Spent fuel rack is the key equipment for the storage of spent fuel after refueling. In order to investigate the performance of the spent fuel rack under the earthquake, the phenomena including sliding, collision, and overturning of the spent fuel rack were studied. An FEM model of spent fuel rack is built to simulate the transient response under seismic loading regarding fluidstructure interaction by ANSYS. Based on D’Alambert’s principle, the equilibriums of force and momentum were established to obtain the critical sliding and overturning accelerations. Then 5 characteristic transient loadings which were designed based on the critical sliding and overturning accelerations were applied to the rack FEM model. Finally, the transient displacement and impact force response of rack with different gap sizes and the supporting leg friction coefficients were analyzed. The result proves the FEM model is applicable for seismic response of spent fuel rack. This paper can guide the design of the future’s fluidstructure interaction experiment for spent fuel rack.
Free standing spent fuel storage racks are submerged in water contained in a pool. During a postulated strong motion earthquake, the water surrounding the racks and fuel assemblies is accelerated and the socalled fluidstructure hydrodynamic interaction is significantly induced between the water and the fuel assemblies, the racks and the pool walls [
The verification of structural integrity and functionality is performed in two steps [
Much attention was paid to the time history analysis of free standing rack. Due to inertia effects and friction, the rack legs may lift off or slide on the pool floor. Such seismic response of the fluidstructurecoupling system may cause the impact between the rack cells and the fuel assemblies that are freely standing inside the rack cells. The complex rack’s movement includes not only the impact between rack and pool wall, but also the impact between rack supporting legs and pool floor. The mathematical models usually used to describe such impact and friction phenomena are materially and geometrically nonlinear, respectively. A 3D nonlinear dynamic time history analysis method is usually needed to solve such nonlinear problems [
According to a report by Westinghouse [
The FSI is characterized by flat structures with large surfaces (area of one rack wall in the order of 10 m^{2}) separated by relatively small water gaps (down to the order of 10 mm). Motion of neighboring racks can consequently lead to a large fluid pressure on the rack walls. Stabel and Ren [
Although much effort has been paid to the seismic behavior of free standing fuel rack, additional research could provide greater confidence in the analysis methods by reducing the levels of uncertainty. In order to assess the sensitivity of response to variations in model parameters, this paper built a simplified mechanical model of spent fuel rack regarding the fluidstructure interaction. The transient response of rack with different values of fluid gap sizes and the supporting leg friction coefficient were analyzed to estimate the most important factor influencing the rack displacement and impact forces.
As reviewed by Ashar and DeGrassi [
A typical fuel rack module.
A high density spent fuel rack installation may include several modules in a pool. The modules could vary in size and storage capacity. To maximize fuel storage, they are arranged in close proximity to each other and to the pool walls. Gaps of zero to two inches are common. A typical AP 1000 spent fuel pool contains three Region 1 rack modules and five Region 2 rack modules. As reported by Westinghouse [
AP1000 spent fuel storage racks data.
Rack  Region  Array size  Storage cell centertocenter pitch (in)  Cell length (in)  Weight (lb) 


