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| 1 1
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| PGandE Exhibit No. 9 j JOSEPH OAT. CORPORA 71CN
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| ' CMEMICAL ENGoNEERS & casarCATORS NUCLEAR POWER COMPCNENTS '
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| 11'aBv5MO r'88 50-275/323- #4// '87 AUG 26 P3131 d'//7/F7 w DOLht . .
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| "e.
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| s h a s,.
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| ~
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| SEISMIC ANALYSIS OF HIGH DENSITY FUEL RACKS l
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| ^
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| for PACIFIC GAS & ELECTRIC for DIABLO CANYOM NUCLEAR POWER STATION Oat Job J-2473 P.O. #2S-5679-AA4 and ZS-5679-AB4 O
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| 1 l
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| 1 l
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| l by Dr. Alan Soler TM Report #779 8709040200 070617 l PDR ADDCK 05000275 C PDR
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| : u. _
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| - - _ - - _j
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| &~----_..
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| MUCLDR RIGULATORT C0fCissf0lt b4b Docket No. 50-W7.0LA p,g,i hk m.
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| 0 q_494 in the matter of -
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| c._ IDENTIFIID _
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| 5tafL. -
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| 1
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| -- _RECEinD _
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| Appficant .-
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| . RillCilD--
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| \ntentnos __
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| Cont's 01f's __
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| ~DATI b'b'kl --
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| Contiactet __ -
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| -~Witnen _
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| Other _ ---
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| AlN Reporter __
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| W TN i
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| i O l l
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| l O
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| = . _ _ _ ,
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| !1! -i
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| .vry1 m., ....r.
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| ;":n tc;. q p.yi.
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| a ., u ,_
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| SEISMtC ,_AM4L 1%f3 CF_ MH D E N %lT N r t. E L RAC 6 k D0GHENT TITIE:
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| roR , P G -. L c pc R n, Ag u c- c As v.M wucma A wcR s:mTicri
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| : LOCIEENT CLASS: A PIPORT NO. 7 '7 Q ,,,,
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| THIS DOCUMENT CONFOPJ15 TO T!iE REQUIRDINT OF THE DESIGN SPECIFICAT APPLICABLE SECTIONS OF THE GOVERNING CODES.
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| b7
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| . dhid
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| . e ....-. - ~...a.6 b1 . p ,
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| ' KRISH.'!?. F '.'. %b
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| - PROFESSIONAL ENGINEER g.y,' WVa.~.;*-G' fp; [N 7
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| .mei
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| ** O n.4 -
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| APPROVALS (Sienatures & Dates)
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| ORIGINAL PIV 1 REV. 2 RhV. 3 REV. 4 RE / . 5 ISSUE DATE l l 2918.6 -
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| 110)66 7. i g.8 6 9 ''
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| AUTHOR h.'nl0$ '3m . So IG 5 ''be Soiet b . Soles REVIEUER h0 0 ,.\ Y f(& WSU f. C O (Method of f, , g _g, ,,
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| Review) (4) ^
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| CERTIFIER j> hh Mc,6 PROJ C E''G . 2 NOTES: !
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| 11)
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| For all Class 1 documents customer has to respond in writing that all i requirements of his design specification pertaining't .the intent of t:his report has been satisfied.
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| (2) Class A: For review end written approval by customer. B: For submittal to customer, Information only, C: Internal document; not to be submitted to cust orner.
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| This docu...ent is proprietary to Oat. Its confidentiality must be observed .;
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| 13]
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| by the designated recipient.
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| (4) Code as follows: '
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| : 1. Revis.wed method of eno. lysin c.nly
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| : 2. Ecviewed wthod of analysis and nim.erical computations -
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| : 3. Veri fled by r.!t t ruate calculation rnethod.
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| 1 l
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| l O .
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| 1 List of Revisions for Revision 1 of Seismic Report f
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| O Revised page 5 I-7.3, I-7.4, I-7.7 O Revised Figures 1.4, 1.5, 1.6 O i i
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| 1 l
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| l LO
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| July -14, 1986 List of Revisions for Revision 2 of Seismic Report l
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| l 0 Added:
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| - page I-7.10a
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| - footnote on page A2-1
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| - page A2-9a
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| - II-42 revised; added II-42a O Section 6.2 completely re-written (Section 6.2 now consists of pages 11-55, 56, 57, 57a, 58, 59, 60, 61, 62a, 62b 0 Revised pages II-66, 67, 69, and 71 0 Added pages II-71a and II-71b 0 Added page 11-75, new section 6.8 0 Modified 2nd page of Appendix I O Added H rack preliminary calcul'ations to Appendix I (handwritten) 0 Added a new appendix, Appendix I-A, with supplementary
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| ~
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| results for archive purposes l
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| 0 Added computer outputs for H rack
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| )
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| 1
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| 'l O -
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| List of Revisions for Revision 3 of Seismic Report O , Revised' Appendix III O Added' Appendix IV ,
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| o Updated Drawing Reference List O Added and/or revised the following pages:
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| II-5 II-17 II-18 II-29a II-34 O II-35 II-39 II-40 II-41 II-61 II-62b II-62c II-62d II-63 II-63a ,
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| II-65 II-67 II-68 II-70 II-71 II-71b II-73 t
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| d O 1 1
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| 1
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| j
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| .O September 12, 1986 ERRATA CORRECTIONS Page Number Correction Item Number I-7.3 Reference to Eq. B-4 deleted 1
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| I-7.5 Reference to Eq. 8-1 deleted 2
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| 3 Fig. 3.1 K is now KR II-30 equation re-written 4
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| 5 II-29a Units in Table 3.1a: "(f in)/ rad" is now "(# in/ rad)"
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| 11-34 wording clarified 6
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| II-35 "i can be either x or y" has 7
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| been added.
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| t 8 11-16 Page is now provided.
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| 9 II-62a & 63 Numbers are now reconciled.
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| Table 6.4 value is 2.905" II-62 7.06x10 is now 7.06x10";
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| 9a 1.33x10 is now 1.33x10" 10 II-63 Run ac33a is now ac33aa 11-64 L/p is now L/A 11 !
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| I 12 11-65 Q(P(NC) is now Q = PxNC !
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| I 13 II-42a Appendix I-A is now IV Table of Contents word " typical" has been added in title l Part II " Typical Outputs from Dyanamics Runs i I (Uncorrected)
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| Drawing Reference Incorporated newest revision numbers.
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| List:
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| placed in order in our Also, the following pages have been O' original: 11-15, 11-28, 11-29, II-65, II-66
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| DRAWING REFERENCES 8
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| D-7704 Rev. 3 D-7705 Rev. 6 D-7706 Rev. 5 0-7708 Rev. 5 D-7711. Rev. 3 D-7712 Rev. 3 0-7776 Rev. 2 D-8033 Rev. 1 E-7702 Rev. 3 E-7703 Rev. 3 l
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| )
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| i l
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| * This report must be revised if a revision to any of the above references affects the calculations' presented herein, i
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| O l 1
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| i
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| .l l
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| A o
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| O .
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| TABLE OF CONTENTS PART I i
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| CONTENTS PAGE SECTION I
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| I-1.1 1 Introduction to Stress Calculations Axial Stress I-2.1 2
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| Torsional Stresses' I-3.1 3
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| 4 Lateral Shear Stress I-4.1 5 Shear Deformation (Timoshenko Shear I-5.1 Correction) 6 Combined Stresses and Corner O.'
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| Displacements I-6.1 I-6.2 A. Within Fuel Rack B. Stresses Within Supporting Legs I-6.5 7 Lifting and Impact Safety I-7.1 APPENDIX 1: TORSIONAL STRESS IN RECTANGULAR SECTIONS A1-1 l
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| ' I APPENDIX 2: SECTIONAL PROPERTIES OF RACK -
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| SAMPLE CALCULATION A2-1 i A. Area A2-1 B.- Moment of Inertia (for distributed area): A2-1 l C. Moment of Inertia for Lumped Area Model A2-4 j D. First Static Moments A2-8 REFERENCES
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| v .
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| TABLE OF CONTENTS i
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| PART II SECTION CONTENTS PAGE 1 Introduction II-l 1.1 Analysis Outline II-l 2 Fuel Rack - Fuel Assembly Model II-3 2.1 Outline of Model II-4 2.2 Model Description (8 DOF Model) II-6 2.3 Development of Model Equations II-8 of Motion
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| (/ 2.4 Coupling Between Fuel Rack and II-16 Fuel Rod Group 2.5 Coupling Between Fuel Rack being II-17 Modelled and Adjacent Structures 2.6 Fluid Mass Effects in Vertical 11-20 Direction 2.7 Calculation of Rack Inertia II-21 Properties 2.8 Force and Moment Resultants II-24 at Rack Base 2.9 Damping II-25
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| ^
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| 3 Detailed Remarks on Non-Linear II-30 Spring Rates and Coupling Coefficients 3.1.1 Impact Elements between Rack II-30 and Rattling Fuel Assemblies O
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| 1 L-__-_______-___-____________-___________-________________-_
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| 4 P\
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| V TABLE OF CONTENTS PART II CONTENTS PAGE SECTION Fuel Rack Base Support II-36 3.1.2 4
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| Elements at z =-h to Simulate II-39 3.1.3 Sliding of Supports Initial Deformation of Rack II-40 3.2 3.3 A Remark on Computation of Input II-41 ,
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| Data 3.4 Effective Rattling Mass for II '
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| Dynamic Analysis Fluid Sloshing Within an Empty II-43 O 3.5 Rack 4 Time Integration of the Equation- II-46 of Motion 4.1 Time History Analysis Using II-46 8 DOF Rack Model 5 Design criteria II-47 5.1 Structural Acceptance Criteria II-47' 5.2 Material Properties II-49 5.3 Stress Limits for Varicus Conditions II-50 5.3.1 . Normal and Upset Conditions II-50 (Level A or Level B) 5.3.2 Level D Service Limits II-52 6 Results II-53 6.1 Remarks II-53 O
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| TABLE OF CONTENTS PART II CONTENTS PAGE SECTION Support Leg Properties II-55 6.2 Tabular.Results for Hosgri II-57 6.3 Earthquake (HE) and for Design Earthquake (DE)
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| Impact Loadings II-64 6.4 Impact Loading Between Fuel II-64 6.4.1 Assembly and Cell Wall 6.4.2 Impact Loading Between Adjacent II-66 O Racks .
