Fluid-coupling coefficients in an array of hexagonal prisms

Nuclear Engineering and Design 92 (1986) 51-59 North-Holland, Amsterdam

FLUID-COUPLING A. P R E U M O N T

COEFFICIENTS

i, p. K U N S C H

51

IN AN ARRAY OF HEXAGONAL

1 a n d J. P A R E N T

PRISMS

*

2

i Belgonuclbaire, Rue du Champ de Mars 25, B-I050 Brussels, Belgium : CEN-SCK, Technology and Energy Department, Boeretang 200, B- 2400 Mol, Belgium

Received 5 August 1985

The study is related to the seismic analysis of fast breeder reactor cores. The first part of the paper describes an efficient method to compute the added mass coefficients in an array of slender bodies immersed in a fluid. It is assumed that the fluid is incompressible and non-viscous and that the flow is bidimensional, The Finite Element model uses one-dimensional elements with parabolic pressure field (three nodes per element) in the areas with small gaps, and two-dimensional elements elsewhere. The method is applied to an array of hexagonal prisms. The second part of the paper deals with the experimental determination of fluid-coupling coefficients in an array of hexagonal prisms. Both inertial and viscous couplings are investigated. It is found that the fluid damping depends critically on the gap size and that the cross-damping terms are small. The added mass coefficients are consistent with the numerical results of the 2D model. They are, however, 20 to 25% smaller, which is likely to result from axial leaks in the tests. Finally, the equation of motion of a seismically fluid-coupled system is briefly discussed. Some difficulties arising from the fluid are pointed out.

1. Introduction

The seismic response of the core of a L M F B R is substantially affected by the fluid that fills the narrow space between the subassemblies. Indeed, the fluid which is trapped between the wrapper tubes induces a strong coupling between the motion of neighbouring assemblies which, in the first place, depends on the gap between the wrapper tubes (typical value of the gap to flat diameter ratio is about 0.05). The aim of this paper is to determine the information required to achieve an approximate representation of the fluid in a structural model of the core. Such a model is mainly concerned with impacts between subassemblies, reaction forces on the supporting structure and the reactivity changes during the earthquake. It does not require a local knowledge of the pressure field, thus allowing the fluid to be represented by added mass and fluid damping matrices. Consider two undeformable bodies immersed in a fluid, the force induced on one body by the harmonic oscillation of the other has, in general, a component in phase with the acceleration of the latter, and a compo* Expanded version of papers presented at the SMiRT-8, Brussels, August 1985 (paper E7/6 *) and at the ASME PV&P Conf., New-Orleans, June 1985.

nent in phase with its velocity. For heavy fluids of light viscosity, such as sodium at high temperature or water, the component in phase with the velocity, that is the damping force, is generally small, and the component in phase with acceleration can be estimated by neglecting the viscosity. Besides, if the frequency of the oscillation is low (which is the case for seismic excitation), the fluid can be assumed incompressible. If the displacements are small, as compared to the characteristic dimension of the flow (here, the gap between the subassemblies), the coupling forces are essentially proportional to the acceleration and the principle of superposition applies. The latter condition can be considered as fairly well fulfilled in a L M F B R core, since the relative displacements of the subassemblies are restricted by spacer pads. Within the framework of the foregoing assumptions (non viscous, incompressible fluid, small displacements), a matrix of influence coefficients connecting the fluid forces to the acceleration of the various immersed bodies can be defined; this is the added mass matrix. Its numerical determination will be discussed in the next section; it will be seen that simplifications arise from the geometry of the core. Though small in amplitude, the forces in phase with the velocity are essentially dissipative in nature; they provide an additional damping which is essential in the

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52

A. Preumont et aL / Fluid-coupling coefficients

limitation of the response amplitude. Their numerical evaluation requires that the viscosity of the fluid be taken into account, which implies a considerable numerical effort [1]. The principle of superposition does not apply anymore in this case. The objectives of the study are as follows: (1) To develop an efficient 2D model for the calculation of added mass coupling coefficients. (2) To validate the model by comparison with experimental results obtained on a simple configuration close to an actual reactor core. (3) To estimate experimentally the fluid damping, as a substitute to a uncertain numerical study.

t

2. Numerical calculations of the added mass matrix

Consider a system of n immersed bodies, the length of which is assumed to be much larger than the transverse dimensions. Under the assumption of a two-dimensional flow, the added mass matrix (of dimension 2(n + 1) × 2(n + 1), including the container) can be computed column by column by studying the pressure field induced in a non-viscous incompressible fluid by a unit acceleration imposed successively on the various immersed bodies. The governing partial differential equation is the Laplace equation, which is very well

