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Seismic analysis of PWR-RCC fuel assemblies
103
Nuclear Engineering and Design 71 (1982) 103- 119 North-Holland Publishing Company
SEISMIC ANALYSIS A. PREUMONT Belgonucleaire
OF PWR-RCC
FUEL ASSEMBLIES
and P. THOMSON
S.A., Rue du Champ de Mars 25, B - 1050 Bruxelles,
Belgium
J. PARENT CEN/SCK, Received
Boeretang 200, B - 2400 Mel, Belgium 9 November
1981
The present paper investigates the dynamic behaviour of PWR-RCC fuel assemblies under seismic excitation. A simple vibrational model of the fuel assembly is proposed, which leads to natural frequencies whose spacing agree with experimental data. Available experimental results are reviewed. Impact characteristics of Zircaloy spacer grids are also discussed. It is proposed that their soundness criteria be expressed in terms of impact energy rather than in terms of impact force. The computer code CLASH is briefly described; it is utilized to perform a sensitivity analysis. An example of application is also given.
1. Introduction In a previous paper by one of the authors [l], an efficient algorithm was presented for the PWR’s core seismic analysis (computer program CLASH). This algorithm performs the time integration of the colliding vibration of a row of fuel assemblies. Its main features are: - The program takes into account a possible difference between the core plates accerelations. - It integrates the equation of motion of each fuel assembly separately, the contact forces (restricted to the spacer grids) being re-evaluated at each time step and treated as external forces. - The fuel assemblies transverse deformations, restricted by the core barrel, are assumed to be small enough to allow a linear dynamic representation of the fuel bundle. Modal superposition can therefore be used. - Two time-steps dt, and dt, (multiple of dt,) are used in the integration, depending on whether or not the fuel assembly analyzed impacts its neighbours. This procedure is made even more profitable by the use of Duhamel’s formula. In fact, in this case, where no error is connected with the integration operator i, the only constraint on the larger time step, d t,, is the
’ More precisely, with its homogeneous 0029-5493/82/0000-0000/$02.75
part, see
[ 151.
0 1982 North-Holland
good representation of the low frequency (below 35 Hz) seismic excitation. Ultimately, dt, could be as large as the core plates accelerograms sampling period. Obviously, the optimum value for dt, (from the CP time point of view) varies from one case to another, depending on the number of fuel assemblies, the gap size, the impact characteristics of the spacer with the time step grids, etc.. . . When integrating dt,, the iterative refinement process is restricted to the fuel assemblies involved in shocks. The whole integration policy is summarized in table 1. Besides the core plates accelerograms (if no available, they can be artificially generated from a core response spectrum), the core motion time integration, as outlined above, requires the following input data (table 2):
Table 1 Two-time
step integration
policy (dt, = k dt,)
104
A. Preumont et al. / Seismic analysis of P WR - RCC fuel assemblies
Table 2 Core seismic analysis, overall process
These various requirements sections.
will be reviewed
in the next
2.2. Fuel assembly static properties
_ a fuel assembly
vibrational model: _ the spacer grids impact characteristics. This is one of the purposes of this paper to discuss how these data can be determined. This matter has already been partially debated in a recent paper [2]. Another aim of this paper is to discuss the soundness criteria of the spacer grids. It is suggested that they be expressed in terms of strain energy rather than in terms of impact force. A sensitivity analysis has also been made with the computer program CLASH; it reveals the predominent role of some parameters (e.g. the modal damping) as compared with others (e.g. the kind of spacer grid dissipation). Finally, an example is fully described.
2. Vibrational model of PWR-RCC
Fig. 1 shows the results of static tests carried out on a 14 X 14 (6 spacer grids) fuel assembly for two types of boundary conditions: (1) simply supported at the top, clamped at the bottom, loaded at the 4th spacer grid from the bottom (referred to as case No. 1); (2) clamped at the bottom and loaded at the top (referred to as case No. 2). The figure displays the equivalent bending stiffness, versus the applied load. For a given force, the vertical scattering corresponds to the results obtained at the various spacer grids. The figure reveals: - The equivalent bending stiffness decreases as the load increases, indicating a softening behaviour of the fuel assembly. This expected non-linear effect is inherent to the fuel assembly design (mainly caused by friction between the fuel rods and the spacer grids) and cannot be accounted for in a linear model. However this effect seems to be small enough in case No. 1 which is close to the in-core boundary conditions, to justify the linearity assumption in this case.
EI
10’ I Fuel
kgmm* Assembly
Equivalent
Bending
Stiffness
fuel assembly
2.1. Requirements A good vibrational model of fuel bundle should fulfill the following requirements. It should: account for the static stiffness properties of the fuel assembly; account for the individual vibrational behaviour of the fuel rods, which make up most of the assembly mass; lead to natural frequencies whose spacing agrees with experimental results within the largest possible frequency range (in fact, even though the seismic excitation contains only low frequency components, higher frequency modes are excited during impacts); be simple, in order to be introduced as such in the core model.
Load 10
20
50
40
Fig. 1. Fuel assembly static tests.
50
60
(kg) 70
F
A. F’reumont et al. / Seismic analysis of P WR - RCC fuel assemblies
- More surprising
is the large dependency of the equivalent bending stiffness on the boundary conditions: the equivalent bending stiffness obtained in case No. 2 is about three times larger than the one obtained in case No. 1. Clearly, this indicates the inadequacy of a beam model to represent the static behaviour of the fuel assembly. However, if one includes the shear deformation, which is much larger with respect to the bending deformation in case No. 1 than in case No. 2, the agreement of a beam model can be considerably improved, provided a large value is adopted for the shear deformation control parameter:
E1 E ( r 12=O12to0 .
a=-----=Gk’7 GO’12
. 15 )
where E stands for the Young’s modulus, G is the shear modulus, 52’= k’A is the effective shear area of the
-1 % f
/ /’
‘\\,,
1 3oLp’
‘\\
\
OL---; f
:
+ Experiment o F.E. case 1
8. F.E. model
lnm (clamped-pinned) I mm(clamped 1 Displacements Fig.
2.
Comparison
element model.
between
the static
tests and
the
finite
105
section, 1 is the length of the beam, and r = fl is the radius of gyration of the cross section. The two-beam model described below includes shear deformation; it can be seen in fig. 2 that it gives results in reasonably good agreement with experiments, for both boundary conditions. It must be understood that for such a large value of the control parameter a, shear deformation makes up most of the total deformation in case No. 1. The boundary conditions of a fuel bundle, in a nuclear reactor core, can conveniently be represented by an embedment at the bottom, and a rotary spring at the top. Actually, the displacements in the central part of the bundle are rather insensitive to the rotary spring stiffness K,, since most of them result from shear deformation. 2.3. Vibrational behaviour of the fuel rodr It is well known that the low amplitude vibrations of PWR’s fuel rods can adequately be represented by a beam model (no shear deformation) with the following properties: - The mass per unit length results from both the fuel pellets and the cladding tube. - The fuel pellets do not contribute to the rigidity of the rod. _ The cladding-to-spacer grid connection is modelled by a rotary spring whose stiffness K, can be determined by a simple static test (fig. 3). It depends on the grid spring force and also, heavily, on the distance between the dimples inside the grid cells. - The fuel rod prestresses induced by the plenum spring are considered in the analysis. This longitudinal preload introduces a rise in the natural frequencies, which is controlled by the ratio of the preload to the Euler’s buckling load of the rod. The suitability of this model is shown on fig. 4 which compares F.E. and experimental results [3]. Although, out of the scope of this paper, it is interesting to note that the vibrational behaviour of the fuel rods is dependent on irradiation, due to: the creep down of the cladding and the resulting fuel-cladding interaction: the gap reduction makes the pellets contribute to the rod rigidity, thus raising the natural frequencies; it also reduces the rattling of the pellets inside the cladding, reducing the damping; the plenum spring relaxation (from six times the fuel weight to zero); the grid spring relaxation: the spring force reduction induced by both the grid spring relaxation and the cladding creep down is responsible for a (slight)
106
Preumonl et ni. / Seismic anatysrs of P WR - RCC fuel
A.
CLAD
TO SPACER
GRID
assemblies
CONNECTION
Model
Dimples
0.6
0
1
Fig. 3. Clad-to-spacer
change in vibration, ting wear, affect the
2 grid connection.
3
4
5
6
Rotary stiffness.
K,. Irradiation dependence of the fuel rod together with other aspects concerning fretis discussed in [3]. It is not expected to bundle properties to a large extent.