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| -II-70 6.5 Weld Smresses 6.6 Local Buckling of Fuel Cell Walls II-72 6.7 Analysis of Welded Joints in Rack II-74 due to Thermal Load 6.8 Definition of Terms Used in Part -II II-75 of the Seismic Report REFERENCES APPENDIX I: PRELIMINARY COMPUTATIONS FOR DYNAMICS DATA FILES
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| -PG&E Preliminary Calculations for j PREDYNA j i
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| Check for Inertia Properties in FR1 ;
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| Input Data for DYNAHIS l
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| \
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| (:) 1 O
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| u i-TABLE OF CONTENTS PART II j
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| ,SECTION CONTENTS 1
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| APPENDIX II: TYPICAL OUTPUTS FROM DYNAMICS RUNS (UNCORRECTED) l 10x11 df=dpga. acorn 10 cof=.8 using v1.2.3 imp springs corner 10x11 df=dpge.aa003, hosgri, cof=.8 empty rack 10x11 df=dpge.aa004, dmp=.001 cof=.2 empty rack l( 6xil df=dpge., ee01, cof=.8 hosgri, full rack 10x11 df=dpge., aa001, hongri cof=.8, ext.sp. dmp=.001 10xil df=dpge., aa002, dmp=.001 cof=.2, hosgri, extspr.
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| l 6x11 df=dpge., aa04, hosgri cof=.8 empty rack 6x11 df=dpge.aa05, hosgri I cof=.8, half rack, pos y 6x11 df=dpge.ee04, hosgri -
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| l cof=.8, empty rack l '6x11 df=dpge.ee05, hosgri dol =.8, half rack, pos y 10xil df=dpge.acl3aal, cof=.8, acornll, f.a. fluid damp.,
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| wall sp 10xil df=dpge.ac33b, cof=.8 Os corner 110 f.a. fluid damp.,obe
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| O -
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| TABLE OF CONTENTS PART II SECTION CONTENTS APPENDIX II: OUTPUTS FROM DYNAMICS RUNS (UNCORRECTED) 10x11 df=dpge.ac33aa cofes.8, corner 11 f.a. fluid damp., obe 10x11 df=dpga.acl3ab, cot!= 2 l corner 11 f.a. fluid damp.
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| 10x11 df=dpge.ac33ab, cof=.2 corner 11, f.a. fluid damping, obe
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| [~\ APPENDIX III: ANALYSIS OF THERMAL STRESSES IN A RACK CELL APPENDIX IV: SUPPLEMENTARY STUDIES
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| : a. Comparisons for Increase in Rack Flexibility - 32 DOF Model to Check Effect of a More Flexible Rack i
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| : b. Comparison of Maximum Displacements, Loads, etc., as Computed from 32 DOF Analysis ands 8 DOF Analysis
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| : c. Addititional Covergence Analyses
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| : d. Tipping Stability of Rack-E (6xil) when 50%
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| Loaded with Fuel (Runs e06, e07 3 /17 /86 outputs)
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| : e. Factor of Safety Against Axial Shear of a Fuel Storage Cell f
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| k i
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| .on e
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| 1
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| . I. GEOMETRIC PROPERTIES AND STATIC ANALYSIS l
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| i l
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| l i
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| l l
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| 1 l
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| i l
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| 1 l
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| l i.
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| I l
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| l 1
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| j l
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| l 1 l
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| 1 i
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| i i
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| I l
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| q NJ ,
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| : 1. Introduction to stress Calculations I The fuel rack is modelled as shown in Fig. 1-1 where the indicated, and the following orientation of the x, y, z axes are dimensions defined.
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| L = height of rack f a = width in x-direction b = width in y-direction c = cell width (distance between cell walls) i t = thickness of cell walls N
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| x = a/c = number of cells in x= direction f
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| N = b/c = number of celis in y-direction y
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| (u,v,w) = displacements in (x,y,z) direction
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| ,F7).=
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| forces on an arbitrary cross-section (Fx,F
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| = m ments on an arbitrary cross-section f (Mx'My'Mz) i stress resultants is the The sign convention used for illustrated in Fig. 1-2 which shows the positive directions of A'B'C' of the f the forces and moments acting on the upper , f ace )
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| beam element (ABC, A'B'C') shown in Fig. 1-2.
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| l l The stress resultants will be found from a dynamic analysis in the rack and used as explained below to calculate the streses and supporting legs. The engineering theory of bending (Section j B
| |
| eg and the
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| : 2) will be used to find the axial bending stress direct stress of'. The shear stresses T T, TT will be found from !
| |
| the St.
| |
| Venant theory of torsion for multicelled thin-walled f
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| structures (Section 3). The primary lateral shear stresses J
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| L L are given by the conventional bending theory for thin-T t x
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| y I
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| i
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| .l V
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| I-1.1 l
| |
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| .. l I
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| D-
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| ~
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| walled sections (section 4).
| |
| The deformation associated with lateral shear (Timoshenko's shear deformation correction) are derived in Section 5. Based. on these .results, the equivalent -
| |
| stress for yielding is calculated in Section 6 ~. Stresses due'to impac t. from a falling object, and those 'due to lifting of the rack, are found in Section 7.
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| 1 9
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| i O
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| O I-1.2
| |
| | |
| l j
| |
| l 1
| |
| i z(w>
| |
| CELL $
| |
| WALLS
| |
| $/(', ,' ' ' ,(' /(',('l /-c t a /, /, .,, , /, / /
| |
| /) /! /l / 8 fi /! / ~~
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| i f /V/1 /\ /1 /t /\ /
| |
| fo /s ll_/t /t /\ /
| |
| 4 =,/ t j ,c/~~ b = N y c j
| |
| _ a= N x c _
| |
| c
| |
| ~
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| OU L
| |
| A B' /y(v)
| |
| RIGID PLATE
| |
| ^ /
| |
| B
| |
| / ir -y (b.=.-J ' :m.x(u)
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| ?
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| 5
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| -A' !
| |
| d ,.= ,= f l l (/
| |
| f 1 l
| |
| / .9f
| |
| / f ,
| |
| SUPPORTS l
| |
| 4
| |
| , O Fl G. l.1 i l
| |
| 1-1.3 l
| |
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| , e e g 4 .. g 4 g 9
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| l
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| . B -
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| l .
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| j i
| |
| . . l F I G,1.2 SIGN C N V E N.'TI O N l 8 0 l % . .
| |
| l e .
| |
| . l
| |
| . I-1.4
| |
| | |
| I 1
| |
| l l
| |
| i O
| |
| * Axial Stress I
| |
| I 2.
| |
| If A = cross-sectional area '
| |
| I x
| |
| = Moment of inertia of cross-section'about x-axis I
| |
| = Moment of inertia of cross-section about y-axis '
| |
| y The axial stress will be M xy Mx Fg
| |
| + -
| |
| y (2-1) o* = A I I x y.
| |
| where the sign convention for stress resultants'is shown in Fig. 1-2.
| |
| It is shown in Appendix 2 that the cross-sectional l properties may be expressed in the form A=F g ct (2-2) /
| |
| =F yx ct 3 (2-3)
| |
| I x
| |
| l I =F 7y ct 3 (2-4) y l following functions of-where F I x, Fry are the FAT in the x and ' y directions, Ng and N y (the number of cells -
| |
| l respectively): I
| |
| = 2NxN +N x +N y (2-5)' !
| |
| F 3
| |
| O j i
| |
| I-2.1 l' '
| |
| 1
| |
| | |
| 1 l
| |
| i i
| |
| ( .
| |
| NN 3 (N + 1) (N + 2) N x
| |
| +1
| |
| =
| |
| xy Y Y + ) (2-6)
| |
| F [ 2 73 2 N t;x y
| |
| l N +1 1
| |
| =
| |
| N Y N*3 (N* + 1)(N* + 2) + Y
| |
| ] (2-7) l F [ 2 g IY 12 N x y .
| |
| j
| |
| 'The corresponding values of A, Ix, Iy are also used in the dynamic analysis which produced the numerical values of the fi 1
| |
| stress resultants (Fx,...Mz).
| |
| l 4
| |
| O i
| |
| l i
| |
| T I-2.2 i
| |
| | |
| i4
| |
| .W
| |
| (
| |
| l
| |
| ^
| |
| /\
| |
| l U l 3. Torsional Stresses ;
| |
| l The torsional analysis is based on the classical analysis of St. Venant, described by Timoshenko and Goodier (1951, pp.
| |
| 258-3151 The governing equations of this theory are:
| |
| V 2 9,B 2,2 + 3 2*
| |
| 2
| |
| -2GB ( 3- 1 )
| |
| ax 3y (3-2) t = constant on Boundaries dy) = 2GOA c (3-3)
| |
| / (Tx dx + T
| |
| = 8* , T =.f8 (3-4)
| |
| T* Y ax ay I
| |
| Mt =
| |
| 2 ff 4 dx.dy (3-5) )
| |
| ^
| |
| O l I
| |
| where I i
| |
| $(x,y): a stress function to be determined G: shear modulus f 0: angle of twist par unit length !
| |
| fc :
| |
| integral around any closed curve C, in the cross-section, which does not cross any boundaries area enclosed by curve C l Ac: \
| |
| shear stress components along x or y axes Tx'Ty:
| |
| Mt twisting moment about z axis We now apply these equations to the rack cross-section.
| |
| Fig. 3-1 shows a typical cell of the rack cross-section in which the square ABCD of side length c represents empty space b'i v
| |
| I-3.1 k--.----___________
| |
| | |
| c ;-
| |
| We denote the surrounded by solid ligaments of thickness t.
| |
| l l
| |
| center of this typical cell by the coordinates.
| |
| xg = ic,:
| |
| l, Y3 '' j c and the constant yalue of the stress function $ along the boundary ABCO is,derIoted by
| |
| ' (3-6) f 1
| |
| $I og,3 -
| |
| 1 Applying the memorane analogy (Timoshenko & Goodier, 1951, pp. 299-302], the stress function $ varies approximately linearly across each ligament so that Eqs..(3-4) may be expressed in the form o -
| |
| o j (3-6a) i'$+i i'$ along DC v; L 33 =
| |
| ay t
| |
| =t a =
| |
| 1,3 i,j-1 along AB (3-6b)
| |
| T x ay t ,
| |
| a
| |
| =t a -
| |
| I 'i , j ~ #i-1,j) along AD (3-6c) 1 T
| |
| Y 3x t
| |
| ,\., p ,e_ (# i+1,3~ i,3 along BC (3-6d) 7 Y 3x t i
| |
| When Eqs. (3-6a-d) are substituted in Eq. (3-3) we find i
| |
| C dy) = _ (4 o g3 - (og,)_y+ og g , ) + - og ,), y
| |
| [ABCDIT x dx + t y
| |
| + 2 (3-7)
| |
| ,. og,y,3)] = 2Gec O :
| |
| p I-3.2 l
| |
| | |
| I From Eqs. (4) and (5) of Timoshenko and Goodier (1951, p.
