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A non linear model for the PWR fuel assembly seismic analysis
Nuclear Engineering and Design 195 (2000) 321 – 329 www.elsevier.com/locate/nucengdes
A non linear model for the PWR fuel assembly seismic analysis B. Fontaine *, I. Politopoulos CEA, CEN-Saclay, Seismic Mechanic Study Laboratory, 91191 Gif sur Y6ette, France Received 20 November 1998; accepted 29 July 1999
Abstract In seismic PWR core analysis, the non linear behavior of fuel assemblies has to be studied taking into account variations of stiffness and damping due to the slippage of rods in spacer grids. Based on the linear CEA’s assembly model, principally composed of two beams representing sets of rods and thimbles, a linear model is described. In the second part of this paper the slippage and loss of contact of rods on a grid relic is analyzed. Based on the Coulomb friction model and plasticity analogy, a grid model constituted of elastoplastic hinge and non linear springs is proposed to simulate the progressive slippage of rods inside the cells. In the third part, a non linear assembly model is designed and validated with experimental data. This model represented with success the increase of damping and decay of frequency when the magnitude of the dynamic loading increases. © 2000 Elsevier Science S.A. All rights reserved.
1. Introduction Reactor cores are one of the most important structures in nuclear power plants from a safety point of view. Their good behavior under seismic loading has to be studied to assess the integrity of the fuel assemblies and the coolability of the core. There exists many fuel assembly models in literature but most of them are linear (Nuno et al., 1973; Kim et al., 1981; Preumont and Van Steenberghe, 1981; Preumont, 1981a,b; Preumont et al., 1982; Johnson, 1983; Leroux, 1983; Leroux et al., 1983; Queval and Brochard, 1988; Queval et al., 1991) and cannot represent the variation of damp* Corresponding author.
ing and frequency observed during vibration and snap back tests (Brochard et al., 1993). The model parameters have to be adjusted a priori for each loading. In the case of seismic excitation, which is unstationary, it is not easy to find the good adjustment of parameters able to represent the behavior of an actual assembly. When non linear models are used (Brochard et al., 1993; Helen and Johansson, 1993), they usually take into account all the contacts between rods and grids and thus, may be too sophisticated to be used in a whole core model for seismic analysis. Some other simpler non linear models are completely issued from global tests and cannot be used in a predictive analysis (Hotta et al., 1985).
0029-5493/00/$ - see front matter © 2000 Elsevier Science S.A. All rights reserved. PII: S 0 0 2 9 - 5 4 9 3 ( 9 9 ) 0 0 2 1 7 - 4
322
B. Fontaine, I. Politopoulos / Nuclear Engineering and Design 195 (2000) 321–329
The aim of this paper is to present a new non-linear model of a PWR fuel assembly which modeled sufficient local phenomena to be predictive but remains simple enough to be used in a whole core model.
2. PWR fuel assembly description
and damping increases when loading becomes higher. This phenomena comes principally from the slippage of fuel rods through spacer grids. During motion, the axial force on rods may become higher than friction forces and then slippage may occur. The slippage tends to reduce the assembly stiffness and creates dissipative phenomena which appear globally as additional damping.
Fuel assembly is composed of two sets: 4. Linear model
2.1. The fuel rods The rods are 3.8 m high tubes containing fuel pellets. Their number depends on the assembly type, but reaches 264 for a 17(17 bundle.
2.2. The squeletton The squeletton which is designed to hold a minimum gap between fuel rods is composed of guide thimbles clamped at both ends on stiff support devices (i.e. top and bottom nozzles). At different levels, grids are welded on the guide thimbles. The grids, in zirconium alloy hold the rods spaced to allow the cooler flow and to contribute to the lateral stiffness of the assembly. The rods are maintained in the grids by springs and dimples. During seismic excitation the rods may naturally slip through spacer grids.
