode, the same value of ~'~, is used for the vertically applied load and the horizontal and rocking motion cases.

2. Applications to liquid-filled shells under horizontal and rocking seismic excitation If the liquid-filled shell is subjected to horizontal and rocking seismic motion, only the translation (n = 1) mode is excited. By neglecting the shear effects, the membrane forces are reduced to

N°(z, O, t) = NZ(l) cos 0, N°(z, O, t ) = 0, Nz°(z, O, t)=0.

(2.1a) (2.1b) (2.1c)

That is, az = a e = be = bze = 0, bz = 1 and Fz(z) = 1. Consequently, the transformed geometric stiffness matrix, G, becomes a symmetric matrix with zero block-diagonals 0

(G)I 2

0

0

0

0

o 0

0

0

o (G).,.+I

o 0 G=

0

v

(2.2)

0

0

(a).,.+l

0

0

0

0

(6)N-~:

0

0

0

0

( G TN - I , N

0

where its components can be written as EGun m =

2

irZ

~zgijnrnlVil

for m = n -- 1, n + 1.

(2.3)

Due to this form of G, modal coupling in both the axial and circumferential directions are observed. Since two axial modes are considered for each n, the governing parameters of the instability regions due to horizontal and rocking ground motion are: (a) circumferential coupling (n, n + 1) for axial mode coupling (1,1) 4~",'-¢ 60~,60~,-+ 1 = (601n "F 601,n+1),

E:cr=

60minGlln,n+l

,

~,,,, o = --(60~,

2 + 601,,+,),

(2.4a)

~'i" (601z,+ °°2z,,+1),"

(2.4b)

60rain

.

(b) circumferential coupling (n, n + 1) for axial mode coupling (1, 2) 4~min¢603n603 n+ l = (601n q'- 602,n+1),

•cr

60minG12n,n+ 1

60rain

(c) circumferential coupling (n, n + 1) for axial mode coupling (2,1) 4 ---- (602. + 601,n+1),

¢cr =

3

3

~min ¢602n601,n + l , 60minG21n,n+l

o=

~min (o,L + 60,L+,); 60rnin

W.K. Liu, R.A. Uras / Transient failure analysis of liquid-filled shells II

145

Table 1 Frequency spectrum and instability regions for a tall shell subjected to axial membrane force due to bending

Frequency

Parameters governing the instability regions 2 2 (m, ~); o--0.0010 (o~.+o)j,n+D ~-.min=0.002; retain= 1.92Hz, *** represents f.~ > 1

axial mode

n!

i=l teln

i=2 tt~

4.13 2.02 1.92 2.35 2.82 3.31 3.80 4.31 4.87 5.43 6.11 6.87 7.71 8.61 9.64 10.69 11.87 13.08 14.31 15.60

12.10

6 7 8 9 l@ 11 12 13 14 15 16 17 18 19 20

b (n,n+l),(l,2)

(m, F.cr)

1

2 4

a (n,n+l),(l,l)

(6.15,***)

c (n,n+l),(2,1)

d (n,n+l),(2,2)

(m, Era)

(~i, r~-,t)

(~, tcr)

(11.44,***)

(14.12,***)

(19.41,***)

(9.23,***) (7.80,***) (7.76,0.80) 48.64,0.75) (9.90.0.76) (11.28,0.80) (12.66,0.85) (14.08,0.95) (15.70,***) (17.29,***) (18.60,1.23) (19.91,1.39) (21.50,1.79) (23.37,2.22) (25.50,3.30) (27.92,***) 00.46,***) (33.16,***) (36.06,***)

(12.77,***) (10.39,***) Q0.27,***) I11.44,0.86) (13.07 0.91) (14.75.***) (16.45.***) (18.24,***) (20.01,***) (21.31,***) (22.19,***) (23.16,***) (24.53,***) (26.30,***) f28.47.***) (31.00,***) (33.71,***) (36.66,***) (39.93,***)

7.31 (3.94,0.77) (7.48,***) 5.46 (4.26,0.54) (6.85,***) 4.94 15.17,0.50) {7.68,***) 5.33 46.13,0.46) 48.93,0.88) 6.11 (7.11,0.46) (10.27,0.89) 6.96 (8.11,0.46) 11.59,0.89) 7.79 (9.18,0.48) (12.97,0.94) 8.65 (10.30,0.53) (14.46,***) 9.59 (11.54,0.54) (15.85,***) 10.42 (12.98,0.53) 117.00,***) 10.89 (14.58,0.61) (18.17,***) 11.30 (16.32,0.76) (19.56,***~ 11.85 (I 8.25,0.99) (21.28.***) 12.68 420.33,***) (23.27,***) 13.63 (22.56,***) ~25.53,***~ 14.84 (24.95,***) (28.03,***) 16.16 (127.39,***) (30.63,***) 17.55 (29.91o***) ,. (33.41,***) 19.10 (32.56,***) (36.43,***)

(d) circumferential coupling (n, n + 1) for axial mode coupling (2,2) 3

3

41,.,.~2."2..+, = (~2. + ~2,.+1),

C~r =

~,.i.G22.,.+1

,

•',.i. (.~. + o~22..+,)o= ~,,,i.

