ML20003H745
| ML20003H745 | |
| Person / Time | |
|---|---|
| Site: | Waterford  |
| Issue date: | 11/30/1979 |
| From: | Scavuzzo R, Stokey W WACHTER ASSOCIATES |
| To: | |
| Shared Package | |
| ML20003H743 | List: |
| References | |
| NUDOCS 8105070384 | |
| Download: ML20003H745 (98) | |
Text
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.i SEISMIC ANALYSIS OF WATERFORD SES SPENT FUEL STO. RAGE RACKS m_.
November, 1979 Revision 2 t
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for
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EBASCO SERVICES, INC.
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WACHTER ASSOCIATES, INC.
I v GIBSONIA, PENNSYLVANIA 9
l' PREPARED BY e dW4 DATE 4 [8/fh bhjbcavuz[o~
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DATE' 4 \\/
p REVIETIED BY
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W. F. Stokey,
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TABLE OF CO!rfENTS 2
]c Page
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Abstract C
1.0 Introduction
'l
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J 2.0 Equi;xnent Description 6
e 3.0 Material Properties and Allowfole Stresses 7
4.0 Analytical Results 8
m I-'
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5.0 Conclusions 9
6.0 References 10 Table I - Su= mary of Rack Assembly Stress 11 3
Intensities Table II-Sn-ary of Rack Assembly Stress 12 Intensities Caused by Spent Fuel
.J d
I= pac:
Appendix A - Static Stress Analysis of Waterford Spent Fuel Storage Racks
( Attachments - Cceputer Output)
~
Appendix B - Seismic Analysis of the Waterford Spent Fuel
-S Storage Racks
( Attach =ents - Cc=puter Output) j Appendix C - Fluid Structure Coupling of Rectangular
-]
~
Modules Appendix D - Experimental Verification of Fluid Coupling
_. j Theory Appendix E - Impact Analysis Waterford Spent Fuel Storage 1
Racks Appendix F - Dynamic Response of a Module Array b
,J Appendix G - Comparison of Finite Ele =ent Plate Model with Finite Ele =ent Beam Model g4.
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..J SEISMIC ANALYSIS OF WATERFORD SES
.J
- g-SPENT FUEL STARAGE RACKS
_. 3
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1.0 INTRODUCTION
f*
1.1 Objective of Analysis
{-
This seismic and mechanical analysis was performed to provide a part of the required documentation for the licens-J ing of new spent fuel racks for the Waterford SES Nuclear i
Power Plant operated by Louisiana Power and Light Company.
[."'
This work, with the addition of separate nuclear, thermo-j, hydraulic and an additional mechanical report will serve as the calculational support for licensing.
1.2 Problem Statement For the equipment and its supports, the ability of the Q
l supported equipment to remain fully functional during and 1
af ter the seismic disturbance must be demonstrated.
The seismic adequacy of this equipment is to be demonstrated by
}
a complete mathematical analysis of the spent fuel racks, subject to both the Safe Shutdown Earthquake (SSE) and the a
Operational Basis Earthquake (OBE).
i-1 1.3 Report Organization
~
Details of this analysis are divided into five Appendices:
~
f" Appendix A - Static Analysis of the Waterford Spent l LJ Fuel Storage Racks
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r-Appendix B - Seismic Analysis of the Waterford Spent Fuel Storage Racks
!..e Appendix C - Fluid Structure Coupling of Rectangular
-A Modules im Appendix D - Experimental Verification *of Fluid Coupling Theory L.*
Appendix E - Impact Analysis - Waterford Spent Fuel l 'w Storage Racks Appendix F - Dynamic Response of a Module Array
'"s-*
Appendix G - Comparison of Finite Element Plate Model I"
with Finite Element Beam Model L.*
1.4 Scope and Limitations of the Report 3
d In general, loading conditions that must be considered are as follows:
(a) internal and external pressur'e i "
(b) impact loads including rapidly fluctuating pressures d
i i
(c) weight of the component and normal contents under rf
_j operating'or test conditions
.~
(d) superimposed loads, such as other components, operating l
equipment insulation, insulation etc.
L-}.
_ wind loads, snow loads, vibrations and earthquake I'
(e)
' loads where specified.
Spent fuel storage racks are subject to impact loads from
~
l spent fuel el'ements (b), the weight load of the stored fuel and earth'uake loads (e).
elements (c) q F1 m.
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J Specifically, this report includes a dynamic stress analysis of the spent fuel storage racks subject to three orth-
.-,. s
- d gonal earthquake spectrum inputs.
Modal stresses are combined c.a asing the SRSS method.
A static stress analysis of the module d
i; fully leaded with spent fuel elements is also included.
This static analysis is used as the basis of the vertical earth-quake loading.
Resulting seismic and static stresses are com-bined absolutely and are compared to allowable values specified in Reference 1.
'L r
.This report does not contain an analysis of the supports of spent fuel racks or of loads on the welds of the rack assembly from the dynamic response.
These topics are the sub-l 23 ject of separate reports.
The thermal and hydraulic analysis is also the subject of a separate report.
d
.f In Appendix A, a static finite element analysis of a storage rack loaded by spent fuel elements is presented.
A dynamic analysis of the racks subject to two lateral orthogonal 1
j seismic loads is presented in Appendix B.
The theory used as the basis of this dynamic analysis l's developed in Appendix C and experimentally verified in Appendix D.
An analysis of the l
l effects of fuel impact is presented in Appendix E.
d l.5 Method of Analysis and -Basic Assumptions E'
The dynamic response of the spent fuel storage racks is 3w based on a finite element analysis using the MCAUTO STRUDL DYNAL 1r.
j j--
program.
Within the program, the seismic response analysis is l
l
,G calculated from a normal node analysis using the pool floor l ;$
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4 response spectra as input.
A computer drawing of one quarter of the module assembly is shown on Figure B-1.
A hybrid element which includes plane stress effects and plate bending effects.
q
_g-has been specified for all elements.
M In this analysis the rack assembly is assumed to be fully loaded.
The weight in the analytical model includes the spent
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fuel, the poison, the structural weight, entrained water and s,
virtual mass effects.
Interaction of the rack with the pool wall through the fluid is included in the analysis.
Details
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of this normal mode analysis, mcdified to include fluid inertial loads, are presented in Appendices B, C and D.
Impact of the spent fuel with the module is considered in a conservative and I
approximate manner.
Details of this analysis are presented in Appendix E.
t" The seismic analysis is based on the 80 module rack rather t
than the lighter 64 module rack.
Forces to the foundation are a
the largest for the 80 module rack.
As a result, stresses in the
,'j support region are the largest for the fully loaded 80 module rack.
l Furthermore, as shown in this report, stresses in WAI designed l
l racks are extremely low when compar'ed to allcwable stresses in 304
-a stainless steel.
Thus, the detailed finite element analysis pre-4 sented in this report may be used for the licensine of both rack i _
sizes.
'~
Partially loaded rac'ks have less weight than a fully loaded rack.
As a result stresses resulting from seismic inputs are y
less in partially leaded racks than fully loaded racks.
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Virtual mass effects are based on the gaps surrounding
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the 16 fuel racks submerged in the pool.
Magnitudes of the m T hydrodynamic masses are based on the work of Fritz (2).
Details
_g of the theory used is presented in Appendices C and D and Re-
)?
ference (3).
In the seismic analysis the rack module is assumed to be l"
fixed at the four corner supports as indicated in Appendix B.
Structural characteristics of the rack module assembly are based on the box walls.
The structure which forms the enclosure for c-.
the poison is not considered as part of the module structure for
,'t strength; however, the weight of this metal is included in the analysis.
1
]
Cne of the basic assumptions made in the finite element analysis is that the racks vibrate in phase.
This assumption is studied in detail in Appendix F.
An array of four modules is analyzed assuming both a fixed a
base and sliding base conditions..Under some extreme conditions racks l
J.._]
may vibrate out-of-phase.
However, relative motions are small l
(= 0.003 inches).
For the case of a weak foundation spring or a sliding base the racks vibrate in phase.
,gj This study indicates that base forces may be increased due
-2 to different loadings in the array.
The maximum calculated in-q, crease over the all full case is a factor of 1.6.
Base shear forces from all-fixed bas'e conditions are well above the more z,
,L realistic sliding base condition.
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E 2.0 EQUIPMENT DESCRIPTION
'.s 2.1 Geometry
-+g The dimensions of the spent fuel pool plan are 390" by 383" for the Waterford SES Unit No. 3.
A total of sixteen i_.
rack assemblies provide storage for 1088 spent' fuel elements.
Twelve racks each have a storage capacity of 64 fuel elements
~
and the remaining four racks each have a capacity of 80 fuel elements.
a The gaps between the storage racks and the. pool walls affect the calculations of the vertical mass.
Along the north,
,.s q
south, east and west walls the gaps are 24.3",
12.8", 13.6" and 37.2" respectively.
Each box in the rack is approximately 10.38" square and 185" high.
All boxes and the bottom plate
.e are fabricated from 304 stainless steel.
The minimum thickness
'n
',J of each box wall is 0.090" and the bottom plate is approximate-ly 0.5".
J 2.2 Rack Assembly Weight The weight of each spent fuel element is considered to be
- l 1500 lbs. and the weight of one poison element is considered
_.o to be 68 lbs.
There are two poison elements per storage box.
1 The structural weight of the 80 module storage rack is approxi-mately 25,800 lbs.
Details of these weight calculations for the dynamic analysis are presented in Appendix B, Section B.3.1.
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Y 2.3 Reference Drawings
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The dynamic analysis is based on the following WAI draw-M ings:
131-1, Rev. G, 7/18/78 and 131-2, Rev. E, 7/18/78.
i.#
3.0 MATERIAL PROPERTIES AND ALLOWABLE STRESSES i_."
Material properties for 304 Stainless Steel are'taken d
from Section III of the ASME Boiler and Pressure Vessel Code,
~
-, j Division 1, Appendices.
The minimum yield strength for Type 304 SA-240 Plate Stainless Steel for tem'peratures less than 300* F is 22.5 KSI per Section III of the Code.
The allowable value of the stress i
intensity is 20.0 KSI for temperatures less than 300 F.
for primary membrane stresses.
Physical test data at room temper-i L.'*
ature on material used for the Waterford spent fuel storage
~
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racks indicate the actual yield strength is above 35 KSI.