Fig. 2. Finite Element model for the computation of the fluid-coupling coefficients of an array of 19 hexagonal prisms.

suited to Finite Element techniques [2,3,4,5]. The corresponding variational problem consists in minimizing the functional

[tax]

+ fpanp

t~yy ] j d x d y

dx,

(1)

where A represents the fluid domain and C denotes the part of the boundary where the normal acceleration a,~ is imposed. It is formally identical to the problem of heat conduction, for which general purpose FE programmes are available. However, for geometries involving thin gaps as the one considered here, the problem can be considerably simplified by noting that (1) In the inner part of the core, where the gap to flat diameter ratio is small ( g / d < 0.05), the pressure is essentially constant over the thickness of the gap [6]. This allows eq. (1) to be integrated over the transverse co-ordinate, leading to

I

jr jJ

Fig. 1. 4 subassemblies in a cylindrical vessel. Assembly no. 2 subjected to a unit acceleration, lso-pressure lines in the vessel and pressure profile in the channels.

,/, = f L [ ~g (t-~x OP ]2 ! + oa.p ] dx.

(2)

where g stands for the thickness of the gap. (2) It is straightforward to demonstrate that minimizing the functional (2) is equivalent to satisfying the

A. Preumont et al. / Fluid-coupling coefficients

53

Table 1 Comparison between 1D and 2D models

g/d = 0.05

2D ref. [5]

Co,~

C~~,

(72~

C2,

0.4575

- 0.0705

0.12

- 0.109

0.4556

- 0.0685

0.1204

- 0.1091

(441 nodes)

1D (present study)

(54 nodes)

Table 2 Added mass coefficients - Finite Element results Gap

C,~o,

C,~'~| x

C,~2 ,

C,~2 ,.

3 5 8

14.19 8.576 5.411

-5.096 - 2.998 - 1.82,4

1.939 1.224 0.8164

-4.062 - 2.438 - 1.524

equation 02p _ oa, OX 2

(3)

g

(this e q u a t i o n is the same as the one-dimensional heat c o n d u c t i o n equation with source term). It follows that if the n o r m a l acceleration a n a n d the gap g are b o t h c o n s t a n t along x, the pressure field must b e parabolic. It results from the foregoing considerations that an o p t i m u m F E discretization of the inner part of the core will be achieved b y one-dimensional elements with parabolic pressure field (three nodes per element). In t h a t way, the size of the mesh inside the core can be as large as the side of the subassembly, as illustrated in fig. 2. A n y sort of two-dimensional elements can be used outside the core (triangular elements have been used in the example illustrated in fig. 2). As first application of the method, consider an array of seven hexagonal prisms surrounded by an impermeable b o u n d a r y ; the gap is c o n s t a n t everywhere *. The mesh consists of one element per side (three nodes), that is a total of 54 nodes. Table 1 compares the numerical values of the added mass coefficients for this model with those of a two-dimensional F E model involving 441 nodes [5]. The table gives the reduced added mass coefficients defined as g

C,,,=-~Ci~,,,,

(a=x,

y),

* This is not necessary for applying the method,

(4)

where C,~,, stands for the inertial c o n t r i b u t i o n of the m o t i o n of the central prism (0) in direction x to the m o t i o n of prism i in direction a **. g and d represent the gap and the flat diameter, respectively. According to the present model, in this particular case, C,, does not d e p e n d on g, nor on d; this is not true for the 2D model. The agreement between the two models confirms the assumption that for small values of g/d, the pressure can be regarded as constant over the thickness of the gap. The second example considers an array of four subassemblies in a cylindrical vessel ( g / d = 0.0058). Fig. 1 shows the isopressure lines in the outer region (discretized with 2D elements) a n d the pressure profile along the one-dimensional elements, when assembly no. 2 is subjected to a unit acceleration. It can be seen that, for the 1D elements, the pressure distribution is parabolic in the channels s u r r o u n d i n g assembly no. 2, a n d linear elsewhere, for the source term disappears from eq. (3). Note that for this value of the g / d ratio, the pressure is one order of magnitude larger in the 1D elements t h a n in the 2D elements. The third example considers an array of 19 hexagonal prisms in a cylindrical vessel, as illustrated in fig. 2, where the F E mesh has been represented. The inner diameter of the vessel is 800 mm; the flat diameter of the prisms is 110 ram; three values of the gap are considered: 3, 5 a n d 8 mm. Some added mass coefficients are given in table 2. The corresponding experimental results will b e given later in the paper.