3. Experimental behavior Many tests have been performed (Preumont and Van Steenberghe, 1981; Preumont, 1981a; Preumont et al., 1982; Queval and Brochard, 1988; Queval et al., 1991) on the single assembly to study its mechanical behavior. During these tests, a fuel assembly is set standing with both ends supported in the same manner as in the actual core and a (static or dynamic) load is applied to the middle grid. Then the deflection at middle grid is measured and is plotted versus load level. By applying sine wave loading, it is possible to determine the resonance frequency and damping of the assembly for each load level. It appears that assembly stiffness (or frequency) decreases
Under low amplitude vibrations. the behavior of the fuel assembly can be considered as linear. The internal forces inside the assembly are assumed to be sufficiently small to prevent slippage between rods and grids: the rods are then considered to be clamped in the grids. Thus a linear model can be used, composed of two main beams, one representing the guide thimble set and the other one the fuel rod set. Between the two beams are located linear springs KI which represent grids. The characteristics of both beams are the following: The bending inertia I eq and I eq (respectively, t r eq eq the cross section S t and S r ) are equal to the sum of the bending inertia It and Ir (respectively, cross sections St and Sr) of the guide thimbles or fuel rods. S eq t = NtSt
(1)
I eq t = NtIt
(2)
eq r
S = NrSr
(3)
I eq r = NrIr
(4)
where Nt and Nr are the number or thimbles and rods in an assembly. In parallel with the beams rotation springs Kt and Kr have been added between the grids to represent the fact that the rotation of the grids induces traction or compression on guide thimbles or rods. Their stiffness is: Kt =
EtSt Nt 2 %d L i i
(5)
Kr =
ErSr Nr 2 %d L j j
(6)
B. Fontaine, I. Politopoulos / Nuclear Engineering and Design 195 (2000) 321–329
with Et and Er the elasticity moduli of thimbles and rods, di is the distance between the ith rod (or thimble) to the axis, and L the height between two grids. Such a model, already described by (Queval et al., 1991) is able to calculate the deflection stiffness and eigenfrequency of an actual assembly.
5. Non linear links between grids and fuel rods The 254 rods, arranged in 17 rows, are maintained in the grids by springs and dimples, with a static force Fs. Each rod – grid link may be represented with a Coulomb friction model. A rod does slip in a grid if the axial force FT acting on it verifies FT ] maFs
(7)
and, in that case, a friction force Fg acts on it Fg = mgFs sgn(z; )
(8)
where (a, (g are, respectively, static and dynamic friction coefficients and z is the relative displacement in the axial direction. If we apply a rotation ( to the grid, perpendicular to the rod bundle axis, the grid undergoes a non linear restoring moment due to the progressive slippage of rods and loss of contact between rods and dimples. One of the consequences of the loss of contact between rods and dimples is the modification of the stiffness of the link between the grid cell and the rod. The resulting moment between rod and grid cell can be computed if we represent dimples by linear springs whereas the cell spring is considered non linear. When the rotation of the rod in the grid cell is small (B 1 mrad), the rod stays in contact with the two dimples and the cell spring. The corresponding stiffness which comes essentially from the dimples is high. For medium rotations (between 1 mrad and 40 mrad) one dimple looses contact with the rod (Fig. 1). As the cell spring is softer than dimples, the stiffness corresponding to this phasis decreases sensitively. For larger rotations, the spring, which is pinched between rod
323
and cell, develops non linear behavior and its stiffness quickly increases (Fig. 2). This variation of stiffness is taken into account in our model with a non linear rotational spring Kc between the rod beam and squeleton represented by the thimble beam. Moreover, the loss of contact between rods and dimples generates an increase of clamping force Fs on the rod in the cell which implies an increase of the slipping threshold. For large rotations between rod and grid, the rod is pinched and slippage becomes almost impossible. In Fig. 3 the variation of the clamping force Fs versus rotation, computed with the cell model used above is presented. The variation of the clamping force on the rod can practically be neglected in the range of rotation we studied (see Fig. 4). If we take into account only the loss of adherence between grid and rods, the restoring moment M(( ) on the grid can be expressed as the sum of the moments due to either adhesive or slipping rods:
Fig. 1. Loss of contact between a rod and a dimple in a grid cell.
B. Fontaine, I. Politopoulos / Nuclear Engineering and Design 195 (2000) 321–329
324
Fig. 2. Grid cell-rod moment vs. rotation.