2.4. Vibrational model, comparison with experimental
data
The preceding considerations lead to the two-beam model of fig. 2a. The first beam includes the shear
deformation and is mainly responsible for the static behaviour of the fuel assembly; its stiffness properties (EZ, a) correspond to the whole assembly (skeleton + fuel rods), as discussed in section 2.2.; the mass per unit length comes from the skeleton only. The second beam simply adds the mass and stiffness properties of the fuel rods. The two beams are connected by rotary springs at the grid levels, whose stiffness is the sum of the individual rod-to-grid connections. If two kinds of these con-
107
A. Preumont et al. / Seismic analysis of P WR - RCC fuel assemblies
SPACER GRIOS
F.E. (Hz) 44.76 49.15
Tests (Hz) 47.6
49.30 53.66
60.4
57.91 61.50
71.6
68.85 71.15 flNlTE -------
75.9
ELEMENT MODEL
EXPERIMENT
79.00 79.81
Fig. 4. Fuel rod vibrational behaviour. Comparison between F.E. model and experiments. For each mode, the two computed frequencies refer respectively to K, = 100 Nm/rad and 200 Nm/rad. The computed mode shapes correspond to Ke = 200 Nm/rad.
nections exist in the fuel assembly (corresponding to various distances between the dimples inside the grid cells, as it is the case in BN’s Zircaloy spacer grids - see fig. 3), then either a mean value is adopted for K, and a two-beam model can be used (with the consequence of introducing a single array of natural frequencies for the fuel rods), or a three-beam model is adopted, with two values K,, and K,, for the clad-to-grid connection stiffnesses. Such a model is shown in fig. 5 for a 14 X 14 (6 spacer grids) fuel assembly. It consists of a 55 degree of freedom (d.o.f.) Finite Element (F.E.) model. Modes 1 to 6 are displayed; their deviation with respect to beam modes increases with the order of the mode.
Modes 5 and 6 are local fuel rod modes (compare with fig. 4); their frequencies depend mainly on the stiffness K,, as can be seen in table3 for the two-beam model. Table3 also shows that the stiffness K, does not influence the results strongly, as already mentioned. Table4 gives experimental natural frequencies (in air) of PWR fuel assemblies [4,5,6,7,8]. Their spacings (ratio fi/fi) are plotted in fig.6, where it can be seen that the various modes are closer than those of a beam clamped at both ends (boundary conditions leading to the closest natural frequencies for a beam). It is this fact, together with the surprising static test results discussed in section 2.2, that suggested the introduction of
108
A. Preumont
r--------
-7
et al. / Seismic analysis of P WR - RCC fuel assemblies
109
A. Preumont et al. / Seismic analysis of P WR - RCC fuel assemblies Table 3 14 X 14 (6 spacer grids) fuel assembly.
Two-beam
K, = lo4 Nm/rad K,=179x300 Nm/rad ’
Mode i
1 2 3 4 5 6
finite element
model natural
frequencies.
K, = 10’ Nm/rad K,=179X300 Nm/rad
Parametric
study on K, and Ke
K, = lo5 Nm/rad K,=179X200 Nm/rad
L (Hz)
h/f,
h (Hz)
L/f,
6.33 14.62 23.72 34.34 50.23 62.62
1.00 2.31 3.75 5.42 7.94 9.89
6.50 14.71 23.84 34.48 50.58 63.20
1.00 2.26 3.67 5.30 7.78 9.72
b
f, (Hz)
X/f,
6.25 14.10 22.90 33.36 47.21 61.08
1.00 1.26 3.66 5.34 7.55 9.77
a 179 fuel rods in a 14X 14 fuel assembly. b Plotted on fig. 6.
shear deformation in the fuel assembly model. For it is well known that shear deformation contributes to bringing the natural frequencies together (see [9], page 318320). The agreement of the F.E. model with experimental results can be judged from the comparison of predicted and experimental spacings f,/fi. Column No. 2 of table 3 has also been plotted in fig. 6. We can see that the F.E. results lie well within the range of the experimental results, except for mode 5. In fact, the comparison is
meaningless for this mode, because it is a local mode of the fuel rod in the F.E. model, but not for the experimental results. The agreement is particularly good with the results of FRAMATOME [7] which have been obtained with the same type of fuel assembly (14 X 14, 6 spacer grids). The ultimate use of such a model being the time history integration of the core motion, it is important to limit its number of d.o.f.. The Guyan’s reduction [lo] can contribute significantly in this way. It expresses the
Table 4 Natural
frequencies
of PWR-RCC
fuel assemblies
- Experimental
data in air
Mode 1
2
3
4
5
6
f;/f,:
1
4
9
f,/f,:
1
2.76
5.4
8.93
13.34
CEA 15x 15 7 sp. grid [4]
f; (f-f+ f;/f,:
3.7 1.0
7.63 2.05
11.67 3.15
18.55 5.01
24.78 6.7
B&W 17x17 8 sp. grids [5]
L/f,:
1.0
1.9
3.0
4.1
5.3
Pinned
at both ends
Clamped
Mitsubishi
at both ends
14X 14 [6]
FRAMATOME 6 sp. grids [7] K.W.U. 16X 16 7 sp. grids [S]
f;/f,
a:
16
25
1.0
2.3
3.6
5.1
6.6
13.21 2.03
22.05 3.39
30.55 4.7
_b
;;“:
6.5 1.0
/I (f&a: J/f,:
3.0 1.0
14X 14
’ Approximate values estimated from graphs. b A fuel rod local mode is missing here.
12.0 4.0
7
20.0 6.6
38.31 9.87
62.16 9.56
78.93 12.14
110
A. Preumont
et al. / Seismic analysis of P WR - RCC fuel assemblzes
Table 5 Comparison of various cases of Guyan’s reductions Nodes (fig. 5)
Independent degree of freedom F.E. model
2 3 5 7 8 9 10 11 13 15 16 17 18 19 21 23 24 25 26 27 29 31 32 33 34 35 37 39 40 41 42 43 45 47 Number of d.o.f.
3
3 3 3 3
3 3 3 3
3 3 3 3
3 3 3 3
3 3 3 3
55
First natural frequency Y , independent d.o.f. clamped
5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5
Guyan’s reduction Case 1
Case 2
Case 3
3 3 3
5 5 5
3 3 3 3
5 5
5
3 3 3 3
5 5 5
3 3 3 3
5 5
5
3 3 3
5 5 5
3 3 3 3
5 5
5
3 3 3
5 5 5
3 3 3 3
5 5
5
3 3 3
3 3
3 3 3
5 5
3 3 3 3
5 5
18
30
30
144.4
174,4
324,5
3 3 3
3 3 3
3 3 3
eigenvalue problem in terms of some selected (independent) d.o.f. by assuming a static relationship for the other (dependent) d.o.f.. It can be shown that this condensation leads to an upper bound for the eigen-fre-
quencies and to a good estimate natural frequencies f, satisfy (.A/%)“~
19
of the modes
whose
(2)
111
A. Preumont et al. / Seismic analysis of P WR - RCC fuel assemblies
2.5. Factors affecting the fuel assembly natural frequencies
fi ffl
::,ment
0
*
0
1
3
Bz V
2.
5
6
Mode
0, Values Finite Element
The vibrational behaviour of a fuel bundle depends on numerous parameters, namely, the amplitude of vibration, the surrounding medium, the temperature and the burn-up. Fig. 7 illustrates the influence of the first three parameters on the first natural frequency according to the results of Stokes and King [5]; these authors’ conclusions follow: - f, decreases when the mid-span amplitude increases; - water induces an approximately 15% drop in f,, which can be accounted for by considering the added masses. Similar results have been recently obtained for higher order modes [8]; - f, does not depend on the water temperature; the drop in the water density is compensated for by the drop of the Young’s modulus; - f, does not depend on the water velocity; - end-of-life first natural frequency is 5 to 10% lower, due to the gridspring relaxation.
model
Fig. 6. Fuel assembly natural frequencies. Comparison between the predicted and experimental spacings.
where rI stands for the first natural frequency of the system obtained by clamping all the independent d.o.f. Table 5 compares three different reductions that have been carried out on the F.E. model of fig. 5. D.o.f. 3 refers to the translation in the “y” direction, d.o.f. 5 refers to the rotation around the “z ” axis. The comparison of cases No. 2 and No. 3 shows that an adequate choice of the d.o.f. can improve the spectral convergence of the reduction considerably: a slightly different choice of the 30 independent d.o.f. raises the frequency Y, from 174.4 Hz to 324.5 Hz. The computation of v, is very helpful in the choice of the independent d.o.f. It must be emphasized that the model which is presented in this paper is intended to represent the overall vibrational behaviour of the bundle, as needed for the core motion integration and the computation of the interassembly impact forces. It cannot be used to estimate the stresses in the fuel assembly under external loads. In particular, the first beam described at the beginning of this section (referred to as type 7 in fig. 2a) has no physical existence and should not be confused with the skeleton of the fuel assembly *. Stress computation requires a more complicated (even non-linear, with friction elements) model which is out of the scope of this paper.
2.6. Modal damping Fig. 8, drawn after Stokes and King [5], illustrates the dependency of the (end-of-life) damping of the first mode on the vibration amplitude, the surrounding fluid and its temperature. Their conclusions are: - the damping is approximately independent of the mode ( ti = tl) and the direction; - .$‘,increases with the amplitude; - .$, is 25 to 100% larger (depending on the amplitude) at the end-of-life;
I
1
04
02
0.
95
1.
mid _ span displacement
1.5 lmm
1
Fig. 7. First natural frequency versus mid-span amplitude. The influence
* Similarly, the distance between the two beams is immaterial.