3. Procedure for the experimental determination of the fluid-coupling coefficients In an attempt to u n d e r s t a n d the fluid-coupling mechanisms, a n experiment has been performed. The mock-up consists of 19 hexagonal prisms, as represented ** The numbering is identical to the one of the seven prisms in the centre of fig. 2.

54

A. Preumont et al. / [7uid-coupling coe:fficients

Tonk

.~5Pc!n~

H:'%Z96

1.d.of

L-z,

,

i

, ,

S~ed

Joint MOCK-UP

Model ests

+

SECTION A-B

Fig. 3. Experimental set-up. 0.6~-

,Og

schematically in fig. 3. The flat diameter is 110 mm while the gap can be varied continuously from 2 to 8 mm. The 12 prisms in the outer row are fixed, whereas the 7 prisms in the centre are mounted on springs, in order to constitute single degree of freedom (d.o.f.) oscillators which can be blocked or freed independently in either of the horizontal directions. The active length is 400 mm. Axial leaks are restricted by staggered joints. Two sets of springs have been used, leading to natural frequencies of 6 Hz and 12 Hz (the former is close to the first natural frequency of an actual subassembly, the latter was used in order to investigate any possible dependence on the frequency). The central prism is excited by an electrodynamic shaker provided with a force transducer. The displacements of the prisms are measured with displacement transducers (more appropriate than accelerometers for that frequency range)• Water, whose hydraulic properties (density and cinematic viscosity) at room temperature are close to those of the sodium at the average core temperature, has been used throughout the experimental programme. To begin with, the in-air properties (natural frequency, damping ratio) of the prisms can be determined from a release test. For each of the in-water tests, only two prisms are freed, the central one, actu-

~ f

Fig. 4. Identification model.

l .Z8

1.93

2.57

x

3.2I OO4O

[918183

q~8[ 5MH XX

, ~001! (~265 i 0022

i

.2

}',:3 190

H~'~% 741

H~ ~

J

o o.

°0'.00 A. Preumont

X

0,64

i

i

t

.28

f

i

[ .93

i"'

~

2.57

i-.:,1

x

3.21

Fig. 5. Comparison between experimental and analytical transfer functions. ated upon by the shaker and one of the first row. The system behaves like a two d.o.f, system, as represented schematically in fig. 4. Experimental transfer functions between the external force f and the displacements of both prisms (denoted H(~ xp and H[ ×p in what follows) are recorded. Details about the experimental procedure can be found in [7]. The identification model is represented in fig. 4. The mass M represents the total mass of the prism, including the added mass; the damping C is the total damping (structural plus fluid); m and d represent the fluid

A. Preumont et aL / Fluid-coupling coefficients coupling coefficients. The difference in the stiffness of the two prisms K - A K and K + A K reflects the fact that (i) Tests have revealed some discrepancy in the .natural frequencies of the prisms, both in air (which results from small differences in stiffness) and in water (here, an additional contribution comes from the fact that the added masses of the two prisms are not exactly the same *). (ii) The prisms behave slightly non-linearly (natural frequency increasing with amplitude).

where the following non-dimensional parameters have been used:

~ =

Y,(x)

K

olx 2

-

2jflx

otx 2

2 jflx

-

(1 - y ) - x

] z + 2j~x

y = A K/K,

2f10~0 = d / M , x = co/o~o.

(6)

In eq. (5), Yo(x), Yl(x) and F ( x ) refer to the Fourier transform of yo(t), yl(t) and f(t), respectively. From eq. (5), analytical expressions for the transfer functions Ho(X ) = Y o ( x ) / F ( x ) and Hl(x ) = Y l ( x ) / F ( x ) are obtained in a straightforward manner. They behave as represented in fig. 5. The qualitative influence of the non-dimensional parameters on the shape of the transfer functions is discussed in [7]. Having measured experimental transfer functions H~Xp(x) and H~Xp(x) in some range [a,b] and estimated the stiffness [ K = Mp(2~rfair) 2, where Mp is the mass of the one prism and fair is the average natural frequency in air], optimum values of the reduced parameters as defined by eq. (6) can be determined by minimizing (in some sense) a functional involving the difference between analytical and experimental transfer functions. The error functional used in this study is given in [7]; the minimization strategy is of the "steepest descent" type. The agreement between experimental and analytical transfer functions is illustrated in fig. 5; it is very good in all cases.