(2.4d)

In tables 1 and 2, frequency spectra and corresponding list of parameters governing the instability regions are given for a tall tank and a broad tank, respectively. Only values of Ccr less than or equal to one are tabulated, with the exception of those printed in bold and bold italic. The stability characteristics for a tall tank are depicted in fig. 1; where fig. 2 presents the same set of parameters for a broad tank. General characteristics of the two tanks can be summarized as follows: (1) Under horizontal and rocking motion, modal coupling in the circumferential direction as well as in the axial direction is observed; (2) For each n, circumferential buckling modes " j u m p " between n and n + 1; and the first axial mode is the most significant mode. The influence of axial modal coupling, i.e. switching between axial mode one and axial mode two, is enhanced only if damping is small (refer to tables 1 and 2); (3) A broad tank is more susceptible to buckling when compared to a tall tank, since the size of instability regions is larger (refer to figs. 1 and 2); (4) The instability regions of the tall tank are sensitive to a small increase in damping (table 1); whereas the broad tank shows similar characteristics (table 2), however, with damping four times that of the tall tank;

146

I,KK. Liu, R.A. Uras / Transient failure analysis of liquid-filled shells II

Table 2 Frequency spectrum and instability regions for a broad shell subjected to axial membrane force due to bending

Frequency

Parametea~governing the instability regions 2 2 (~, ¢.~); o=0.0018 (~,+toj,n+l) ~mi.~ 0.0065; ¢-Omin=3.61Hz. *** represents ~. > 1 except those in bold italic and bold

axial mode

n

i=l O)ln

i=2 ¢~n

1 13.98 29,2~ .2., 3 4 5i 6i 7 8 9 10 I 11 I

8.71 5.77 4.27 3.67 3.61 3.88 4.30 4.79 5.30 5.84 12 6.42 13 7.00 14 7.70 15 8.41 161 9.23 17 I 10.11 lfl 10.99 19 11.92 20 12.96

23.55 18.08 14.68 12.55 11.11 10.16 9.69 9.68 10.03 10.65 11.41 12.24 13.12 14.02 14.94 15.90 16.92 17.98 19.15

a (n,n+l),(1,1) (~, ~ ) (22.69,*** I

(14,48,11.0) (10.04,*** I (7.94,*** I (7.28,0.51) ~7.49,0.501 (8.18,0.51) (9.09,0.49~ (10.09,0.49) (11.13,0.52) Q2.26,0.56)

b (n,n+l),(l,2)

c (n,n+l),(2,1)

d (n,n+l),(2,2)

(m, ~ ) (~, e~) (37.54,***~ (37.95,***) i (26.79,***) (29,32 57.4) (20.45r***) (22.35,***) (16.82, *'~*) (18.35,***) (34.78,*** I (16.17,*** ! (13.77~1.44) (14.99,***)

113.57,1.12~ (13.98,0.98)

(14.82,0.92) (15.95,0.88) (17.24,0.91) ~13.42,0.611 (18.67,0.94) (14.70.0.691 (20.12,0.97) (16.11~0.78~ 121.72,***) (17,64,0.91) (23.35,***) {19.34"**)] (25.13,*** I (21.10,***) (27.02,***) (22.92,***) (28,98rl.70) ~24.88,***I ~31.07,***I (27.01,***) . (33.36,***)

(m, ~r) I52.79,***) (41.64,***) (32.76,***) (27.23,0.84) ~23.67,0.94) (21.27,***) ~14.46,1.16t (19.84,***) (14.47,0.95) (19.36.**) 114.98~0.82) (19,71,*** I (15.87¢0.82~ (20.68,*** I (17.07,0.85) (22,06,***) (18.41r0.90) (23.65,*** I (19.94,0.97) (25.36,*** I (21.53,***) (27.14.*** I (23.25,***) 1(28.96,1.19) ~25.05,***) ~30,8S,1.23) (26.90,***) (32.82,***) ~'~$,84,1.71) 134.90,***) ~30.94,***) (37A3r***) (33.20,***) (39.55,***)

possible bucklingmodes: cos70, cos80 and cos90 oosl00 and cos110

possible bucklingmodes eos20, cos30 and cos40 - - ~

F

1.2 ......... ez

n=2,3,4

n=7,8,9 n=10,11

1.0' 0.8'

0.6' 0.4' 0.2' 0.0

f' i n=l i=l

5

First axial mode

n=l i=2

10

Second axial mode

|

15

20

co(Hz.)

Fig. 1. Overall stability spectrum for a tall shell subjected to axial membrane force due to bending.

W.K. Liu, R.A. Uras / Transient failure analysis of liquid-filled shells H

147

possible buckling modes: cos80 and cos90 eosl20, eosl30 and eos140 1.2 n=8,9

8 z

n=12,13,14

vV

1.0'

Ill I

0.8' 0.6' 0.4' 0.2'

n=l i=1

n=l i=2 Sectmd axial mode

0.0

|

5

|

10

15

|

20

|

25

o(Hz.) First axial mode Fig. 2. Overall stability spectrum for a broad shell subjected to axial m e m b r a n e force due to bending.