WATER Thii curve is temperature independent I 2@C _ 32O’C
0.6
of water is temperature
King (51).
independent
(after Stokes and
A. Preumont et al / Seismic analysis of P WR - RCC fuel assemblies a LMFBR, the thin coolant layer which is trapped between the wrapper tubes, induces a strong interaction between the fuel assemblies. Usually, it is assumed that this interaction can be expressed in terms of a coupling added mass matrix (part of the coupling forces in phase with the accelerations) and a coupling fluid damping matrix (part of the coupling forces in phase with the velocities). This problem has been analysed in [ 111. Control rods do not affect the bundle vibration characteristics [5].
I 0.5
15
1.
mid. spm
displacement
I 2.
1mm)
Fig. 8. Damping of the first mode versus mid-span amplitude. In air and still water (temperature ranging from 20°C to 32OT) (after Stokes and King [5]).
- 5, increases considerably (+ 120%) with water; - 5, does not depend on the temperature; - 5, increases strongly (+60 to 400%) with the axial flow velocity. Larger amplitudes have been considered by Wehling et al. [8]; they reveal the following damping values: 0.05
CO.08
for air,
0.12<(,
CO.30
for still water.
3. Spacer grids impact behaviour 3.1. Spacer grid dynamic model In the core model, the spacer grids are modelled by impact elements consisting of a spring, a viscous damper and an hysteretic damper (fig. 9a); so that it contains the two classical impact elements of figs. 9b and 9c as particular cases:
(4 Spring and viscous damper (fig. 9b): its constitutive equation
reads:
f = kx + c,ic. The restitution by: r= -V/&‘o-e-“c,
(3) coefficient
is approximately
(.$Kl)
where 5 = c/26.
(4)
(5)
It depends
2.7. Inter-assembly jluid coupling - Control rods At this point, it should be mentioned that off-diagonal coupling coefficients (mass as well as damping) are not considered in the core model, meaning that hydraulic coupling between the fuel assemblies is neglected. This assumption is partly justified by water loop tests reported in [5], which indicate that the natural frequency drop can be accounted for by added masses corresponding to the displaced liquid, meaning that a fuel assembly in a square channel does not behave in a significantly different manner than a fuel assembly in an infinite fluid. Physically this is a consequence of the fact that without a wrapper tube, the fluid can flow from one side of the assembly to the other, across the fuel assembly rather than around it. From the foregoing observation, it is the authors’ belief that the cross-coupling terms neglected in the analysis should be small. On the contrary, it is obvious from the well known results on concentric cylinders that, when the fuel assemblies are provided with wrapper tubes as they are in
given
(b)
explicitely on the impacing mass m and, consequently, varies from one spacer grid to another. Spring and hysteric damper (fig. 9~): this element has been used by Nuno et al. [12]. Its constitutive equation reads: for120, (k-h)x
forf1<0.
Its restitution coefficient enjoys property of being only dependent element itself:
the remarkable on the impact
r does not depend on the initial velocity VO, nor on the impacting mass m; it is constant for all spacer grids. A sensitivity analysis presented later in the paper reveals that, if the viscous damping coefficient c is chosen in order to lead to the same value of r for the
A. Preumont et al. / Seismic analysis of P WR - RCC fuel assemblies
-I
dash-pot
impact drrnent a. Spacer Grid Representation in the Core Model
b. Spring and Viscous Damper
f .kx+ci r-i-e-6V VO
C
-nt
I--
2Jiz
oLw X
c.
Spring and Hysteretic Damper.
(k+h)x f’lk-h)x
(i-0) tic* 0)
r=_V-_
JR-
vo
Fig. 9. Impact element.
Fig. 10. 14X 14 Zircaloy spacer grid. Static compression
c-r,.“,
“0
.a
14114 Zr apacer grid
tests.
1
6.
r
.6
.5
.4
maximum mass translational d.o.f. of a spacer grid as the one given by eq. (7) for the hysteretic damper, only slight differences are observed in the grid displacements and impact forces. Constants k, c and h depend on the stiffness and dissipation properties of the spacer grids. These properties are examined in the next section. 3.2. Static and dynamic tests on spacer grids
.3
.2 I-.1 t
I 0.
i -
1
-.5*
I 1.
1":
Fig. 10 shows the results of static compression tests performed on BN’s 14 X 14 Zircaloy spacer grids. The
Fig. 11. 14x 14 Zircaloy spacer grid. Impact tests. Restitution coefficient
and transverse deformation
versus impact velocity.
114
A. Preumont
et al. / Seismic analysis oj P WR - RCC fuel assemblies
following features are worth noting: _ the loading characteristic is non linear. This makes the stiffness k of the impact element rather difficult to choose; _ the ruin occurs by instability; - the grid stiffness is much larger when provided with fuel rods, as compared with a grid with empty cells. The easiest way to quantify experimentally the spacer grid dissipation during impact is to measure the restitution coefficient r. Once r is known and an impact model has been adopted, a straightforward calculation through formula (4) or (7) leads to the damper characteristic c or h. Fig. 11 shows a plot of the coefficient of restitution r versus the impact velocity Vc. One sees that in the elastic range, below a critical velocity V:, r is almost independent of V, (r - 0.60 to 0.65). This agrees with the eqs. (4) and (7). S;, a measure of the spacer grid permanent transverse deformation after three successive impacts at Va, is also plotted on fig. 11. We can see that below a critical velocity V:, no permanent deformation occurs. For velocity V0> V:, the transverse permanent deformation increases with the number of impacts and, sharply, with the impact velocity. Note that the critical velocity V,* corresponds to a drop in the restitution coefficient (first impact).
0
10
20
30
40
50
60
Time (ms) Fig. 12. 5 X 5 Zircaloy grid. Velocity diagram during impact. Comparison of impact No. 2 and impact No. 13 on the same sample grid.
Fig. 12 shows the experimental velocity diagrams of the impacting mass for two of a succession of impacts (impact No. 2 and impact No. 13) on the same sample grid. Although obtained on a 5 X 5 (empty cells) spacer grid, these results are indicative of the plastic behaviour of the spacer grid when the number N of impacts increases: (1) The contact duration (i.e. the time required for the velocity to reverse) increases, meaning that the impact becomes softer (k decreases). (2) The reverse velocity V (after the shock) increases, meaning a rise of the coefficient of restitution r. 3.3. Earthquake
design criteria
According to the Standard Review Plan (section 4.2., Fuel System Design), the following capabilities must be maintained, under all conditions: (a) The fuel damage is never so severe as to prevent the control rod insertion. (b) The fuel assembly retains its rod-bundle geometry with adequate coolant channels to permit removal of the residual heat. In structural terms, this implies that the spacer grids do not buckle under the earthquake induced impact forces. Besides the above requirements, two possibilities can be considered for the OBE (Operating Basis Earthquake), depending on economic considerations: (i) the core must remain undamaged in order to restart the reactor without unloading, the assembly must remain within the elastic range; (ii) it is acceptable to waste the whole core and the unloading capability must only be assured. Recent studies have shown that irradiated Zircaloy remains ductile [13]. Spacer grid brittle fracture can therefore be ruled out and the analysis can be confined to the dynamic buckling for which experimental results obtained on non-irradiated spacer grids are essentially applicable. In fact, buckling loads are mainly dependent on the Young’s modulus which is not affected by irradiation. This, of course, is a rough statement that should be confirmed experimentally on irradiated spacer grids. The following considerations have led to us to adopt an energetic criterion to quantify the damage to the spacer grids: the stiffness k of the impact element is difficult to choose, because of the spacer grid non-linear characteristic in the elastic range (fig. 10) and its softening behaviour in the plastic range (fig. 12). On the other hand, as will be seen in section 4, the computer results are less sensitive to k when expressed in terms of impact element maximum strain energy Ed than in terms of maximum impact force F,,,, these quantities being
115
A. Preumont et al. / Seismic analysis of P WR - RCC fuel assemblies
severe impacts at V, > Vz. Although large (maximum transverse deformation = 15 mm), the deformation is uniformly distributed throughout the spacer grid. Moreover, the guide tubes remain undamaged, which is essential to the control rod insertion. 4. Sensitivity analysis
Fig. 13. 14~ 14 Zircaloy grid (provided with claddings) several severe impacts. Maximum transverse deformations mm. The guide tubes remain undamaged.
connected
after - 15
by:
The energy with which Ed must be compared in order to verify the soundness criterion still has to be found. If we impose that the impacts remain in the elastic range, two values of this energy can be defined, respectively from the static and the dynamic tests: (1) The shaded area under the static load diagram (fig. 10) represents the maximum elastic energy. E,, that can possibly be stored in the spacer grid before instability. Ed c E,/2 is therefore a possible criterion2. . (2) Probably more realistic is the following criterion, defined from the dynamic tests: we have seen in section 3.2. that below a critical impact velocity Vt, no permanent deformation occurs. From Vz and the impacting mass, we can define the critical energy: E,* = fmV*i.