Y°(x)) =

+ y ) - x Z + 2j~x

= K/M,

a m/M, 2 ~ coo = C / M ,

The transfer matrix of the system is

=1((1

55

- 1

) (5)

* Strictly speaking, a uncertainty A M on the mass should also be introduced. However, numerical results with the model of fig. 2 indicate that, for g = 3 mm, the discrepancy between the added mass coefficients of prisms nos. 0, 1 and 2 is smaller than 5%.

• Finite Element

X

,.k e x p e r i m e n t

Cmox

15

(fair- 5.9 Hz)

x

a

b

Cmie x Cmox

A

10

0,4

zx z~

--6 . . . . . .

i= 1.(t =x ,_ / ......

i=20'=y

0.2 .

.

.

.

i=2.cr =x

% 0.02

0.'04

0.06

0.68 =

~_

0.02

0.04

0.66

% 0.08-

Fig. 6. (a) Added mass coefficient C~o.~ as a function of a gap to diameter ratio (g/d). (b) Relative amplitude of the cross-coupling coefficients vs. g/d.

A. Preumont et al. / Fluid-coupling coefficients

56

4. Experimental results Once the optimum values of the reduced parameters (6) have been obtained, the mass coupling coefficients can be determined as follows: denoting by Mp, the mass of one prism and by/x its apparent density (/z = M p / M f , where M r is the mass of the displaced liquid), one has by definition: M = Mp + C,~',o.,.Mf = Mp(1 + C,~oUlZ).

(7)

F r o m the first of eqs. (6), the ratio of the natural frequencies f~i~ and fo = °~o/2~r can be expressed as _ _

=

fo

•

~

.1/2

5. Damping

= (1 + c,.o,/~ )

The expression for C~1~ follows:

(8) Similarly, from the definitions of a and the mass coupling coefficients Cm",/~ (/3 = x, y, depending on the direction of vibration of prism i),

a = ~=

C,;mMr/tM~

,

tions which has not been used) is not provided. These results can be compared to the numerical results of table 2. This is done in fig. 6 as a function of g / d . It can be seen that (a) Although 20 to 25% smaller, the experimental values of C~0 ` are consistent with the numerical values obtained with the plane model: the discrepancy is likely to be attributed to axial leaks in tests *. A similar discrepancy of 30% was observed in [8]. (b) The relative values of the cross-coupling coefficients (-,i /f,~ ,,,,,/'~,,0, are in very good agreement; they are approximately independent of the gap.

p + C,~0,.Mf).

It follows that (:~,B = a ( , + Cm~0~).

(9)

The above procedure has been applied for three values of the gap (3.5 and 8 mm) and two values of the spring stiffness (f,i~ = 5.9 Hz and f~i~ = 12.1 Hz). The full experimental results corresponding to the case f,,~ = 5.9 Hz are reported in [7]. The added mass coefficients are given in table 3. Only part of the tests have been reproduced for fair = 12.1 Hz; they are given in brackets in table 3; they are in close agreement with those corresponding to f~i,- = 5.9 Hz. Note that the sign of the coefficients (contained in the phase of the transfer func-

As discussed in [7], for the same value of the gap, the damping ratio ,~ does not change appreciably from one configuration to another. Nor does the coupling damping ratio/3. The average values are reported in table 4, for the two values of f, ir. It can be seen that: (a) The damping ratio ~ is strongly dependent on the gap size, as illustrated in fig. 7. This was also observed in other studies [9]. (b) The coupling damping coefficient fl is generally small (5 times as small as ,~). * Actually, the staggered joints, aimed at preventing axial leaks, were not very efficient. This was due to the fact that no mechanical interaction between the prisms could be allowed.

• fair =5.9Hz fa~, = 12.1Hz

oo,, 0.06

¥ Table 3 Added mass coefficients - Experimental results, f,,i~ = 5.9 Hz (figures in brackets correspond to f,i~ = 12.1 Hz) Gap (mm)

Cr~,o,

Cml,-

Cm2,

C;~2,

3

10.58 (10.18)

3.89 (3.48)

1.22 (1.20)

3.24

5

6.96 (6.26)

2.396 (2.29)

0.893 (0.835)

1.96

4.27

1.505

0.526

1.20

8

•

i

0.04

.It 0.02

g/ 0.02 Fig. 7. Damping ratio.