(5) When a fluid-structure system is subjected to seismic excitation, the cos0-type mode is the dominant mode (Shih and Babcock [4]), and the initiation of instability is expected at this frequency. The possible circumferential buckling modes due to ithe first and second cos0-axial mode excitations are also depicted in figs. 1 and 2 for the tall tank and the broad tank, respectively; (6) For a tall tank, within a + 5% margin of the (1, 1) cos0-mode frequency, i.e. ton = 4.13 Hz, the tank will buckle with %r < 1 at the following mode shapes and frequencies (see table 1 and fig. 1): (a) axial mode (1, 1) and circumferential mode coupling (2, 3) and at frequencies (2.02 and 1.92 Hz); (b) axial mode (1, 1) and circumferential mode coupling (3, 4) and at frequencies (1.92 and 2.35 Hz); (7) For a tall tank, within a + 5% margin of the (2, 2) cos0-mode frequency, i.e. ,o22= 12.10 Hz., the tank will buckle with ccr < 1 at the following mode shapes and frequencies (see table 1 and fig. 1): (a) axial mode (1, 2) and circumferential mode coupling (7, 8) and at frequencies (3.80 and 7.79 I-Iz); (b) axial mode (2, 1) and circumferential mode coupling (8, 9) and at frequencies (7.79 and 4.87 Hz); (c) axial mode (1, 1) and circumferential mode coupling (10, 11) and at frequencies (5.43 and 6.11 Hz); (8) For a broad tank, within a + 5% margin of the (1,1) cos0-mode frequency, i.e. ton = 13.98 Hz., the tank will buckle with %~ < 1 at the following mode shapes and frequencies (see table 2 and fig. 2): (a) axial mode (1, 1) and circumferential mode coupling (12, 13) and at frequencies (6.42 and 7.00 I-Iz); (b) axial mode (1, 1 ) a n d circumferential mode coupling (13, 14) and at frequencies (7.00 and 7.70 I-Iz); (c) axial mode (1, 2) and circumferential mode coupling (8, 9) and at frequencies (4.30 and 9.68 Hz); (d) axial mode (2, 1) and circumferential mode coupling (8, 9) and at frequencies (9.69 and 4.79 Hz); (9) For a broad tank, within a + 5% margin of the (2,2) cos0-mode frequency, i.e. to22= 29.24 Hz., the tank will not buckle with %~ < 1.

W.K. I t'u, R.A. Uras / Transient failure analysis of liquid-filled shells H

148

3. Applications to liquid-filled shells under shear loading Under shear loading, the membrane forces take the following form

N°(z, O, t)= O,

(3.1a)

N°(z, O, t)= O, N~°(z, O, t) = NZ°(t) sin O.

(3.1c)

The matrix G and its components are given as

0

(G)12

(a) 2 o

o

o

0

(G T

0

0

0

G=

0

0

0

0

o

o o

o o

o

0

0

0

0

(3.2)

(G)N-1,N 0

T (G)N_I,

N

0

and zO IrzO cGijn, n = -- ~CzogijnmlVia cGijnm =

~T

zO

. ~zO

-~%ogij,,#vfl

for m = n - 1,

(3.3a)

for m = n + 1,

(3.3b)

respectively. Since G takes the same form as in eq. (2. 2), the same types of modal coupling and instability regions are obtained. The four instability regions corresponding to each n and the two axial modes are given in eqs. (2.4). The governing parameters, ~, %r and o for a broad tank are tabulated in table 3. In this particular type of loading the instability regions for eqs. (2.4a) and (2.4d) disappear from the w-c plot since %o is greater than 1 even when a small amount of damping is introduced. This dictates that only axial mode-coupling (1, 2) and (2, 1) are important in dynamic buckling. Also for this reason, it is very difficult to identify the buckling modes and frequencies in an experiment. The overall stability spectrum is given in fig. 3. The possible circumferential buckling modes due to the first and second cos0-axial mode excitations are also depicted in fig. 3. The stability characteristics of a broad tank under shear loading can be summarized as follows: (1) In contrast to the case of horizontal and rocking excitation, the axial and the circumferential couplings play a significant role in dynamic buckling; (2) The first and second axial modes (i.e. (1, 1) and (2, 2)) can only be excited if the system damping is extremely small (refer to columns a and d in table 3); (3) Half of the damping in the previous axial loading case is needed to obtain similar criteria for a broad tank (compare table 3 with table 2); (4) The second axial shear mode, i.e. n= 1 and i = 2, is relatively unimportant; however, it is more important than that of the bending case (compare table 3 with table 2 for n -- 18). (5) Within a + 5% margin of the (1, 1) cos0-mode frequency, i.e. ton = 13.98 Hz., the tank will buckle with %r < 1 at the following mode shapes and frequencies (see table 3 and fig. 3): (a) axial mode (1, 2) and circumferential mode coupling (6, 7) and at frequencies (3.61 and 10.16 Hz.);

W.K. l_~u, R.A. Uras / Transient failure analysis of liquid-filled shells H Table 3 Frequency spectrum and instability regions for a broad shell subjected to shear force

regions 2 2 (m, r~); o=0.0009 (0-~n+mj,n*0 ~nin= 0.0032; mmi~ffi3.61Hz. *** represents ecr > I except those in

Frequency

Parameters governing the insmbilky

axial mode

bold italic and bold i=l co,u

iffi2 ¢e2n

1 13.98

29.24

2 3 4 5 6 7

23.55 18.08 14.68 12.55 11.11 10.16 9.69 9.68 10.03 10.65 11.41 12.24 13.12 14.02 14.94 15.90 16.92 17.98 19.15

n

8

9 10 11 12 13 14 15 16 17 18 19 20

8.71 5.77 4.27 3.67 3.61 3.88 4.30 4.79 5.30 5.84 6.42 7.00 7.70 8.41 9.23 10.11 10.99 11.92 12.96

a (n,n+l),(l,l)

b (n,n+l),(l.2)

c (n,n+l),(2,1)

d (n,n+l),(2,2)