M(u)=
% adherence
FTini di + % Fgnj dj
%adherence d 2i (9)
%all d
where FTi is the adherence force of ith row of rods and may be expressed as, FTi =KTuni di
(10)
(11)
di is the distance between the ith row and the axis, ni is the rod number in the ith row, Let us call Melas the ‘elastic’ moment: Melas(u)= % −KTuni d 2i
2 i
(13)
sliding
The sliding rod rows are those satisfying Coulomb’s adhesion condition: KTudi \ maFs
(14)
which is equivalent to:
with KT the axial stiffness of a rod between two grids. Fg is the slippage force of an ith row given by the Coulomb relationship, Fg = mgFs Sgn(x; )
Melas + % mFsni di
M=
sliding
(12)
all
which represents the moment on the grid obtained if no rod is sliding in the cells. Eq. (9) may be written in terms of elastic moment:
maFs %all d 2i di \
Melas
(15)
This inequality shows that only the external rods are slipping whereas the internal ones remain clamped in the grid. For further developments, we assume that the static and dynamic friction coefficient are identical. ma = mg
(16)
Relationship expressed with elastic moment, only depends on the geometry of the assembly (di, ni ), the clamping force Fs and friction coefficients. In Fig. 5, the variation of restoring moment M is presented versus elastic moment Melas. Below 100 Nm, no rod is slipping in the grid: the behav-
B. Fontaine, I. Politopoulos / Nuclear Engineering and Design 195 (2000) 321–329
ior of the rod grid junction is linear. Beyond this value, rods become to slip. The maximum moment, which corresponds to a global slipping of rods, is equal to 127 Nm.
325
Consequently, the junction between the rod beam and the grids is modeled in our assembly model with an elastoplastic hinge whose behavior corresponds to Eq. (13).
5.1. Elastoplastic analogy In our assembly model, each row of rods inside a grid works as a linear spring and a friction element in series. This description corresponds to the Saint-Venant model used to represent in an analogic way a perfect elastoplastic solid behavior (see Fig. 6). With this small model, strains o are related to material stresses ( through following equations: s (17) s Bss o= oe = E s s =ss Sgn(o; ) o = +op(arbitrary) (18) E where (s, and E are, respectively, the stress threshold and elasticity modulus. In this case many of these models are associated in parallel, as it occurs in our grid model, the generalized SaintVenant model for elastoplastic solid with Mroz hardening are obtained. With that analogy, Fig. 5 becomes a sort of moment – relative rotation relationship of an elastoplastic hinge.
6. Implementation of the model in CASTEM 2000 This non-linear model (Fig. 7) has been programmed in the CEA general purpose code CASTEM 2000. In the ground referential, the dynamic equation of the assembly is: MX8 + CX: +K(X)X=Fex
(19)
where, M, C and K (X) represent mass, damping and non-linear stiffness matrices of the assembly, X its relative displacement, Fex the external forces. In this equation stiffness K (X) has to be updated for each time step. By separating the stiffness matrix in two parts, one linear Kl and one non linear Knl, the dynamic equation may be expressed as: MX8 + CX: +KlX= Fex + Fnl
(20)
where non linearities are treated as external forces.
Fig. 3. Application force on a rod vs. rotation.
B. Fontaine, I. Politopoulos / Nuclear Engineering and Design 195 (2000) 321–329
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Fig. 4. Forces on rods in a spacer grid.
Fnl = − Knl(X)X
(21)
This equation is integrated with the modal superposition method, as this method combined with external forces depending on the displacement and velocity fields has shown its ability to reduce the computation time for some non linear problems, particularly when non linearities are located in few points of a structure (Strincklin and Haisler, 1977). The modal basis of the assembly model is computed after deactivation of non-linearities: rotational non linear springs and elastoplastic hinges are removed from the model. Then the following eigen problem is solved where (i and (i are the circular eigenfrequency and eigenvectors. (Kl −v 2i M)fi =0
(22)
Then the relative displacement of the structure can be decomposed on the orthogonal modal basis: X=% aifi
(23)
i
where (i is called the generalized coordinate. The dynamic equation written in the modal basis becomes a set of equations: mia¨ i + 2bi mi wia; i +mi w 2i a; i =f ti Fex +f ti Fnl
(24)
where mi, (i and wi are modal mass, damping and polar frequency of the ith mode.
At each time step, displacement is computed with Eq. (23) to calculate the non linear forces through Eq. (21). The resolution of Eq. (24) by an explicit numerical integration method, such as the Fu Devogelaere algorithm, leads to the generalized coordinates, and consequently, the displacement of the structure.