(9)
The soundness criterion becomes Ed G E,*/2. As expected, E,* is larger than E,. Experimental results on 14 X 14 Zircaloy spacer grids have shown: E,=
1.6 Nm,
A sensitivity analysis has been made with the computer program CLASH for a row of 5 fuel assemblies. The fuel assembly model is displayed on fig. 5 (Guyan’s reduction No. 3, in table 5). 15 modes are considered in the integration (f,, = 135 Hz). The excitation corresponding to a 0.1 g max ground acceleration consists of the first 5 s of the accelerogram shown in fig. 5 of [l]. The excitation has also been doubled and trebled to simulate, respectively, a 0.2 g and a 0.3 g max ground acceleration. The results are summarized in table 6. The following conclusions can be drawn: - the comparison of cases No. 1 and No. 2 shows that doubling k induces a 30% rise in F,, and a 12% drop in Ed. This confirms the lower sensitivity of the results to k, when expressed in terms of maximum strain energy than in terms of maximum impact force. This has already been stressed in section 3.3; - the comparison of cases Nos. 1, 3 and 4 indicates that the maximum impact strain energy E,, does not depend heavily on the coefficient of restitution r of the spacer grids 3. Moreover, the results obtained in case No. 7 with a viscous damper are not widely different from those obtained in case No. 6 with an hysteretic damper. Therefore, it can be said that the computer results are only slightly sensitive to both the kind and the magnitude of the spacer grid dissipation; - the comparison of the results obtained in cases No. 5 and No. 6 indicates a heavy dependence on the modal damping 5,. Note that the reduction in CP time, for ti = 0.05, corresponds to a smaller number of impacts; - cases 7, 8 and 9 indicate that both the impact energy and the CP time increase greatly with the maximum ground acceleration. More details are available in [ 141.
E,*=4to5Nm.
Curves 8; similar to 8; shown in fig. 11 can be used to define other critical energies, leading to a given amount of transverse deformation after N impacts. Fig. 13 shows a Zircaloy spacer grid after numerous
5. Example
’ The factor l/2 results from the representation grid by two impact elements.
3 Note however the unexpected
of the spacer
As an application, we consider the Belgian reactor DOEL l-2. Its core is represented schematically in
rise in Ed.
trend: a decrease
of r induces a
116
A. Preumont
et al. / Seismic analysis of P WR - RCC fuel assembhes
_
I,
A. Preumont
et al. / Seismic analysis of P WR - RCC fuel assemblies
117
i,
H
+ B
3 ” i
E
I
F
I I
I
I
118
A. Preumont
et al. / Seismic analysis of P WR - RCC fuel assemblies
Table 6 CLASH,
sensitivity
Case No.
analysis
Max. ground act.
Time steps (ms)
(g)
dt,
0.1
0.5 0.5 0.5 0.5
0.1 0.1 0.1 0.1 0.1 0.1 0.2 0.3
1 1 1 1 1
Impact k
h
c
dt,
(N/m)
(N/m)
(Ns/m)
0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25
2x10’ 1x 10’ 2x10’ 2x 10’ 2x107 2x10’ 2x107 2x10’ 2x10’
2x 106 1x106 4x lo6 6X 1oh 4x IO6 4x 106 0 0 0
0.0 0.0 0.0 0.0 0.0 0.0 3500 3500 3500
a Based on m =31 kg (mass matrix table 5). h According to eq. (8). ’ CYBER 174, Scope 3.4.
Table 7 Application
to the DOEL
- Summary
Lx 0’0
Ed (N.m) h
(s)
3705 2192 4200 4464 4212 3541 3023 5344 6110
0.312 0.354 0.367 0.383 0.369 0.26 1 0.228 0.714 0.933
3.11 3.11 3.1 I 3.11 3.11 3.12 3.12 3.56 3.74
to a translation
CPU ‘ (5)
d.o.f. of a spacer
t
grid: node
576 621 591 596 502 369 372 799 1096
18, d.o.f. 3, in
of the results CPU a
(N)
Ed
3483 10 140
0.2g
PJm)
t (s)
0.253 2.14
174, Scope 3.4. (this figure would be reduced
Impact
5,
0.02 0.02 0.02 0.02 0.02 0.05 0.05 0.05 0.05
0.904 0.904 0.816 0.734 0.816 0.816 0.802 a 0.802 0.802
term corresponding
Most severe impact
(s)
O.lg
Table 8 Impact forces between of occurence
r
Modal damping
Most severe impact F,,,
Baffle Assy 1 Assy 2 Assy 3 Assy 4 Assy 5 Assy 6 Assy 7 Assy 8 Assy 9 Assy 10 Baffle
larger diagonal
1-2 reactor
Maximum ground acceleration
a CYBER
element
1350 2945
4.477 3.125 by a factor
10 on a CYBER
the spacer grids of the 3rd level ( amax =0.2g,
F,,,
176).
= 10 140 N, Ed = 2.14 Nm). Each cell contains
forces (W F,,,)
10
20
30
40
50
60
70
24 19 35 20 30 22 24 28 27 32 16
6 15 15 14 15 16 18 17 11 16 24
9 4 6 8 6 8 8 7 9 7 10
6 5 3 3 3
3 4 3
4
1
80
90
1
1 2 3 4 5 4
2 2 6
100
1 1
the number
A. Preumont et al. / Seismic analysis of P WR - RCC fuel assemblies
fig. 2 of [ 11; the maximum number of fuel assemblies in a row is 13. Computations have been carried out for a row of 10 fuel assemblies. The 10 s duration core plate accelerogram is the one displayed on fig. 5 of [l] for a 0.1 g maximum ground acceleration or its double, corresponding to a 0.2 g maximum ground acceleration. The fuel assembly dynamic model is the one in fig. 5, except for the fluid added masses which slightly reduce the natural frequencies. The other data are the same as those corresponding to case No. 6 of the sensitivity analysis (section 4). Note that the modal damping (& = 0.05) can be regarded as very conservative (see section 2.6.). The main results are summarized in table7. Figs. 14 and 15 display the motion of the 3rd level of spacer grids (from the bottom), respectively for a 0.1 g and a 0.2 g maximum ground acceleration. The distances between the axis represent the gaps between fuel assemblies (0.73 mm between inner assemblies and 1.48 mm between outer assemblies and baffle); curve crossings indicate shocks. The advantage of the two-time step algorithm is quite clear from fig. 14 where one sees long periods without shocks for this level of spacer grids. For this example, dt,/dt, = 4. Table 8 gives the impact history for the 0.2 g case; each cell contains the number of occurence of the impact force between the 3rd spacer grids of the various assemblies in the row. For example, 24 impacts between the 3rd spacer grid of assembly 1 and the baffle have a maximum force G O.lF,,; 6 impacts have a maximum force between 0.1 F,, and 0.2Fm,, etc... Note that the largest impact forces involve peripheral (fresh) fuel assemblies. From table7, one sees that the maximum impact element elastic strain energy Ed remains below the critical energy E,*/2 = 2 to 2.5 Nm, even for a 0.2g maximum ground acceleration. No plastic strains are therefore to be expected. Of course, several core plate accelerograms should be considered to achieve a complete safety assessment.
6. Conclusions A simple vibrational model of the PWR-RCC fuel assembly has been proposed that gives a good account of the natural frequency spacing. Such a model has not been available up to now. It can be introduced without further modification into the computer program CLASH which studies the colliding vibration of a row of fuel assemblies.
119
From a sensitivity analysis performed with the program CLASH and impact tests on Zircaloy spacer grids, it is suggested that the spacer grid soundness criterion be expressed in terms of strain energy. Critical energy obtained from dynamic tests is almost three times larger than that obtained from static tests.
Acknowledgement The authors wish to thank their co-workers, Messrs Nysten, Oris and Delcon of the Technology Department, CEN/SCK, Mol, whose contribution to the experimental program was essential.
References [l] A. Preumont, Nucl. Engrg. Des. 65 (1981) 49-62. [2] A. Preumont and T. van Steenberghe, Smirt-6, Paris, 1981, Paper D 3/8. [3] A. Preumont, On the vibrational behaviour of PWR’s fuel rods, to be published in Nuclear Technology. [4] Y. Tigeot and P. Buland, Analyse du comportement vibratoire des assemblages combustibles destines a equiper les reacteurs de puissance du type ‘eau ordinaire sous pression’, BIST, CEA No. 213 (April 1976). [S] F.E. Stokes and R.A. King, PWR fuel assembly dynamic characteristics, BNES, Vibration in Nuclear Plants, Keswick, UK (May 1978). [6] H. Nuno et al., SMiRT4, San Francisco, 1977, paper D 4/6. [7] Aimablement communique par la societe FRAMATOME (siege de Lyon). [8] H.J. Wehling et al., SMiRT-6, Paris, 1981, paper D 3/7. [9] R. Clot& and J. Penzien, Dynamics of Structures (MC Craw Hill, New York, 1975). [lo] R.J. Guyan, Reduction of stiffness and mass matrices, AIAA Journal, Vol. 3, No. 2 (1965). [ 1 l] C.I. Yang and T.J. Moran, Calculations of added mass and damping coefficients for hexagonal cylinder in confined viscous fluid, Trans. ASME, Journal of Pressure Vessel Technology, Vol. 102, May 1980. [12] H. Nuno et al., SMiRT-2, Berlin, 1973, paper K 6/10. [13] F. Garzarolli et al., SMiRT-6, Paris, 1981, paper C 2/l*. [14] A. Preumont, Analyse seismique du coeur d’un reacteur PWR, These de doctorat, Universite de Liege (1981). [ 151 A. Preumont, Frequency domain analysis of time integration operators, to be published in Eurthquake Engineering and Structural Dynamics.
