coupled vibration analysis of a cylindrical shell ... · the analytical methods of coupled free...

Coupled vibration analysis of a cylindrical shell including a liquid, connected with piping and attached unaxisymmetic structures

K. Fujita Mechanical Systems Engineering, Graduate School of Engineering, Osaka Prefecture University, Japan

Abstract

Recently, great earthquakes have occurred in the world. The damages due to these earthquakes can be said to become more serious as our social systems become larger and complicated. In this paper, the vibration analysis of a liquid storage container fixed on a foundation with piping and attached structures is taken into consid- eration. The coupled vibration analysis among the cylindrical shell, the liquid contained in the shell, the attached mass due to unaxisymmetric attached structures and the attached stiffness due to the flexibility of piping is performed. The fixed-free cylindrical shell is dealt with by the Fliigge's shell theory analytically, the liquid by the potential flow theory and the unaxisymmetric attached structures as added mass and the piping as added stiffness. Following these treatments, the coupled equations of motion are obtained analytically. Further, the seismic response analysis of the liquid storage container with unaxisymmetric attached mass and stiffness subjected to seismic excitation is performed. And, a part of the present numerical analysis is compared with the experiment in my former report.

1 Introduction

Great earthquake have occurred in recent years. In 1994, the Northridge earthquake in the U.S.A. occurred. In the following 1995, the Hyogoken-Nanbu earthquake occurred in Japan. In 1999, a series of earthquakes, that is, the Kocaeli earthquake in Turkey, the Chi-chi earthquake in Taiwan, furthermore up to nowa- days, often huge earthquakes have occurred. Many people have died and huge sum of properties have been destroyed due to these earthquakes. The damage caused by earthquakes can be said to become more serious as our social systems

Transactions on the Built Environment vol 71, © 2004 WIT Press, www.witpress.com, ISSN 1743-3509

130 Fluid Structure Interaction 11

become larger and more complicated. The necessity requiring integrity of impor- tant industrial facilities against earthquakes rises year by year as huge earthquakes occur world-widely.

A liquid storage container can be considered to be one of the severe damaged structures. In this paper, I consider the vibration analysis of a liquid storage container fixed on a foundation including piping and attached structures. The beginnings of earthquake-proof design method for a liquid storage container have proposed by Housner [l]. In this design method, the liquid pressure due to seismic excitation is evaluated assuming a container wall as a rigid. Hereafter, many earthquake-proof design methods have been proposed by using a beam theory, a shell theory based on analytical solution or finite element method in which liquid pressures are considered to be dynamic. Also, the author have already published many papers 121-[l01 concerning with a cylindrical liquid storage tank.

Here, a liquid storage container to be a thin cylindrical shell with a liquid, and moreover connected with pipes, and attached with unaxisymmetric structures such as a manhole, various kind of flanges, ladders and meters is taken into con- sideration. Following the above-mentioned requirement for investigation, the free vibration analysis method and the seismic response analysis method is proposed as a forced vibration of a thin cylindrical shell with unaxisymmetric attached mass and stiffness, which have the clamped-free support conditions

Weingarter [l11 , Takahashi [l21 have reported the analytical solutions on the free vibration of a clamped-free supported thin cylindrical shell without a liquid. However, in these papers these are some indistinct points, and further they are not always sufficient for applying them to the fluid structure interaction analysis here. So the eigenmode functions of a thin cylindrical elastic shell is obtained by solving the eight-order equation based on the Fliigge's shell theory. The liquid in a container is considered to be described by the velocity potential theory, and the solution of velocity potential is determined by using the boundary conditions at the side wall and the bottom of a container as presented in the former reports[4]-[a].

Moreover, the coupled vibration mode due to the unaxisymmetric attached mass and stiffness is assumed to be expressed by superposing the eigenmode functions of a vacant shell. The analytical methods of coupled free vibration and of coupled forced vibration are proposed as seismic response analysis when a cylindrical shell is coupled with a liquid, attached mass and stiffness. Furthermore, the comparis6n with the former reported solutions and experiments[3], [4] is performed, and the notice on earthquake-proof design are investigated.

2 'Jkeatment of a thin cylindrical shell

Fig.1 shows the modeling of a free-clamped cylindrical shell including liquid and with attached mass and stiffness. When the vibrational characteristics such as natural frequencies and eigenmodes are necessary to be obtained for seismic design, the finite element method is useful in practical use. However, it is not useful in order to get physical meaning concerning the coupling effect among a shell, liquid, attached mass such as manholes, ladders, and measuring instruments and attached stiffness such as pipings. So, we introduce analytical solution in this paper.