7. Static analysis Static analysis and tests have been performed on a single assembly to study the variation of its deflexion stiffness. In the first test several static loads, with a small magnitude, are applied on the middle grid, and displacement is measured on the same grid. In the second test the assembly is submitted to larger loads. In Fig. 8 the results of computation are plotted, compared with the experimental measures of lateral force versus assembly deflection. It appears that the analysis is able to predict, with rather good accuracy, the behavior of the assembly both in the linear and in the non linear domain. For the model, the transition between linear and non linear domain is clearly visible for 3 mm displacements. This transition phasis, which is piloted by non linear springs Kc rather than the elastoplastic hinges, corresponds to the moment where rods begin to loose contact with dimples inside grid cells.
B. Fontaine, I. Politopoulos / Nuclear Engineering and Design 195 (2000) 321–329
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Fig. 5. Restoring moment on the grid versus elastic moment.
8. Dynamic analysis Dynamic tests and analysis have been also performed on a fuel assembly. They consisted of the excitation of the middle grid by an electric actuator. The applied excitation is a sine wave loading with constant magnitude and slowly moving frequency. The magnitude of the force is changed after each test. For each force level, the grid displacement is plotted versus the excitation frequency. The maximum displacement is then obtained for the assembly resonance frequency. Despite the analyzed system having a non-linear behavior, the equivalent damping from the bandwidth of the resonance peak was computed. In Fig. 9 the resonance frequency versus maximum displacement obtained for each test are plotted. These measures are compared with the computation performed with our non-linear model. The decrease of the frequency due to loss of stiffness appears both in the tests and calcula-
tion. The model gives a rather good estimate of the evolution of the frequency with the displacement. The non-linear springs located at grid level and the elastoplastic hinges allow correct modeling of the loss of stiffness during dynamic motion. Fig. 10 presents the variation of damping with respect to the middle grid maximum displacement. The damping increases with the displacement because the more the rods are moving the more energy is dissipated by friction in grid cells. In the model the energy is mainly dissipated in the hysteresis loops of the elastoplastic hinges. The values of the damping coefficient are correctly estimated in the analysis for a displacement smaller than 15 mm. Beyond this displacement the
Fig. 6. Saint Venant model.
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B. Fontaine, I. Politopoulos / Nuclear Engineering and Design 195 (2000) 321–329
Fig. 8. Static force deflection relationship.
9. Conclusion A non linear dynamic model of PWR fuel assembly has been proposed based on obvious data such as geometry and cell-rods forces. This model represents the progressive slippage of rods inside grids and the loss of contact between rods and dimples with an elastoplastic hinge analogy and non linear springs.
Fig. 7. Non-linear assembly model.
model predicts a low decrease of damping. That may be explained by the fact that dissipated energy by friction increases lower than kinetic energy does. This phenomenon does not occur during tests because displacements remain limited.
Fig. 9. Frequency vs. displacement relationship.
B. Fontaine, I. Politopoulos / Nuclear Engineering and Design 195 (2000) 321–329
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Framatome and Electricite De France.
References
Fig. 10. Damping vs. displacement relationship.
The comparison of the static and dynamic behavior of this model with experimental data performed on actual assembly is in good agreement. We observe the reduction of frequency and the increase of damping when the amplitude of the assembly displacement increases. Moreover, the values issued from numerical analysis remain close to the experimental data. The next step of the project will be the application of this model to a complete seismic analysis, with time-history comparisons. Another possible utilization of this model is for the prediction of the dynamic behavior of assemblies at the end of their life, by replacing spring properties, rods geometry and other parameters by values measured on actual assemblies.
Acknowledgements This work has been partially supported by
.