!4 J l
Allowable values of stress in this report are based on s
- f__,
the requirements of Reference (1).
In this specification allowable values for the OBE are to be based on the AISC Manual.
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This manual restricts stresses to 6'O percent of the yield strength.
Furthermore, Reference (1) requires that allowable
~
stresses for the SSE be limited to 90 percent.of the yield f,
strength.
These values are 13.5 KSI and 20.3 KSI respectively,
(
l for temperatuYes below 300* F.
Both of these allowable stresses
- n
[-
are more conservative than the NRC position paper for fuel lr]
storage racks.
,1J l"
The elastic ~ and shear moduli used in this report are 27.76 x 6
6 10 psi and 10.67 x 10 psi, respectively.
These values are based on the ASME Code.
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."1-a c Jd 4.0 ANALYTICAL RESULTS me 4.1 Calculated Stresses
-- T.
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The highest stresses in the spent fuel storage racks exist in the lower portions of the box walls and in the bottom plate around the supports.
Stresses in these areas are calcu-lated in order to determine peak values of the stress intensity.
Calculated stresses in all sections of the spent fuel storage racks are low.
As a result, the peak stress intensity from each loading condition is added absolutely.
The location 2 -
a of the peak stress intensity is not taken into account.
As a
-4 result, the ccabined stress is ccnservative.
If high stresses were calculated in the rack, a more exact method of combining stresses would be applied.
Stresses considered applicable to the module are the verti-
- d cal seimic input, the north-south seismic input, the east-west i
seismic input and the static loading.
Ther=al' stresses within the rack assembly are not significant.
l,_
Values of peak stress intensit'y for each loading are listed in Table I, for the OBE and SSE seismic conditions.
Static 1
13
~
values are also listed in Table I.
1 An analysis of impact of spent fuel against the rack during l
seismic loading-is consid'ered in Appendix E.
A conservative i l' L,,
estimate of the increase in OBE stresses and SSE stresses'is 43 i; -
percent in the North-South direction and 43 percent in the East-t.
i West direction.
Resulting stresses are listed in Table II.
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Rev. 1
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9 Using the allowable values listed in 3.0 the lowest resulting factors of safety are 2.l and 2. 4 for the OBE and SSE, respE 1-
, ~.
tively.
~
4.2 Calculated Frequencies
.a The fundamental frequencies of a fully loaded spent fuel storage rack are 14.7 Hz and 11.5 Hz in the North-South and East-West directions, respectively.
Details of the analyti-
~
cal model are presented in Appendix B.
Computer results are attached to the Appendix.
- 4 4.3 Factors of Safetv The factors of safety based on the calculated stresse, summarized in Table I and the allowable' stresses in Section a
3.0 are 2.6 and 3.0 for the OBE and SSE respectively.
er 2*
With impact of the fuel against the modules, the factors s
of safety are 2.1 and 2.4 for the OBE and SSE respectively.
.a l
l
5.0 CONCLUSION
S l
It is concluded from this analysis that seismic stresses 1
in the spent fuel storage racks caused by seismic and other applied loads (Section 2.1) are below the required allowable stresses.
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6.0 REFERENCES
2 1.
EBASCO SPECIFICATION, Waterford SES Nuclear Power Plant, d
Operating Company, Louisiana Power and Light Co.,
February 22, 1977.
^-
,e 2.
R. J. Fritz, "The Effect of Liquids on the Dynamic Motions of Immersed Solids," ASME Transactions, Journal of Engineering for Industry, February 1972, pp 167-173.
w 3.
R.
J. Scavuczo, W.
F. Stokey and E.
F.
Radke.
Dynamic 4
Fluid Structure Coupling of Rectangular Modules in Rectangular Pools".
ASME Special Publication PVP-39, 1979, pp. 77-86.
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~iw Table I
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SUMMARY
OF RACK ASSEMBLY STRESS INTENSITIES
.,,a Factor North-South East-West Vertical Static Combined of l
2 e
Condition Seismic Seismic Seismic Loading Stress Safety g
OBE 1100 psi 1700 psi 300 psi 2100 psi 5200 psi.
2.6 SSE 1600 psi 2500 psi 500 psi 2100 psi 6700 psi 3.0 a
- 11 a
8 OBE values are obtained by taking 15 percent of the static s'
loading and SSE values are obtained by taking 25 percent of the static loading.
a 1
2 Values listed in Table A-1 (Appendix A) are in ceased by a factor of 4 9 ' 27 9 - 4 0 ' 112 =
1.3 to account for structure 30,H0 and poison weight.
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,5 Table II
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SUMMARY
OF RACK ASSEMBLY STRESS INTENSITIES CAUSED BY SPENT FUEL IMPACT
- 5 8
2.
Factor Ncrth-Scuth8 East-West Vertical Static Combined of i
Condition Seismic Seismic Seismic Loading Stress Safety
...a
-,, 4
.1 e
OBE 1600 psi 2400 psi 300 psi 2100 psi 6400 psi 2.1 SSE 2300 psi 3600 psi 500 psi 2100 psi 8500 psi 2.4 l
4 4
.J 2
Stresses increased by 43 percent from impact.
2 Stresses increased by 43 percent-from impact.
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A.1 Rev. 2 I
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APPENDIX A
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STATIC STRESS ANALYSIS OF
'E WATERFORD SPENT FUEL STORAGE RACKS A.1.0 Objective It is the objective of this Appendix to present a static
~
structural analysis of stress in the spent fuel racks of the
- u
, 54 Waterford S.E.S. Unit 3 Nuclear Power Plant. caused by weight
~
of the spent fuel assemblies.
Results are based on finite r;A element calculations using the NCAUTO STRUDL DYNAL program.
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A.2.0 Analytical Model The one-fourth model of the 80 module storage rack used b~
for the seismic analysis (Appendix B) was also used as the r'
basis of the static analysis.
The weight of the fuel assem-blies were concentrated on the bottom plate joint coordinates.
l.
Each outside corner node carried one-fourth of the weight (325 lbs.) of a spent fuel assembly.
The other outside node L"
points carried one-half of the weig'ht (750 lbs.) and the inside-
!3 nodal points the total weight of one spent fuel assembly bs3 (1500 lbs.).
~
~
Planes of symmetry were assumed along the Z = 0 plane.
o (Figure A-1) "a'nd along the X = 51.9 inch plane.
All nodes
'i i
1 along these two planes ~were assumed to be supported.
Symmetry 7.}
was obtained by releasing the XY forces and Z moment along the w
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Rev. 2 A.2 s
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.J Z = 0 plane and the YZ forces and X moment along the X = 51.9 m
- ~
inch plane.
These conditions are indicated in the attached
-]
program output.
d All elements are assumed to be plate elements which in-clude both membrane and bending effects and are specified PBSQ2 and PBST2 for the rectangular and triangular elements, respectively.
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A.3.0 Program Outlet Forces and stress on each element are listed in the en-closed output.
Stresses are calculated in the lower portion 1
of the rack, the bottom plate, and support plates.
Maximum stresses in each area of the rack are listed in Table A-1 in each plate.
d A. 4. 0 -
Conclusion 1,-
All stresses are less than 2000 psi.
Values'are consider-
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ed negligible.
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Table A-1 1.*
MAXIMUM CALCULATED STATIC STRESSES 7
m Stress Element Intensity Number Area SI, osi I
3 l
l 268 0.090" Plate at Support 1600.
250 0.5" Bottom Plate 200."
225 0.75" Bottom Support 900.
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APPENDIX B SEISMIC ANALYSIS OF THE WATERFORD
- e3.j SPENT FUEL STORAGE RACKS B. l. 0" Objective It is the objective of this Appendix to present a seis-mic analysis of the spent fuel storage racks for the Waterford S.E.S. Unit 3 Nuclear Power Plant operated by the Louisiana Power and Light Company.
The seismic response analysis is based on floor response spectra provided in Reference (B.1).
Finite element calcu-1 lations were made using the MCAUTO STRUDLE DYNAL program.
..d B.2.0 Analytical w
A computer drawing of the finite element model of 1/4 of the 80 module storage rack is shown on Figure B.l.
The bottom plate is
~
}
drawn on Figure B.2.
All other planes in the structur'e are presented on Figures B.3 to B.13.
-i.
The rack assembly is assumed to be fixed at the base support
.'M (Nodes 172 to 176, Figures B.4, B.5, B.ll, B.12).
Furthermore, nodes
.)
129 and 140 cannot move in the z-direction and nodes 121 and 122 can-not move in the x-direction.
F]
For the North-South (X) seismic input the plane Z=0 is assumed to be a plane of symmetry.
The forces in the x and y directions and
.I the 2 moments were specified to be zero (Figures B.1 and B.9).
All t__.
- #7
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other degrees of freedom along this plane ~ (displacement z and rota-tions xy) were fixed.
Along the x = 51.9 inch plane (Figure B.8) the forces in the x and z directions and moment in the z direction were nade equal to zero.
The y displacement and the x and y rota-tions were fixed.
l
, Similarly, for the East-West (z) seismic input the forces in
,j the x and z directions and the x moment were made equal to zero along the z = 0 plane.
Displacement y and the rotations y and z are zero along this plane.
Along plane x = 51.9 inches, the forces y and z and the mcment x are zero.
Displacements in the x direction and rota-tions in the y and z direction are zero.
These boundary conditions insured that the symmetric vibration modes in the direction of loading are excited in the dynamic analysis.
c Member density of the structure was increased to account for the weight of the fuel poison and enclosed water.
Thus, the mass of the fuel is lumped at the node points along the box walls.
This l
lumped mass assu=ption is usually made in dynamic analyses.
The influence of the virtual mass is added by adding external weight to
~
j the outer node points in the direction of motion.
Because of the 1
l different clearances and dimensions, these masses varied and two dy-namic analyses have been conducted; one in the z-direction and one l
in the x-direction (Figure B.1).
All elements are assumed to be l
[
plate elements which include both mambrane and bending effects and l
are specified PBSQ2 and PBST2 for square and triangular elements,
~
i respectively.
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+
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.,j B.3.1 Program Input Calculations are based on 1/4 of the total rack (20 modules).
in B.3.1.1 Fuel Weight The weight of each spent fuel element is 1500 lbs.
(20)(1500) = 30,000 lbs.