(m,e~) (m.~ ) (m,e~) (37.54,***) (37.95,***) (52.79~***) {ld.d&r529.) (26.79,***) 429.32.42.2) (41.64,***) 410.04,***) (20.45,***7 422.35~***) 432.76,***) (7.94,***) (16.82,***) 418.35,'**) 427.23,***) (7.28~***) 414.78,***) 416.17~***1 423.67,***) (7.49,***) 413.77~0.84) 414.99,***) 421.27,***) (19.84,***) (8.18,***) i(13.57,0.65) (14.46,0.80) 49.09r***) ](13.98,0.58 I (14.47~0.69) (19.36,***) (10.09~***) 04.82~0.57) 414.98,0.63) 09.71,***) (11.13,***) (15.95,0.567 05.sv,o.e4? (20.68,***) 412.26,***) 417.24,0.58) (17.07,0.65) (22.06,***) (13.42,8.73) (18.67~0.60) 18.41,0.69) (23.65,***) (14.70,***) (20.12,0.62) 419.94~0.70) 425.36,***) 416.1lt***) (21.72,0.69) (21.53~0.727 427.14,***) (17.641***) (23.35,0.68) (23.25,0.79) (28.96,10.6) 419.34~***) (25.13T0.78) 425.05~0.77? 00.851"**) (21.10,***) (27.02,0.83) (26.90,0.85) 432.82,***) (22.92,***) (28.98,0.88)[ (28.84,0.89) (34.90,***) (24.88,***) (31.07,0.92) (30.94,0.94) (37.13,***) (27.01,***) (33.36,***) (33.20,0.95) (39.55,***) (m,~) (22.(0r***)

possible bucklingmodes: cos60, cos70, co580 and oos90 - ~

possible bucklingmodes: cos189 and coslgO "~

1.2 n 67.8. ,9b*

e~

n=18,19

1.0' 0.8'

0.6'

0.4'

nll i 1

0.2"

nffil iffi2 Seccmdaxial mode

0.0

|

0

10 ~

15

|

i

20

25 W (l'Iz.)

Fwst axialmode

Fig. 3. Overall stabifity spectrum for a broad shell subjected to shear force.

149

150

W..K. I_,iu, R.A. Uras / Transient failure analysis of liquid-filled shells II

(b) axial mode (1, 2) and circumferential mode coupling (7, 8) and at frequencies (3.88 and 9.69 Hz.); (c) axial mode (1, 2) and circumferential mode coupling (8, 9) and at frequencies (4.30 and 9.68 Hz.); (d) axial mode (2, 1) and circumferential mode coupling (7, 8) and at frequencies (10.16 and 4.30 Hz.); (e) axial mode (2, 1) and circumferential mode coupling (8, 9) and at frequencies (9.69 and 4.79 Hz.); (6) Within a + 5% margin of the (2, 2) cos0-mode frequency, i.e. o~22= 29.24 Hz., the tank will buckle with %r < 1 at the following mode shapes and frequencies (see table 3 and fig 3): (a) axial mode (1, 2) and circumferential mode coupling (18, 19) and at frequencies (10.99 and 17.98 Hz.); (b) axial mode (2, 1) and circumferential mode coupling (18, 19) and at frequencies (16.92 and 11.92 Hz.). Remark: Looking at the possible buckling frequencies listed in items 5 and 6, we see that the possible post-buckling frequencies can jump from a low frequency to a relatively higher frequency, e.g. from 3.61 Hz to 10.16 Hz (Case 5a). This is because cross-couplings among axial modes 1 and 2 and circumferential modes n and n + 1 are likely to occur.

4. Applications to liquid-filled shells under combined bending/shear loading If the liquid-filled shell is subjected to combined bending/shear loading due to horizontal and rocking seismic motion, the membrane forces become

N°(z, O, t) = NZ(t)bz cos O,

(4.1a)

N~(z, O, t ) = O,

(4.1b)

N ° ( z , 07 t) = NZ°(t)b~e sin O.

(4.1c)

The values of b,o and b~ determine the relative weight of the bending and shear effects. Five cases are studied: (b~, bz0) = (1.0, 0.0), (0.0, 1.0), (1.0, 0.5), (1.0, 1.0) and (0.5, 1.0). Additionally, the normalized membrane force amplitudes % and cze are set equal to c since the relative importance of bending and shear is determined by b, and b,o. The transformed geometric stiffness matrix, G, becomes a symmetric matrix with zero block-diagonals

0

(6)12

0

0

0

0

o

(6).,.+1

0

0 0

0

o

=

0

0

0

0

(6 T n,n+l

0

0

0

0

0

0

(4.2)

(6)u-l,N

0

T

0

(6)u-~,N

0

where its components can be written as

¢£Gijnm z z , G i j n m ---- 2 ¢ ( b z g i j n m N l l +

OzogijnmlVil ) -

zO

,,,zO\

Ozogijnml¥il ]

for

m = n - 1,

(4.3a)

for

m = n + 1.

(4.3b)

The same types of modal coupling and instability regions are obtained since G takes the same form as in eq. (2.2). The four instability regions corresponding to each n and the two axial modes are given in eqs. (2.4).

W..K. Liu, R.A. Uras / Transient failure analysis of liquid-filled shells II

151

Table 4 Frequency spectrum and instability regions for a broad shell subjected to bending and shear forces (b z = 1; bze = 1)

Parameters governing the instabifity regions 2 2

F~quency

(m, e~); ~=0.0018 (mi,+o>j,n+l) ~min=0.0065; e0min=3.61Hz. *** ~ t s ~ • 1 except those in bold italic and bold

axial mode

n

i=l COin

i=2 ~

a (n,n+l),(1,1) (m, ~ )

1 13.98 2 9 . 2 4 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

8.71 5.77 4.27 3.67 3.61 3.88 4.30 4.79 5.30 5.84 6.42 7.00 7.70 8.41 9.23 10.11 10.99 11.92 12.96

23.55 18.08 14.68 12.55 11.11 10.16 9.69 9.68 10.03 10.65 11.41 12.24 13.12 14.02 14.94 15.90{ 16.92 17.98 19.15

{22.69,***)

{14.48,11,2) {10.04,***) {7.94,'**) 17.28,0.64) {7.49,0.52) {8.18,0.49) {9.09,0.49) (10.09,0.47) {11.13,0.47) 02.26,0.51t

{13.42,0.54) (14.70,0.61) {16.11,0.69) (17.64,0.79) 09.34,0.93) 21.10,***) (22.92,***) (24.88,***) (27.01,***)

b (n,n+l),(l,2) (m, ~ )

(37.54.***) {26.79,***) (20.452**) (16.82,***) I14.78,***)

c (n.n+l),(2,1)

d (n,n+l),(2,2)

(m, e~.)