Brochard, D., Jedidia, A. Ben, Gantenbein, F., Gibert, R.J., 1993. Dynamic modelling of PWR fuel assembly for seismic behaviour Proceedings of 12th SMiRT Conference. Helen, J., Johansson, A., 1993. Non linear analyses of the buckling load and the eigenvalues in a 17(17 PWR fuel assembly, Proceedings of 12th SMiRT Conference. Hotta, A., Sato, T., Niborii, H., Kawahara, H., 1985. The advanced model for PWR fuel assembly seismic analysis including the non-linear damping mechanisms Proceedings of ASME PVP Conference. Johnson, M.A., 1983. Impacting of fuel assemblies in horizontal seismic analysis of reactor core, Proceedings of ASME PVP Conference. Kim, J.D., et al., 1981. Comparison of theoretical and experimental examinations on PWR fuel assemblies under external loads, Proceedings of 6th SMiRT Conference. Leroux, J.C., et al., 1983. Structural analysis of PWR fuel assemblies in LOCA and seismic conditions, Proceedings of 7th SMiRT Conference. Leroux, J.C., 1983. CASAC: a code computing the LOCA and seismic loading on PWR fuel assemblies, Proceedings of 7th SMiRT Conference. Nuno, H., et al., 1973. A method of colliding vibration analysis of PWR fuel assemblies under seismic conditions, Proceedings of 2nd SMiRT Conference. Preumont, A., et al., 1982. Seismic analysis of PWR-RCC fuel assemblies. Nucl. Eng. Des. 71, 103 – 109. Preumont, A., Van Steenberghe, T., 1981. Dynamic characteristics of PWR fuel assemblies, Proceedings of 6th SMiRT Conference. Preumont, A., 1981. Analyse sismique du coeur d’un reacteur nucle´aire PWR, PhD Thesis, Belgonucleaire. Preumont, A., 1981b. A two-time step algorithm for seismic analysis of PWR core. Nucl. Eng. Des. 65, 49 – 62. Queval, J.C., Brochard D. 1988. Analysis of impact phenomena between PWR fuel assemblies in earthquake situations. Proceedings of ASME PVP Conference. Queval, J.C., Gantenbein F., Brochard D., Jedidia, A. Ben, 1991. Seismic behaviour of PWR fuel assemblies model and its validation Proceedings of 11th SMiRT Conference. Strincklin, J.A., Haisler, W.E., 1977. Computers and Structures, pp. 125 – 136.
A non linear model for the PWR fuel assembly seismic analysis B. Fontaine *, I. Politopoulos CEA, CEN-Saclay, Seismic Mechanic Study Laboratory, 91191 Gif sur Y6ette, France Received 20 November 1998; accepted 29 July 1999
Abstract In seismic PWR core analysis, the non linear behavior of fuel assemblies has to be studied taking into account variations of stiffness and damping due to the slippage of rods in spacer grids. Based on the linear CEA’s assembly model, principally composed of two beams representing sets of rods and thimbles, a linear model is described. In the second part of this paper the slippage and loss of contact of rods on a grid relic is analyzed. Based on the Coulomb friction model and plasticity analogy, a grid model constituted of elastoplastic hinge and non linear springs is proposed to simulate the progressive slippage of rods inside the cells. In the third part, a non linear assembly model is designed and validated with experimental data. This model represented with success the increase of damping and decay of frequency when the magnitude of the dynamic loading increases. © 2000 Elsevier Science S.A. All rights reserved.
1. Introduction Reactor cores are one of the most important structures in nuclear power plants from a safety point of view. Their good behavior under seismic loading has to be studied to assess the integrity of the fuel assemblies and the coolability of the core. There exists many fuel assembly models in literature but most of them are linear (Nuno et al., 1973; Kim et al., 1981; Preumont and Van Steenberghe, 1981; Preumont, 1981a,b; Preumont et al., 1982; Johnson, 1983; Leroux, 1983; Leroux et al., 1983; Queval and Brochard, 1988; Queval et al., 1991) and cannot represent the variation of damp* Corresponding author.
ing and frequency observed during vibration and snap back tests (Brochard et al., 1993). The model parameters have to be adjusted a priori for each loading. In the case of seismic excitation, which is unstationary, it is not easy to find the good adjustment of parameters able to represent the behavior of an actual assembly. When non linear models are used (Brochard et al., 1993; Helen and Johansson, 1993), they usually take into account all the contacts between rods and grids and thus, may be too sophisticated to be used in a whole core model for seismic analysis. Some other simpler non linear models are completely issued from global tests and cannot be used in a predictive analysis (Hotta et al., 1985).
0029-5493/00/$ - see front matter © 2000 Elsevier Science S.A. All rights reserved. PII: S 0 0 2 9 - 5 4 9 3 ( 9 9 ) 0 0 2 1 7 - 4
322
B. Fontaine, I. Politopoulos / Nuclear Engineering and Design 195 (2000) 321–329
The aim of this paper is to present a new non-linear model of a PWR fuel assembly which modeled sufficient local phenomena to be predictive but remains simple enough to be used in a whole core model.
2. PWR fuel assembly description
and damping increases when loading becomes higher. This phenomena comes principally from the slippage of fuel rods through spacer grids. During motion, the axial force on rods may become higher than friction forces and then slippage may occur. The slippage tends to reduce the assembly stiffness and creates dissipative phenomena which appear globally as additional damping.