0D4

0.06

0.08

A. Preumont et al. / Fluid-coupling coefficients

57

Table 4 Damping ratios - Experimental results

g

g/d

(mm)

)r,i~ = 12.1 Hz

fai~ = 5.9 Hz

/0=

~

B

/0

~

-

0.0035

5.54 6.44

0.053 0.034

B

~%/2~r Air

0.003

Water

3 5 8

0.0273 0.0455 0.0727

2.66 3.12 3.67

0.0725 0.0435 0.0244

(c) ~ is a decreasing function of f,~, the natural frequency in air. On the contrary, the viscous damping coefficient C-~'~00 is an increasing function of f~i~, suggesting a damping mechanism involving a power law with a power coefficient larger than 1.

6. Equation of motion The equation of motion of a structure subjected to a multi-support seismic excitation is well known [10] and will not be reproduced here. The alterations arising from the fluid-coupling forces will only be discussed. Consider a structure immersed in a container filled with liquid and subjected to a seismic excitation )~0- The inertia force fl in the equation of motion can be derived from Lagrange's equation: fl-

d OT dt Ok

OT i)x'

(10)

where T is the kinetic energy of the system, which has two contributions 7~, and Tf coming from the structure and the fluid, respectively. Upon partitioning the absolute displacements x into the constrained (support)

0.0144 0.0093 0.0039

and unconstrained d.o.f., x 0 and x~, respectively, both T~ and Tf have the form

I ,T • =~[X1AllXl-I-x0TA01kl +jgTA10-1f0-I-k0TA00x0],

(11)

where A stands either for the structural mass matrix, M, or for the hydrodynamic mass matrix, F, obtained as discussed in the beginning of the paper. A m = A]III, for both M and F are symmetrical. In assuming the form (11) for the fluid kinetic energy, it has been implicitely assumed that the container displacements are kinematically related to the support displacements x o. Upon introducing T = ~ + Tf in eq. (10), one gets the inertia forces associated with the unconstrained d.o.f.: f , = (M,1 + F l l ) ~ , + (M10 + r l 0 ) ~ 0 .

(12)

Following [10], the absolute structural displacements x~ can be decomposed into their dynamic and quasi-static components according to

xl = y + Tq~xo,

(13)

where Tq.~ is the quasi-static transmission matrix, connecting the absolute displacements of the unconstrained d.o.f, to the absolute displacements of ~he supports (the columns of Tq~ are the static absolute displacements induced in the structure by unit displacements of the various supports). Combining eqs. (12) and (13), one gets:

f,=(M+F))'+[(M+F)Tq~+M,o+FIo]i,,,

Fig. 8. 1 d.o.f, system immersed in a rigid container.

0.007 0.003

(14)

where the subscript 11 has been deleted for the part of the structural and hydrodynamic mass matrices relative to the unconstrained d.o.f. Eq. (14) must be substituted to the corresponding inertia forces appearing in the equation of motion without fluid-coupling (see [10], for

A. Preumont et al. / Fluid-coupling coefficients

58

example). The new equation of motion reads:

( M + F ) ~ , + ( C + Cr)j,+ Ky = - [(M+

F)Tq~ + M,o + Fro] if'0,

(15)

where a viscous structural damping has been assumed and a fluid damping contribution of the form C~ j~ has been added, to take into account the energy dissipation in the fluid *. The equation of motion without fluid can be retrieved from eq. (15), by cancelling out the hydrodynamic mass matrices F and F10, and the fluid damping matrix C r. Note that M,~ disappears for a lumped mass structural system and is frequently neglected. This cannot be done with F~0, as we shall see presently. To understand the physical meaning of the various hydrodynamic contributions, consider the simple c a s e of 1 d.o.f, system in a rigid container represented in fig. 8. The full hydrodynamic mass matrix reads [11]

connected with the closure of gaps; the impact locations are known a priori: (2) the structural system is made up of a large number identical substructures, the subassemblies ( periodic structure). Such systems are well suited to the so-called pseudo-force approach [t2,13], according to which the non-linearities are dealt with in the right hand side of the equation of motion, that is as external forces. As a result, the left hand side of eq. (15) remains essentially linear, while an additional non-linear term representing the impact and friction forces, f ( y , j', g), is added to the right hand side, where g stands for the gap distribution inside the core, dependent on the thermal state and the irradiation history. Upon neglecting - M , ) , the modified equation of motion reads