Region 1  9 × 9  0.277  5.07  13200 

Region 2  12 × 11  0.229  5.07  11200 

12 × 10 (−2) Region 2  12 × 10 (−2)  0.260  5.07  11400 
According to the design of AP1000, free standing rack is supported by 4 legs as shown in Figure
The simplified rack analysis model is assembled in one uniform crosssection beam having the same mass of the rack. The parameter of the beam can be obtained by adjusting the equivalent section moment of inertia of the beam to make natural frequency of the simplified model equal to a detailed FEM rack model. The bottom plate is made of relatively thick stainless steel; consequently, it is modeled by a rigid Xshaped frame.
The contact between the leg and the floor is simulated by Coulomb friction model. The static friction coefficient and kinetic friction coefficient were equal. The friction coefficient was set to 0.3 in the water obtained by the vibration test [
The impact between the rack and the wall is simulated by gap spring model. This element behaves like a stiff spring until the force reaches a limiting value equal to the specified friction coefficient times the normal force. The spring stiffness can be derived by Hirtz contact theory, which could be automatically calculated by the ANSYS.
When the rack is placed in water, the FSI have to be taken into consideration. According to Ashar and DeGrassi’s study [
The main aim of the paper is to study the sliding, overturning, and impact of the rack. The gaps between the assemblies and the rack are only about 5 mm, relatively smaller than the gaps between the rack and the wall. Consequently, the authors think if the assemblies and the rack are merged together, the sliding distance will be larger and more conservative than that of the coupling one because the distributed mass model ignores the impact and the friction. Consequently, the fuel assemblies are simplified into three distributed masses attaching on the beam of rack illustrated in Figure
Model of free standing fuel rack.
Fluid added mass.
In this paper, the AP1000 rack in Region II is chosen as an example to validate the model. The rack data are listed in Table
Parameters of rack adopted in the simulation.
Rack size 

2.57 m; 2.8 m 
Rack height 

5.07 m 
Gap size 

0.05 m; 1.00 m 
Fluid added mass 

3.30 × 10^{5} kg 
Friction coefficient 

0.3 
Mass of rack 

1.12 × 10^{4} kg 
Mass of total fuel assemblies 

1.15 × 10^{5} kg 
Corresponding to the model illustrated in Figure
FEM model of free standing fuel rack.
In the analysis, the spent fuel pool is fixed so it is represented by several nodes which are connected with the conta178 elements and Combin40 elements. The boundary condition for this model is that all the nodes that represent spent pool are fixed. In the time history analysis, transient acceleration loading is input as the inertia; moreover, the gravity is also included.
In order to improve the understanding of the rack motion under earthquake, the rack is assumed to be a rigid body to sketchily provide the critical sliding and overturning accelerations.
First of all, the rack is regarded as a rigid body with planar motion. The rack of mass
Under the earthquake, the friction and the fluid forces apply on the rack. Here, the fluid force
Establish the force and momentum equilibriums using method of dynamic equilibrium. The momentum equilibriums are written as
We obtain
From the above equations, if
The above equation shows that the critical overturning acceleration is
From Figure
The rigid rack model.
Assume there is relative sliding between the rack and floor. When Coulomb friction model is introduced,
The above equation shows that the critical sliding acceleration is
In this system, friction is driving force for the rack motion and fluid force is resistance. According to the Coulomb model, if there is relative sliding, the maximum friction is the sliding friction. Consequently, this means the acceleration of rack can never surpass the critical sliding acceleration. If critical overturning acceleration is larger than critical sliding, the rack will not overturn.
The previous analysis assumes that the rack is a rigid body and the dose is not taken into consideration of the impact. The natural frequency of the loaded rack is above 10 Hz; the rack is not so flexible. This simplification has limited influence on the sliding and overturning effect of the rack. Consequently, the critical sliding and overturning accelerations can be used to test the FEM model discussed in Section
Knowing the critical sliding and overturning acceleration, as listed in Table
Description of loading cases.
Loading number  Description  Acceleration amplitude 

1  Uniform acceleration amplitude  0.06 g 
2  Uniform acceleration amplitude  0.10 g 
3  Uniform acceleration amplitude  0.16 g 
4  Sine wave with frequency of 1 Hz  0.40 g 
5  Seismic loading  0.40 g 
Figures
Uniform acceleration amplitude loading: 0.06 g.
Uniform acceleration amplitude loading: 0.10 g.
Uniform acceleration amplitude loading: 0.16 g.
Sine acceleration amplitude loading: 0.4 g, 1 Hz.
Test Loading 5: seismic loading with peak ground acceleration 0.40 g.
Figure
Figure
Figure
Figure
Figure
Seismic wave.
Finally, these transient tests prove that the FEM model in Section
In this section, different gap sizes and friction coefficients are discussed to investigate the rack response based on the FEM model in Section
The maximum results of the 9 sets of parameters combinations.
Case number  Gap sizes/mm 