(ca, ~.) {52.79,***) {41.64,***) {32.76,*** I {27.23,0.84) (23.67,0.93) (21.27,***) (19.84,***) {19.36,***) {19.71,0.98) 420.68,0.97) (22.06,1.00) {23.652**) (25.36,***) (27.14,***) 28.96,1.13) (30.85,***) {32.82,***) (34.90,***) {37.13,***) (39.55,***)

{37.95,110.) (29.32r33.1) {22.35,***) (18.35,***) {16.17,***) {14.99,***)

{13.77,0.75) {13.57,0.581 {14.46,5.25) (13.98,0.52) (14.47,3.55) 04.82,0.49) (14.98,***) {15.95,0.48) {15.87,***) {17.24,0.49) {17.07,***) (18.67,0.51) {18.41,***) {20.12,0.53) (19.94,***) {21.72,0.59) {21.53,*** t {23.35,0.60) 123.25,***) 125.13,0.69) (25.05,***) 127.02,0.76) (26.90,*** I (28.98,0.83)(28.84,43.4) (31.07,0.90) (30.94,***) {33.36,***) (33.20,***)

The same m o u n t of damping is used for the five loading cases. Under this condition, bending effect totally dominates the stability characteristics. Case 1: bz = 1; bze ~- O, refer to table 2. Case 2: b~ = 0; b~0 --- 1, shear effect is totally negligil~le, since all values of %r > 1. The corresponding table is excluded. Case 3: b~ = 1; bz0 = 1, refer to table 4, very small contribution from shear is observed. Case 4: b~ = 1; bze --- 0.5, refer to table 5, bending dominates the spectrum. Case 5 bz = 0.5; bz0 = 1, refer to table 6, bending effect is comparable to shear effect. In summary, a broad tank will possibly buckle due to axial forces induced by bending.

5. Applications to liquid storage tanks under vertically applied load In a seismic event, the axial stress induced by the breathing mode is very small as compared to that caused by the translational mode, i.e., az << b~. On the other hand, the hoop stress due to the n = 0 mode can be of the same order of magnitude of the axial stress due to the n = 1 mode. Nevertheless, in the response spectrum of interest, this circumferential stress is negligibly small in comparison to the critical buckling hoop stress. Hence, the stability in the low frequency range is mainly governed by the axial stress induced by the horizontal and rocking excitations. In this analysis, a~ is included to obtain a better

W.K. Liu, R.A. Uras/ Transientfailure analysisof liquid-filledshells II

152

Table 5 Frequency spectrum and instability regions for a broad shell subjected to bending and shear forces (bz = 1; b~0= 0.5)

Frequency

Paramete~ governing the instability regions 2 2 (m, ecr); O=,,0.0018(o.h.+coj.n+l) ~i.= 0.0065; ohni.= 3.61Hz. *** rewesents p.~ > 1 except those in bold italic and bold

axial mode

n

i=l ¢01n

i=2 a O>Zn (n,n+l),(l.l)

1 13,98 2 9 , 2 4 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

8.71 5.77 4.27 3.67 3.61 3.88 4.30 4.79 5.30 5.84 6.42 7.00 7.70 8.41 9.23 10.11 10.99 11.92 12.96

23.55 18.08 14.68 12.55 11.11 10.16 9.69 9.68 10.03 10.65 11.41 12.24 13.12 14.02 14.94 15.90 16.92 17.98 19.15

b (n.n+l),(l.2)

c (n,n+l),(2,1)

(~, ~ ) (m, e~) (m, ~ ) (22.69,***~ (37,54,***) (37.95,***) (/4.48r11.1 } (26.79,***) (29.32.***) ( 10104r***) (20A5,***) (22.35.***~ (7.94~***) (16.82,***) ~18.35,***) (7.28,0.63) (14.78,1.72) 06.17,***) (7.49,0.52) (13.77,0.99) (14.992"'t (8.18,0.50) (13.57,0.76)(14.46,1.90) (9.09r0.50) (13.98,0.68) ~14.47,1A9) (10.09,0.48) (14.s2,0.64) (14.98,***) (11.13,0.48) (15.95.0.62) 115.87,***) (12.26,0.51) 117.24,0.64) 117.07,***) (13.42,0.557 I18.67.0.66) I18.41,***) (14.70,0.61) (20.12,0.68) (19.94,***) (16.11,0.69) 121.72,0.77) (21.53,***) (17,64r0.79) (23.35,0.78) (23,25r***) (19.34,0.92) (25.13,0.91) (25.05,***~ (21.10~***) (27.022**) (26.90,***) (22.92,*** I (28.98,1.12)(28,84,3.57) (24.88~***) (31.07,***) (30.94,***) (27.01,***) (33.36,***) (33.20,***)

d (n,n+l).(2,2) (m, ~ ) (52.792**) (41.64.***) (32.76.***) (27.23,0.84) (23.67,0.94) (21.27,***) (19.84,***) (19.36,***) (19.71,***) (20.68,***) (22.06,***) (23.65,***) (25.36,***) (27.14,***) 28.96,1.16) (30.852**) (32.82,***) (34.90,***) (37.13,***) (39.55,***)

understanding of the relative significance of a~ versus bz. Since only the breathing (n = 0) m o d e is excited b y vertical ground motion, the m e m b r a n e forces b e c o m e

N f ( z , O, t ) = NZ(t),

(5.1a)

NeO(z, O, t ) = O,

(5.1b)

N ° ( z , O, t ) = O.