( M + F) y ' + ( C + C,)~,+ 19' =f(y,.i,,g)-[(M+F)Tq,

Fol

F00

m~ + m h )

m 1 +m2+m

h

'

(16) where m h is the added mass, m~ is the mass of displaced fluid, and m 2 is the mass of the fluid that would fill the container in the absence of the immersed body. Substituting (16) in eq. (15), one gets:

( g + m h ) Y ' + ( C + Cr)5'+ Kv = - ( M - m , ) 2

o. (17)

We note that the added mass appears only in the left hand side of the equation and that the seismic excitation is decreased by an amount equal to the buoyancy force.

7. The pseudo-force approach Although out of the scope of this paper, we would like to point out some of the problems associated with eq. (15) in the context of the seismic analysis of FBR cores. First, although highly non-linear, the structural system has the essential features that (1) the non-linearities are exclusively geometrical and

* Although the energy dissipation in the fluid is an important contributor to the energy balance in the system, the form assumed here is merely a mathematical convenience. It allows, when further decomposing the motion in the normal modes of the fixed base structure, to use the experimental information discussed in section 5.

+F,o]2 o.

(18)

In order to appreciate the changes introduced by the hydraulic mass matrix F. it must be realized that the matrices M, C and K are block-diagonal, each block corresponding to one subassembly, and that large commerical FBR may contain more than 1000 subassemblies. In the absence of fluid forces, this fact can be used to solve eq. (18) extremely fast, since using the normal modes of the subassemblies as Ritz vectors to reduce the size of eq. (18) leads to uncoupled equations of motion in modal co-ordinates (providing appropriate assumptions are made on the damping matrix). This method has been used in [14] to estimate the seismically induced reactivity change of the future SNR-2 FBR On the contrary, the hydrodynamic matrix F is full and the preceeding method no longer applies. In that case, efficient techniques for solving eq. (18) remain to be found. Further discussion of this topic is out of the scope of this paper.

Acknowledgment The authors want to thank their collegues, M. Delcon and A. Raymackers for their contribution to this • study.

References [1] C,1. Yang and T.J. Moran, Calculations of added mass and damping coefficients for hexagonal cylinders in a confined viscous fluid, J. Press. Vessel Technol. 102 (1980) 152-157.

A. Preumont et al. / Fluid-coupling coefficients [2] O.C. Zienkiewicz, The Finite Element Method in Engineering Sciences (McGraw Hill, New York, 1971). [3] S. Levy and J.P.D. Wilkinson, The Component Element Method in Dynamics, Chapter 9 (McGraw-Hill, New York, 1976), [4] T.J. Chung, Finite Element Analysis in Fluid Dynamics (MacGraw Hill, New York, 1978). [5] J.A. Essers, Von Karman Institute, Research Report CR 1982-14 (March, 1982). [6] A. Preumont et al., Fluid-coupling effects in LMFBR core seismic analysis, SMiRT-7, Chicago, 1983, paper E6/7. [7] A. Preumont and J. Parent, Experimental determination of fluid-coupling coefficients in an array of hexagonal prisms, Proc. of ASCE Conf. on Structural Engineering in Nuclear Facilities, NCSU, Raleigh, September, 1984. [8] R.J. Fritz, Theeffect of an annular fluid on the vibrations of a long rotor, Part 2: Test, J. Basic Engrg. (December 1970) 930-937.

59

[9] S.S. Chen et al., Added mass and damping of a vibrating rod in confined viscous fluids, J. Appl. Mech. (June, 1976) 325-329. [10] R.W. Clough and J. Penzien, Dynamics of Structures (MacGraw Hill, New York, 1975). [11] R.J. Fritz, The effect of liquids on the dynamic motions of immersed solids, J. Engrg. for Industry (February 1972) 167-173. [12] A.J. Molnar et al., Application of normal mode theory and pseudoforce methods to solve problems with non-linearities, J. Press. Vessel Technol. (May 1976) 151 156. [13] J.S. van Kirk et al., Methods for minimization of solution costs for transient dynamic analysis of non-linear periodic structures, J. Press. Vessel Technol. 104 (August 1982) 159-160. [14] A. Preumont and A. Pay, SNR-2 preliminary design. Seismic evaluation of various core support concepts, to be issued.