W =
F B.3.1.2 Poison Weight There are two 68 lb. poison elements per module
~
W (20) (2) (68) = 2720 lbs.
=
p B.3.1.3 Enclosed Water j
The volume inside of one module is (less wall for poison)
V=
(10.38 -.27)2(185) = 18,909 in 3 4
Assuming a specific gravity of 10 for the fuel and d
5 for the poison, the weight of the enclosed water
,3 is J
1500 _ (2) (68)
N=VY-
~
10 5
-8 W = 682.8 - 150 - 27.2 a
W = 505.6 lb/ module JL Thus, for 20 modules W
= 10112 lbs.
a W
B.3.1.4 Rack Structural Weight a
~
The weight of one module (neglecting the bottom
.1l' plate) is determined as follows:
f w
w
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s.4 s
1 = (10.38)2 (10.2)2 = 3.7044 in2 i
A 2
\\
'A (2) (8. 57) (.09)
= 1.5426 in
=
2
-tx 2
(4 ) (1. 6) (. 09)
A =
.576 in
=
3 AT " 5*023 i"
Thus for one module 185 in. long, the metal volume is 3
V = 1077.3 in For twenty modules the weight is W=
(20 (y) (V) j
= 6140 lbs.
6 1
The bottom plate weight is
'.1 (0. 5) (41. 5 2) (51. 9) v =
p V = 1077.4 in i...
P rs The weight neglecting the holes is W = 307 lbs Thus the total structure weight for 20 modules is W3.= 6447 lbs B.3.1.5 Effective Weight Density I I53 L3 The volume of metal of the one quarter model is.
V=
(20)(185)(3.7044) + 1077.4
'L~.
3 V a.14784 in i
Th'e total weight is
~
l W = 30,000 F
2,720 W
=
p W = 10,112 g
6,447 W
=
3 W = 49,279 lbs.
7 i
. _ _ ~.
a nuv.
j, B.5
,.2 -
m.M Thus, the effective weight density is
~'
T 49279 Y
Y " I " 14764 3
! ;3 y = 3.333 lb/in
- g B.3.2 Virtual Mass Effects
~'
B.3.2.1 East-West (Z) Input As described in Appendix C, the hydrodynamic mass M
is determined as follows:
H For the east side of the module assemblies, the gap is 13.6".
_C 3
(1851 3(51.9)
H1, yh b, 62.4
.'4 12G 1723 (12) (13. 6)
MH1 = 72,710 lbs.
For the opposite side the gap thickness is 37.2"
.as
- J MH2 = 26,580 lbs.
es
- Thus, 0,*
MH"MH1 + b2 = 99, 90 lbs.
i.:
L*
Since there are four racNs in a line, the virtual r+)
weight per 1/2 module is
,2 M
= 24,823 lbs.
~
~
H
'4
- i__,
Because of symmetry only one half of this value is
~
input' 1
M
= 12,410 lbs.
7 H If the nodes 5, 30, 127, and 171 are assumed to carry a one-fourth load each, nodes 10, 15, 20, 4
elmuS em e-...w - - __
Rev. 2
- p B.6
- f, e-
.ue 25, 35, 60, 65, 90, 95, 120, 139, 153, 161, and
=E 166 are assumed to carry one-half loads and the
.e remaining nodes are assumed to carry one full share;
.s there are 20 full shares to be considered.
At each full share node the input is i,
M
= 620. 6 lbs.
NODE "
2 This mass is included in the model on the INERTIA OF JOINTS ADD input of STRUDL (attached output).
e
'd B.3.2.2 North-South (X) Incut For this direction en the 12.8" side of the pool
-i 2
3' 1723)g(185)3(41.521)
, yh b,
62.4 g
3H1 12G (12. 8) (12)
M
= 61,800 lbs.
H1 9
and for the other side (G = 24.3 in.)
ad MH2 = 32560 lbs.
Since there a;3 four modules the effective hydro-dynamic mass per 1/2 module is M = 23590 lbs.
-y H
i LJi Because of symmet'y only 1/2 of this value is input r
l to STRUDL 1800 lbs.
M n,
7 H
=
L On this side, the nodes 1, 5, 123 and 127 receive one-fourth of a load, nodes 2, 3, 4, 31, 35, 61, map
Rev. 2
~1 B.7
.y.
4 65, 91, 95, 124, 125, and 126 receive one-half of j
a load and nodes 32, 33, 34, 62, 63, 64, 92, 93, 4
94 receive one load.
N
=d Thus, for each full node, the added mass is M,= 118
= 737.2 lbs.
6 fM,=368.6lbs.
Im --
1 M = 184.3 lbs.
T a
.s 1
B.3.3 Participation Factor Modification B.3.3.1 East-West (Z) Input
'q As described in Appendix C and verified in Appendix D,
the fluid surrounding the module reduces the relative motion of'the module.With respect to the pool.
This reduction in dynamic response is modeled
[.cd as a factor C where m-M 1
C =,,g
.d H
i where L-M
= hydrodynamic weight H
L_:2 m = structural weight p
My = weight of displaced fluid MH.* 12410 lbs.-
fl m = 49,279 lbs.
t M1
- Y LHW re M
y=y (185)(41.52)(51.9) r-l M1 = 14400 lbs.
m e
Rev.'2 C.8 Thus for a one-fourth model
~
49279 - 14400 C = 12,410 + 49279
- r*
C = 0.565 an This value is input as a factor in the spectrum
- ,.e response curve input of the program.
This factor is gC.
i _a Factor = 218.
M B.3.3.2 Ncrth-South (X) Input For this case
'l C = 49279 - 14400
,l 49279 + 11800 C = 0.571 Thus the 5 actor is "8
Factor = gC = 220
..A
'-i B.4.0 Program Output
, L.
A copy of positions of the program output for the North-l South seismic input are attached to this' Appendix.
Seismic stres-
,g ses, deflections, element load frequencies, modal participation factors and mode shapes are included.
'w B.4.1 Fundamental-Frequencies" b'
Ls*
The fully loaded modu'le under water has a fundamental fre-quency of 11.5 Hz in the East-West (Z) direction and 14.7 Hz in the North-South (X) direction.
i d
I--
.w w
w-
Huv. e O
B.9
- t.
- r,
. e*
B.4.2 Calculated Stresses Effective seismic stresses calculated by the SRSS method,
-1 labled "RMS" in the program output, and the absolute modal sum method, e
labeled " PEAK" in the program output are listed in the attached out-
[
put.
Stresses determined by both methods are in close agreement because dynamic forces are concentrated.in the first mode.
The maxi-mum effective peak stress which is twice the maximum shear stress from the North-South input is 1100 psi and 1600 psi for the OBE and SSE',
respectively.
These stresses occur in element 275 which is adjacent to the support.
In the East-West (:-input) direction, the maximum s
" PEAK" stresses of 1700 psi and 2500 psi were calculated for the OBE
~;
and SSE respectively in element 272.
B.4.3 Relative Deflections a
Calculated relative SRSS deflections for both input directions
-d are listed in the attached output.
For X and Z SSE inputs, the maximum deflections are 0.0166 inches and.0198 inches.
B.5.0 Conclusions t
- _ame j
By combining seismic stresses with static values, the total
_.s stress in the module assembly is well below allowable values.
~
e
+M f
- 1 f
-..e t ~
4 J
l L..
~' '
Rev. 2
~ j, D.10 e
n s
REFERENCES
" m.
...J i
EBASCO Services Inc., "EBASCO SPECIFICATION - Spent Fuel Storage w
i.
Racks," for Louisiana Power and Light Co., Waterford SES.
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.... s
- ..;. b c.,s M,49
.,y lc,-.-
- 3. '.. g f 3 *-
.m u.
+;g y.'...
3.,, s. -
~, s,.~
~
a. o;g, 'r,.
c,. w '-
.c - *rn f...
.<..r.
-- i..
- g
~-
.p.s,, y*.
..x
, cc
-. l' 4-
,T P,.
.s e,,
,W.
i
... v
.c3 s
s tR-5 5
t 8
1 n.
'. {
G.,.. :. '.,
t
.e.
l
.e' s
'.l4
=
pgs 3
ed 3
8 h
'h h
+'
,f r-8 W
v y
M OOI 0J G3 00 0'9 00 07 C0 02 00-I
- b. lad.:
l>-
- m i
w c.
w
m-3
=
+
- I APPENDIX C
==
t !
- 1...
, FLUID STRUCTURE COUPLING OF RECTANGULAR MODULES -
~
.u
'. ~
4, w
a
=
g..
C.l.0.Obiective 7
It is the. objective of this Appendix to present the fluid-
~
structure coupling theory used in,the analysis of spent fuel
,e y, C '.
. modules. 'The Theory section is divided into two main sec-h,,, j.
J tions:
normal mode analysis and the derivation of the effective, o
- s.. -
hydrodynamic mass matrix.
, f....y :
a t
..,+,r.-
y
- w.,
4 3:,.
~
-un
.i.,, 3 In.the normal mode theory.section,- equations are. derived which'
.L
. couple the structure through the, fluid to a moving pool wall...
m
.g
~ p;' +,r~....' The basic., assumption made in this analysis is. that the poolg'. "
i.
.',' walls move with the same seismic motion as the' floor.
- Thus,
,, p g.,((.. i s -
thel pool walls become an extension of the. foundation.
Hydro ~
4 dynamic masses are derived for an' infinitely long. rigid x,
( i;.. ;'. rectangular box in 'a rigid infinitely long rectangular pool.
g m
.1 In addition, the hydrodynamic mass matrix for five rectangular u
taLp.' T. '... boxes in a4line in a rectangular pool'. are pi esented.
In,this,. -
s
-C...: '. :' derivation.'effect of fluid moving in the vertical direction
~
m c. -,.
p f,.
is neglected.
The solution for,the fluid moving vertically
'~
A d d -l ':,' -. is] presented by Fritz. (Cl)#.
aand '[ncl$ded in Section'. C. 3. 3.1. E.~;r G,~
t
. r ~r,
[
W I.,,
[
'Fo'r. design the smallest.hy'drodynamic mass of these twotca'ses'
<- [ is. applied >.
'.. if.y..'~5M. J.pl.. :o :. f i *.2;;.
~
F N n: ? ?
$ :.W.. u.- x5[.] t 2.:i */ h..-l.