Nuclear Engineering and Design 71 (1982) 103- 119 North-Holland Publishing Company
SEISMIC ANALYSIS A. PREUMONT Belgonucleaire
OF PWR-RCC
FUEL ASSEMBLIES
and P. THOMSON
S.A., Rue du Champ de Mars 25, B - 1050 Bruxelles,
Belgium
J. PARENT CEN/SCK, Received
Boeretang 200, B - 2400 Mel, Belgium 9 November
1981
The present paper investigates the dynamic behaviour of PWR-RCC fuel assemblies under seismic excitation. A simple vibrational model of the fuel assembly is proposed, which leads to natural frequencies whose spacing agree with experimental data. Available experimental results are reviewed. Impact characteristics of Zircaloy spacer grids are also discussed. It is proposed that their soundness criteria be expressed in terms of impact energy rather than in terms of impact force. The computer code CLASH is briefly described; it is utilized to perform a sensitivity analysis. An example of application is also given.
1. Introduction In a previous paper by one of the authors [l], an efficient algorithm was presented for the PWR’s core seismic analysis (computer program CLASH). This algorithm performs the time integration of the colliding vibration of a row of fuel assemblies. Its main features are: - The program takes into account a possible difference between the core plates accerelations. - It integrates the equation of motion of each fuel assembly separately, the contact forces (restricted to the spacer grids) being re-evaluated at each time step and treated as external forces. - The fuel assemblies transverse deformations, restricted by the core barrel, are assumed to be small enough to allow a linear dynamic representation of the fuel bundle. Modal superposition can therefore be used. - Two time-steps dt, and dt, (multiple of dt,) are used in the integration, depending on whether or not the fuel assembly analyzed impacts its neighbours. This procedure is made even more profitable by the use of Duhamel’s formula. In fact, in this case, where no error is connected with the integration operator i, the only constraint on the larger time step, d t,, is the
’ More precisely, with its homogeneous 0029-5493/82/0000-0000/$02.75
part, see
[ 151.
0 1982 North-Holland
good representation of the low frequency (below 35 Hz) seismic excitation. Ultimately, dt, could be as large as the core plates accelerograms sampling period. Obviously, the optimum value for dt, (from the CP time point of view) varies from one case to another, depending on the number of fuel assemblies, the gap size, the impact characteristics of the spacer with the time step grids, etc.. . . When integrating dt,, the iterative refinement process is restricted to the fuel assemblies involved in shocks. The whole integration policy is summarized in table 1. Besides the core plates accelerograms (if no available, they can be artificially generated from a core response spectrum), the core motion time integration, as outlined above, requires the following input data (table 2):
Table 1 Two-time
step integration
policy (dt, = k dt,)
104
A. Preumont et al. / Seismic analysis of P WR - RCC fuel assemblies
Table 2 Core seismic analysis, overall process
These various requirements sections.
will be reviewed
in the next
2.2. Fuel assembly static properties
_ a fuel assembly
vibrational model: _ the spacer grids impact characteristics. This is one of the purposes of this paper to discuss how these data can be determined. This matter has already been partially debated in a recent paper [2]. Another aim of this paper is to discuss the soundness criteria of the spacer grids. It is suggested that they be expressed in terms of strain energy rather than in terms of impact force. A sensitivity analysis has also been made with the computer program CLASH; it reveals the predominent role of some parameters (e.g. the modal damping) as compared with others (e.g. the kind of spacer grid dissipation). Finally, an example is fully described.
2. Vibrational model of PWR-RCC
Fig. 1 shows the results of static tests carried out on a 14 X 14 (6 spacer grids) fuel assembly for two types of boundary conditions: (1) simply supported at the top, clamped at the bottom, loaded at the 4th spacer grid from the bottom (referred to as case No. 1); (2) clamped at the bottom and loaded at the top (referred to as case No. 2). The figure displays the equivalent bending stiffness, versus the applied load. For a given force, the vertical scattering corresponds to the results obtained at the various spacer grids. The figure reveals: - The equivalent bending stiffness decreases as the load increases, indicating a softening behaviour of the fuel assembly. This expected non-linear effect is inherent to the fuel assembly design (mainly caused by friction between the fuel rods and the spacer grids) and cannot be accounted for in a linear model. However this effect seems to be small enough in case No. 1 which is close to the in-core boundary conditions, to justify the linearity assumption in this case.
EI
10’ I Fuel
kgmm* Assembly
Equivalent
Bending
Stiffness
fuel assembly
2.1. Requirements A good vibrational model of fuel bundle should fulfill the following requirements. It should: account for the static stiffness properties of the fuel assembly; account for the individual vibrational behaviour of the fuel rods, which make up most of the assembly mass; lead to natural frequencies whose spacing agrees with experimental results within the largest possible frequency range (in fact, even though the seismic excitation contains only low frequency components, higher frequency modes are excited during impacts); be simple, in order to be introduced as such in the core model.
Load 10
20
50
40
Fig. 1. Fuel assembly static tests.
50
60
(kg) 70
F
A. F’reumont et al. / Seismic analysis of P WR - RCC fuel assemblies
- More surprising
is the large dependency of the equivalent bending stiffness on the boundary conditions: the equivalent bending stiffness obtained in case No. 2 is about three times larger than the one obtained in case No. 1. Clearly, this indicates the inadequacy of a beam model to represent the static behaviour of the fuel assembly. However, if one includes the shear deformation, which is much larger with respect to the bending deformation in case No. 1 than in case No. 2, the agreement of a beam model can be considerably improved, provided a large value is adopted for the shear deformation control parameter:
E1 E ( r 12=O12to0 .
a=-----=Gk’7 GO’12
. 15 )
where E stands for the Young’s modulus, G is the shear modulus, 52’= k’A is the effective shear area of the
-1 % f
/ /’
‘\\,,
1 3oLp’
‘\\
\
OL---; f
:
+ Experiment o F.E. case 1
8. F.E. model
lnm (clamped-pinned) I mm(clamped 1 Displacements Fig.
2.
Comparison
element model.
between
the static
tests and
the
finite
105
section, 1 is the length of the beam, and r = fl is the radius of gyration of the cross section. The two-beam model described below includes shear deformation; it can be seen in fig. 2 that it gives results in reasonably good agreement with experiments, for both boundary conditions. It must be understood that for such a large value of the control parameter a, shear deformation makes up most of the total deformation in case No. 1. The boundary conditions of a fuel bundle, in a nuclear reactor core, can conveniently be represented by an embedment at the bottom, and a rotary spring at the top. Actually, the displacements in the central part of the bundle are rather insensitive to the rotary spring stiffness K,, since most of them result from shear deformation. 2.3. Vibrational behaviour of the fuel rodr It is well known that the low amplitude vibrations of PWR’s fuel rods can adequately be represented by a beam model (no shear deformation) with the following properties: - The mass per unit length results from both the fuel pellets and the cladding tube. - The fuel pellets do not contribute to the rigidity of the rod. _ The cladding-to-spacer grid connection is modelled by a rotary spring whose stiffness K, can be determined by a simple static test (fig. 3). It depends on the grid spring force and also, heavily, on the distance between the dimples inside the grid cells. - The fuel rod prestresses induced by the plenum spring are considered in the analysis. This longitudinal preload introduces a rise in the natural frequencies, which is controlled by the ratio of the preload to the Euler’s buckling load of the rod. The suitability of this model is shown on fig. 4 which compares F.E. and experimental results [3]. Although, out of the scope of this paper, it is interesting to note that the vibrational behaviour of the fuel rods is dependent on irradiation, due to: the creep down of the cladding and the resulting fuel-cladding interaction: the gap reduction makes the pellets contribute to the rod rigidity, thus raising the natural frequencies; it also reduces the rattling of the pellets inside the cladding, reducing the damping; the plenum spring relaxation (from six times the fuel weight to zero); the grid spring relaxation: the spring force reduction induced by both the grid spring relaxation and the cladding creep down is responsible for a (slight)
106
Preumonl et ni. / Seismic anatysrs of P WR - RCC fuel
A.
CLAD
TO SPACER
GRID
assemblies
CONNECTION
Model
Dimples
0.6
0
1
Fig. 3. Clad-to-spacer
change in vibration, ting wear, affect the
2 grid connection.
3
4
5
6
Rotary stiffness.
K,. Irradiation dependence of the fuel rod together with other aspects concerning fretis discussed in [3]. It is not expected to bundle properties to a large extent.