| |
| that Eq. (3-7) is the finite difference 462] it follows representation of the equation V
| |
| 2
| |
| =-2 ( 1 G ) 9 = -2G,0 (3-8) c
| |
| \
| |
| where the " equivalent shear modulus" G is defined by G
| |
| e E1G (3-9) c' A comparison of Eqs. (3-8) and (3-1) shows that the function a satifies the same differential equation as does the function 4 provided that G is replaced by Ge.
| |
| Furthermore, the boundary condition
| |
| $ = constant on boundaries (3-10) is also satisfied.
| |
| In short, we see that for a fine mesh (Nx and Ny >> 1), the stress function for the gridwork of Fig. 3-1 is found by solving the classical St. Venant torsion problem (Eqs. 3-8 and 3-10) for a ho.nogeneous solid cross-section whose shear modulus is given by Eq. (3-9) as Ge = (t/c) G.
| |
| For a solid rectangular cross-section with modulus Ge the twisting moment'ie given [T & G, p. 2781* by M
| |
| * 3 1 G e a be = k t G (t/c) a be 3
| |
| (3-11) t Henceforth, "T & G" = Timoshenko and Goodier, 1951. Note
| |
| - that T& G represent the side lengths (a and b) by (2a) and (2b).
| |
| 1-3.3
| |
| -_--__--_ - - ~
| |
| | |
| i l
| |
| where k i 'i s , a tabulated function ;1 the rati'c b/a, given below (in Table 3-1). The torsional rigidity of the " grid" structure is, therefore M
| |
| 1 Ga b (t/c) (3 ,
| |
| (Kf) grid'A 1
| |
| 1 This is seen to differ from the torsional rigidity of a solid rectangle with modulus G by the factor (t/c); i.e.
| |
| l (K tI grid = (t/c) (k t I solid (3-13) l T c>. find the stresses in the grid we note from Eq. (3-6b) that l aleng ligament AB:
| |
| a 1,3 ai , 3_~ 1 ai ,3.- air 2 I
| |
| )c ,cU (3-14) y . .
| |
| ( , _
| |
| t c t t ax i l
| |
| (
| |
| That is c (3-15a)
| |
| ( 'x ) grid ={(7x) solid similarly (3-15b)
| |
| (t ) grid " ITyI solid where (t) solid is the stress calculated for a given distribution a ( x ,y;) in a solid cross-section. But, from Eq.
| |
| (3-5), we cee that a given distribution of 9, gives a unique twisting mcment 1
| |
| de will refer to the network of orthogonal plates
| |
| , constituting the rack cross-t.<3ction as the " grid" structure. J
| |
| \
| |
| 1-3.4 i 1
| |
| | |
| I O Mt = 2. Jf t'dx dy a 2 [et,) c 2
| |
| (3-16) l Therefore, for a given moment Mt, the stresses in the grid structures are increased over those in the solid structure by a factor of (c/t).
| |
| The peak stress in a solid cross-section occurs at the mid points of the longest side, i.e.
| |
| ccurs at points P and P' in Fig . 3-2 (tx) max (ty) ,, x occurs at points O and O' in Fig. 3-2 These stresses for th'e solid cross-section can be expressed in the form M
| |
| = (3-17a)
| |
| (Tx) max 2
| |
| 2P a3 1
| |
| = (3-17b)
| |
| (ty) max k ab2 20 where the coefficient k20 ic tabulated by T & G [p. 277] for different values of b/a (what we call k20, T & G call k 2) .
| |
| The coefficient kp 2 does not seem to have been published before, so it has been calculated here. As shown in Appendix 2, this coef ficienti is given by the infinite series k = Tanh a- Tanh 3 a + Tanh 5a 2P 1.- Tanh 7 a + ... (3-18) I O
| |
| I-3.5
| |
| | |
| O where
| |
| =2$ (3-19')
| |
| a 2 a This is a rapidly codverging series from which the values of
| |
| ~k2P shown in Table 3-1 have been calculated. Also shown in tabulated by T and G Table 3-1 are the values of k i and k20 (p. 277].
| |
| l TABLE 3-1 k k k 20 2P b/a 1 1.0 0.1406 0.208 0.208 1.2 0.166 0.219 0.235 1.5 0.196 0.231 0.269 2.0 0.229 9.246 0.3097 O 2.5 0.249 0.263 0.258 0.267 0.3357 0.3543 3.0 4.0 0.281 0.283 0.3785 5.0 0.291 0.291 0.3919 10.0 0.312 0.312 0.4202
| |
| = 0.333 0.333 0.4489 Multiplying the stresses in the solid section by the factor c/t, we see from Eq. (3-17) that, for the grid cross-section, the peak stresses are:
| |
| M
| |
| * (at point P) (3-20a)
| |
| (Tx) max t k 2P ab 2
| |
| i
| |
| ) (at point 0) (3-20b)
| |
| (Ty) max " ( t k 2 ab 20 l
| |
| O where the coefficients (ki, k2P, k20) are given in Table l 3-1.
| |
| I-3.6
| |
| | |
| n . . .
| |
| 3 1 .
| |
| ^ .
| |
| n n, ,
| |
| { --+ + ,#-
| |
| J. . .
| |
| : g. .
| |
| 1 -
| |
| ' . 1
| |
| . \
| |
| \
| |
| g.I,j. t l. .
| |
| e\ .
| |
| ( .
| |
| O c - -
| |
| a g
| |
| g..
| |
| g.l-l .il g.3+l,J
| |
| : t. -
| |
| Il s c
| |
| _1 .
| |
| l A B b s t -
| |
| e
| |
| .t[. ..
| |
| , 4
| |
| . e e
| |
| 5 L.' 1 i '
| |
| <*> . m *.
| |
| * - -- C --
| |
| )
| |
| . i l
| |
| FIG. 31 .
| |
| . 9 t
| |
| e G
| |
| e 9
| |
| 1 .
| |
| e e
| |
| _________.___.__________.__._________.____n O. *
| |
| | |
| /
| |
| K.: 2 A
| |
| 0 K w
| |
| T,
| |
| /
| |
| K.
| |
| w <._
| |
| jw 2 L v d,/M I k j FvvvJg .
| |
| 2, l
| |
| Gap Elements To Simulete Inter-Rack Impacts (4 f or 2-D Motion ' Rack Centroid 16 for 3-D Motion) (Assumed At H/2)
| |
| O H
| |
| G V
| |
| [ ' Rigid fiack
| |
| -Bacepiato K K I
| |
| j e j,.w( g ._-- f.. ,
| |
| Vg Kg ,
| |
| j K 1 I i
| |
| s y' v u!/ v - . .,_r. . n , ; 3, - .. . - . '. -
| |
| _> K -
| |
| K, ',' K ,, l' 7 R h brr :
| |
| FIGURE 3.1 Spring M n; Rimulr >!on Foi Twa-Dim:oi::!aru MuHc a
| |
| .]
| |
| l l
| |
| | |
| 1 .
| |
| Y
| |
| .y . 1
| |
| : p. . . >
| |
| 3 j,. # a i i i i i I L
| |
| < ::, c Q Q -
| |
| x '
| |
| , .o x b ' '
| |
| b'
| |
| * 1 11 1 . . ,e l y .
| |
| y: '' .
| |
| . _ P, . : p
| |
| . , %C _
| |
| n 0 -
| |
| ~ ,, .
| |
| .- :SO LI D . GRID 4
| |
| F G, 3.2
| |
| ~
| |
| e g
| |
| l j
| |
| 1 e
| |
| 3
| |
| . . s
| |
| . 's p - -
| |
| a .
| |
| ( .
| |
| i e e 1-3.8 .
| |
| b o
| |
| : 4. Lateral Shear Stress _
| |
| From an analysis of I-beams, channels and other thin-walled beam sections, it is readily shown [see Timoshenko 1955, pp. 123 and 238] that the lateral forces on such beams are carried j primarily by approximately uniform distributions of shear stress In addition, in the thin webs parallel to the lateral loads.
| |
| since the bending moments are resisted mainly by axial stresses in the , flanges, it has become common practice [see Megson, 1974, Chap. 1 or Kuhn 1956, Chap. 1) to model thin-walled beams as The an assemblage of stringers interconnected by shear panels.
| |
| stringers carry the axial stresses on the cross-section and the stresses which are uniformly distributed panels carry shear between stringers.
| |
| For the case of the rectangular gridwork of the fuel rack Fig. 4-la, the stringers are cross-section, illustrated in O concentrated at the node points and the shear panels are the plates connecting the nodes.
| |
| The area of one stringer includes the area of all attached Thus, f plates within one half cell width of the node in question.
| |
| there are three distinct types of stringer areas which crise, designated by l
| |
| (4-la) !
| |
| A, = 2 c t !
| |
| (4-lb) l A = ct T 2 i
| |
| =
| |
| (4-lc)
| |
| Ag = ct 1 1
| |
| form of +
| |
| Accordingly, the tnoutary area takes the (interior nodes), T (edge nodes), or L (corner nodes) .
| |
| I-4.1
| |
| | |
| To find the ' primary shear. -stress Ty due to the force Fy acting along , the y axis, we ' consider . an isolated' " vertical"'
| |
| parallel to y axis) strip of ' stringers and use. the (i.e.
| |
| well-known formula-[Timoshenko, 1955, p. 238).
| |
| F q=tt = S (4-2).
| |
| x x
| |
| where b/2 S
| |
| g j ydA (4-3)
| |
| Y is the first static moment (about the x-axis) of all the axially stressed area above the point where Ty is being evaluated. In this expression dA includes only the areas of the stringers above the current y coordinate.