' 2 YI.w$$?.Mp.<
..c~
. w.: - x -
m'-~
~
.."'.. c GWW f.%.W..YJ:"Y.. A' ~,W~. w.;.U. :.~. * ;.
l Gi
_.l ?
J&y y e. 'i '-['
- 2;is '.: -+
7:..
. ~ ~.?v:[;'}'
.'?;':' ~ W:
' s.c t.
2'.'
.r4 y.
i nJ
'* % s 4,
=
f
. A *v. g..--
' y; ".
,f' &..
- i
,j, _. -
& ? '.',. ^
'. 4 ;-,-
k
<s
- f,
. om : %'..
6,...~-t f
.*G
.vr
~
J
- 4
? :.-
.~
- J.,.,
E.
- n
'4 I"
1*
.s s.
s J,
- s C-s g
I.
d.
,, 8
+*"* b oN N
,'. j d,
x.e2*iya..m.. " y, ; y 4 h e, ' ~~
~ ~ '
..'c, :. p.u -+ $~ f,1...,l.
V5 2I,h*#
~ ' [ y, (( ~
t,v
,. a.. n.y
.x-
- ~
-g,i.
.y.
O s
.a..
r h
q'-
i A
,\\
g-Rey, 1
C2..
s.
C.2.0 Normal Mode Theory g
j i.
C.2.1 Equations of Motion t
,0.'.
...,1 t
- . '. c'..g..
- . a.; : ~.Q o.c.
an.,.
- h..
....7f.the viscous forces.and' wave. effects are neglectedi the
.g
'Q-coupling between~an elemsst of a rigid body and an element-of a rigid foundation through a fluid can be written as
~
v.
(Refs..Cl, C2).
g
~
f s
-'p r
......f.~.*-
Y.'
1 I
,3 r.-
l l
.. t r 1,, ;.
y
_g g
L y.
g
-e i
Hi,
li.+
i 1
T<
.,c.
' hu
[.dNl..r
/
)
)
?
'"( 13 1
. - Q p
d. ;'
- e
.d c',,
p l
. Mli +
iv
'y; ;. i $',
..e-
~
+M
'"-a fi (M'~i.+M21 i
Hi) 0 Z
F..t.. ", <.
..;. e, ;,,
1
- b c..... y.n.,e
.. g g
)
i
.s
-r w
e; ?..
s 2
- .. ~. ~.,
- 1. ::.,=.,..,";. ';. - where- (Fig. C.1. 0).... _.,
c.
"c cf.
4
--w :..,.. <:..
. fy = Force 'cting'on the pool wall.in the;i,th
- f..
.. ~.
3.
g : %.,,.U. ; V.<
. element.;
t h. %.;' W, Q,.
'. g g./ F a
b.f *. <., g.i.
,,c...,..
,, 31
.;,,,,,, l p;p; M ' tJ
. h ;Fyi = Force' acting;'on.hc.,.
r.
th I DF
' i ' ',' 'U wbr the i mass. -.'
.v..
.c
.[ !
i j
~
_.m pl. -
69
-;7 t MHi = Hydrodynainic $5s's ~ ssociated 'wiiih'm.1 'V.. '
a
>w i
g a.....
.,4 -
r_ c.,,c 1,,c.. -;;.
-..-^-3.,.-
~.
~
L
...c.,...e.
..n
.p.:
m-
. J Ml..[,.[i b,.
Y
.l.h Y 'i = Mass If.; the fl'u'id ' dis'placsd"bYm ~. - (Tab 1'e"C.1. 0)..
Q. <.f. $...;.'.Wl;. h ' '(;'
1
- e.:. % v : i.;'.
.y
'~
21 = Mass of!fl'u).d 3,,.2ased,on'the'.vo'lume enclosed by the'
. '. t..
J M
~
y c.,..,.
.(.N*. (c1.I
'.Mf.?.. -
Pool. wall ~which is coupled' to m[. -(Table C.l.0). :
MM.th,.*
.. s
. _; b.
5...
,,~**.
- ..n.y'.' *. '.. '. %
,c
.s t
5_~.
.~...-.~.u.".
,' U, b. Y ( t ) = Motidn.o$Im Nelativ'e' to the pool wal'l' f.ourida' tion..
.~
.,. ?
.t t.
h*f.'.';~YfAYf$Q. " '?
f.'{. $$..
a h 1.hhf,. Q.r *-
. $. Z (t) =, Absolute. foundation m.otion.'.*-;.;,VL,'.
+.": -
.4 J
-4, ; '.i:.. -
t
.?..
s U..)'..:.
uw '...
l A ' j.)* * ** *
.hf,.; f..j '*;. t.; 4 h fh U.;$,;-
r.s..
k.
~
.)"; 4
+., ;..
~
- e_: ~ (.
. Th'e'r' fore' the fluid force,on e'ach" mass is, f.* r.
.. a; -
l e
, ~
l
- 4. -
,y 7,,
--c q i. '.. e
.J F
=-
.-Y'+M Z'~
c 'm.
(2)
>. g. '
y1 1
i, 11 masses (
?RWes. M.S,:8. : i*.'~vA.4lU.';T UE' ".
'A.
.r
.&'n.,1:i:b.h}i.for jall.
- .I.5L -
O. % A :,..
e ' _. ' 4. -
or
. e i, $'
- %.s
, q,.,e
.e
,s..
.y 1
-a.
r -
(..
F
=-
M y Y
+
M Z
(3) y H
1
-3
)
i Where denotes a diagonal matrix.
^
,t l-i o
1
C3, i
Ig 1.1 If ~[K] *is the stiffness matrix and (m] the mass matrix of J
the structure the equation of motion is
[yf' + '
' (K]'
(i)$
m
+
=
F QC. y
...,s,
- ,h U.
.,.7. f.c
.i-
..2 '
..,q. 4.
W '.'.
.'Thus, 'subsd.ituting fo[,F
- f..
~
I.
c3 y;
o r-t-
m+M y o +-, (K).
=-
m-M z p (5) y yq J.-
a s
u..
~ ' '
L--
.C.'2.27 Modal Displacements
..s
..e
... ~ -
- The eicle' values,. w 2,. and eigenvectors e
" '.t.
n Y
of.the homo..
...f
.%; ', ' i,,
a a.
genous equation can be. determined.in the. usual' manner. (..,~...
- .. -.., ly
- 1..W.
0 '. +.
. ' Let -
q
~
- 1 9...'..
=
y )qa (t)
..y.
- - ^-
",.. s..;.:....a,.
y-a
. f...,,
..f,. ;
y
.. a :
.(6).
,v..,. -
a,
.,a
.,....s.
. s.,.,
.., ;; s g.
'.and.
.:.v:.
u..a
. aer.
.. ~
~
~
.. i. c g'
i.,
..F,,
o jN a
_.Y, 7
- m,6, 7 -
m+M Yg ) c =. 0, L
t
- (7),
. H a.
.c;
^
r ).-.:.
1,;
c '
'. s.. #,.';
,..~.._f...,-
.s Thus j.'Q r.h ' *y. q
.rj
'.?? '. '.4
.,q.
,,.y,
- (0W,.
.f 3.r.)..?.2 L,. ?. ; -.,c,..
.%' s.., - ~
. Mi 1.h, :,j-
.U' T. '
n 'c '...
s 1.
~-
q 4.w2 -q % =
-P Z.
- ,,;f * :.
.o 2 -
(8).
a a
a a
~~
. ".s o g.L.tt.i ;
- g..
. '._y' CV * ~
a.
er--
,, ; "i
- Q v
.'.i,,,.?
wliere."
'.T
~ '
i.-.
- J..,- **.
T.; \\ev.'. '
i
- t.... i. '-
.v.;,
.rt y
.1 2 x. ML
- '". ~.. c: ~n ' awr -
1.is. ~
a h.1 ).1 m-
......e p
a.,
,~
- s-a c;
.df4*-
, f ;>.,.
. '?$
' Y.c$ '- '
?-.4.1.. f. ~ (9)'. E. 7'. ~.
>y.'..=...-
p
=
- T l '-
.. E ;1 - w 1,,
a
.r ' m +. M
,a' a
Y Y
c ~ :1' '.
.~
.~
.v.,..,.. '; w. -
\\
n.
r
~
..n?.&.
- ;~e ~ ~. -
s.
i:. ns h..a.~my,u 1
t..
H,
h.
- >... e. m.. n -.
.,i
.. g...m,,.n...,:.y y4,r.s..v,x.
e.
+.;.
,.2..,,..'...
c.
y
.. s. ~:
,7 f... -
..3
. -3 s +-. ir.
.( -
e..
.e 5
j.
. ; 4. '
- f.. +.J P.
h ? R,, 'W ' U ' The' influence 'of the water coupling,between the structure " '
e I
s.
and foundat.i.o.n is. to reduce ~ tihe modal participation factor 2
.as. compared to-the uncoupled structure..The magnitude of-the 3
7 d-change depends upon'(M ]'and.[M
].. Theseeffectivemasses}..
y H
." depend on the niass ' f the displaced fluid ~and clearances in..
o E.. ?.. -
c
_3 the coupled system.
e Rev. 1 m
i
. C4 a
If the initial conditions are zero (q,(0) = q,(0) = 0)
F p
+*
j~
q,(t)
=-
a 2 (t) sin (w, (t-T)di (10)
"a J 0
o or Y
9a max "
aa (11)
"a 17 where V, is the spectrum velocity at the circular frequency, w 3
},
C.2.3 Modal Forces r.
The forces acting on the structure can be written as 4
fF
= (K] fy (12) q.3c or fF
=[X fY, q,
(13) a But
[K]
Y, w2 m+M Y
(l4)
=
H a
,(
Thus fY q,
(15) w2 m+M F
=
y a
The force acting on each mass, m, in each mode is a g
4
,]
F Y,g V,w (16)
Y""
I"
- MHi)
P i
}
In terms of a modal acceleration in g's, G,,
equation (16) becomes I
(mg+MHi a
ai a
Fy,=g g
C5 t
==
l The modal or spectrum acceleration is related to the spec-me trum velocity as follows:
a
-m G
=
s4 Vw a
aa
,. s (18) 9 b
f6 I,:
g~
d u
i D
I
'! m NA
'P>
9
+
'a 1
.e 9
C6
_3 C.3.0 Hydrodynamic Masses d
Hydrodynamic masses for a square rigid module in a square pool are derived.