(5.1c)

T h e t r a n s f o r m e d geometric stiffness matrix, G, b e c o m e s block-diagonal

G=

(G)n 0

0 (G)22

0

0 0

0

0

0

0

0

0

0

0

0

0

0

(a)..

0 0

0 0

0 0

0 0

0

'

(5.2)

0

0

where its c o m p o n e n t s can be written as

eGij.,, = vtezg~.,mN;l

for

m = n.

(5 3)

153

W.K. Liu; R.A. Uras / Transient failure analysis of liquid-filled shells H Table 6 Frequency spectrum and instability regions for a broad shell subjected to bending and shear forces (b~ = 0.5; b~o = 1)

Frequency

Panunete~ govea'ningthe inslabilityregions 2 2 (m, eer); o=0.0018 (¢em+cej,~.t) ~mm= 0.0065; o~ia= 3.61Hz. *** t'~-'tgesentse.~ > 1 except those in

axial mcxle

bold italicand bold i=l cox.

n

i=2 to2n

a b c d (n,n+l).(l,l) (n,n+l),(l.2) (n,n+l),(2.1) (n,n+l),(2,2)

(m, e=)

(m, e=.)

(=, e=.)

(m, e=)

1 13.98 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

29.24 (22.69r***) (37.54~***) (37.95,***) (52.79,***) 8.71 23.55 (14.48,22.6) (26.79,***) 129.32,46.6) {41.64,***) 5.77 18.08 (10.04,***) (20.45,***) (22.35,*** I (32.76,***) 4.27 14.68 (7.94,***) (16.821***t (18.35.***) (27.23,***) 3.67 12.55 (7.28,***) (14.78,***) (16.17,***) (23.67,***) 3.61 I1.11 (7.49,***) (13.77,1.01) (14.99,***) 121.27,***) 3.88 10.16 (8.18,0.97) (13.57,0.78)(14.46,4.18) (19.84,***) 4.30 9.69 (9.09,0.94) (13.98,0.70) (14.47,4.05) (19.36,***) 4.79 9.68 (10.09,0.91) (14.82,0.671 (14.98,***) 09.71,***) 5.30 10.03 (11.13,0.921 (15.95,0.65) (15.87.***) (20.68,***) 5.84 10.65 (12.26,0.99) (17.24,0.68) (17.07,***) (22.06,***) 6.42 11.41 (13.42,1.04) (18.67,0.70) (18.41,***) I23.65,***) 7.00 12.24 (14.70~***) (20.12,***) (19.94,***) (25.36,***) 7.70 13.12 06.11,***) (21.72,0.81) (21.53,***) (27.14.***) 8.41 14.02 (17.64,***) (23.35,0.81) (23.25***) (28.96,2.13) 9.23 14.94 09.34,***I (25.13,0.94) 125.05,***) (30.85,***) 10.11 15.90 (21.10,***) (27.02,***) (26.90,***) (32.82,***) 10.99 16.92 (22.92,***) (28.98,1.11) 28.84,3.17) (34.90,***) 11.92 17.98 (24.88r***) (31.07,***) (30.94,***) (37.13,***) 12.96 19.15 (27.01,***) (33.36,***) (33.20,***) (39.55,***)

When two axial modes are used in the axial direction, then for each n, three instability regions are observed (note that Gijnm- Gijm), the governing parameters of the instability regions due to vertically applied load are given by (a) circumferential coupling (n, n) for axial mode coupling (1, 1) = 2(.Oln,

Ecr =

4~min~n OOminGllnn ,

0

= 2¢o2, ~r,,i,,., £Omin

(5.4a)

(b) circumferential coupling (n, n) for axial mode coupling (2, 2) = 2602n,

Cer =

¢X3minGz2nn,

(7 = 2 ¢ 0 2

OJmin~rai,n

(5.4b)

(c) circumferential coupling (n, n) for axial mode coupling (1,2) i4' -~"m -~'n "l/

= (,01. + o , 2 . ) ,

'or =

3 3 ¢'01n~O2n

,0.,. V G12..o2,..

~

o =

~min ¢,Omin

+

(54¢)

Tables 7 and 8 include frequency spectra for tall and broad tanks, respectively. Figure 4 depicts the plot for the tall tank, whereas the instability regions of the broad tank are depicted in fig. 5.

o~-e

154

W.K. Liu, R.A. Uras / Transient failure analysis of liquid-filled shells I I

Table 7 Frequency spectrum a n d instability regions for a tall shell subjected to vertically applied load

Frequency axial mode

i=l ¢01.

i=2 ¢02n

Parametersgoverningthe instabilityregions 2 2 (~, ecr); o= 0.0010 (O~in+O)jn) *** representse.~ > I exceptthosein bold italic and bold a (n,n),(1.1)

b (n,n),(2,2)

c (n,n),(l,2)

(~, ~ )

(m, e~)

(~, ~r)

0 6.01 16.71 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 i 17 I 18 19 20