| |
| In . applying Eq . . (4-2). we must deal with two types of vertical strips (Fig. 4-lb). Strips within the interi~or will-be called Type "A" strips, and those on the edges of the cross-section will be called "B" strips, With the nodes numbered k=0, 1, 2... along any vertical strip as shown in Fig . 4-lb, the static moment Sx for that part of strip just below the level k is given (see Appendix 2) by k < y - b/2 < k + 1 (4-4a)
| |
| S (y) =S kA A(k) for type A strips S (y) =S kB
| |
| = t K (k) for type B strips -(4 4b)
| |
| B where Kg(k) = $ N' + k (n' - k.- 1) (4-Sa)
| |
| \
| |
| O I-4.2 1
| |
| ---_-_--____-____A
| |
| | |
| ' +3 k (N Y k- 1) (4-5b)
| |
| =N1 -
| |
| K0(k) 2 Y 4 for the value k =
| |
| The maximum value of these coefficients occur M, just above the x-axis, i.e.:
| |
| (4-6a)
| |
| K gg E max Kg(M = KA(M)
| |
| (4-6b)
| |
| K BM E max KB(k) =KB(M)
| |
| - 1)/2; i.e.:
| |
| where M is the integer part of 'the term (Ny Y (4-7)
| |
| M E INT ( )
| |
| 2 f
| |
| It is also shown in Appendix 2 that for the lumped area model of Fig. 4-1, the moment of inertia about the x-axis is given by ct3 (4-8)
| |
| I =F X 7x where (4-9a)
| |
| F Ix
| |
| = (N x
| |
| : 1) [34 N2+E]+N2y y+ 3.2 E N -1 (N Y
| |
| : 3) (N y + 1)
| |
| (4-9b)
| |
| K = Y
| |
| [l+ ] ;
| |
| 2 3 i
| |
| I Using the above notation, the primary shear stress distribution given by Eq. (4-2) can be expressed in'the form
| |
| . ct2 F y
| |
| K A(k) q^ = t T A,p KA(k) , (4-10a)
| |
| Y Y F 7x ct3 c F l 7x F KI I qB= t T B=l B (4-106)
| |
| Y c F 7x O
| |
| I-4.3
| |
| | |
| where thp superscripts (A, B) denote A or B strips. For a sufficiently large number of cells, the distribution given by .
| |
| Eq. ( 4-10) is practically a parabolic - distribution as predicted by the elementary theory for a uniform vertical strip. To ' 'see this we have plotted in Fig. 4-2 the quantities A K( A( '
| |
| =
| |
| A = (4-11a)
| |
| (TA) max Kg(M) Kg l
| |
| l T
| |
| B K( B B
| |
| = = (4-llb)
| |
| (TB) max K B BM along the depth of the beam for the case of 13 cells along the vertical axis (Ny= 13). For the sake of comparison we have also plotted the parabola l
| |
| O T
| |
| =1- ( Y
| |
| )
| |
| 2
| |
| =1- (1 b )
| |
| T max b/2 N y
| |
| predicted by classical theory for a strip of uniform thickness.
| |
| It can be seen in Fig . 4-2 that for Ny= 13, the present lumped area model predicts a piecewise constant distribution of lateral shear, which represents an almost perfect parabolic distribution. Note too, that except for the outermost cell, the j form of the shear distribution is virtually identical for both A )
| |
| and B strips (interior and edge' strips).
| |
| The primary shear Tx due to the vertical force Fy is i l
| |
| assumed to be zero everywhere, i
| |
| By symmetry, the primary shear Tx due to a horizontal l
| |
| lateral force Fx are found from the above formulae merely by interchanging the symbols x and y wherever they occur.
| |
| 1 I-4.4
| |
| | |
| 1 O From ,Eq. (4-10), the peak lateral shear stresses at the x axis are given by F K y Kg(M) = _y AM (4-12a)
| |
| (r A y
| |
| ),,x = F ct F ct F Ix 7x F F K B y KB(M) =
| |
| y BM (4-12b)
| |
| =
| |
| (Ty ) max ct- F 7x ct F yx For a uniform rectangular beam of width t and depth b the max I shear stress predicted by Eq. (4-2) is the classical result.
| |
| I F F
| |
| ( tb/2) (b/4 ) 3 y (4-13) v =1 yS = _y = - -
| |
| 3 max t I x
| |
| t tb /12 2 bt That is ax , ' max = 1 (for a solid section) (4-14) 2
| |
| *AV F /A where A is cross-sectional area.
| |
| For the case of a gridwork, the corresponding ratio predicted by Eq. (4-12) is F K gg K F I
| |
| max , y g , AM A (4-15)
| |
| -c t F yx F F gx r
| |
| gy y 1
| |
| where, by Eqs. (1-2) and (2-5),
| |
| = N +N x +N y) ct (2N xy (4-16)
| |
| A= F get i
| |
| many cells (Nx +=, Ny+=) it follows >
| |
| When the gridwork has from Eqs. (4-16), (4-9), (4-5) and (4-6) that I-4.5
| |
| _ _ _ _ _ - . _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ . _ _ _ _ _ _ _ _ _ _ _ _ _ _ . _ _ _ _ _ _ _ _ A
| |
| | |
| T (Ny/2) 2(2N Nx y) +3 (4-17) max ,
| |
| T AV (NxN /6) grid i Comparing Eqs. (4-17) and (4-14) we see that for the l
| |
| cross-section the max shear stress approaches 3 times the~ average applied shear stress rather than 3.r -
| |
| g as for the solid uniform 2
| |
| cross-section. The reason is clear; only half the area of the Therefore, grid cross-section is aligned with the vertical load.
| |
| the effective shear carrying area is half the cross-section of for a the grid, and the corresponding shear stress is twice that solid cross-section of the same area.
| |
| j Numerical Example For the case of N x =N = 13, we find *
| |
| = 5551 ;
| |
| Kg = 45.75, F = 364, F yx A
| |
| 1 Therefore, Eq. (4-15) gives l ,
| |
| F AM A , (45.75)(364) ,AV
| |
| ,3h (4-18)
| |
| ( ,Ay ) max y AV 5551 A Ix
| |
| ~
| |
| ""'""""" '"" '" """" " '' ' " " ' " "'' ''~''''
| |
| O ;
| |
| l i
| |
| I-4.6 l
| |
| l
| |
| \
| |
| | |
| B P o1 -
| |
| m .} 0 EI R = = - a P T k k k Y
| |
| T S g [
| |
| )
| |
| t AP - S
| |
| * ) S EI _ _
| |
| PR c b E YT _ _ _ (
| |
| N TS
| |
| - K C
| |
| I H
| |
| T 1
| |
| E
| |
| - V 4 A H
| |
| O E R S L
| |
| U L G
| |
| I A
| |
| x F W L
| |
| = L L
| |
| A E F
| |
| cT -
| |
| t C
| |
| L L
| |
| r F _
| |
| _ _ M T A
| |
| : _ A (
| |
| c T
| |
| l
| |
| =_ f '
| |
| c x h H -
| |
| _ L_. 9
| |
| )
| |
| a N _ _ _
| |
| (
| |
| l ' I
| |
| = _.
| |
| a Tr }
| |
| _ t
| |
| ~ F p"'L l pl pl A
| |
| O ~
| |
| t e
| |
| " .a i !
| |
| | |
| Y
| |
| .f c
| |
| O es -
| |
| x= 0 .
| |
| 2-4 es
| |
| ' 6- '
| |
| K=1
| |
| ' \ ' MAX II I MAX l
| |
| 5- \N '
| |
| O PARABOLA' )
| |
| K=2 ..
| |
| 'N 4
| |
| l ,
| |
| \
| |
| i .
| |
| K=3 -
| |
| l
| |
| .3 - -
| |
| .h .
| |
| i K=4 - -
| |
| 2- .
| |
| k=5' I- .- .
| |
| * [
| |
| : .. K= 6 ..
| |
| . .n :
| |
| : 0. .
| |
| ,t~
| |
| . .. g. . .
| |
| . .O .2 4 6 . 8- ,
| |
| Z~ MAX
| |
| -l.
| |
| 2 ,. ._: .
| |
| . . . o . .,,,-
| |
| O. -
| |
| 4 . 1
| |
| -2* - * ,.
| |
| ( -
| |
| eam *
| |
| -3 -
| |
| 4 l
| |
| -4 i
| |
| ~5 f y -
| |
| .6 - -
| |
| .i
| |
| - 6.5 -
| |
| . . l FIG. 4 2 .
| |
| DISTRIBUTION O F LATER AL. SHEA'R STRESS 4 s e
| |
| * e 1.*4 , 8
| |
| - - - - _ _ _ _ _ _ _ _ _ _ _ _ _ _ . _ _ _ _ _ _ _ _ _ _ . _ _ _ _ _ _ _ _ . _ m _ _ _ . _ . ,
| |
| | |
| .- i
| |
| : 5. Shear Deformation (Timoshenko Shear Correction)
| |
| Consider a short beam segment of length AL, cross-sectional area A and shear modulus G. Due to the transverse shear force *'
| |
| Fy 5 0, the central axis will undergo a shearing strain Y as-4 shown in Fig. 5-1.
| |
| l The associated contribution h=y (5-1) to the slope of the deformed beam is frequently neglected in simple beam theory, but Timoshenko (1921] showed that the term could be important for a very short beam or for beams vibrating at extremely high frequencies, and suggested that y be found'from the expression f 0 = kGYA (5-2) where k is a numerical factor that depends upon the cross-sectional shape. One way to find k (following the example of Timoshenko (1955, p. 3181 for the case of a rectangular cross-section) is to equate the external work of the end load 0 (5-3)
| |
| : b. OYAL = b 0 ( ) AL 2 2 kGA with the strain energy AL dA = b f f' T dA 2 (5-4) l U= f f .1.2 2G 2G Throughout this section we assume thatx-axis, the applied force is
| |
| _ it is only l
| |
| along the y-axis. For a force on the necessary to interchange the symbols x and y in the principal results of this section.
| |
| O I-5.1
| |
| : l. i
| |
| | |
| O Equating , expressions (5-3) and (5-4) gives the following alternative expressions.for 1/k:
| |
| ,, 2GA ( U/ AL) (5-Sa)
| |
| ,l_
| |
| k 02 (5-5b) 1 'A ff 7 2dA k =h dA = R.M.S. of (
| |
| T
| |
| .) (5-5c) 1 =1 .
| |
| ff(
| |
| T
| |
| ) Tgy
| |
| *k A Q /A i'
| |
| Eq. (5-Sa) agrees with the alternative derivation of Boley
| |
| ( p. 259),
| |
| [1955), Eq. (5-5b) agrees with that of Washizu (1968, and Eq. (5-5c) shows that 1/k is the root mean' square value of ratio of shear stress to average shear stress over the the cross-section.