It is assumed that the gap surrounding 9
the module is narrow so that approximations used for fluid d
velocities are justified.
Furthermore, it is assumed that these gaps are uniform and the geometry is symmetric so that
.=
the flow velocity at the middle of gap is zero. (Fig. C.2.0)
Og.
In addition, flow in the axial direction is neglected.
As a result, the magnitudes of the calculated hydrodynamic masses exceed the actual value.
G7 In all calculations, the hydrodynamic mass calculated for plane motion is compared to the value determined assuming
.g only vertical fluid motion for a plane near an infinite wall, b5 (Ref. Cl, Case 13).
The minimum of these two values is used as input.
C.3.1 Fluid Velocities mm k 4 The fluid velocity in the y direction of the top gap (Fig.
"i C.2.0) is Ix
~ *1 2
y y
y (19) i "1
i The magnitude is identical in the bottom gap.
The velocity in the gaps parallel to the x-axis is 2 ~ *l I (20)
[
V
=
x b
i 2
j l
_2 m.
-i C7 d
C.3.2 Kinetic Energy d
The Kinetic energy of the fluid in the channels caused by relative motion is av j
Ib T=4 y2 w h dy o
k.
(x
- *1) 2
+20 b ch (21) 2 2
' ~, ;,
Thus
..e g
l.-
T=2 (x
- *1) 3 bh q
2 3
+
'd "1
1l {}
+
20 (x
- l) b ch 2
(22) w2 C.3.3 Fluid Force Matrix
~d a
O Since the fluid forces can be determined from the following (Ref. Cl) :
I I
Fy=
t a.
'I 1
L A.
the forces from relative motion become L,
rp 3
N c3 x1 H
-M X
H 1
h
- 1... j t
s j
f F
- 2{1 LM M
X (24)
- H y
L H
2; where 4
bsh 40 ba h c
H"1E
+
w w
(25) y 2
If b = c and wy=w2""'
u _ in usu
1 C8 j
C.3.3.1 Vertical Fluid Motion - Hydrodynamic Mass If fluid is assumed to move vertically with no horizontal as motion, the hydrodynamic mass is (Ref. Cl) :
n j
M
=phb 3
H 12w (27)
I The minimum value determined from either equation (26) or equation.(27) is used as input in design calcolations.
+
C.3.4 Total Hydrodynamic Mass Matrix CE By considering absolute motions of the system, two additional hydrodynamic masses are developed (Ref. Cl):
_.N~
My = 4p (b - w/2)
(c - w/2)h (28) mO and M
= 4p (b + w/2)
(c + w/2)h (29) 2
{.d
, dvd Thus, including these effects F
y+M2+
-(My+M y
xy I
k(30) 4
=
~~
F
" IN1+MH H
2 l
x2 5
g L
4 V4 k0i C.3.5 Hydrodynamic Mass of a System of Modules In addition to using energy methods, hydrodynamic masses can be derived -by calcula".$ng the pressures acting on each f ace l[. _.
of a module.
Usins tl s r6ethod, the hydrodynamic mass matrix of five modulef aae ving pool was determined.
This result is presented in Pable C,1.0.
i i-
~
5 5
C9 m
a sg TA8tE C.1.0 D.drodyngmig Mass,$_tr.J.s,pf,fjy_,5q,uare Mules in a %ing Pool With a Uniform G p, 3
[,
_...h2bM F
28 J
2b
+
,5.
8s xg x2 8
g xg as 3
v
- p Lutryp) e g
i
=
- qs W
"xi y
(FI*(M]()where(X)=(*x2 )
H H
,as
=
M s%y-"t
-[
M 0
0
'O g
-h N
-h
- l%-M 0
0 i
8\\~"3 0
M 0
g--
g
.y.
N N
6 ~
0 0
M g-g g--
8"H "I 0
0 0
8 "H
"h
~
k'$
-fM-M;i - lM -M ;
0-M -M ; - h -M ; gk -M ; 4%+5M +M g
g i g i i
i2 r'[
a.
- M,=fa"
~~
,Mi =.4eb2hM2 = 4cBL'.
g m
1 I
e 1
a v
C10 t
- l. -
0 l sa d
- s..
(.J
....m I
Y
- \\4
(-
N 2..d
/
4 Le Nd T,.J 8
N g2 y2 Tw b
['8 I
h' I
(
11 b
-*- Z( t)
V-C L Floor-Wall Motion Ff0. C.I.0 a
k d
1 I
A.
~
C11
.w med ed
- se
~
t j
$ age j,81
>4 i
Wand 2b L,]
1 nr
=.*
i C:
l L.s j
J "2
lJ 2c
'~
W W2 3
- d
-a i
Y o
L n
4 WI l
Fig. C.2.0 Rectangular Module In a Square Pool WI
'4 1
~
e i
~
t-
__1
'e m
-~
---4
f C12
[.
References ll Cl R. J. Fritz, "The Effect of Liquid on the Dynamic Motions of Immersed Solids," Transactions ASME Journal P "*
of Engineering for Industry, February, 1972.
w1
' _ -s K
C2 W.
F. Stokey and R. J. Scavuzzo, " Normal Mode Solution
'd of Fluid-Coupled Concentric Cylindrical Vessels," ASME F.
Publication 77-PVP-37, 1977.
SmW F*
1---.f.
l'.
c E' 1
~
h'd hbN
[':
a V
6 I--J Ehl
~
.\\
~
=
MS
=i 1
5 3
APPENDIX D
~
EXPERIMENTAL VERIFICATION OF FLUID COUPLING THEORY 88 D.l.0 Objective S
It is the purpose of this Appendix to present results of mm an experimental investigation which verifies the basic A
fluid-structure coupling theory developed in Appendix C and the hydrodynamic mass of a rigid square box in a square pool with two-dimensional fluid motion in the g"'
U horizontal plane.
(Ref. Dl)
The experimental model was excited with an electrodynamic shaker.
Changes in the resonant response were used to i
=a
' 'i.
determine the hydrodynamic mass.
The coupling theory was u
evaluated by comparing the measured amplitude ratio with
~h calculated values.
w P
d l N" i
}
N Eh a
O O e
a' W8
-s
3 D2
-s D.2.0 Model Description In this experimental study a 4 x 4 x 3/16 structural tube as eight inches long was'used to model the rigid box.
.a (Figure D.l.0)
This tube was supported by two high strength L
steel cantilever beams 3/32 x 1 x 4 which served as a
springs.
Because of the parallelogram effect, there h
was no significant rotation of the tube during horizontal a
motion.
Seals at the top and bottom of the tube restricted j
vertical fluid flow.
Originally the fit of the teflon u "
seals was tight.
However, in order to eliminate friction
~
(and thus coulomb damping) a small clearance was provided.
"2 mc As a result, some vertical fluid motion was observed.
However, the basic motion of the fluid during harmonic excitation was in the horizontal plane.
A plexiglass pool surrounded the tube with a gap of 0.5 inches on all sides (Figure D.l.0).
The foundation of the pool was fabricated from 1/4" steel plate.
This
~"
plate was reinforced with two steel angles.
The entire
[en pool assembly was supported by cantilever-beams from a
- d rigid foundation.
An electrodynamic shaker was used to
{,
input a sinusoidal motion to the pool foundation plate.
Um The effect of fluid coupling between the pool walls and rigid box was studied by measuring the response of the system with and without water in the pool.
The center of the structural tube was dry and not filled with water for a
all measurements.
Durihg preliminary testing of the system it was found that additional weight had to be added to the structural tube in order to obtain relative motion
-d C:
between the pool and tube.
Without this additional weight the tube moved with the motion af the pool walls.
The total weight of the tube, additional weight, seals, bolts, etc. was 15.0 lbs.
me I
.j D3 Motions of both the pool foundation and tube were measured with piezoelectric accelerometers.
Output was recorded on b
a storage oscilloscope.
By tch.ing a ratio of the response ku to input, the ratio of absolute displacements was obtained at each input frequency.
k 6
s,
'w L.e "e
4
~ ;
4 mb
-2 i
I-M4 Go 7
-}
e e o
M
'e
d 4
4 D4 l
D.3.0 Experimental Results a
y First, hydrodynamic mass effects are described.
- Then, response curve shapes are discussed.
P"Y
%d ~
D.3.1 Hydrodynamic Mass Effects
["
The amplitude ratio in air is presented on Figure D.2.0.
The natural frequency of the system was found to be 15.2
,p, Hz.
The damping ratio of the system was found to be approximately 5%.
This frequency was used to determine the effective stiffness of the cantilever beams.
2 K = 4n g2 979 (1) d
-j where t r' f = 15.2 Hz w = 15.0 lbs.
d t
g = 386 in/sec u
Q fs K = 354 lbs/in.
This value is approximately half of the calculated value of c.
This stiffness is based on formulas for two fixed -
K fixed beams.
?
K
= 2[
)
(2) c i
3 3
-=
14 33 bh3 y,
12 p
b = 1 in.
h =,,3/32 in.
8 E = 28.5 x 10 psi I-*
L = 4 in.
.2 4
K = 734 lbs/in.
s c
This difference is caused by the fact that the ends of the beams rotate slightly and, therefore, equation (2) will over-i estimate the stiffness.
The stiffness based on the measured l
l L
a
-)
DS frequency is used to evaluate the hydrodynamic mass.
After filling the pool with water, the response shown in d
Figure D.3.0 was measured.
The natural frequency was p_ o found to be approximately 9.7 Hz.
If the hydrodynamic mass
- End is based on the box dimensions (rather than the pool center-line) good agreement between theory and experiment is obtained.
For this case 3
F 16 yhb II
[,,
H 3
w
~. j h = 8 in.
"A b = 2 in.
w = 1/2 in, bd y = 62.4 lb ft 3 r1 kg.
Thus W
= 24.65 lbs.
H The calculated natural frequency is determined as follows 1
(
i 5) f=
K4
'~
2n $
W+W I4I H
K = 354 lb/in.
- g, g = 386 in/sec.2 f
W = 15.0 lbs.
W
= 24.65 lbs.
l H
f = 9.3 Hz
~~~'
This value. compares well with the experimental value.
It l
should be noted that if center-line theory were used the
~
E-.
calculated frequency decreases to 8.3 Hz.