2.4. Vibrational model, comparison with experimental
data
The preceding considerations lead to the two-beam model of fig. 2a. The first beam includes the shear
deformation and is mainly responsible for the static behaviour of the fuel assembly; its stiffness properties (EZ, a) correspond to the whole assembly (skeleton + fuel rods), as discussed in section 2.2.; the mass per unit length comes from the skeleton only. The second beam simply adds the mass and stiffness properties of the fuel rods. The two beams are connected by rotary springs at the grid levels, whose stiffness is the sum of the individual rod-to-grid connections. If two kinds of these con-
107
A. Preumont et al. / Seismic analysis of P WR - RCC fuel assemblies
SPACER GRIOS
F.E. (Hz) 44.76 49.15
Tests (Hz) 47.6
49.30 53.66
60.4
57.91 61.50
71.6
68.85 71.15 flNlTE -------
75.9
ELEMENT MODEL
EXPERIMENT
79.00 79.81
Fig. 4. Fuel rod vibrational behaviour. Comparison between F.E. model and experiments. For each mode, the two computed frequencies refer respectively to K, = 100 Nm/rad and 200 Nm/rad. The computed mode shapes correspond to Ke = 200 Nm/rad.
nections exist in the fuel assembly (corresponding to various distances between the dimples inside the grid cells, as it is the case in BN’s Zircaloy spacer grids - see fig. 3), then either a mean value is adopted for K, and a two-beam model can be used (with the consequence of introducing a single array of natural frequencies for the fuel rods), or a three-beam model is adopted, with two values K,, and K,, for the clad-to-grid connection stiffnesses. Such a model is shown in fig. 5 for a 14 X 14 (6 spacer grids) fuel assembly. It consists of a 55 degree of freedom (d.o.f.) Finite Element (F.E.) model. Modes 1 to 6 are displayed; their deviation with respect to beam modes increases with the order of the mode.
Modes 5 and 6 are local fuel rod modes (compare with fig. 4); their frequencies depend mainly on the stiffness K,, as can be seen in table3 for the two-beam model. Table3 also shows that the stiffness K, does not influence the results strongly, as already mentioned. Table4 gives experimental natural frequencies (in air) of PWR fuel assemblies [4,5,6,7,8]. Their spacings (ratio fi/fi) are plotted in fig.6, where it can be seen that the various modes are closer than those of a beam clamped at both ends (boundary conditions leading to the closest natural frequencies for a beam). It is this fact, together with the surprising static test results discussed in section 2.2, that suggested the introduction of
108
A. Preumont
r--------
-7
et al. / Seismic analysis of P WR - RCC fuel assemblies
109
A. Preumont et al. / Seismic analysis of P WR - RCC fuel assemblies Table 3 14 X 14 (6 spacer grids) fuel assembly.
Two-beam
K, = lo4 Nm/rad K,=179x300 Nm/rad ’
Mode i
1 2 3 4 5 6
finite element
model natural
frequencies.
K, = 10’ Nm/rad K,=179X300 Nm/rad
Parametric
study on K, and Ke
K, = lo5 Nm/rad K,=179X200 Nm/rad
L (Hz)
h/f,
h (Hz)
L/f,
6.33 14.62 23.72 34.34 50.23 62.62
1.00 2.31 3.75 5.42 7.94 9.89
6.50 14.71 23.84 34.48 50.58 63.20
1.00 2.26 3.67 5.30 7.78 9.72
b
f, (Hz)
X/f,
6.25 14.10 22.90 33.36 47.21 61.08
1.00 1.26 3.66 5.34 7.55 9.77
a 179 fuel rods in a 14X 14 fuel assembly. b Plotted on fig. 6.
shear deformation in the fuel assembly model. For it is well known that shear deformation contributes to bringing the natural frequencies together (see [9], page 318320). The agreement of the F.E. model with experimental results can be judged from the comparison of predicted and experimental spacings f,/fi. Column No. 2 of table 3 has also been plotted in fig. 6. We can see that the F.E. results lie well within the range of the experimental results, except for mode 5. In fact, the comparison is
meaningless for this mode, because it is a local mode of the fuel rod in the F.E. model, but not for the experimental results. The agreement is particularly good with the results of FRAMATOME [7] which have been obtained with the same type of fuel assembly (14 X 14, 6 spacer grids). The ultimate use of such a model being the time history integration of the core motion, it is important to limit its number of d.o.f.. The Guyan’s reduction [lo] can contribute significantly in this way. It expresses the
Table 4 Natural
frequencies
of PWR-RCC
fuel assemblies
- Experimental
data in air
Mode 1
2
3
4
5
6
f;/f,:
1
4
9
f,/f,:
1
2.76
5.4
8.93
13.34
CEA 15x 15 7 sp. grid [4]
f; (f-f+ f;/f,:
3.7 1.0
7.63 2.05
11.67 3.15
18.55 5.01
24.78 6.7
B&W 17x17 8 sp. grids [5]
L/f,:
1.0
1.9
3.0
4.1
5.3
Pinned
at both ends
Clamped
Mitsubishi
at both ends
14X 14 [6]
FRAMATOME 6 sp. grids [7] K.W.U. 16X 16 7 sp. grids [S]
f;/f,
a:
16
25
1.0
2.3
3.6
5.1
6.6
13.21 2.03
22.05 3.39
30.55 4.7
_b
;;“:
6.5 1.0
/I (f&a: J/f,:
3.0 1.0
14X 14
’ Approximate values estimated from graphs. b A fuel rod local mode is missing here.
12.0 4.0
7
20.0 6.6
38.31 9.87
62.16 9.56
78.93 12.14
110
A. Preumont
et al. / Seismic analysis of P WR - RCC fuel assemblzes
Table 5 Comparison of various cases of Guyan’s reductions Nodes (fig. 5)
Independent degree of freedom F.E. model
2 3 5 7 8 9 10 11 13 15 16 17 18 19 21 23 24 25 26 27 29 31 32 33 34 35 37 39 40 41 42 43 45 47 Number of d.o.f.
3
3 3 3 3
3 3 3 3
3 3 3 3
3 3 3 3
3 3 3 3
55
First natural frequency Y , independent d.o.f. clamped
5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5
Guyan’s reduction Case 1
Case 2
Case 3
3 3 3
5 5 5
3 3 3 3
5 5
5
3 3 3 3
5 5 5
3 3 3 3
5 5
5
3 3 3
5 5 5
3 3 3 3
5 5
5
3 3 3
5 5 5
3 3 3 3
5 5
5
3 3 3
3 3
3 3 3
5 5
3 3 3 3
5 5
18
30
30
144.4
174,4
324,5
3 3 3
3 3 3
3 3 3
eigenvalue problem in terms of some selected (independent) d.o.f. by assuming a static relationship for the other (dependent) d.o.f.. It can be shown that this condensation leads to an upper bound for the eigen-fre-
quencies and to a good estimate natural frequencies f, satisfy (.A/%)“~
19
of the modes
whose
(2)
111
A. Preumont et al. / Seismic analysis of P WR - RCC fuel assemblies
2.5. Factors affecting the fuel assembly natural frequencies
fi ffl
::,ment
0
*
0
1
3
Bz V
2.
5
6
Mode
0, Values Finite Element
The vibrational behaviour of a fuel bundle depends on numerous parameters, namely, the amplitude of vibration, the surrounding medium, the temperature and the burn-up. Fig. 7 illustrates the influence of the first three parameters on the first natural frequency according to the results of Stokes and King [5]; these authors’ conclusions follow: - f, decreases when the mid-span amplitude increases; - water induces an approximately 15% drop in f,, which can be accounted for by considering the added masses. Similar results have been recently obtained for higher order modes [8]; - f, does not depend on the water temperature; the drop in the water density is compensated for by the drop of the Young’s modulus; - f, does not depend on the water velocity; - end-of-life first natural frequency is 5 to 10% lower, due to the gridspring relaxation.
model
Fig. 6. Fuel assembly natural frequencies. Comparison between the predicted and experimental spacings.
where rI stands for the first natural frequency of the system obtained by clamping all the independent d.o.f. Table 5 compares three different reductions that have been carried out on the F.E. model of fig. 5. D.o.f. 3 refers to the translation in the “y” direction, d.o.f. 5 refers to the rotation around the “z ” axis. The comparison of cases No. 2 and No. 3 shows that an adequate choice of the d.o.f. can improve the spectral convergence of the reduction considerably: a slightly different choice of the 30 independent d.o.f. raises the frequency Y, from 174.4 Hz to 324.5 Hz. The computation of v, is very helpful in the choice of the independent d.o.f. It must be emphasized that the model which is presented in this paper is intended to represent the overall vibrational behaviour of the bundle, as needed for the core motion integration and the computation of the interassembly impact forces. It cannot be used to estimate the stresses in the fuel assembly under external loads. In particular, the first beam described at the beginning of this section (referred to as type 7 in fig. 2a) has no physical existence and should not be confused with the skeleton of the fuel assembly *. Stress computation requires a more complicated (even non-linear, with friction elements) model which is out of the scope of this paper.
2.6. Modal damping Fig. 8, drawn after Stokes and King [5], illustrates the dependency of the (end-of-life) damping of the first mode on the vibration amplitude, the surrounding fluid and its temperature. Their conclusions are: - the damping is approximately independent of the mode ( ti = tl) and the direction; - .$‘,increases with the amplitude; - .$, is 25 to 100% larger (depending on the amplitude) at the end-of-life;
I
1
04
02
0.
95
1.
mid _ span displacement
1.5 lmm
1
Fig. 7. First natural frequency versus mid-span amplitude. The influence
* Similarly, the distance between the two beams is immaterial.