| |
| Eq. (5-5) predicts the values of.k given in Fig . 5-2 for i various cross sections, To find k for a rectangular grid we recall (see Fig. 4-2) that the lateral shear is distributed essentially parabolica11y over each vertical strip. Therefore, the shear stress Ty in a typical strip i is given by l t
| |
| t
| |
| = T mi [l- (
| |
| b/2
| |
| )) (5-6) where the maximum stress in the. strip is given by Tmi and the net force in the strip is j bt (5-7)
| |
| Og = fTt dA = .
| |
| 3 T
| |
| mi ,
| |
| i From Eq. (5-Sa), the strain energy for the strip, per unit beam length I.5-2 l
| |
| | |
| 2 0 14
| |
| 'U t 1 Ot 6 6-(2 r
| |
| * bt )
| |
| kg 2GA g 5 2Gbt 10Gbt 3 AL 4 bt 2
| |
| = _ . _ _ t, g 15 G since k = 5/6 for.a rectangle (case 1 of Fig. 5-2) .
| |
| ~
| |
| the
| |
| -Therefore, the strain energ'y for the entire cross-section of
| |
| -)
| |
| grid is given by SU i 4 bt 2 (5-8)
| |
| U
| |
| ___ = = { r,g AL AL 15 G i and the shear coef ficient for the entire cross-section is given by Eq. (5-Sa) as 4 bt 2 1
| |
| -
| |
| * 2GA 2
| |
| [U- )" 2GA 2 G
| |
| ETmi k 0 AL 0 15 i T
| |
| btA 2. , 8 bt y( mi (5-9) l_ = 8_ ,, )
| |
| k 15 02 1 15 A i O/A Since b E Ny c (see Fig. 1-1) and A = F Act (Eq. 2-2) 2
| |
| * (5-10) b = -_. [ ( )
| |
| l I k 15 Fg i 0/A From Eq. (4-12) we see that ,
| |
| T T
| |
| mi ,
| |
| mi ,7 AM (for Type A strips) (5-11a) .
| |
| 0/A Fy /(F get) F Ix T FKg gg mi ,
| |
| mi , (for Type'B strips) (5-11b)
| |
| O/A Fy /(F get) F yx I-5.3 l
| |
| | |
| f From Fig. (4-1) we see that there are (Nx-1) interior (Type A)
| |
| 'A Therefore, Eq. (5-10) takes strips and 2 end (Type B) strips.
| |
| the form 2 2 7^g "
| |
| 1=8 N
| |
| y
| |
| [ (3 x _ 1) cAF F K AM
| |
| )+2 ( F
| |
| ))
| |
| k 15 F 7x 7x A
| |
| ^ [ (N x- 1) (Kgg ) 2 + 2K (5-12) 1=8 --
| |
| Y BM. }
| |
| k 15 (F7x) where it may be recalled from Eqs. (2-5) and (4-5) .- (4-9) that
| |
| +N y (5-13) x N y' +N x F = 2N 3 _
| |
| = 3
| |
| +M (N M) (5-14a)
| |
| Kg N'y (5-14b)
| |
| K BM
| |
| =bN Y +3 M (N Y
| |
| - 1 - M) 2 Y (5-14c)
| |
| .M E INT ( )
| |
| 2 2 (5-15)
| |
| F 7x
| |
| = (N x - 1) [3 g2+E]+N + E
| |
| -1 (N - 3) (N Y + 1)
| |
| N Y Y ] (5-16)
| |
| E E [l+ 3 2 l NUMERICAL EXAMPLE .
| |
| Let N x *N y
| |
| * II M = INT ( 13 - 1 )=6 2
| |
| l l
| |
| F =2 (13)(13) + 13 + 13 = 364 l (
| |
| l L I-5.4
| |
| | |
| )
| |
| l (13) +6 (13 6) = 45.'75' !
| |
| K gg = _3 4
| |
| K gg = 1._ (13) +3_4 (6) (13 6) = 33.50 2
| |
| g , 13 - l' [y (10)(14) ] = 236 2 3 2 2 3
| |
| + 266] + (13) +._ (286) a-5551, F
| |
| 7g =.(13 - 1) [34 (13) 2 1=8 _
| |
| (13)(364) ((13 - 1) ( 45.75) 2 + 2 (33.50).21 = 2.24 k 15 (5551)2 k = 0.446
| |
| )
| |
| Note on Limiting Case For grids with a great many cells the value of k should approach a limiting value as N y+= and Nx +". In the limit ;
| |
| Fg +2NN xy M +1N Y 2
| |
| K gg +M (N y - M) +N /4 R+bN3Y 6
| |
| I Ix
| |
| +N x Ny,/6 l
| |
| l l
| |
| Had we used the "= continuous area" model for Fx I (Eq. .2-6),
| |
| we would find Fx I 5521 and k = 0.441. This value of k l differs by only-l% from the." lumped area model" prediction.
| |
| O I-5.5 I
| |
| f
| |
| ____________u
| |
| | |
| 1 l
| |
| : p. 2 Q 1 +
| |
| 8 Ny I yI IN x (
| |
| 2
| |
| ))= . (5-17) 2 4 5 k 15 /6 (N Nx y_)
| |
| This limiting value is exactly twice the value (6/5) for a homogeneous rectangular cross-section. This . result is reasonable because, as we have seen in Eq. (4-17), the peak shear stress corresponding value for a solid approaches twice the cross-section and the corresponding shear deflection should be twice as great for a given shear force.
| |
| i Note on Cross Section Warping The previous calculation of a shear deformation coef ficient for the rack implicitly assumed that the beam could warp freely.
| |
| We now estimate the effect of cross section restraint on the calculated shear displacements. Reference 1 gives results for
| |
| /]
| |
| tip deflection of an end loaded cantilever beam of rectangular cross section including shear deformation effects calculated with and without warping restraint. Generalizing these results to the case of non rectangular solid sections we obtain (5-18) 6 3=
| |
| PL (1 + .98 [ AL 12I ))
| |
| 2 3EI 1 l.5 12I 127 6 2= (1 + 71
| |
| [ AL2
| |
| )~*1 AL
| |
| )
| |
| 3EI where 6 1 is the solution allowing cross sections to warp freely, and 6 2 is the solution where warping restraint is included.
| |
| I For a typical fuel rack configuration, the ef fective moment 213,300 in.", the effective A =
| |
| 347.4 in.2, and i of inertia I = 2 ,
| |
| This yields I/AL the cantilevered length L = 182.71875".
| |
| .01839. Therefore, assuming free warping root cross section yields i
| |
| I-5.6 ,
| |
| 1
| |
| ---------_______________a
| |
| | |
| _q 3
| |
| (5-20) 6 j, = b ( 1. 216 3 )
| |
| 3EI at the base yields while assuming complete warping restraint b
| |
| (5-21) 5 2= (1.1463) 3EI 1
| |
| Therefore 0
| |
| (5-22)
| |
| -.2 = .94 og Thus, the neglect of root warping restraint, as we have done in' this analysis,
| |
| :s conservative in that it generally results in larger tip deformation.
| |
| e It will be seen in the results section of Part'II of this in the rack
| |
| ( ]/
| |
| s-report, that the gross cross section shear stresses are low. Therefore, any warping of the rack cross section away g from the root will have negligible influence on the numerical results.
| |
| O I-5.7 i l
| |
| | |
| O 1
| |
| /
| |
| \
| |
| /
| |
| /
| |
| i 3AL ;
| |
| z N
| |
| + e
| |
| _x-
| |
| ~
| |
| i g yFy =Q
| |
| /
| |
| = AL =
| |
| O hv FIGURE i 5.1 O ^
| |
| 1 4
| |
| | |
| ~
| |
| L .
| |
| _k 6 4 A
| |
| )
| |
| a y a 1
| |
| N 5 4 2 O T
| |
| (. H + 0 i
| |
| =- k E I
| |
| . T)
| |
| + / 'C 6'
| |
| .' F k 'l t . E
| |
| ' F j I
| |
| O R5 ,
| |
| +-
| |
| i 8
| |
| H R 4 G
| |
| ) -
| |
| 9 3 ( O ,
| |
| C2 1._' a .
| |
| 2 3
| |
| (
| |
| ~ . .
| |
| L '
| |
| t_ 0 =
| |
| ~
| |
| RS
| |
| = k J.
| |
| _k
| |
| / -
| |
| AE ES 1 ~
| |
| ' . i
| |
| ~
| |
| HA SC J
| |
| 'S R
| |
| ~
| |
| - l O "
| |
| : 0. , ~ ) OO K F.
| |
| . ~
| |
| 5 2 N 3
| |
| ) 1
| |
| = E
| |
| $ 0 k 6 .
| |
| ~
| |
| =
| |
| / 6 H0
| |
| (, k S 1
| |
| =
| |
| 2 9,
| |
| O O 5
| |
| Ml T
| |
| 5 I I
| |
| 1 7 2 P
| |
| TA
| |
| )
| |
| 5 1
| |
| ' R*
| |
| 8 1
| |
| = P
| |
| _2 0 k / 2
| |
| ( = 8 .S
| |
| /
| |
| k 1 6 5 'N ,
| |
| 9 1 G.O S
| |
| ~
| |
| I 3 U FS l
| |
| ) ,
| |
| 3 Z O l
| |
| 8 l P
| |
| Q 2 H
| |
| ( S (
| |
| O 6
| |
| =
| |
| 'l A
| |
| /
| |
| 5 k
| |
| =
| |
| (
| |
| W -
| |
| = ./
| |
| k 1
| |
| , .~ -
| |
| )
| |
| I
| |
| | |
| 1 O 6. Combined Stresses and Corner _ Displacements i
| |
| The cross-sectional properties and the Timoshenko shear l correction factor calculated in the previous section are to be j fed into a dynamic analysis of the system shown in Fig . 1-l'with l a specified ground motion simulating earthquake loading.. From dynamic analysis, the stress. resultants (Fx, F y, Fz, the 1-2, are computed for Mx, Mye Mz), acting as shown in Fig.
| |
| at the root a large number of times t = Lt, 2 At , ... etc.,
| |
| cross-section. The displacements (u x, uy, u) z at selected nodal points on the z axis are also provided by the dynamic analysis, as well as rotations (e,x e, y e) z of the cross-sections at the nodes. Stresses are calculated at the critical root section of the rack, and in the rack supports.
| |
| At each point of the cross-section there exists a normal (axial) stress a= o, z and a shear stress T which is the result of both lateral forces (F 2, Fy) and a twisting- moment (Mz). The severity of their simultaneous action is measured by the maximum stress a 2, 4t2" # ~ (6-1) o,q E M " MIN theory of failure *,
| |
| According to the maximum stress yielding occurs when i
| |
| aeq (6-2). l
| |
| " 8YP when Syp is the, yield point in simple tension.