4
_1 Because some fluid bypasses the seals and moves vertically, the measured hydrodynamic mass is expected to be less than measured values.
Agreement between experiment and theory is felt to be very good.
_7 D 6, D.3.2 Response Curve Shape Because of the relatively small magnitude at resonance of is the ratio of amplitudes, it was originally assumed that there was very high damping of the structure caused by the g
fj fluid motion.
Furthermore, the fact that the amplitude ratio approached one as the forcing frequency increase could not be explained.
However, by applying the theory developed in Appendix C, both'of these phenomena could be explained with a relatively low damping ratio.
u-*
D.3.2.1 Theory
,p
.b
=
For a one mass system,.the equation o2 motion of the structure can be written as
-O
~;
mx + c (x - z)
+k (x - z)
=F (t)
(4)
M;
~
where A
F (t) = -MH*
H+
1 Thus (m + M I..*+
- +
- " I H+
1 y + cy + ky (5)
H By dividing by (m + M ), equation (5) becomes l
q H
"H +
1 x+2w ncx+w a
2 x=,,g y+2w C Y + "n y (6) n n
Let y = Ye "
~
(7) fl.
and x = Xej.(wt - Y) -
(8) m Thus MH+M1 w
+ 2w w c j 2
"n m+M
~
X
.=
H
~
- g
-j?
gg)
_]
}e J
(w
-w ) + 2 w "5 n
n I
D eee 9
D7 a
Therefore the amplitude ratio is 7p
-2 2
1 luj2
+ [2
~
"H
+N "n (
l~
~
1 1
ul
(
f
- b "U
m 's
=
(10)
I;2 u
2 f *n#,
+
1-
,2 C
W J u
,L n
J Where i
I-a" =
g
\\m+M II1)
H L ;
t_;
Note as u becomes large, equation (10) converges to the following mass ratio
\\y "H
- N1 p
= 0.793 (12)
=
go m
+M U
For the uncoupled case, the ratio goes to zero with increasing frequecev (Fig. D.2.0).
Results of equation (10) are plotted on Figare D.4.0.
It should be noted that the damping ratio is approximately 6.2% for a curve with the same magnification kd at resonance.
Furthermore, the graph has the same basi: shape as the experimental curve and the ratio increases with gy frequency.
D.3.2.2 Damoing Since the damping ratio is based upon the structure mass and 4
y hydrodynamic mass, the damping constant C i,ncreases by more than 1%.
Since C = CC
,c where C
4n (m + M I
=
c H
n (14) 5 u
ass 0
v
.A D8 For the uncoupled case (no water) as 386\\l5.2 15 55 C
= 4n c
g,sq E.,
and
['
C = 5.3%
i m
Thus; t 1 lb-sec C = 0.393 in.
For the coupled case 4n [ 39.6519.7 C
=
386 j c
r.
bd and C = 6.2%
.s
- Thus,
' US C = 0.777 lb-sec 1:
. in.
.F' Dly As a result, the damping constant, C, is almost doubled.
For a give amplitude, the peak velocity of the coupled system vibrating with sinusoidal motion is decreased by a factor of 4
1.56.
Thus, peak damping forces are increased by 26% for the coupled system.
Thi-s estimate is made~as follows:
l
/0.777 9.7 j= 1.26 w,
Fd
=
Fd
-(0.393 15.2 1 1 ?,
1
-m where 1
l l
Fd the damping force with fluid coupling.
=
Fd the damping force without fluid coupling.
=
s
==
D9 D.4.0 Conclusions nei The results of this experimental investigation verify the e
calculated hydrodynamic mass for a square rigid structure in F,,
g a square pool.
Furthermore, by comparing the measured and w
calculated response curves of the fluid coupled system, the basis theory developed in Appendix C is verified.
F The damping ratio of the coupled system was found to
'"~
increase by only a small amount -(ul%).
However, this damping 4
ratio of the coupled system is based on the sum of the u -)-
structure mass and hydrodynamic mass.
As a result, peak damping forces acting.on the coupled system for a given gjj amplitude were shown to increase by 26% above values acting on the uncoupled system.
"T-3
>}$
'e mE hdld L
mi u
w= d G.
i i _ _.
l l
1 D10 References m
D1 Edward F. Radke, " Experimental Study of Immersed f.4k Rectangular Solids in Rectangular Cavities" Project for Master of Science Degree, The University of Akron, Akron, Ohio, 1978.
T'k*
t-
~.. g F3 P
M
. 4 a
D. '
F-l b
IL
,1 Ii-Y 4 x A TUBE
/
. 7 g
PLEXIGLAS.G Mqu 5
.d b
evATER 4. EVE /.
A l
t
??OTION OVERL. APING 7E*FLON
_9CCELEROMETER5 SEALE
~
e kl.-i u-r l
3HAKER
'C STEEL. ~ ~..
l L"
sparnas
~
udU D
U~
lR o
--S TEE L.
~
Id 5UPPDRT SPRZA/55 b
J L
tu s
a
- o.,;a a.,
.s
=,a b
4
.' Y lT 'r
- ' ' O '
G,
,o 1
4 2
g
_' 4-
'A l
?
A J
,'.,A
.4 c
's
.o 2,
s s
CDNC47TE l
BLOCK
[_
Figure D.l.0 Experimental Apparatus h
ase e e I
4
"'1 4
D13 a
d h
h8
> I
- o. -,
am og l
)=s O
P, n
o m y
2 be
=
r 4
0 O1 1
sM
=.d4 "6
i N
C
- , e*
1 3
M
-6g
.u.
-o
=3 a
.e. M E
Cd f~
4 b1
~~
N
'l 3=
l
- -d O
l
+
43
,, o aC e
e,
G i
e b
3 we l
E l
l o.
N
-e o
.e l
=
l o,,
e m
m o
o e
u r'
z l '
O =
r* umut e
e.
CH3 ifm FTG ff"3 T3 i
r("1 FFp c9 Ei i
E5T 1
(
L_
ml L.
Lu LE.
L
.A tr i 4.
t i
L k.
\\1 i
i;..
11 10 9
8-
- oa]
7_
Experimental Result g
Theoretical Curve f or' C =.062 a
6-Using Theoretical f, w
b 5-4-
3-2-
i t
1-s
\\..
s
-1 4
5 10 15 20 25 30 35
{
- f. (Hz) l l
Fig. D.3.0 Measured and Calculated Absolute Anip11tude Ratio of the Structure in Water i
l l
m A
EiEE Q
EG 2Tl1 i
PrTI ml r'n ESSE M1 I
hl 1
l L+3nJ 1
1 J<
LE L..
LA l;
(..
t t
L. p.
(t; L
Lw,.
10.
,9-8-
7.
s O
lil I
ua 5-4.
4 C=0 3-T C =.062 C *. 28 i.
I i
i
=_
'n Fig. D.4.0 Calculated Absolute Amplitude Ratio of the Structure in Water
Rev. 2 4
E.1 i
-e APPENDIX E as w
IMPACT ANALYSIS
-- a j
WATERFORD SPENT FUEL STORAGE RACKS E.1.0 Introduction In this Appendix the effect of fuel impact on the structural
_.p.
adequacy of the spent fuel racks is investigated.
Each fuel assembly
[.
is simply supported at the bottom edge.
Duria.g seismic excitation 6
the rack will move and impact'the fuel.
As noted from the equations j
presented in Appendix C and discussed in Appendix D, an empty rack
_4 will not have any significant relative motion with respect to the pool walls.
Thus, it will move with the wall and impact the fuel.
Hydrodynamic effects associated with underwater impact are a-o nonlinear.
Gaps between the walls of the rack assembly and spent V4
.,41 fuel elements go to zero.
Linear theory, normally used to calcu-late hydrodynamic masses, predicts an infinite mass as the gap goes to zero.
This nonlinear effect is not considered in this analysis.
{
Also, most spent fuel elements tend to warp slightly and are not l
free to impact the walls in phase with each other.
As a result, i
FJ i;gl impact loads. estimated in this Appendix are felt to be conservative and over-estimated.
(_
t l
Analytical procedures used in this Appendix follow guide-lines published by the NRC.
It is assumed that one-half of the I
weight of the spent fuel elements impact the top of the spent fuel l
rack. assembly.
Inplane flexibility of each spent fuel element and I
-s I
Rev. 2
.j E.2 local flexibility of the rack are not considered.
Kinetic energy of g
this mass is assumed to be totally absorbed as strain energy of the
-- spent fuel rack assembly.
Any loss associated with impact is neg-
-d I
g, =5 lected.
The kinetic energy of the fuel mass is based on the SSE y
pool floor spectrum velocity.
_e E.2.0 Impact Analysis a,"
E.2.1 Empty Spent Fuel Rack Assembly Frequency
=.;
u "*
E.2.1.1 North-South Model j
The structural weight of one-fourth of the assembly,
,4 including-fuel is 49,279 lbs.
(Appendix B.)
The 9
North-South hydrodynamic weight is 11,800 lbs. and the fuel weight for one-fourth of the assembly is 30,000 lbs.
Thus, the natural frequency is approxi-mately f'
61,079 f
=
- Pd n
31,079 D where f
= the fundamental frequency of the rack (14.7 Hz) 3
_s f
= 20.6 Hz n
N I E.2.1.2 East-West Model In this direction the hydrodynamic mass is 12,400
~~
lbs.
- Thus, f'=_/61,679 f n
V 31,679 n
Rev. 2 i
j E.3 where I
f
= 11.
Hz n
~
-3 f
= 16.0 Hz.
P.
kd E.2.2 Spectrum Velocity h*
In this frequency range, the lateral G-load of the SSE is 0.3 g.
Thus, the spectrum velocity can be calculated from the following equation g,
V=b w
t N3 C"
E.2.2.1 North-South Model PQ g;
For the SSE
}
V=
.89 in./sec.
=
(2 4
E.2.2.2 F,ast-West Model i
For the SSE F
V = j, '(
(
}
UM 1.15 in./sec.