Fluid Structure Interaction I1 13 1

Figure 1: Analytical model of a thin cylindrical shell.

2.1 Fliigge's equation

The solution which satisfy the Fliigge's equation is assumed to be expressed in circumferential direction term, axial direction one and time function separately as follows;

U = U (t) cos(n6) (A COS wt + B sin wt)

v = V([) sin(n0) (A cos wt + B sin wt) (1)

W = W(<) cos(nO)(A cos wt + B sinwt)

Substituting eqn.(l) into the Fliigge's equation, the following simultaneous equations is obtained;

l - v U" - - 2

(1 + r2) n 2 u + yw2u + l + u n ~ ' 2

-- l - v

+ u n ~ ' + _I_ (1 + 3n2) V" - n 2 v 2

+ W - yw2 W + rr2 (W"" - 2n2 W" - 2n2w + n4 + W) = 0 (4)

As U, V, W in eqns.(2)-(4) are the required mode functions, let consider these U, V, W to be expressed as U = cleAc, V = c2eAf, W = c3eXc. Substituting these relations into eqns.(2)-(4),

eAc [I: 2 2 ] [i: ] = e A c [ ~ ] [i: 1 =G ( 5 ) W1 W2 W3

is obtained, where Ul - U3, Vl - V3, Wl - W3 are functions of W, X. The conditions of free vibration in eqn.(5) is that cl, Q, c3 must have not zero solutions.



so, det[A] = %X8 + alX6 + azX4 + a3X2 + a4 = 0 (6)

has to be satisfied, where a0 a4 are functions of W. Solving eqn.(6), Xj(i = 1 . - . ,8) are obtained, then the general solutions concerning U, V, W are given as follows;

where k, (Xj), k, (Xj) is given

2.2 Boundary conditions

When a thin cylindrical shell

(7) as functions of cl /c3, c2 /c3 by solving eqn. (5).

has the boundary conditions in which the bottom is clamped and the top is free, the 8 equations of condition can be given. That is, substituting eqn.(7) into these equations of boundary conditions, the following equation is obtained;

where Q is the matrix having 8x8 elements which are functions of W,>. The solutions of X in eqn.(6) are complex solutions consisting of complex conjugates. So, we introduce the following vector dconsisting of 16 elements;

Equation(8) is modified by using das follows;

Equation(l0) is expressed by a matrix 8 X 16. Besides, [c?) . . . in eqn.(8) are complex conjugates because X in eqn.(8) have complex conjugates. Then, as the real parts are equal and the imaginary parts have equal absolute values but not equal, the 8 following equations are given;

[R@= a (11)

Combining eqns.(lO),(ll) together, the 16 equations are obtained as follows;

For the condition that dhave not trivial solutions, the following equation must be satisfied;

det[S] = 0. (13)

Solving the simultaneous equations consisting of eqns.(6) ,(13),w, X can be obtained. Furthermore, substituting W, X into eqn.(l2), the vector dcan be given by getting


Fluid Structure Interaction I1 133

the nontrivial solutions. Following the above-mentioned procedure, we can get wks, Xks in k-th circumferential mode and in s-th axial mode. When we set A k s j = dj + jd j+8( j is G) using the vector $determined by wk,, Xks, the eigenmode functions in k-th circumferential mode and in s-th axial mode are expressed finally as follows;

3 Coupled fiee vibration analysis among a shell, liquid, attached mass and attached stiffness

The eigenmode functions in a vacuum in the axial direction for each circumferential mode can be obtained by solving the Fliigge's equation. The displacement of an empty shell U, v, W are expressed as follows;

3.1 Cylindrical shell

The kinetic energy and the strain energy stored in a cylindrical shell can be expressed as follows;

where Mks is the modal mass of a shell.

3.2 Liquid included in a shell

When a liquid included in a shell is assumed to be governed by the velocity potential @ as reported in reference[4], iP is given as follows;

where

The kinetic energy of a liquid in shell KA can be expressed as follows;



3.3 Attached mass

The kinetic energy of attached mass K, is expressed as follows;

where m, is the mass of attached mass, and the subscript a means the attached mass on a shell. When the co-ordinates of attached mass location is substituted into U, v, W of eqn.(l5), the velocities of attached mass can be obtained.