Fuel assembly is composed of two sets: 4. Linear model
2.1. The fuel rods The rods are 3.8 m high tubes containing fuel pellets. Their number depends on the assembly type, but reaches 264 for a 17(17 bundle.
2.2. The squeletton The squeletton which is designed to hold a minimum gap between fuel rods is composed of guide thimbles clamped at both ends on stiff support devices (i.e. top and bottom nozzles). At different levels, grids are welded on the guide thimbles. The grids, in zirconium alloy hold the rods spaced to allow the cooler flow and to contribute to the lateral stiffness of the assembly. The rods are maintained in the grids by springs and dimples. During seismic excitation the rods may naturally slip through spacer grids.
3. Experimental behavior Many tests have been performed (Preumont and Van Steenberghe, 1981; Preumont, 1981a; Preumont et al., 1982; Queval and Brochard, 1988; Queval et al., 1991) on the single assembly to study its mechanical behavior. During these tests, a fuel assembly is set standing with both ends supported in the same manner as in the actual core and a (static or dynamic) load is applied to the middle grid. Then the deflection at middle grid is measured and is plotted versus load level. By applying sine wave loading, it is possible to determine the resonance frequency and damping of the assembly for each load level. It appears that assembly stiffness (or frequency) decreases
Under low amplitude vibrations. the behavior of the fuel assembly can be considered as linear. The internal forces inside the assembly are assumed to be sufficiently small to prevent slippage between rods and grids: the rods are then considered to be clamped in the grids. Thus a linear model can be used, composed of two main beams, one representing the guide thimble set and the other one the fuel rod set. Between the two beams are located linear springs KI which represent grids. The characteristics of both beams are the following: The bending inertia I eq and I eq (respectively, t r eq eq the cross section S t and S r ) are equal to the sum of the bending inertia It and Ir (respectively, cross sections St and Sr) of the guide thimbles or fuel rods. S eq t = NtSt
(1)
I eq t = NtIt
(2)
eq r
S = NrSr
(3)
I eq r = NrIr
(4)
where Nt and Nr are the number or thimbles and rods in an assembly. In parallel with the beams rotation springs Kt and Kr have been added between the grids to represent the fact that the rotation of the grids induces traction or compression on guide thimbles or rods. Their stiffness is: Kt =
EtSt Nt 2 %d L i i
(5)
Kr =
ErSr Nr 2 %d L j j
(6)
B. Fontaine, I. Politopoulos / Nuclear Engineering and Design 195 (2000) 321–329
with Et and Er the elasticity moduli of thimbles and rods, di is the distance between the ith rod (or thimble) to the axis, and L the height between two grids. Such a model, already described by (Queval et al., 1991) is able to calculate the deflection stiffness and eigenfrequency of an actual assembly.
5. Non linear links between grids and fuel rods The 254 rods, arranged in 17 rows, are maintained in the grids by springs and dimples, with a static force Fs. Each rod – grid link may be represented with a Coulomb friction model. A rod does slip in a grid if the axial force FT acting on it verifies FT ] maFs
(7)
and, in that case, a friction force Fg acts on it Fg = mgFs sgn(z; )
(8)
where (a, (g are, respectively, static and dynamic friction coefficients and z is the relative displacement in the axial direction. If we apply a rotation ( to the grid, perpendicular to the rod bundle axis, the grid undergoes a non linear restoring moment due to the progressive slippage of rods and loss of contact between rods and dimples. One of the consequences of the loss of contact between rods and dimples is the modification of the stiffness of the link between the grid cell and the rod. The resulting moment between rod and grid cell can be computed if we represent dimples by linear springs whereas the cell spring is considered non linear. When the rotation of the rod in the grid cell is small (B 1 mrad), the rod stays in contact with the two dimples and the cell spring. The corresponding stiffness which comes essentially from the dimples is high. For medium rotations (between 1 mrad and 40 mrad) one dimple looses contact with the rod (Fig. 1). As the cell spring is softer than dimples, the stiffness corresponding to this phasis decreases sensitively. For larger rotations, the spring, which is pinched between rod
323
and cell, develops non linear behavior and its stiffness quickly increases (Fig. 2). This variation of stiffness is taken into account in our model with a non linear rotational spring Kc between the rod beam and squeleton represented by the thimble beam. Moreover, the loss of contact between rods and dimples generates an increase of clamping force Fs on the rod in the cell which implies an increase of the slipping threshold. For large rotations between rod and grid, the rod is pinched and slippage becomes almost impossible. In Fig. 3 the variation of the clamping force Fs versus rotation, computed with the cell model used above is presented. The variation of the clamping force on the rod can practically be neglected in the range of rotation we studied (see Fig. 4). If we take into account only the loss of adherence between grid and rods, the restoring moment M(( ) on the grid can be expressed as the sum of the moments due to either adhesive or slipping rods:
Fig. 1. Loss of contact between a rod and a dimple in a grid cell.