=
.0 L
E.2.3 Estimated Stiffness a
I_"J Assuming a one mass system In" l>
)
Thus, the efYective stiffness is (2nf '2 t
i ts n
K=
9 a
l
=
l
Rev. 2 d
E.4 a
E.2.3.1 North-South Model 5
For this case
~
""I (2nl4.7) (4)(61079)
I=d K=
386 K = 5.34 +6 lb./in.
.a E.2.3.2 East-West Model F
he-For this case r
L: *
(2x 11. 5) 3 (4 ) (616 7 9) g, 386 K= 3.337+6 lb./in.
ff4 E.2.4 Dynamic Forces
.N The kinetic energy of the spent fuel elements is 1
2 KE = y m V g
and the strain energy of the racks is b
F 1
d SE = 7 g er,--
ll Thus e
Fd" f
~
In both cases, the total weight o'f the fuel in the 80 module rack M~ '.
is 120,000 lbs.
One-half of the weight is used to calculate m.
~
a-g
=
E.2.4.1 North-South Model In this case
. s F, = /q2o;;;;,-> <5. 34.e) (0. ee)
Fd = 2 5, 600 lbs.
I
$ $t W J G'
5,'h0b2 5??. $$$
i3. % ?-
laQ^H h.* ; V :,'.:_. Q g j_-
Rev. 2 8
E.5
.us E.2.4.2 East-West Model i
e0h In this case i
d" 120,000- ) ( 3,337+6) (1 15 )
F (2) (386)
{
F
= 26,200 lbs.
d E.2.5 Effective G-Load i
-..a l~
By dividing these dynamic forces by the total module weight
,.i an effective G-load can be calculated 4
E.2.5.1 North-South Medel hh
'~
For this case th'e effective G-load for the SSE is e
25,600 (4)(49279) =.13 g "ud E.2.5.2 East-West Model p
..]
For this case the effective G-load for.the SSE is
= 0.13 9' (4
49 9)
FW E.3.0 Conclusion Since the G-load acting on the module assembly is 0.3 for the SSE in the frequency range oE interest, the impact forces f
increase stresses by 43 percent in the North-South direction and I
by 43 percent in the East-West direction.
Because calculated
~
seismic stresses are not significant, these increases caused by 43 impact are not considered to be harmful to the module assembly
(
)
structure.
This same percentage increase can also be applied to
[
the OBE stresses.
l l
I$
F-1 lj APPENDIX F FLUID COUPLING OF A MODULE ARRAY 2
9 F.1.0 Introduction It is the purpose of this study to determine the dynamic response of a row of spent fuel racks subject to seismic time-history motion.
Maximum relative motions between racks will be assessed to determine if impact will occur.
~
Hydrodynamic effects couple the module to each other and to "j
the wall of the pool.
It is assumed that the pool walls and founda-N tion have identical motion.
The ra-:.2 are first assumed to be fixed at the base.
Forces to the base are cilculated.
Sliding is first
> i.d simulated by reduction of the stiffness of the module orders of mag-nitude.
Then an analysis with sliding is conducted.
,-I N C
Computations were made both with the ICES STRUDL program and a special purpose program developed for WAI.
Results of the two y
programs were compared for a nonsliding case in order to verify the special purpose program.
Even though different numerical procedures were used, calculated deflections were almost identical
(= 1 percent g
difference).
The special purpose program was used for the sliding analysis.
In all cases except one the first five seconds of the DBE a
m j
H, East-West Input for the Waterford spent. fuel pool was used as input 1
u j
to the dynamic system.
In the last case the first ten seconds of 4 {r_
the East-West DBE was used.
The dynamic response of the array will not be significantly affected by the particular tim'e-history input.
I In Appendix C, a hydrodynamic mass matrix is developed assum-ing that the fluid,between module moves only in the horizontal plane.
l-Vertical motion is.not considered.
However, for the geometry of this s.,
system, the hydrodynamic mass matrix asscciated with in-plane motion is much larger than that calculeted for vertical fluid motion.
In any physica.1 system the fluid will move in the path of least resis-tance and, thus, the smaller mass matrix associated with vertical l
~
_'l F-2
.a motion must be applied.
The hydrodynamic mass matrix for this type w
of coupling is presented in this Appendix.
g An array of four modules is considered.
The arrangement is M
shown in Figure F.1.
i eu_
dd d
b 7,
am e L
E L
L n
.3 lea
.t NIa t
t i
E
(
.l.
2 1
3L 0
L 3
/
ML 5
1 1
1 4
w1 x
+
S 4
K l
I1 C
a_
A Re R
E 4
G 4
A l
R L
i F
3 M
O x
T s
3 L
.Et I
l E
/ )t U
F
(
z T
L
/
N E
3 P
1 1
S i
2 N5 x
F O
6
=
=
2 Y
m A
W..
s RR T
A J
2 1
l n7 1
i x
M F
k; e
i E s.
r E
y u
m g
C3
(
i F
J, Ti y
L 3
M J
0.-
s Y
/
li J'
s eL
' i i
i,
'k F-4 w
F.2.0 Theory a
d The equations of motion for the module array in Figure F.1
- ~"can be written as follows:
C*'
[m] [{h} +{l)$]
+ [K]{y} = {F}
(1) where
[m] - diagonal mass matrix of structural masses m,
y N
m ' "3, and m4 2
w~
[K] - diagonal stiffness matrix
..,l 2
- absolute pool foundation and wall motion
{x} - absolute motion of the racks p,
{y} - motion of racks relative to pool wall, (y1 = x1 w
- z) f.==,
and the hydrodynamic forces are related to the accelerations as follows:
1
-MHl "H2
+M 0
-0 t
~
H2 "H2 H2 H3 H3
-M 0
~
L I'F} =
t 0
+MH3
~"H3-NH4
+ H4 0
0 MH4
~"H4~ H5 l
- h F
~
f.. 9 y1 1
MH1 +
1 5
5 1
+M (p)+(y)z 2
y x
+
+M E
(2) y
.3 1
I 4
s s
HS + "1 Q
Y, u
Substitution of equation (2) into equation (1) yields
.[ml)1y} + [K) {y) = - [m-M 1 {ld (3)
((M ] +
g 1
l r-l %
where l
3 man
t n
F-5 MHl+NH2+"1
-M 0
0 H2 1
-M NH2+"H3+"2
~N 0
u,2 H3
((M I + I*lI "
H M
"N
!!3+N '4 +"3
~M H3 H
H4
_0 0
-M MH4+NH5+"4.
g4 E
(4) k and L
~
m -M 0
0 0
y y
O m -N 0
0 2
1
[m-M ]
=
y (5) 0 0
m -M 0
3 y
0 0
0 m -M 4
y _j 51
.a F.3.0 Input Preparation For this investigation, the structural characteristice of the spent fuel storage. racks designed for the Waterford Nuclear Power Plant, Unit 3 are used as the basis of the input.
Time-history input in the East-West direction for the DBE is specified.
Results of this investi-
)
gation are typical for all rack assemblies.
L F.3.1 Stiffness Matrix Assuming that the mass of the rack is concentrated at the 1 i
! U.
center of gravity the stiffness of the K=
= 1,82 x 10.lb/in (6) a L
j where 6
E = 27.76 x 10 pgy 4
I = 172846 in L = 92.5_in
i F-6 F.3.2 Hydecay_ r.ic ttasses It is ast.umed that there is a 0.25 in. gap between racks.
'8 Actual talues of the walls are used.
"H " Y 2
].
h = rack height = 185 in.
G - wall gap (see Table) 3 b - rack width = 103.8 in.
3 r
y-fluid density = 62.4 lb/f t MH G
Weight
- 'J (in.)
(lb.)
bi M
37.21 5.315 x 10 4 H1 6
N 0.25 7.911 x 10 H2
~
6-M 0.25 7.911 x 10 H3 6
M 0.25 7.911 x 10 g4 5
M 13.63 1.451 x 10 HS Ri 5
In each case fi is calculated as follows:
j M
= y Volume j
~ M)
(10.38)2(8)(10)(185)
=
4 M; = 5.76 x 10 lbs F.3.3 Structural Mass The total weight of a full 80 module rack including 'the
.a j
enclosed water is 197,100 lbs (Appendix 8).
Since the fuel weighs 120,000, the weight of an empty rack is 77,100 lbs.
I-
F-7 F.4.0 Numerical Results as F.4.0.1 Calcu' lated Displacements g.
~4 Calculated displacements are presented on Figures L
2 to 5.
On Figure 2, the dynamic displacement of the four modules in air is compared to the dynamic displacement of the four modules in fluid.
As expected, the modules vibrate in
[-
phase for both cases.
The fluid reduces the dynamic response 7.a from almost 10 mils to 1.8 mils, u
'l On Figure 3, module 2 is assumed to be fully loaded.
E*len$
All other modules are assumed to be empty.
This arrangement
{
was chosen in order to obtain the maximum relative motion.
It should be noted that the modules become out-of-phase at 4.5 sec.
However, the maximum relative motion is only 0.0045 inches in the 0.250 inch gap.
EW l
In order to increase the amplitude of vibration and PWI to simulate sliding, the magnitude of the stiffness of all a
modules was reduced by a factor of 10.
Results are plotted on Figure 4.
In this case the modules vibrated in phase j
with no significant relative motions.
E On Figure 5, calc'ulated displacements for the case of friction forces between each module and foundation is Ai'o~, for this case module 2 was full and others studied.
s F
empty.
The coefficient of kinetic friction was taken to be 0.09 and the static value was taken to be 0.10.
Calculations d
L
-me e
e 4
r-s a
indicate that all modules move in-phase with no significant
,,j,
relative motion, as indicated on Figure 5.0.
It should be
~~~~~
noted that the time history was conducted for 10 seconds.
W F.4.1 Calculated Foundation Shear Forces P'
~
Shear forces calculated with these array models are listed I
on Table F.1.
The effect of fluid is to decrease these forces by a factor of 5.6 when compared to the in-air solution.
By consider-ing only one rack assembly to be full of spent fuel elements, forces a
are increased by a factor of 1.6 over the case of all racks full.
E='
In all cases base shear forces are well above that obtained for the Obi sliding case.
Thus, rack' stresses are conservatively determined with a fixed base response analysis.
Also, the base shear (SRSS
~
value) compare well with the time-history solution, considering all modules to be full.
_.m 4-la La "J
m1 N
s w&
R!
~_
6
F-9 m
F.5.0 Conclasions d
Based on the results presented in this Appendix and other pc hreliminarystudiesmadeforWAI, it can be concluded that arrays of closely spaced spent fuel racks tend to vibrate in-phase with
~
no significant relative motion.
If the original gap is less than b.
1/4 in., the hy'drodynamic coupling is increased.