| |
| i
| |
| " octahedral shear stress", or " distortion Also known as energy", or " von Mises" Theory.
| |
| O l I-6.1
| |
| | |
| A. Within Fuel Rack According to Eq. (2-1) o varies linearly over the cross-section and achieves its extreme value at one of the four corners of the rack. The shear stresses due to torsional loads (Mz) achieve their extreme values near the middle of each side
| |
| .(see Section 3). The shear stresses due to lateral forces (Fx,
| |
| , Fy) will achieve their extreme values at the center of the cross-section or at the middle of each side (see Section 4).
| |
| f Thus, candidates for the most critical point on any section will be the points labelled 1,2,.. 9 in' Fig. 6-1.
| |
| We now define the " direct stress" oo and " bending stress" OBx < OBy by 00= Fg/A EC pg F (6-3) 3 l
| |
| My a/2 f C = EC g My (6-4) i By j I y Y <
| |
| l Mx b/2 (6-5) l C = EC g Mx i
| |
| Bx I x i
| |
| x where using Eqs. ( 2- 2 ) - ( 2- 4 )
| |
| C pg =1= 1 (6-6) l A Pg et ,
| |
| U' .\
| |
| a .
| |
| x (6-7a) i C = !
| |
| l Y 2Iy 2c 3 tF yx Nc y b = (6-7b)
| |
| C"*= 2Ix 3
| |
| 2c tF l Iy l
| |
| b\
| |
| U 1
| |
| I-6.2 i
| |
| u
| |
| | |
| ' ,i i,
| |
| 1
| |
| ..t .5 We also define r.hplateralshears TLX r TLy by, I; (6-Ba) :.
| |
| t =C Fy p y ,? . [7 Ly
| |
| -( 6-8b )-
| |
| T =C px F x
| |
| Lx
| |
| .'j where (see.Eqs. (4-15) and'(4-18))
| |
| K^"F K^M 3 (6-9a)
| |
| C Y
| |
| = ( A4)=
| |
| ctF yx
| |
| >A F
| |
| 7x A 3 s (6-9b)
| |
| C Fx = _.
| |
| 3 1
| |
| The torsional stresses tTx t tty are given by' (6-10c) t.rx = C,px M g (6-10b)
| |
| T =C Ty M z
| |
| -Ty p
| |
| where (see Eq. (3-21))
| |
| 1 (c/t)
| |
| (6-11) 'e C = 1C/DI =
| |
| Tx 32 N 2P"2 k 2P cN x y (c/t) , (c/t) (6-12)
| |
| C,ry = k20a b 2 cN32x g y k
| |
| 20
| |
| \
| |
| and k2P, k20 are givent in Table 3-1 (a<b)
| |
| Using the above notation, it may be seen.that the normal, stresses- r
| |
| -;s t at the points 1,2,.. 9, in Fig. 6-2 are o and net shea ogx (6-13a) , >[ ,
| |
| a (1) = co - epj,j + .,
| |
| 'e 6M 3 b )- l e (2) =
| |
| co + ogy + e Bx J l i
| |
| ~ff
| |
| ' ' I-6.3 ,
| |
| I 8
| |
| ).-
| |
| ~ _ _ - _ _ _
| |
| 1
| |
| | |
| a i
| |
| )
| |
| j c (3) = 00 + eBy (6-13c)'
| |
| l
| |
| - /* h o (4) = (6-13d) ]
| |
| co 5. oBy ~ "Bx 1'
| |
| q (6-13e)
| |
| $y' o (5) = 00 -
| |
| (6-13f) !
| |
| od6) '= co + ogx l
| |
| j
| |
| ' (
| |
| i 9<' (6-13g) l o -(j )' k 'o [ + g y .
| |
| 7
| |
| . o (8) =, c of- eBx (6-13h)
| |
| 'l o (9).= 00 (6-131)
| |
| (6-14a) j r (1) =(0 (6-14b)
| |
| T (2) = 0 r (3) =0 (6-14c) t (4) = 0 (6-14d)
| |
| = + T (6-140) r (5) gy Ly r (6) = r Tx T .(6-14f)
| |
| * Lx
| |
| = (6-14g) t (7) gy + Yty (8) = T T (6-14h) f
| |
| ,t Xy + Lx t (9) = T gx + T (6-141) gy
| |
| .3 Corner Displacements kinematics it follows that the. components of From simple displacement (u,v) in the (x,y) directions at'each of the corners 8 is given by: ,
| |
| I u (1) = u x -
| |
| bez /2 (6-15a)-
| |
| u (2) = u (1) (6-15b) u (3) = u + bez/2 (6-15c) x 1
| |
| _O ,,
| |
| ' 'y j
| |
| w I ac t, !
| |
| I-6.4 (wT I b;'
| |
| t
| |
| | |
| ~ - -
| |
| (6-15d) u (4) = u (3)
| |
| (6-15e) u (5) =u-x (6-15f);
| |
| u (6) = u (1)
| |
| (6-15g) _
| |
| u (7) =u x
| |
| . (6-15h) u (8) = u (3)
| |
| ~
| |
| u (9) =u x (6-16a) v (1) =u y + aez/2 (6-16b)-
| |
| v (2) =u y - a eg /2
| |
| -(6-10c) v (3) = v (2)
| |
| (6-16d) v (4) = v (1) v (5) = v (1)
| |
| (6-16e) v (6) = u y (6-16f)
| |
| (6-16g) v (7) = v (2) v (8) = u y (6-15h) v-(9) = u y (6-161)-
| |
| l I
| |
| i B. Stresses Within Supporting Legs tubular section shown in Fig. 6-2, the For the square
| |
| [-
| |
| normal stress is F My Mx 2 x Y (6-17a)-
| |
| o = _._ + -
| |
| 1 A I I K X. .)
| |
| (/ l l
| |
| I-6.5 ,
| |
| | |
| i e.
| |
| The torsional stress is 2 , (6-17b)
| |
| T.,,
| |
| i
| |
| = Mz/She is found The shear stress rty 'due to lateral' loading F y by applying cq. (4-2) to find the shear distribution shown in Fig. 642b whfch takes the form:
| |
| For e < x < - et -
| |
| (6-18) x F
| |
| y ch = .{xl
| |
| ,,. hI x w
| |
| where
| |
| + 2 (1ehe2 ) . M he 3 (6-19)
| |
| I*= 12 3 .
| |
| l A=4 1(2e)h) = 8eh (6-20)
| |
| Fe 2 y y Y 3 =,3 y (6-21)
| |
| Ti = = _. _.E.
| |
| I 16 eh 2 A x
| |
| For -e < y < e:
| |
| =
| |
| ti+ _ n(e-y) . 1 = ': [1 + 1/2 (1 -
| |
| 2 j )] (6-22) vL,f hI 2 .
| |
| e x
| |
| The peak lateral, shear stress ;. occurs at the middle of the side
| |
| - parallel t o the load (x = .1 e , y = 0 ) , and is given by i
| |
| 3 T 1=
| |
| 9 Y (6-23)
| |
| T 2 '" -
| |
| j 2 4 A l
| |
| LO i
| |
| I-6.6 .
| |
| I
| |
| | |
| O Similarly,- 'a force Fx in the x-direction - produces the state of stress shown in Fig. 6-2, but rotated through 90', i.e.
| |
| T = at all corners (6-24)
| |
| Lx 2 A-T gx =b at'(x=0, y= t n e) (6-25) 4 A We now use the notation' F2 80 E ECp, F -(6-26a) 3 A
| |
| Me x 5 EC gx M g '(6-26b)-
| |
| %x I x
| |
| Me E Cgyy M (6-26c)
| |
| 'By 7 Y
| |
| M T (6-26d)
| |
| = CAz M 2 T " 8e h 2 F
| |
| T = Chy F y (6-26e)
| |
| Ly1 =
| |
| T F (6-26f)
| |
| Ly2 = = Ch2 y T
| |
| Lx1
| |
| =3.F x =Cj,1 Fx (6-26g) 2 A F
| |
| 9 x (6-26h)
| |
| T Lx2
| |
| * J A Ch2 E x l
| |
| lO .
| |
| 9 I-6.r
| |
| | |
| l
| |
| /~N (m) .
| |
| where
| |
| =
| |
| 1 (6-27a)
| |
| C'z F = 1A 8eh
| |
| = 3 (6-27b) j C'x M = C'y M = *I 16he 2 x
| |
| l 1 (6-27c)
| |
| C'*= 8he 2
| |
| Using this notation, denoting the net ' shear at point i by t( i )
| |
| and letting Cc = 1/2 (6-28)
| |
| We find the stresses at the 8 points shown in Fig. 6-2 to be (D
| |
| \) Eq.
| |
| l Point (i) o (i) r (i)
| |
| -C c (6-29a) 1 ao + eBx
| |
| ~
| |
| 'By T T+C c T Ly2 T
| |
| Lx2
| |
| \
| |
| T -C c T Ly2 - C c T Lx2 (6-29b) 2 oo + eBx + 'By T l
| |
| T -C c T Ly2 + C c T Lx2 (6-29c) I 3 oo + oBx + "By T (6-29d) 4 ao , oBx ~
| |
| "By I T+C e T Ly2 + C c T Lx2 ao - e By (6-29e) 5 T T+ T Ly2 I
| |
| 6 co + o gx T ~
| |
| T Lx2 (6-29f) l T
| |
| < ('~}-
| |
| g.g g
| |
| | |
| - w. <
| |
| l 0 7 00+ e By T T
| |
| ~ T Ly2 (6-29g) i (6-29h) 8 ao - agx T T+ T Lx2 i
| |
| The equivalent stress at these points is calculated by Eq. (6-1).
| |
| the gross cross section, of The maximum stresses, acting on
| |
| *either the root of the rack, or the rack supports,. are also checked for compliance with the relevant sections of the ASME Code at each time step during the elastic ant.'ysis, Part II of '
| |
| this report describes, in detail, the Code checks carried out.
| |
| l I
| |
| i O
| |
| I' I-6.9
| |
| | |
| 4 b
| |
| 4 l g i
| |
| 1 ,. .
| |
| 6 -
| |
| .b . ,
| |
| 9 .
| |
| s .
| |
| ~
| |
| @ i4 a,
| |
| G -
| |
| =
| |
| d
| |
| . . ~ . .
| |
| " FIGURE 6.1 .