B. Fontaine, I. Politopoulos / Nuclear Engineering and Design 195 (2000) 321–329
324
Fig. 2. Grid cell-rod moment vs. rotation.
M(u)=
% adherence
FTini di + % Fgnj dj
%adherence d 2i (9)
%all d
where FTi is the adherence force of ith row of rods and may be expressed as, FTi =KTuni di
(10)
(11)
di is the distance between the ith row and the axis, ni is the rod number in the ith row, Let us call Melas the ‘elastic’ moment: Melas(u)= % −KTuni d 2i
2 i
(13)
sliding
The sliding rod rows are those satisfying Coulomb’s adhesion condition: KTudi \ maFs
(14)
which is equivalent to:
with KT the axial stiffness of a rod between two grids. Fg is the slippage force of an ith row given by the Coulomb relationship, Fg = mgFs Sgn(x; )
Melas + % mFsni di
M=
sliding
(12)
all
which represents the moment on the grid obtained if no rod is sliding in the cells. Eq. (9) may be written in terms of elastic moment:
maFs %all d 2i di \
Melas
(15)
This inequality shows that only the external rods are slipping whereas the internal ones remain clamped in the grid. For further developments, we assume that the static and dynamic friction coefficient are identical. ma = mg
(16)
Relationship expressed with elastic moment, only depends on the geometry of the assembly (di, ni ), the clamping force Fs and friction coefficients. In Fig. 5, the variation of restoring moment M is presented versus elastic moment Melas. Below 100 Nm, no rod is slipping in the grid: the behav-
B. Fontaine, I. Politopoulos / Nuclear Engineering and Design 195 (2000) 321–329
ior of the rod grid junction is linear. Beyond this value, rods become to slip. The maximum moment, which corresponds to a global slipping of rods, is equal to 127 Nm.
325
Consequently, the junction between the rod beam and the grids is modeled in our assembly model with an elastoplastic hinge whose behavior corresponds to Eq. (13).
5.1. Elastoplastic analogy In our assembly model, each row of rods inside a grid works as a linear spring and a friction element in series. This description corresponds to the Saint-Venant model used to represent in an analogic way a perfect elastoplastic solid behavior (see Fig. 6). With this small model, strains o are related to material stresses ( through following equations: s (17) s Bss o= oe = E s s =ss Sgn(o; ) o = +op(arbitrary) (18) E where (s, and E are, respectively, the stress threshold and elasticity modulus. In this case many of these models are associated in parallel, as it occurs in our grid model, the generalized SaintVenant model for elastoplastic solid with Mroz hardening are obtained. With that analogy, Fig. 5 becomes a sort of moment – relative rotation relationship of an elastoplastic hinge.
6. Implementation of the model in CASTEM 2000 This non-linear model (Fig. 7) has been programmed in the CEA general purpose code CASTEM 2000. In the ground referential, the dynamic equation of the assembly is: MX8 + CX: +K(X)X=Fex
(19)
where, M, C and K (X) represent mass, damping and non-linear stiffness matrices of the assembly, X its relative displacement, Fex the external forces. In this equation stiffness K (X) has to be updated for each time step. By separating the stiffness matrix in two parts, one linear Kl and one non linear Knl, the dynamic equation may be expressed as: MX8 + CX: +KlX= Fex + Fnl
(20)
where non linearities are treated as external forces.
Fig. 3. Application force on a rod vs. rotation.
B. Fontaine, I. Politopoulos / Nuclear Engineering and Design 195 (2000) 321–329
326
Fig. 4. Forces on rods in a spacer grid.