Results indicate m
L;,
that racks will not have relative motion large enough to impact 7,
each other.
Base shear forces may be increased by various loading 4
conditions.
Based upon this study, a factor of two increase P. ' '
is an upper bound, assuming th'at the foundations are fixed.
Fixed A
foundation forces are well above those obtained from the more 7,z a;;
realistic sliding model.
wi W5 L
- I l
M t
l l
[.
F 10
.e Y
Table F.1 9
.)
RACK SHEAR FORCES (2) s
Case Maximum Maximum Description (
Deflection Shear Force No.
In.
Lbs.
g _.
I No fluid - Racks in 0.00978 178,000 j.;
Air ("#
II All Racks Full 0.00175 31,900 III Module 2 Full 0.00297 54,100 Others Empty ( }
gg
.: 5 O#
IV Module 2 Full 0.168 30,600 Others Empty K Reduced by Factor
=*
of 100 V
Sliding, p
= 0.09 0.124 13,500 lbs n'
S (1)
Input based on DBE East-West time-history for the Waterford Plant e-l (2)
Maximum SRSS shear force based on East-West DBE spectrum response (Appendix G) analysis,is 26,700 lbs.
(3)
K = 1.82 x.10 lb/in w-
~_
t*
kW I
i m
1
d i; l
l kL
(
t,k 1
I Je L
L s
l 10 3 l
5 No Fluid I
N M
6 1 I
o i
g 4
x i
m r
~ L y
e
=
u y.
./
f f.
o
.j 4/.
,Q/-
L\\
\\
,}
1
.\\
c Time l
0 h
(sec)
Fluid Present o
t-to g
l i-a.
m *
.a O
Figure 2.0 DISPLACEMENT OF FULLY LOADED MODULES IN AIR AND IN WATER
.g i
H H
Q Mile F"4 F84 M
A rTf]
I U C::3 C
C t..
L1 l_.
L_..
til L..
I,._,
K.i L
lu m i
2.
L4 1
L.
L t. k l.
L I.1..
I I
t Full No. 2
- 3
'I
.# ~~
Module I
i t
e.
v,1 o
a i
j I
x m
Module i
- i..
- l.,.
. n o
l.
1
.c
~1
,e I*'
- r
. Td.
- .O
,n
., : \\,
'>. J W: :r 0
A-..,.
- e. g'r'= x.-
n 1
i
.\\
i,_-
- . u ;
y.
. y p w U:
.?
...,~..,,i s-I I,,i:-l ' :l1.1
,.i :
1v J
ki.
Time
. G,,i/3 1(J ii ',
i c
(seet 3
0 4:
\\.,
iIe i ou
/-
e
)
I l'
l\\
6 i l-en
- I s
i s
y a.
I m
.,4 o
Module 3
3 i-2s-Figure 3.0 MODULE 2 FULLY LOADED AND OTilERS EMPTY s
w bJ
[
th t
l L.
L u.
L Lw L
L L
L
'1 uE L-t.
t j, ;- -<-
?g
,. n.-
.4 I
e k.--
t r
e,
-j-O I
~
1 i,
.?
6 x
--.. y i
I.
[
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'a APPENDIX G r'.
COMPARISON OF FINITE ELEf;ENT PLATE MODEL I,*
WITII FINITE ELEMENT BEAM MODEL G.l.0 w
Introduction
?,
If the spent fuel rack assembly is mod l d which include shear deformation basic dyna i with beam elements ee 7
ture can be easily and quickly calcul t d m c forces,in the struc-ae to the detailed results determ'ined f These forces are compared d
based on hybrid plate elements which in l drom the finite element a j
- offects, c u e bending and tensile a
G.2.0 Input Preparation j
In this section, input for a dynamic an l rock based on beam elements is developed a ysis of the spent fuel G.2.1 Section Properties
\\
G.2.1.1
?
Box Properties The outside' dimension of the square box is 10.38" with a 0.090" wall.
Thus -
i
'7,J10.38)f__10.20 4 l
12 12 I = 65.378 in4 A = 3.704 in r
1
Ii I=
l l5 G.2.1.2 G-2 Section Properties W
of 80 NE G.2.1.2.1 Module Box M
North-South Motion For one column -
I=
(10)I CG + AE II }
r (h) + (f)
~+ (h)
- I}
u
]
b W)
EIs _
gp I=
(10)(65.378) m.
+ 2 (3. 704) (10. 38) 2(41 2 LU I = 35,578 in 4 5)
F:
For,el Columns -
IT = 268,626 in L.
4 G.2 1.2.2 East-West Motion For one row -
U$
I=
th (8)ICG + ^
II I
+
~
I=
I}
+ II (8)(65.378)
+II}
+ (2) (3. 704) (10.38)
I = 17,285 in 4 (21)
For ten rows -
l I
IT = 172,846 in 4 G.2.1.2.3 Section Properties E
For this of 64 Module Box t
case the are identical. north south and east we ties st proper-Using the previous If = (8) (17,285) section
\\
= 138,277 in 4 l
- w.,
i G-3 G.2.2 Shear Areas
'o es G.2.2.1 Shear Area of an 80 Module Box u.
f, 3= (h) (80) (3.704)
A Ag = 148 in l
G.2.2.2 Shear Area cf a 64 Module Box
= (h) (64) (3.704)
L A3 Ag = 118 in G.2.3 Effective Weights p -s L.J G.2.3.1 Total Weight - 80 Modules - Based on B.3.1.5 W =. ( 4 ) ( 4 9.,2 79 ) = 197,116 lbs g
G.2.3.2 Total Weight - 64 Modules a
Wg = 157,693 lbs G.2.4 Hydrodynamic Masses - Single 80 Module Box G.2.4.1 East-West Motion L
For the 37.21 inch gap -
W
=Y
= 5.315 x 10 lbs.
4 H1
.3 h = 185" l
f, b = 103.8" C
G.,=,37.21" f_
Y = 62.4 lb/ft For the 13.63" gap W
= 1.451 x 10 lbs H2 1
M l
l
N
~
c_4
.a Thus MH1 + MH2 4
W
=
= 4.956 x 10.lbs gg I'[
G.2.4.2 North-South Motion b
For the 24.3" gap -
4 WH1 = Y
= 6.511 x 10 lbs b = 83.04" C'p h = 185" G = 24.3 y = 62.4 lb/ft For the 12.8" gap -
5 5-*
W
= 1.236 x 10 lbs H2 T3 Since there are four modules in a row 3
' '~
, 1.236 x 100 + 6.511 x 104 g
HNS 4
4 W
4.718 x 10 lbs HNS f*
G.2.5 Underwater Modification Factor The mass M is calculated as follows (Appendix C) y y
Wy = YbhW =
- 8. (8)(10)(10.38)2 (185) 4 W = 5.758 x 10 Rs G.2.5.1 East-West Motion W -W 3
y C
= 0.566
=
g gS,gHEW
..A, G.2.5.2 North-South Motion W -W CNS " WS+WHNS
= 0.571 1A I
r
,a G-5 G.2.6 Base Plate Flexibility j
Based on experience, the flexibility of the base plate can
-- be' estimated assuming the plate is a short beam with shear deforma-l h' tion.
Note that 1 is the half span.
4 A"L L
~
3EI AG g
I I
1=
(5) (10. 38 ) = 51.9 in.
i 7, bh (0.5) (83.04)3 d
p 12 12
'1.$
4 t s I = 23860 in I
6 E = 27.76 x 10 p,1 (h) (0. 5) (83. 04 )
3
= 27.68 in A
=
3 i o l k, (51.90)3 51.9 3,
4 6
6 (3)(27.76)(10 )(23860)
(27.68)(10.67)l0
-7
-s A = 2.46 x 10 in/lb 6
S K = 4.06 x 10 lb/in 1
E-G.3.0 Dynamic Model la"
~
A six mass five beam element dynamic model is used in this study (Figure F.1.).
Each beam element includes both bending and l
shear effects.
Beam properties are calculated in Section G.2.
Virtual mass effects are included in this model as described in Appendix C.
k i
9 ee l
.j l
l 2
.J4 G-6 4
m I
I..N 11
_w, m/2
's-_
Ns s
~"
i N
- 1. 4
.\\.,
r M,,
L' m
l
.t u_
M r
11 g.;
he m
(3 4
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,_.. C 4
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.._ s 1 I
L
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r
~
~~
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-- s
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=
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L.
s.
a 1
Figure G.I.
DYNAMIC MODEL
1 G-7 a
ems G.4.0 Comparison of Results m
~
Program output for this lumped mass is attached to this ud Appendix.
Results include calculated frequencies, element modal
="
forces and momenta and SRSS values.
The. maximum moment occurs at the base of element 1 (Figure F.1.); the maximum shear force occurs in the bane spring K which is element 6 in the program output.
j, G.4.1 Comparison of Frequencies The East-West fundamental frequency calculated from the plate j
finite element model (Appendix B) is 11.5 Hz.
This relatively simple beam model has a fundamental frequency of 10.5 Hz.
G.4.2 Comparison of Overall Bending Stresses 73 The maximum SRSS bending moment calculated with this model is a-hLd g
2.79 x 10 lb-in. The resulting bending stress is
, MC, (2.79)(106 ) (4 ) (10. 38 )
g I
172846 l
c = 670 psi i
p' Stresses in elements 181 and 180 of the plate element model are 6-~
450 psi and 640 psi respectively.
Because of the detail of the
- base support, this comparison is felt to be good.
g, EJ G.4.3 1
R
_ Comparison with Base Shear Forces i
O The base shear force from the beam model is 26700 lbs.
Using elements 222, 223, 226, 227, and 201 the total shear force to the' l
Lj base is 20,300 lbs.
Force in Ele'm'ent 201 = 2137 l;_
Force in Elements 222, 223, 226, 227 = 2 926 lbs Total x 4 = 20,300 lbs l
It is believed that this 6,000 lb. difference comes from the fact that l
shear forces perpendicular to elements 224, 225, 228, and 229 are not included.
lI l
.j G-8
=
G.5.0 Conclusion Based on the comparisons shown above, basic ~ dynamic forces from seismic loading can be determined from a simple beam model.
P5 If detailed stresses in the vicinity of a support is required, a static finite element analysis could be used for engineering pur-poses.
The fundamental frequency is estimated adequately with the
' simple beam model.
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