4.13 2.02 1.92 2.35 2.82 3.31 3.80 4.31 4.87 5.43 6.11 6.87 7.71 8.61 9.64 10.69 11.87 13.08 14.31 15.60

(12.02,***) (33.42,***) (22.72,***) 12.10 /8.26,***) (24.20,***) (16.23,30.3) 7 . 3 1 (4.04,0.53) (14.62,0.49) (9.33,***) 5 . 4 6 13.84,0.27) (10.92,0.60') (7.38,0.77) 4 . 9 4 (4.69,0.26) (9.87,0.57,) (7.28,0.48) 5 . 3 3 I5.65,0.23) (10.67,0.43) (8.16,0.40) 6.11 ~6.62,0.23) (12.21,0.40) (9.42,0.39) 6 . 9 6 /7.60,0.23) (13.93,0.48') (10.76,0.41) 7 . 7 9 (8.63,0.23) (15.$8,0.52) II2.10,0.42) 8 . 6 5 ~9.73T0.25') (17.31,0.62) (13.52,0.47) 9 . 5 9 /10.86r0.28') (19.18~0.81') /15.02,0.56) 10.42 (12.22,0.26) (20.84r0.76) (16.53,0.58) 10.89 /13.74,0.27) /21.78,0.55/ (17.76,0.58) 11.30 ~15.41,0.34') I22.60,0.52) (19.01,0.64) 11,85 ~17.22,0.42) I23.71,0.53) (20.46,0.74) 12.68 (19.28,0.58/ (25.35,0.58) /22.32,0.95) 13.63 (21.38.0.76) ~27.26,0.681 /24.32,***/ 14.84 (23.75,***) (29.69,0.78) (26.72,***) 16.16 (26.16,***) (32.31,0.77) (29.24,***) 17.55 (28.61,***) (35.10,0.92) (31.86,***) 19.10 (31.21,***) (38.21,***) (34.71,***) possible buckling modes: cos80 and cosg0 cosl00, cos110 and cos l20

possible buckling modes cos50 and cos60

f

cos130 and cos140

1.2 £z 1.O"

0.8"

0.6"

0.4"

0.2"

0.0

0 Fn-st axial mode

Second axial mode

Fig. 4. Overall stability spectrum for a tall shell subjected to vertically applied load.

W.K. Liu, R.A. Uras / Transient failure analysis of liquid-filled shells II Table 8 F r e q u e n c y s p e c t r u m a n d i n s t a b i l i t y r e g i o n s for a b r o a d shell s u b j e c t e d to v e r t i c a l l y a p p l i e d l o a d

Frequency axial mode

n

i=l col,

i=2 co2,

Paramctcxs governingthe instabilityregions 2 2 (~,e~r);o= 0.0018 (Oin+o~jn) *** represents ~ > 1 except those in bold italic and bold a (n,n).(1.1) (m, ~ )

0 14.65 3 3 . 3 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

13.98 8.71 5.77 4.27 3.67 3.61 3.88 4.30 4.79 5.30 5.84 6.42 7.00 7.70 8.41 9.23 10.11 10.99 11.92 12.96

29.24 23.55 18.08 14.68 12.55 11.11 10.16 9.69 9.68 10.03 10.65 11.41 12.24 13.12 14.02 14.94 15.90 16.92 17.98 19.15

I29.30,***) (27.97,***)

(17.42,10.9) (11.54,***) (8.54,0.75) (7.33,0.36) (7.23,0.26) (7.76,0.25) (8.61,0.25) (9.57,0.25) (10,60,0.24) (11,67,0.25) (1Z85,0.27) II 3,99,0.29)

(15.40,0.33) (16,81,0.36) Q 8.46,0.42) (20.22,0.49) (21.99,0.57) (23.84,0.66) (25.91,0.81)

b (n,n),(2,2)

c (n,n),(l,2)

(t0, ~cr) (m, ~ ) (66.60,***) (47.95,***) (58.51,***) (43.23,***) (47.11,***) (32.26,16.6) (36.17.0.83) (23.85,***) (29.36,0.44) (18.95,***) (25.11,0.39) (16.22,***) (22.22,0.51) (14.72,0.92) (20.31,0.62) (14.03,0.62) (19.37,0.59) (13.99,0.51) (19.35,0.52) (14.46,0.45) (20.06,0.51) (15.33,0.42) (21.30,0.52) (16.48,0.43) (22.82,0.55) (17.83,0.45) (24.49,0.55) (19.24,0.47) (26.24,0.55) (20.82,0.50) I28.04,0.62) 122.43,0.57) (29.88,0.57) (24.17,0.59) (31.81,0.66) (26.01,0.71) (33.83,0.70) (27.91,0.80) (35,96,0.74) (29.90,0.91) (38.30,0.77) (32.11,1.06)

possible bucklingmodes: cosl00, cos110 and cosl20 oasl40 and cosl50

1.2 n=14,15

Ez

n=lO,l1,12

[

1.0"

1

0.8" 0.6" 0.4" 0.2" 0.6

|

5 First axial mode

to(Hz.)

Fig. 5. O v e r a l l s t a b i l i t y s p e c t r u m f o r a b r o a d shell s u b j e c t e d to v e r t i c a l l y a p p l i e d l o a d .