Fnl = − Knl(X)X
(21)
This equation is integrated with the modal superposition method, as this method combined with external forces depending on the displacement and velocity fields has shown its ability to reduce the computation time for some non linear problems, particularly when non linearities are located in few points of a structure (Strincklin and Haisler, 1977). The modal basis of the assembly model is computed after deactivation of non-linearities: rotational non linear springs and elastoplastic hinges are removed from the model. Then the following eigen problem is solved where (i and (i are the circular eigenfrequency and eigenvectors. (Kl −v 2i M)fi =0
(22)
Then the relative displacement of the structure can be decomposed on the orthogonal modal basis: X=% aifi
(23)
i
where (i is called the generalized coordinate. The dynamic equation written in the modal basis becomes a set of equations: mia¨ i + 2bi mi wia; i +mi w 2i a; i =f ti Fex +f ti Fnl
(24)
where mi, (i and wi are modal mass, damping and polar frequency of the ith mode.
At each time step, displacement is computed with Eq. (23) to calculate the non linear forces through Eq. (21). The resolution of Eq. (24) by an explicit numerical integration method, such as the Fu Devogelaere algorithm, leads to the generalized coordinates, and consequently, the displacement of the structure.
7. Static analysis Static analysis and tests have been performed on a single assembly to study the variation of its deflexion stiffness. In the first test several static loads, with a small magnitude, are applied on the middle grid, and displacement is measured on the same grid. In the second test the assembly is submitted to larger loads. In Fig. 8 the results of computation are plotted, compared with the experimental measures of lateral force versus assembly deflection. It appears that the analysis is able to predict, with rather good accuracy, the behavior of the assembly both in the linear and in the non linear domain. For the model, the transition between linear and non linear domain is clearly visible for 3 mm displacements. This transition phasis, which is piloted by non linear springs Kc rather than the elastoplastic hinges, corresponds to the moment where rods begin to loose contact with dimples inside grid cells.
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Fig. 5. Restoring moment on the grid versus elastic moment.
8. Dynamic analysis Dynamic tests and analysis have been also performed on a fuel assembly. They consisted of the excitation of the middle grid by an electric actuator. The applied excitation is a sine wave loading with constant magnitude and slowly moving frequency. The magnitude of the force is changed after each test. For each force level, the grid displacement is plotted versus the excitation frequency. The maximum displacement is then obtained for the assembly resonance frequency. Despite the analyzed system having a non-linear behavior, the equivalent damping from the bandwidth of the resonance peak was computed. In Fig. 9 the resonance frequency versus maximum displacement obtained for each test are plotted. These measures are compared with the computation performed with our non-linear model. The decrease of the frequency due to loss of stiffness appears both in the tests and calcula-
tion. The model gives a rather good estimate of the evolution of the frequency with the displacement. The non-linear springs located at grid level and the elastoplastic hinges allow correct modeling of the loss of stiffness during dynamic motion. Fig. 10 presents the variation of damping with respect to the middle grid maximum displacement. The damping increases with the displacement because the more the rods are moving the more energy is dissipated by friction in grid cells. In the model the energy is mainly dissipated in the hysteresis loops of the elastoplastic hinges. The values of the damping coefficient are correctly estimated in the analysis for a displacement smaller than 15 mm. Beyond this displacement the
Fig. 6. Saint Venant model.
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Fig. 8. Static force deflection relationship.
9. Conclusion A non linear dynamic model of PWR fuel assembly has been proposed based on obvious data such as geometry and cell-rods forces. This model represents the progressive slippage of rods inside grids and the loss of contact between rods and dimples with an elastoplastic hinge analogy and non linear springs.
Fig. 7. Non-linear assembly model.
model predicts a low decrease of damping. That may be explained by the fact that dissipated energy by friction increases lower than kinetic energy does. This phenomenon does not occur during tests because displacements remain limited.
Fig. 9. Frequency vs. displacement relationship.
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Framatome and Electricite De France.
References
Fig. 10. Damping vs. displacement relationship.
The comparison of the static and dynamic behavior of this model with experimental data performed on actual assembly is in good agreement. We observe the reduction of frequency and the increase of damping when the amplitude of the assembly displacement increases. Moreover, the values issued from numerical analysis remain close to the experimental data. The next step of the project will be the application of this model to a complete seismic analysis, with time-history comparisons. Another possible utilization of this model is for the prediction of the dynamic behavior of assemblies at the end of their life, by replacing spring properties, rods geometry and other parameters by values measured on actual assemblies.
Acknowledgements This work has been partially supported by
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