WATER Thii curve is temperature independent I 2@C _ 32O’C
0.6
of water is temperature
King (51).
independent
(after Stokes and
A. Preumont et al / Seismic analysis of P WR - RCC fuel assemblies a LMFBR, the thin coolant layer which is trapped between the wrapper tubes, induces a strong interaction between the fuel assemblies. Usually, it is assumed that this interaction can be expressed in terms of a coupling added mass matrix (part of the coupling forces in phase with the accelerations) and a coupling fluid damping matrix (part of the coupling forces in phase with the velocities). This problem has been analysed in [ 111. Control rods do not affect the bundle vibration characteristics [5].
I 0.5
15
1.
mid. spm
displacement
I 2.
1mm)
Fig. 8. Damping of the first mode versus mid-span amplitude. In air and still water (temperature ranging from 20°C to 32OT) (after Stokes and King [5]).
- 5, increases considerably (+ 120%) with water; - 5, does not depend on the temperature; - 5, increases strongly (+60 to 400%) with the axial flow velocity. Larger amplitudes have been considered by Wehling et al. [8]; they reveal the following damping values: 0.05
CO.08
for air,
0.12<(,
CO.30
for still water.
3. Spacer grids impact behaviour 3.1. Spacer grid dynamic model In the core model, the spacer grids are modelled by impact elements consisting of a spring, a viscous damper and an hysteretic damper (fig. 9a); so that it contains the two classical impact elements of figs. 9b and 9c as particular cases:
(4 Spring and viscous damper (fig. 9b): its constitutive equation
reads:
f = kx + c,ic. The restitution by: r= -V/&‘o-e-“c,
(3) coefficient
is approximately
(.$Kl)
where 5 = c/26.
(4)
(5)
It depends
2.7. Inter-assembly jluid coupling - Control rods At this point, it should be mentioned that off-diagonal coupling coefficients (mass as well as damping) are not considered in the core model, meaning that hydraulic coupling between the fuel assemblies is neglected. This assumption is partly justified by water loop tests reported in [5], which indicate that the natural frequency drop can be accounted for by added masses corresponding to the displaced liquid, meaning that a fuel assembly in a square channel does not behave in a significantly different manner than a fuel assembly in an infinite fluid. Physically this is a consequence of the fact that without a wrapper tube, the fluid can flow from one side of the assembly to the other, across the fuel assembly rather than around it. From the foregoing observation, it is the authors’ belief that the cross-coupling terms neglected in the analysis should be small. On the contrary, it is obvious from the well known results on concentric cylinders that, when the fuel assemblies are provided with wrapper tubes as they are in
given
(b)
explicitely on the impacing mass m and, consequently, varies from one spacer grid to another. Spring and hysteric damper (fig. 9~): this element has been used by Nuno et al. [12]. Its constitutive equation reads: for120, (k-h)x
forf1<0.
Its restitution coefficient enjoys property of being only dependent element itself:
the remarkable on the impact
r does not depend on the initial velocity VO, nor on the impacting mass m; it is constant for all spacer grids. A sensitivity analysis presented later in the paper reveals that, if the viscous damping coefficient c is chosen in order to lead to the same value of r for the
A. Preumont et al. / Seismic analysis of P WR - RCC fuel assemblies
-I
dash-pot
impact drrnent a. Spacer Grid Representation in the Core Model
b. Spring and Viscous Damper
f .kx+ci r-i-e-6V VO
C
-nt
I--
2Jiz
oLw X
c.
Spring and Hysteretic Damper.
(k+h)x f’lk-h)x
(i-0) tic* 0)
r=_V-_
JR-
vo
Fig. 9. Impact element.
Fig. 10. 14X 14 Zircaloy spacer grid. Static compression
c-r,.“,
“0
.a
14114 Zr apacer grid
tests.
1
6.
r
.6
.5
.4
maximum mass translational d.o.f. of a spacer grid as the one given by eq. (7) for the hysteretic damper, only slight differences are observed in the grid displacements and impact forces. Constants k, c and h depend on the stiffness and dissipation properties of the spacer grids. These properties are examined in the next section. 3.2. Static and dynamic tests on spacer grids
.3
.2 I-.1 t
I 0.
i -
1
-.5*
I 1.
1":
Fig. 10 shows the results of static compression tests performed on BN’s 14 X 14 Zircaloy spacer grids. The
Fig. 11. 14x 14 Zircaloy spacer grid. Impact tests. Restitution coefficient
and transverse deformation
versus impact velocity.
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A. Preumont
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following features are worth noting: _ the loading characteristic is non linear. This makes the stiffness k of the impact element rather difficult to choose; _ the ruin occurs by instability; - the grid stiffness is much larger when provided with fuel rods, as compared with a grid with empty cells. The easiest way to quantify experimentally the spacer grid dissipation during impact is to measure the restitution coefficient r. Once r is known and an impact model has been adopted, a straightforward calculation through formula (4) or (7) leads to the damper characteristic c or h. Fig. 11 shows a plot of the coefficient of restitution r versus the impact velocity Vc. One sees that in the elastic range, below a critical velocity V:, r is almost independent of V, (r - 0.60 to 0.65). This agrees with the eqs. (4) and (7). S;, a measure of the spacer grid permanent transverse deformation after three successive impacts at Va, is also plotted on fig. 11. We can see that below a critical velocity V:, no permanent deformation occurs. For velocity V0> V:, the transverse permanent deformation increases with the number of impacts and, sharply, with the impact velocity. Note that the critical velocity V,* corresponds to a drop in the restitution coefficient (first impact).
0
10
20
30
40
50
60
Time (ms) Fig. 12. 5 X 5 Zircaloy grid. Velocity diagram during impact. Comparison of impact No. 2 and impact No. 13 on the same sample grid.
Fig. 12 shows the experimental velocity diagrams of the impacting mass for two of a succession of impacts (impact No. 2 and impact No. 13) on the same sample grid. Although obtained on a 5 X 5 (empty cells) spacer grid, these results are indicative of the plastic behaviour of the spacer grid when the number N of impacts increases: (1) The contact duration (i.e. the time required for the velocity to reverse) increases, meaning that the impact becomes softer (k decreases). (2) The reverse velocity V (after the shock) increases, meaning a rise of the coefficient of restitution r. 3.3. Earthquake
design criteria
According to the Standard Review Plan (section 4.2., Fuel System Design), the following capabilities must be maintained, under all conditions: (a) The fuel damage is never so severe as to prevent the control rod insertion. (b) The fuel assembly retains its rod-bundle geometry with adequate coolant channels to permit removal of the residual heat. In structural terms, this implies that the spacer grids do not buckle under the earthquake induced impact forces. Besides the above requirements, two possibilities can be considered for the OBE (Operating Basis Earthquake), depending on economic considerations: (i) the core must remain undamaged in order to restart the reactor without unloading, the assembly must remain within the elastic range; (ii) it is acceptable to waste the whole core and the unloading capability must only be assured. Recent studies have shown that irradiated Zircaloy remains ductile [13]. Spacer grid brittle fracture can therefore be ruled out and the analysis can be confined to the dynamic buckling for which experimental results obtained on non-irradiated spacer grids are essentially applicable. In fact, buckling loads are mainly dependent on the Young’s modulus which is not affected by irradiation. This, of course, is a rough statement that should be confirmed experimentally on irradiated spacer grids. The following considerations have led to us to adopt an energetic criterion to quantify the damage to the spacer grids: the stiffness k of the impact element is difficult to choose, because of the spacer grid non-linear characteristic in the elastic range (fig. 10) and its softening behaviour in the plastic range (fig. 12). On the other hand, as will be seen in section 4, the computer results are less sensitive to k when expressed in terms of impact element maximum strain energy Ed than in terms of maximum impact force F,,,, these quantities being
115
A. Preumont et al. / Seismic analysis of P WR - RCC fuel assemblies
severe impacts at V, > Vz. Although large (maximum transverse deformation = 15 mm), the deformation is uniformly distributed throughout the spacer grid. Moreover, the guide tubes remain undamaged, which is essential to the control rod insertion. 4. Sensitivity analysis
Fig. 13. 14~ 14 Zircaloy grid (provided with claddings) several severe impacts. Maximum transverse deformations mm. The guide tubes remain undamaged.
connected
after - 15
by:
The energy with which Ed must be compared in order to verify the soundness criterion still has to be found. If we impose that the impacts remain in the elastic range, two values of this energy can be defined, respectively from the static and the dynamic tests: (1) The shaded area under the static load diagram (fig. 10) represents the maximum elastic energy. E,, that can possibly be stored in the spacer grid before instability. Ed c E,/2 is therefore a possible criterion2. . (2) Probably more realistic is the following criterion, defined from the dynamic tests: we have seen in section 3.2. that below a critical impact velocity Vt, no permanent deformation occurs. From Vz and the impacting mass, we can define the critical energy: E,* = fmV*i.