3.4 Attached stiffness

The potential energy of attached stiffness Up is obtained as follows;

where kpu, kp,, kpw are spring constants in U, v, W directions respectively, at the location of attached stiffness on a shell. When the co-ordinates of the attached stiffness location is substituted into u,v, W of eqn. (15) in the same way, Up can be obtained.

3.5 Equations of motion for the coupled fkee vibration

Substituting the kinetic energy of a shell KE, the strain energy of a shell UE, the kinetic energy of a liquid KL, moreover the kinetic energy of an attached mass K, and the potential energy of an attached stiffness Up into the Lagrange's equation, the following equation are obtained;

Equation(22) can be rewritten as follows;

where [Ml] is a mass matrix of a shell, [M2] is a mass matrix of a liquid, [M,] is a mass matrix of attached mass, [Ks] is a stiffness matrix of a shell and [Kp] is a stiffness matrix of attached stiffness, q' is the time function of the coupled vibration. When eqns.(16),(19),(20), and (21) are substituted into eqn.(22), we can orthogonize the circumferential modes for KE, KL, UE, but not for K,, Up. Therefore, at the time when the eigenvalue analysis is performed, the following vectors q'must be defined;

where q'is a vector having (kM X sM) elements. The coefficient matrices of (23) have ( k ~ X SM) X ( k ~ X sM) elements. Substituting q'= Gcdwct into eqn.(23), the



following equation of eigenvalue is obtained;

Solving eqn.(25), eigenvalues W , and eigenfunctions in which a shell, liquid, attached mass and attached stiffness are coupled are obtained. Then the mode functions of the coupled vibration system are expressed using GC as follows;

where U,, v,, W , are the coupled mode functions of U, v, W directions respectively.

4 Seismic response analysis of coupled vibration system

In this section, we investigate a seismic response analysis as forced vibration. When the clamped kase of a cylindrical shell is subjected to horizontal seismic velocity t, the velocity V of arbitrary points of a cylindrical shell is expressed in rectangular coordinates as follows;

The total kinetic energy of a shell is given as follows;

where FE is the exciting energy due to seismic motion for a cylindrical shell. Moreover, the velocity potential of a liquid which satisfies the boundary conditions is expressed from the reference 141 as follows;

where

00 2 - ) I ( ) Eliz 'l0 = - C ~ ~ 2 1 ; (y) cos (,) COS(^).

a=o

The total kinetic energy of liquid subjected to seismic motion can be given as follows;



Furthermore, the total kinetic energy of attached mass subjected to seismic motion can be given as follows;

Substituting KTE, KTL, KTa and UE, Up into the Lagrange's equations, the following equation can be obtained.

where F is the forcing vector due to a seismic excitation. When the seismic response analysis of time history is performed, the Rayleigh damping [C] = a[M] + b[K] is adopted, where [C] is a damping matrix, and a, 6 can be determined when 1st and 2nd modal dampings are given.

5 Numerical calculations and considerations

Table 1. Dimensions of numerical model[4].

0.0254[m]

V 0.3

Figure 2: Eigenfrequencies of an empty cylindrical shell.

5.1 Vibrational characteristics of an empty cylindrical shell

Figure 2 shows the eigenfrequencies of an empty shell. In this figure, n means the wave number of circumferential direction, and m means the mode number of axial direction. Furthermore, Fig.3 show the 3-dimensional mode shapes in order to investigate the deformation of a thin cylindrical shell concerning the axial modes m = 1 , 2 for each circumferential mode n = 1,2.

5.2 Coupling vibration between a shell and liquid

Figure 4 shows 3-Dimensional mode shapes to investigate the effect on depth of liquid on the deformation of a thin cylindrical shell when the depth of a liquid is



Figure 3: Three-dimensional mode shape of an empty cylindrical shell.

changed. From this figure, it is observed that the wet side wall area of a cylindrical shell deforms heavily comparing with the dry side wall area. And Table 2 shows the comparison among the present solutions and the former solutions[4] based on the finite element method. It can be said that they have good agreements each other.

Figure 4: Relation between depth of liquid and eigenfunction (n = 1).