155

W.K. Liu, R.A. Uras / Transient failure analysis of liquid-filled shells H

156

Basic characteristics for tall and broad tanks under vertically applied obtained under horizontal and rocking ground motion. However, larger vulnerability of broad tanks to buckling. Even at a relatively large value instability regions remain in the buckling spectrum (see table 8). The basic difference in characteristics between vertical and horizontal

load are quite similar to those instability regions increase the of damping a large number of excitations can be summarized

as:

(1) (2) (3) (4)

Only axial coupling is observed, and it appears to be crucial in the overall stability of the broad tank; Failure modes for a tall tank under vertically applied load are cos5O and cos68; The possible buckling modes of the broad tank are cos108 through cos120 and cos14Othrough cos158; When the same amount of damping is used in each excitation case, the number of regions under vertically applied load are considerably larger; (5) When a fluid-structure system is subjected to vertically applied load, the breathing (n = 0) mode is the dominant mode, and the initiation of instability is expected at this frequency; (6) For a tall tank, within a + 10% margin of the (1,1) breathing (n = 0) mode frequency, i.e. t011 = 6.01 Hz, the tank will buckle with Ecr _~< 1 at the following mode shapes and frequencies (see table 7 and fig. 4): (a) axial mode (1, 1) and circumferential mode coupling (5, 5) and at frequency (2.82 Hz); (b) axial mode (1, 1) and circumferential mode coupling (6, 6) and at frequency (3.31 Hz); (7) For a tall tank, within + 10% margin of the (2, 2) breathing (n = 0) mode frequency, i.e. to22= 16.71 Hz, the tank will buckle with %r < 1 at the following mode shapes and frequencies (see table 7 and fig.

4): axial mode (1, 1) and circumferential model coupling (13, 13) and at frequency (7.71 Hz); axial mode (1, 1) and circumferential mode coupling (14, 14) and at frequency (8.61 Hz); axial mode (2, 2) and circumferential mode coupling (8, 8) and at frequency (7.79 Hz); axial mode (2, 2) and circumferential mode coupling (9, 9) and at frequency (8.65 Hz); axial mode (1, 2) and circumferential mode coupling (10, 10) and at frequencies (5.43 and 9.59 Hz); (f) axial mode (1, 2) and circumferential mode coupling (11, 11) and at frequencies (6.11 and 10.42 Hz); (g) axial mode (1, 2) and circumferential mode coupling (12, 12) and at frequencies (6.87 and 10.89 Hz); (8) For a broad tank, within a + 10% margin of the (1, 1) breathing (n = 0) mode frequency, i.e. tOll = 14.65 Hz, the tank will buckle with ~cr ~ 1 at the following mode shapes and frequencies (see table 8 and fig. 5): (a) axial mode (1, 1) and circumferential mode coupling (14, 14) and at frequency (7.70 Hz); (b) axial mode (1, 1) and circumferential mode coupling (15, 15) and at frequency (8.41 Hz); (c) axial mode (1, 2) and circumferential mode coupling (10, 10) and at frequencies (5.30 and 10.03 Hz); (d) axial mode (i, 2) and circumferential mode coupling (11, 11) and at frequencies (5.84 and 10.65 Hz); (e) axial mode (1, 2) and circumferential mode coupling (12, 12) and at frequencies (6.42 and 11.41 Hz); (9) For a broad tank, within a + 10% margin of the (2, 2) breathing (n = 0) mode frequency, i.e. tO22= 33.30 Hz, the tank will buckle with %r < 1 at the following mode shapes and frequencies (see table 8 and fig. 5): (a) axial mode (2, 2) and circumferential mode coupling (3, 3) and at frequency (18.08 Hz); (b) axial mode (2, 2) and circumferential mode coupling (17, 17) and at frequency (15.90 Hz); (c) axial mode (2, 2) and circumferential mode coupling (18, 18) and at frequency (16.92 Hz); (d) axial mode (2, 2) and circumferential mode coupling (19, 19) and at frequency (17.98 Hz). (a) (b) (c) (d) (e)

W.K, Liu, R.A. Uras / Transient failure analysis of liquid-filled shells II

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6. Concluding comments The dynamic buckling characteristics of liquid-filled shells subjected to horizontal and rocking motion, shear loading, bending/shear combined loading, and vertically applied loads are investigated in this paper. It is found that in the design of anchored liquid storage tanks, horizontal and rocking seismic excitations appear to be the dominant cause of dynamic buckling. For a tall tank, both the first and second axial beam modes (cos0) will lead to the following buckling modes cos20 through cos40 and cos70 through cosll0, respectively. Similarly, a broad tank would possibly buckle at cos68 through cos90, and cosl20 through cos140. Under a shear load, if the system damping is small, buckling can occur only with axial and circumferential cross-couplings (i.e. (1, 2) and (n, n + 1)). When combined bending and shear loads are applied onto the tank, axial forces induced by bending appear to be dominant. The major change in stability characteristics under vertically applied load is identified as a shift of buckling modes to higher circumferential modes. However, under seismic events, vertically applied loads are negligibly smaller than the bending forces induced by horizontal and rocking seismic motions, and hence induced seismic loads due to vertical seismic excitation are not likely to be responsible for the buckling damage of liquid-storage tanks.

References [1] W.K. Liu and R.A. Uras, Transient failure analysis of liquid-filled shells - Part I: Theory, Nucl. Engrg. Des. 117 (1989) 107-140, preceding article in this issue. [2] C.S. Hsu, On the parametric excitation of a dynamic system having multiple degrees of freedom, Journal of Applied Mechanics 30 {1963) 367-372. [3] C.-E. FrSberg, Numerical Mathematics Theory and Computer Applications (The Benjamin/Cummings Publishing Company, Inc., Menlo Park, 1985). [4] C.F. Shih and C.D. Babcock, Buckling of oil storage tanks in SPPL tank farm during the 1979 Imperial Valley earthquake, J. of Pressure Vessel Technol. ASME 109 (1987) 249-255.