| |
| * I i
| |
| 1
| |
| .. . 'I I
| |
| \
| |
| .. l 1
| |
| . 1
| |
| . : i
| |
| . l
| |
| . I i
| |
| 1
| |
| * 1 I-6.ll .
| |
| 1
| |
| . . . . , ,,_m _ _ _ .
| |
| | |
| 7- .
| |
| . D C :
| |
| y r
| |
| r W I -
| |
| T
| |
| :b . .
| |
| S
| |
| .y ;
| |
| S p -
| |
| 3 E .
| |
| R Fy
| |
| - ~
| |
| . T SE -
| |
| C R R
| |
| ~ A O E F H
| |
| 'SA
| |
| '- ' ~
| |
| )
| |
| b
| |
| (
| |
| O i
| |
| E h...
| |
| k -
| |
| B
| |
| _U
| |
| .T E
| |
| R
| |
| * A U
| |
| y - -
| |
| F Q
| |
| @ n/ -
| |
| 2
| |
| : e. )
| |
| G
| |
| . S
| |
| ( )
| |
| h ,
| |
| a
| |
| , (
| |
| + 4 -
| |
| j h hr .
| |
| 6 2
| |
| E R
| |
| U WN i
| |
| i G
| |
| O I
| |
| F
| |
| , ~
| |
| ~
| |
| ~ * , - -
| |
| | |
| .)
| |
| e Liftino and Impact Safety p 7.
| |
| k A.- Rack Stress Due to Uplift Force P
| |
| : 1. Bendinc ' Action :
| |
| A vertical force P, acting at the rack corner (x=a/2, y=b/2) produces bending moments of magnitude M = Pb/2 (7-1) x My = Pa/2 (7-2)'
| |
| For cross sections at depths greater than 2a beneath the loaded surface, the corresponding normal stress on the rack cross section is given by the simple beam formula Myx Mxy o* = p + - (7-3)
| |
| A I x
| |
| I y
| |
| The absolute maximum value of o g occurs at the' corner and is given as j
| |
| b2 3 2 e max = P [1A + + ) (7-4)
| |
| * 4I g 4I y
| |
| For the rack types studied in detail, we have 2
| |
| 3 2 j 4= [1+b A 4I 4I x y Rack Type 6 10x11 .018 6x11 .028 Therefore, we obtain the following values for due to an uplift load P = 4400 lbs.
| |
| olz
| |
| ! Rack Type oz (psi) 10x11 79.2 1 1
| |
| 6x11 123.2 j
| |
| \
| |
| I-7.1 1
| |
| /
| |
| | |
| We see that the stress on the rack corner is negli' ' tor the stated uplift force.
| |
| : 2. Local Loading Effects Suppose the uplift force P = 4400 lbs. is applied uniformly over a segment of length e on the edge of a thin p?, ate of thickness t. 'The corresponding applied
| |
| , stress is q = P/(et)
| |
| Due to this load, the most critical principal stresses occur under the load and have the values
| |
| * l l c1 = q, 02= q, 03 = 0 where 03 is the stress normal to the plate. According to the maximum shear stress criterion, yielding will occur when at- a3 =q=L =Y et where Y is the yield stress.
| |
| Therefore no yielding can occur unless the extent of the loaded region is less than
| |
| * min "
| |
| For P = 4400 lbs, Y = 23,150 psi, t = 0.08 inches.
| |
| 4400 = 2.38 inches e =
| |
| *" (0.08)(23,150)
| |
| I
| |
| * Timoshenko and Goodier, 1951, p. 95.
| |
| I-7.2
| |
| | |
| i f' . Thus, no ' yielding will occur if an uplift force is spread over a distance greater than 1.19 in. along the edge of any vertical plate in the rack since there are twe plates at any edge. Also, yielding, if it occurs, would be cocifined to - a region of the same order of magnitude as the loaded region.
| |
| l B. Falling Object of Known Energy
| |
| : 1. Direct and Bending Stresses:
| |
| The rack behaves like a large spring which absorbs the energy of the falling object. To find the maximum stresses in the rack due to this energy absorption, let P be the peak compressive force exerted by the falling object which strikes a corner (x = a/2, y.= b/2)-of the rack. This force produces bending moments M
| |
| x = P,b/2 M
| |
| y
| |
| = P,a/2 The on all horizontal cross-sections of the rack.
| |
| corresponding strain energy absorbed is therefore g* 2 g g 2g U=b[p
| |
| * 2n
| |
| + + Y
| |
| ]
| |
| 2 AE EI x EI y
| |
| ' 2 2
| |
| , m g 1 + b . 3 2E A 4I x 4I y 4
| |
| Thus the peak force is 2 2 1/ 2 a
| |
| l P, = [2EU j [ 1, 4 b ,
| |
| )
| |
| L A 4I 4I x y O and the corresponding peak stress in the rhek (at the corner x = a/2, y = b/2) is given in the form I
| |
| I-7.3 j
| |
| | |
| ,m 1 2 2 2 /
| |
| 1 o = P, ( _1 +b - +a -
| |
| ) = [ 2EU r ]
| |
| max A 4I x 4I y L A where 2 2 2 1
| |
| =_
| |
| 1
| |
| +a_ ( b /a + _y )
| |
| A A 4 I x I y
| |
| The strain energy absorbed by the rack is set equal to the kinetic energy of the falling object. Including a assuming it equal to the mass of virtual mass by (cd " 1) 1 displaced water, and including town ding' we can write a simple computer code to evaluate the final object speed at the end of a drop. The details of the method are outlined in part D of this section (which is concerned with dropping of a fuel assembly G all the way to the base plate). For the case considered here, the drop is 36" and the final velocity 1
| |
| l approximately 125.33 in/sec for a 1616 lb. weight. 3 I
| |
| For the given data the largest value of 1/A = .028 in 2 for the 6x11 rack. The length of the rack L =
| |
| 165", and the energy U is l
| |
| 1616 U=bx x 125.33 2= 32846 in# ;
| |
| 2 386.4 so that 1 1 2 6 7 x 32846 ix .0280) = 17636 psi !
| |
| c)***
| |
| =
| |
| [ 2 x 27.9 x165 10 is an estimate of the maximum stress at the rack base due to the impact at the top of the rack. This stress n is less than the specified yield stress. Since we have l () neglected any strain rate effects and used a lower
| |
| ~
| |
| bound on damping coefficients, the actual stress will be lower than the value reported.
| |
| | |
| u <[
| |
| ~
| |
| . Local Impact Effect If- the kinetic energy .U is thel result of a ' weight.W falling through.at distance h, the impact velocity V is )
| |
| ' given by' 13v 2=U 2 g V = -(2g U/W) !
| |
| When a rigid body, moving with velocity V, strikes the-it may' be shown-edge of an elastic- rod or plate, (Timoshenko and Goodier, 1951, pp'. 441-4423 that ' . a' .
| |
| stress is developed of magnitude og = F.V/c where E is the elastic modulus'and-
| |
| - c = (E/p) ! is the-acoustic. velocity.
| |
| Therefore, the stress is given as oy = E pV Using V = 125 in/sec yields ,
| |
| 1 6
| |
| x .283 o= [ 27.9 x 10 /)2x 125 = 17878 psi 386.4 which is less than the material yield stress.
| |
| To agsess' the local effect of 1100# applied C..
| |
| horizontally, we examine Figure . 7-1 which represents a fuel cell wall above' the' active fuel area.
| |
| We examine the depth b required ' to prevent plastic collapse under a pressure load distributed over the area bL equivalent to 1100# total load. We assume that l
| |
| I-7.5 I
| |
| | |
| the section acts as a-beam of-width b, thickness t and length L that is clamped at each end. The collapse.
| |
| theorem gives P = 4Mp(26 6L L
| |
| -)
| |
| 2 where bt 2 ,p=-
| |
| p M p = oy 4 bL Therefore b=1 ,,,, 4 PL
| |
| )
| |
| 2t o y
| |
| O For t = .080", L = 8.85", P = 1100#. oy = 23150 psi, we obtain b = 4.05" Since the active fuel is at least 9" below the top of rack, we can see that all plastic ' action is the confined to a region above the active fuel.
| |
| D. Dropping of a Fuel Assembly Assume that a fuel assembly drops into a ' storage cell from a height of h inches above the fuel pool floor, all the way to the bottom of the- rack, and impacts on the baseplate. wa now show that the base j plate will not be perf orated.
| |
| I-7.6 L__-_-____:_______-___________-
| |
| | |
| O Let:
| |
| W = weight of fuel' assembly.(1616 lb) h =. height above base. plate (201 in) i l
| |
| *a = side of (square) effective solid cross-section of fuel assembly = 8.548" T = thickness of base plate (0.625 in)
| |
| Y = Yield stress'of base plate material (23,150. psi) 6 = distance that fuel assembly penetrates-into the base plate Figure .7 -2 shows the final deformed configuration of the base plate with fuel assembly sitting on it, i The work dissipated in plastic deformation is 1
| |
| F6 I
| |
| where F is the average resisting force exerted on the fuel assembly by the base plate. ,
| |
| i U, sing the model which has proved accurate for the punching of slugs out of plates {see Paul and: Zaid (1958)] the force F may be conservatively estimated by 1
| |
| F=( " Y) 4aT i
| |
| O I-7.7
| |
| \
| |
| | |
| O' J where /3 . Y is the yield stress in shear.according to
| |
| 'the distortion energy theory of yielding.
| |
| Then the work dissipated in plastic deformation is F6= YaT6
| |
| [3 Upon equating this _ work to the kinetic energy -gained during the fall, we find-i YaT 6 = W _ VI2
| |
| /3 2g so that 2
| |
| V W f 6 = [/3YaT 2g To calculate the final velocity of a body dropping through a channel, we account for virtual mass, gravity and fluid drag. We assume that the virtual mass is the- buoyant mass, and that the drag equal to coefficient is based on exposed frontal area of the fuel rods. The governing equation for a mass element l I
| |
| in free f all subject' only to gravity and drag effects is l
| |
| C*
| |
| (M + My) v+ pA y
| |
| V E= (M - My)g 2 l 1
| |
| where Cg = effective drag coefficient due to all con-l tributing effects, and My = virtual mass of object. l If c = Wy/W, then P 9
| |
| ( 1- c) W D^*
| |
| ;= g-(1+ c) 2 (1+ c)
| |
| Ro7 '' - _ - - - - . _ _ _ _ _ _
| |
| | |
| subject to y = 0 at x =-0 v=v g at x = h In' finite diffarence form 2
| |
| , $, C{A*vg g- ]
| |
| avi,y = at [ 1+ c 2w (1+ct )
| |
| v t,1 =v g + av g 1 ; x g ,1 =x 1 + (vg +vg ,y)at/2 For a rod like body with characteristics similar-to a fuel assembly, Cg a 1.0 + A where - A represents the incremental increase in ef fective drag coef ficient due to the fluid being confined in a narrow channel.