(9)
The soundness criterion becomes Ed G E,*/2. As expected, E,* is larger than E,. Experimental results on 14 X 14 Zircaloy spacer grids have shown: E,=
1.6 Nm,
A sensitivity analysis has been made with the computer program CLASH for a row of 5 fuel assemblies. The fuel assembly model is displayed on fig. 5 (Guyan’s reduction No. 3, in table 5). 15 modes are considered in the integration (f,, = 135 Hz). The excitation corresponding to a 0.1 g max ground acceleration consists of the first 5 s of the accelerogram shown in fig. 5 of [l]. The excitation has also been doubled and trebled to simulate, respectively, a 0.2 g and a 0.3 g max ground acceleration. The results are summarized in table 6. The following conclusions can be drawn: - the comparison of cases No. 1 and No. 2 shows that doubling k induces a 30% rise in F,, and a 12% drop in Ed. This confirms the lower sensitivity of the results to k, when expressed in terms of maximum strain energy than in terms of maximum impact force. This has already been stressed in section 3.3; - the comparison of cases Nos. 1, 3 and 4 indicates that the maximum impact strain energy E,, does not depend heavily on the coefficient of restitution r of the spacer grids 3. Moreover, the results obtained in case No. 7 with a viscous damper are not widely different from those obtained in case No. 6 with an hysteretic damper. Therefore, it can be said that the computer results are only slightly sensitive to both the kind and the magnitude of the spacer grid dissipation; - the comparison of the results obtained in cases No. 5 and No. 6 indicates a heavy dependence on the modal damping 5,. Note that the reduction in CP time, for ti = 0.05, corresponds to a smaller number of impacts; - cases 7, 8 and 9 indicate that both the impact energy and the CP time increase greatly with the maximum ground acceleration. More details are available in [ 141.
E,*=4to5Nm.
Curves 8; similar to 8; shown in fig. 11 can be used to define other critical energies, leading to a given amount of transverse deformation after N impacts. Fig. 13 shows a Zircaloy spacer grid after numerous
5. Example
’ The factor l/2 results from the representation grid by two impact elements.
3 Note however the unexpected
of the spacer
As an application, we consider the Belgian reactor DOEL l-2. Its core is represented schematically in
rise in Ed.
trend: a decrease
of r induces a
116
A. Preumont
et al. / Seismic analysis of P WR - RCC fuel assembhes
_
I,
A. Preumont
et al. / Seismic analysis of P WR - RCC fuel assemblies
117
i,
H
+ B
3 ” i
E
I
F
I I
I
I
118
A. Preumont
et al. / Seismic analysis of P WR - RCC fuel assemblies
Table 6 CLASH,
sensitivity
Case No.
analysis
Max. ground act.
Time steps (ms)
(g)
dt,
0.1
0.5 0.5 0.5 0.5
0.1 0.1 0.1 0.1 0.1 0.1 0.2 0.3
1 1 1 1 1
Impact k
h
c
dt,
(N/m)
(N/m)
(Ns/m)
0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25
2x10’ 1x 10’ 2x10’ 2x 10’ 2x107 2x10’ 2x107 2x10’ 2x10’
2x 106 1x106 4x lo6 6X 1oh 4x IO6 4x 106 0 0 0
0.0 0.0 0.0 0.0 0.0 0.0 3500 3500 3500
a Based on m =31 kg (mass matrix table 5). h According to eq. (8). ’ CYBER 174, Scope 3.4.
Table 7 Application
to the DOEL
- Summary
Lx 0’0
Ed (N.m) h
(s)
3705 2192 4200 4464 4212 3541 3023 5344 6110
0.312 0.354 0.367 0.383 0.369 0.26 1 0.228 0.714 0.933
3.11 3.11 3.1 I 3.11 3.11 3.12 3.12 3.56 3.74
to a translation
CPU ‘ (5)
d.o.f. of a spacer
t
grid: node
576 621 591 596 502 369 372 799 1096
18, d.o.f. 3, in
of the results CPU a
(N)
Ed
3483 10 140
0.2g
PJm)
t (s)
0.253 2.14
174, Scope 3.4. (this figure would be reduced
Impact
5,
0.02 0.02 0.02 0.02 0.02 0.05 0.05 0.05 0.05
0.904 0.904 0.816 0.734 0.816 0.816 0.802 a 0.802 0.802
term corresponding
Most severe impact
(s)
O.lg
Table 8 Impact forces between of occurence
r
Modal damping
Most severe impact F,,,
Baffle Assy 1 Assy 2 Assy 3 Assy 4 Assy 5 Assy 6 Assy 7 Assy 8 Assy 9 Assy 10 Baffle
larger diagonal
1-2 reactor
Maximum ground acceleration
a CYBER
element
1350 2945
4.477 3.125 by a factor
10 on a CYBER
the spacer grids of the 3rd level ( amax =0.2g,
F,,,
176).
= 10 140 N, Ed = 2.14 Nm). Each cell contains
forces (W F,,,)
10
20
30
40
50
60
70
24 19 35 20 30 22 24 28 27 32 16
6 15 15 14 15 16 18 17 11 16 24
9 4 6 8 6 8 8 7 9 7 10
6 5 3 3 3
3 4 3
4
1
80
90
1
1 2 3 4 5 4
2 2 6
100
1 1
the number
A. Preumont et al. / Seismic analysis of P WR - RCC fuel assemblies
fig. 2 of [ 11; the maximum number of fuel assemblies in a row is 13. Computations have been carried out for a row of 10 fuel assemblies. The 10 s duration core plate accelerogram is the one displayed on fig. 5 of [l] for a 0.1 g maximum ground acceleration or its double, corresponding to a 0.2 g maximum ground acceleration. The fuel assembly dynamic model is the one in fig. 5, except for the fluid added masses which slightly reduce the natural frequencies. The other data are the same as those corresponding to case No. 6 of the sensitivity analysis (section 4). Note that the modal damping (& = 0.05) can be regarded as very conservative (see section 2.6.). The main results are summarized in table7. Figs. 14 and 15 display the motion of the 3rd level of spacer grids (from the bottom), respectively for a 0.1 g and a 0.2 g maximum ground acceleration. The distances between the axis represent the gaps between fuel assemblies (0.73 mm between inner assemblies and 1.48 mm between outer assemblies and baffle); curve crossings indicate shocks. The advantage of the two-time step algorithm is quite clear from fig. 14 where one sees long periods without shocks for this level of spacer grids. For this example, dt,/dt, = 4. Table 8 gives the impact history for the 0.2 g case; each cell contains the number of occurence of the impact force between the 3rd spacer grids of the various assemblies in the row. For example, 24 impacts between the 3rd spacer grid of assembly 1 and the baffle have a maximum force G O.lF,,; 6 impacts have a maximum force between 0.1 F,, and 0.2Fm,, etc... Note that the largest impact forces involve peripheral (fresh) fuel assemblies. From table7, one sees that the maximum impact element elastic strain energy Ed remains below the critical energy E,*/2 = 2 to 2.5 Nm, even for a 0.2g maximum ground acceleration. No plastic strains are therefore to be expected. Of course, several core plate accelerograms should be considered to achieve a complete safety assessment.
6. Conclusions A simple vibrational model of the PWR-RCC fuel assembly has been proposed that gives a good account of the natural frequency spacing. Such a model has not been available up to now. It can be introduced without further modification into the computer program CLASH which studies the colliding vibration of a row of fuel assemblies.
119
From a sensitivity analysis performed with the program CLASH and impact tests on Zircaloy spacer grids, it is suggested that the spacer grid soundness criterion be expressed in terms of strain energy. Critical energy obtained from dynamic tests is almost three times larger than that obtained from static tests.
Acknowledgement The authors wish to thank their co-workers, Messrs Nysten, Oris and Delcon of the Technology Department, CEN/SCK, Mol, whose contribution to the experimental program was essential.
References [l] A. Preumont, Nucl. Engrg. Des. 65 (1981) 49-62. [2] A. Preumont and T. van Steenberghe, Smirt-6, Paris, 1981, Paper D 3/8. [3] A. Preumont, On the vibrational behaviour of PWR’s fuel rods, to be published in Nuclear Technology. [4] Y. Tigeot and P. Buland, Analyse du comportement vibratoire des assemblages combustibles destines a equiper les reacteurs de puissance du type ‘eau ordinaire sous pression’, BIST, CEA No. 213 (April 1976). [S] F.E. Stokes and R.A. King, PWR fuel assembly dynamic characteristics, BNES, Vibration in Nuclear Plants, Keswick, UK (May 1978). [6] H. Nuno et al., SMiRT4, San Francisco, 1977, paper D 4/6. [7] Aimablement communique par la societe FRAMATOME (siege de Lyon). [8] H.J. Wehling et al., SMiRT-6, Paris, 1981, paper D 3/7. [9] R. Clot& and J. Penzien, Dynamics of Structures (MC Craw Hill, New York, 1975). [lo] R.J. Guyan, Reduction of stiffness and mass matrices, AIAA Journal, Vol. 3, No. 2 (1965). [ 1 l] C.I. Yang and T.J. Moran, Calculations of added mass and damping coefficients for hexagonal cylinder in confined viscous fluid, Trans. ASME, Journal of Pressure Vessel Technology, Vol. 102, May 1980. [12] H. Nuno et al., SMiRT-2, Berlin, 1973, paper K 6/10. [13] F. Garzarolli et al., SMiRT-6, Paris, 1981, paper C 2/l*. [14] A. Preumont, Analyse seismique du coeur d’un reacteur PWR, These de doctorat, Universite de Liege (1981). [ 151 A. Preumont, Frequency domain analysis of time integration operators, to be published in Eurthquake Engineering and Structural Dynamics.