The comparison between the present numerical analysis and the former reports[3], [4] for the sinusoidal resonance excitation is performed. The dimensions of a liquid storage cylindrical shell are R = 0.55m, .e = 2.0m1 h = 0.002m, E = 3.14 X 109Pa, U = 0.3, p = 1.40 X 103kg/m3, p~ = 1000kg/m3, which were the experimental plastic model shell conducted by the former reports[3], [4]. The damping ratios adopted in the present numerical analysis are 1.63% for the vacant shell and 4.75% for the full shell H = 192cm which were measured in the experiments of a plastic model shell. The natural frequencies in the present analysis are 56.4Hz in the vacant and 7.4Hz in the full H = 192cm which correspond to the 47.7Hz in the vacant and



Table 2. Influence of depth of a liquid to the eigenfrequency (n = 1).

axial mode number

( f i l l ) Former[4] 6.31 11.54 15.55 18.58 21.47 H = 0.75! Present 13.42 17.54 20.7 23.93

Former[4] 7.72 13.88 18.54 22.63 27.02 H = 0.5e Present 9.87 16.96 22.13 27.44 34.09

1 (Empty) I Former[4] 1 34.1 1 44.02 1 44.86 1 45.69 ] 46.94 1

H = 0.25l

H = O

7.OHz in the full H = 192cm obtained by experiments[3] respectively. They are found to show good agreements considerably.

5.3 Coupling vibration among a shell, liquid, and attached mass

Former[4] Present Former[4] Present

When a cylindrical shell couples with attached masses and stiffnesses, the orthog- onality between different modes in circumferential direction is not satisfied. So, these coupling modes can not be described in each circumferential and axial mode separately and become 3 dimensional vibration mode shapes ultimately. Especially, the more locations the attached masses and stiffnesses install at, the more complex the vibrational modes become.

Figure 5: Influence of attached mass (eigenfrequenq 6.24[Hz]).

10.2 15.4 16.6 34.0

Figure 5 shows the coupled eigenfrequency and coupled vibration mode shape in which the ratio of 1st axial and 1st circumferential mode of a shell is predominant, and which is considered to have much influence on seismic response. Moreover, this figure also shows the mode shape of the section at X = 0.51. Where, there is an attached mass of 104[kg] at X = 0.51,B = 0, against the empty shell of 2.8 X 105[kg] and liquid's mass of 1.3 X 107[kg]. It is found from these figures that the coupling among circumferential modes is recognized considerably, and the coupled mode becomes much complex, though the circumferential modes of a shell never couple in case of a coupling with a liquid alone. When an uniform cylindrical shell is subjected to a seismic excitation, the response of a shell coupled with a liquid

18.11 27.17 33.69 43.71

24.75 34.42 36.69 44.34

34.11 41.18 44.29 44.73

37.47 43.96 45.17 45.29



alone depends on the 1st circumferential vibration mode alone, however that of a shell coupled with a liquid and unsymmetric attached mass shows much complex deformation due to the coupling among circumferential modes and axial modes in an empty uniform cylindrical shell.

5.4 Coupling vibration among a shell, liquid, and attached stiffness

Figure 6 shows the coupled vibration mode of the full shell with an attached stiffness k p = 108[N/m] at X = 0.51,0 = 0 against empty container's stiffness 4.2 X 1Ol1[N/m] calculated roughly as a bending beam. This is the coupled mode

Figure 6: Influence of attached stiffness (eigenfrequency 6.29[Hz]).

in which the ratio of 1st axial mode and 1st circumferential mode of an empty shell is predominant, and is considered to have much influence on seismic response. Moreover, Fig.6 also shows the mode shape of the section at X = 0.51 where an attached stiffness is installed. It is found that the coupling among circumferential modes appears, and the coupled mode becomes much complex as well as the coupling among a shell, liquid and attached mass.

5.5 Seismic response

The seismic response calculations is performed by using the dimensions of a shell as shown in Table 1 as well as the free vibration. The El-centro recorded earthquake wave 360gal (3.6[m/s2]) is used for a numerical time history analysis calculation in order to compare with the former report results.

Figure 7 shows the comparison between the present numerical analysis and the former reported experiment[3], [4] for the seismic excitation. The dimensions of a shell are the same as the sinusoidal resonance excitation. Also, the damping ratios are the same as the measured data in a plastic model experiment. They are found to show good agreements comparatively. However, there are some differences in the response acceleration of a full tank top. This is considered to be due to the stiff flange installed at the top of the experimental shell model though the analytical model does not have it.