Critical sliding acc./g  Critical overturning acc./g  Bottom disp./mm  Top disp./mm  Bottom impact force/N  Top impact force/N 

1  20  0.2  0.028  0.070  20.2  50.1  3.29 
3.24 
2  20  0.3  0.042  0.070  20.2  50.1  1.21 
5.38 
3  20  0.8  0.111  0.070  20.1  50.1  2.13 
2.89 
4  40  0.2  0.047  0.118  40.1  63.9  5.10 
— 
5  40  0.3  0.070  0.118  40.2  43.85  2.461 
— 
6  40  0.8  0.186  0.118  35.5  70.1  —  2.70 
7  60  0.2  0.060  0.152  60.1  69.9  4.92 
— 
8  60  0.3  0.090  0.152  50.2  52.57  —  — 
9  60  0.8  0.240  0.152  54.8  56.11  —  — 
In the previous section, the critical sliding acceleration is determined by
However, there are some exceptions for this correlation. For example, the maximum bottom impact force for Case 2 is smaller than that for Case 3. These exceptions may be caused by the randomness of the seismic wave. Here is the possible reason. For Cases 1 to 3, the critical sliding accelerations are 0.028 g, 0.042 g, and 0.11 g, respectively. If the peaks of earthquake wave that surpass the critical accelerations are in the same direction, the rack will move closely to the wall and may impact the wall. On the contrary, if the peaks of earthquake wave that surpass the critical accelerations are in different directions, the rack may just remain still. For the same seismic wave, if the friction coefficient changes, the peaks that surpass the threshold change. Consequently, the impact forces calculated with smaller friction coefficient are possibly smaller than the larger ones in a single transient test.
As the statement in Section
Another finding is that when the gap between rack and the pool wall is larger, the rack is less likely to impact the wall. For Cases 1, 4, and 7, with increasing gaps, the impact forces decrease. This phenomenon is obvious because larger gap means larger sliding distance; the friction force will dissipate more energy with larger sliding distance; therefore, it is less possible to impact with larger gap.
This paper concentrates on the seismic response of spent fuel rack including phenomena like sliding, overturning, and impact between rack and spent fuel pool wall. The AP1000 spent fuel rack is analyzed by an FEM model with ANSYS codes regarding effects including friction and impact. The FEM model regards the assemblies as a part of rack and ignores the impact between racks and assemblies. Based on D’Alambert’s principle, the equilibriums of force and momentum were established to obtain the critical sliding and overturning acceleration. Then, 5 characteristic transient loadings designed according to the critical sliding and overturning acceleration were applied to the rack FEM model to test the sliding and overturning. A discussion on fluid gap sizes and supporting leg friction coefficient was carried out to estimate the most important factor influencing the rack displacement and impact forces.
The detailed conclusions are listed below:
The fluid added mass can decrease the critical sliding and overturning acceleration.
There is a positive correlation between friction coefficient and the critical sliding acceleration.
When friction coefficient increases, the sliding distance and impact force tend to be smaller. However, there are some exceptions, possibly on account of the randomness of the seismic waves.
If the critical sliding acceleration is larger than critical overturning acceleration, the rack is much more likely to overturn than slide.
When the gap between the rack and the pool wall is larger, the rack is less likely to impact the wall.
This paper can guide the design of the future’s fluidstructure interaction experiment for spent fuel rack. With the proper set of parameters including loading amplitudes and gap sizes, the nonlinear phenomena including the friction effect, the impact effect, and the FSI effect can be evaluated independently with higher accuracy.
The authors declare that they have no competing interests.
The project was sponsored by National Science and Technology Major Project of the Ministry of Science and Technology of China (2015ZX06004002003).