Figure 8 shows the seismic response time history calculations of a full cylindrical shell without attached mass and stiffness. On the other hand, Fig.9 shows that with attached mass and stiffness a t xp = 0.51,0 = 0 on a shell. The direction of 0 = 0 shown in Fig.8(a) and Fig.g(a) correspond to the exciting direction of seismic input



( simulation I

maximum response maximum response experiment acceleration of tank top: experiment acceleration of tank top: m 346gal (3.46m/s2) m 106gal (8.06m/s2)

(a) Vacant shell (H = 0) (b) Full shell (H = 1.92m) Figure 7: Comparison between simulation and experiment[3].

wave, and that of B = 7r/2 shown in Fig.8(b) and Fig.S(b) are perpendicular to the exciting direction of seismic wave. From these figures,. the acceleration response in radial direction at B = 7r/2 of a shell with the attached mass and stiffness become larger even if the response direction at B = 7r/2 is perpendicular to the exiting direction. On the other hand, the response a t B = 7r/2 of a shell without attached mass and stiffness shows nothing. It is because of that the circumferential higher modes than 1st mode are not excited in the coupling only between a shell and a liquid, but that they are exited in coupling among a shell, liquid, and attached mass and stiffness.

1 , --, L 0 1 2 3 4 5 6 0 1 2 3 4 5 6

amdrssl me[-]

(b) 2 + W at X = 0.51,8 = 7r/2 (b) 2 + W a t X = 0.51, B = 7r/2

Figure 8: Response acceleration of Figure 9: Response acceleration of a a shell without attached shell with attached mass mass and stiffness (H = 1). and stiffness (H = l).

6 Conclusions

The following conclusions may be drawn from the above-mentioned considerations.


Fluid Structure Interaction I1 14 1

0 The eigenfunctions of a free-clamped thin cylindrical shell is clarified to be obtained by solving the eight-order equation based on the Fliigge's shell theory, and the coupled vibration analysis method among a shell, liquid, attached mass a,nd stiffness is proposed.

0 It is confirmed that the coupled vibration modes are much affected by the depth of a liquid

0 The vibration modes become more complex and 3-dimensional due to the coupling in circumferential modes and also in axial modes when attached masses and stiffnesses are installed on a shell.

0 Moreover, the seismic response analysis method as a coupled forced vibration is proposed. It is confirmed that attached masses and stiffnesses have much influence on seismic responses as shown by numerical parameter studies.

References

[l] Housner, G.W., Bull. of the Seismologl Soc. of Amer, 1957. [2] Fujita, K., Report on the Hanshin-Awaji Earthquake Disaster, Mechanical

Engineering Volume, Damage and Failure of Machines and Industrial Equip- ment, JSME, 1998.

131 Fujita, K. & Shiraki, K., Approximate seismic response analysis of self- supported thin cylindrical liquid tanks, Proc. of 4th Structural Mechanics in Reactor Technology, in San Francisco K5/4, pp. 1-12, 1977.

[4] Fujita, K., A seismic response analysis of a cylindrical liquid storage tank, Bulletin of the JSME, 24(192), pp. 1029-1036, 1981.

[5] Fujita, K., A seismic response analysis of a cylindrical liquid storage tank including the effect of sloshing, Bulletzn of the JSME, 24(195), pp. 1634-1641, 1981.

[6] Fujita, K., A seismic response analysis of a cylindrical liquid storage tank including the effect of sloshing, 2nd report: analysis based on energy method, Bulletin of the JSME, 25(204), pp. 977-985, 1982.

[7] Fujita, K., A seismic response analysis of a cylindrical liquid storage tank on an elastic foundation, Bulletin of the JSME, 25(210), pp. 1977-1984, 1982.

[8] Fujita, K., Seismic response analysis of a cylindrical liquid storage tank, Proc. of the ASME PVP, 77, pp. 78-85> 1983.

[g] Fujita, K., Vibration analysis of fluid-coupled two coaxial axisymmetric shells containing fluid, Bulletin of the JSME, 29(248), pp. 516-524, 1986.

[l01 Fujita, K. et. al., Seismic response of liquid sloshing in the annular region formed by coaxial circular cylinders, Bulletin of the JSME, 29(258), pp. 4318- 4325, 1986.

[ll] Weingarten, V.I., Free vibration of thin cylindrical shells, AIAA Journal, pp. 717-722, 1964.

1121 Takahasi, S., et. al., Free vibration of thin cylindrical shells, J. of Japan Soc. Mech. Eng. (in Japanese), 35(275), pp. 1412-1416, 1969.

[l31 Fujita, K., Saito, A., Coupled vibration analysis of a cylindrical shell with liquid, piping and unaxisymmetric structures, Proc. of 6th Biennial Confer- ence on Engineering Systems Design and Analysis (ESDA 2002), CD-ROM Proceedings APM-026 2002.


coupled vibration analysis of a cylindrical shell ... · the analytical methods of coupled free...

Documents