| |
| Consider Figure 8-3.
| |
| V A*v = A 3v3; A cell f*AY3 3 Therefore v3" UVf Vg " u1 V where A A u=- ; ul =
| |
| A3 A cell Assuming that the expansion at A 3 is to a very large area, then the pressure p3 is essentially equal to the fluid pressure outside of the cell. Neglecting any depth effects, an energy balance yields the pressure difference across the fuel assembly as 2
| |
| a pv W
| |
| Ap = 2 (v 3 2_y g2 ) ,
| |
| (92_ pt 2) l 5 2 2 g
| |
| I-7.9 o-_______. _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
| |
| | |
| O v as long as p 2 ut. The effective incremental resist-ing force due to cell geometry is AF = opA* so that
| |
| *2 A" M ~
| |
| "A "
| |
| 2 [ 1 - ^3 /^ cell b A3 The BASIC program on the next page has been written for an IBM micro computer.- The accuracy of the.results can easily be verified by considering the drag free case with vg = /2gh for any typical set of' input data.
| |
| For the Pacific Gas and Electric rack and cell geometry, for a fuel cell drop to the base of the_ rack, h = 165.0" + 36" = 201" O and.the final velocity at impact with the rack base is V1 = 202 in/sec Therefore, the maximum depth of penetration is 2
| |
| 1 =
| |
| .443 x 1616 x 202 6= .443 YaT 2g 23150 x 8.548 x .625 x 386.4 x 2 6= .306" < .625"
| |
| \
| |
| ThereLore, the base plate will not be penetrated by the drop. i I
| |
| i l
| |
| i O
| |
| I-7.10
| |
| | |
| ~.
| |
| A distance between adjacent storage The center-to-center cells is not dependent on the presence or absence of support from
| |
| .the base plate. The purpose of the above calculation is only to show that there is no danger to the liner.. In the event of a dropped fuel assembly, it is correct to say that the base plate will separate from the tube in the immediate vicinity of the affected tube. While this would result in base plate plastic bending, it would not affect center-to-center spacing since there would be no effect on the welds between adjacent tubes nor on the base plate-to-tube welds away from the immediately vicinity of the dropped assembly. The base plate, even with plastic bending I occurring, will not touch the liner floor in the event of a dropped fuel assembly hitting the base plate.
| |
| O
| |
| | |
| ' R E '*1 - Fuel rac< creo calcula;iens INPU' " Cel1 d i fd. . Q a o . hi;il e . F 3 c l ue " ! C D . 9 0, *- 9
| |
| ... AC=CD*CD: AS=(CD-GP)*(CD-GC)
| |
| * a3= 2. A L139*47+di 13 t.DAINT" acell=",AC."astar=".CS .
| |
| 14 _ PAINT "A3= ",A3 15.LM=(AS/A31"2*(1-(A3/CC)^2)
| |
| -16 PRINT "im = . J' 17 PRINT "inout abcve value for 1rn or inout 0 if drag coef =1.0" 18 INPUT LMsCD=1?+LM 20 INPUT "1,w,dt,xf";L,W,DT,XT 21 LPRINT " fuel length =" L 28 LPRINT " weight =",W." time stec=".DT 23 LPRINT "droo cistance=".XC 24' INPUT" READ IN FUEL SOLIDITY";FR 1.
| |
| l 25 EP=FR*AS*L*64/1728*1''/W 26 LPRINT "eos=",EPLPRINT "ed=",CD 40 C2=(1-EP)/(1+EP) 50 C1=64t/1728!*CD/2!*AS/(W*(1+Eo))
| |
| 60 X1=0 70 Vi=0:X=0 80 V=V1:X=X1 90 DV=DT*(386.4*C2-Ci*V*V) 100 Vi=V+DV:X1=X+.5*(V+V1)*DT 105 PRINT "x1=".X1," v1=", V1 109 DAINT "x1=",X1," v1=",Vi
| |
| _30 IF X 1) =XF TWEN GOTC 340 E8.SE GOTO 80 1 PRINT "x1=".XI." v 1 = " , V ?.
| |
| O LPRINT "x1=",X1," v1=",V1 150 LPRINT" " LPRINT" 200 END
| |
| ~
| |
| I-7.ll m
| |
| o e
| |
| | |
| ~. ,
| |
| k
| |
| .L1= 73, 2 2 :" '
| |
| 50.26544 aatar= /O.9638 s1 ' a n ;1 : - = 165 i
| |
| .001
| |
| *etget= 1616 t t 'a e steo=
| |
| 36 Mref de ''
| |
| droa distarce=
| |
| eos= ,25825A I8f 8 - I'8L C cd= 1 xim 36.05A1 .v1= 125.3354 7 5 . *. 4 5 6 9 star = 7J,.82906 i acell= -eel' h we n a3= 28.27451
| |
| ' iel lanct,u '63 ho verdwal inut.Ss ga/ Cg = e
| |
| ,.atgats i E '. f. t i ne <s t e c = . O t n. e QQst, a
| |
| .c r c ;.3 : s t a r.c ar' 36 1
| |
| sos =
| |
| : v. = tg h 1= :s. .xs s v1= ae. m.:
| |
| V: /66,925 ,
| |
| for 11 x 3 6 o SW 7 ' '
| |
| m 0 IC .
| |
| i 1
| |
| l l
| |
| l l
| |
| l I.7.12 )j l
| |
| t
| |
| | |
| ac ar= 7 1 . 0 6 6 3 '.
| |
| .1= '78. J2251 NL5P E0*El"A1 c,ce hko
| |
| 'uel lance-= 165 001 0 h rA c IC welgnt= 16'.6 t i ra e steo=
| |
| 20; dr'oc distance =
| |
| eps= .276317?
| |
| cd= 6.6081 201.1655 vi= 202.3124 x1=
| |
| \
| |
| 6 I-7.13
| |
| | |
| O e
| |
| ' t i
| |
| b P
| |
| \ /
| |
| L.
| |
| O
| |
| / -
| |
| FIGURE 7.1 W
| |
| e
| |
| .j 4
| |
| O I-7.14 i
| |
| | |
| 's' O .
| |
| t l .i 8-f l .
| |
| * I, I FUEL ASSEMBLY O 6-p .
| |
| 7W '/ Z
| |
| \
| |
| 'Q .
| |
| (f. 'fi (
| |
| '* r, i BASE. . PL ASTIC PLATE D EFOR M ATIO N FIGURE 7.2 O ' - ' "
| |
| 1
| |
| | |
| l O 'A 3
| |
| h A3
| |
| ~
| |
| ~ -
| |
| l l
| |
| e l
| |
| i c n a m
| |
| ~ _A -
| |
| f v
| |
| =
| |
| O 3 7
| |
| h m
| |
| E R
| |
| /[ p a
| |
| g
| |
| . U G
| |
| I F
| |
| .A .
| |
| h 9L .
| |
| h
| |
| - 4 _
| |
| A. _
| |
| u O 7"" 2 Il.llfllll11 I
| |
| ~ m
| |
| | |
| APPENDIX 1 TORSIONAL STRESS IN RECTANGULAR SECTIONS Side of a Maximum Shear-Stress on the Short Twisted Rectanole_
| |
| For the maximum rectangular cross-section shown in Fig.
| |
| Al-1 (where A'< B) the peak torsional stresses occur at points.O (1951, p.
| |
| and O' and have been given by Timoshenko and Goodier 276).
| |
| However, the shear stresses along the shorter sides of~ the have not been given rectangle, which peak at points P and P',
| |
| explicitly by Timoshenko and Goodier. To find these stresses we use the following expression for the stress function $ given by Timoshenko and Goodier on page 276.
| |
| "~1 32GOA 2
| |
| ~
| |
| (,7) T- , Cosh (nwy/2A) ]Cos n wx (gy,y) 9 7 2A w n Cosh (nwb/2A) n = 1,3,5...
| |
| From Eq. 3-4, we have Eq. (3-4) n-1 I
| |
| . 3* 32G0A 2
| |
| 1, g, )2 [ _ nu CoshSinh (nny/2A) ] Cos " "*
| |
| 7 _
| |
| 2A (nwb/2 A) 2A ay w 3 n=1,3,...
| |
| (Al-2) l i
| |
| At point P , x = 0, y = -B , and n-1 2 T nw n wB 32G0A # !
| |
| rp= (Tx) max
| |
| * 3 b "1
| |
| ~~J 2A 2A l w n I O 1,3...
| |
| l e
| |
| l l l l
| |
| 1 Al-1 l
| |
| | |
| (Al-4) 16 GBA Ct rp = T n-1 (Al-5)
| |
| Tanh na C1= { 12 (-1) 1,3...
| |
| (Al B =y b as1 2 a
| |
| 2A i ting Timoshenko and Goodier (p. 2781 have also expressed the tw s moment in the form (Al-7)
| |
| M t =ki G8 (2A)3 (2B) hence Eq. (Al-4) can be expressed as (Al-8)
| |
| O rp =
| |
| Mt .
| |
| = _ ___.~
| |
| .k 2P ab k (2B) 2P( 2 A) where 2 (gy_9)
| |
| W kt k _,
| |
| 2P 8 C1 has been evaluated from Eq. (Al-5) for various f The coefficient Ct l values of (b/a); ki has been found from the tables of Timoshenko f and Goodier, and k2P has been evaluated from (Al-9) with the results given in Table 3-1 of Section 3 of tnis report.
| |
| 1 O l Al-2 j
| |
| | |
| r- u -
| |
| m i l O I
| |
| + \
| |
| : h. .,
| |
| I i
| |
| a, ;
| |
| i . [
| |
| ,, 'u 1 n a ,
| |
| <f' s,
| |
| s Y
| |
| h h 'p' B
| |
| b o o' ho x a ._..
| |
| B l
| |
| u o P l
| |
| .- {
| |
| *-- A = -
| |
| A+ .
| |
| 1 i
| |
| = a r l
| |
| FIGURE A1-1
| |
| .i 4
| |
| ~
| |
| I t
| |
| Note that A r e p r e s e n t s h a l f t h'e s h o r t e r t side le ngt h (a = 2 A, b = 2 8)' i I
| |
| 4 t ,
| |
| t
| |
| - Al-3 s n
| |
| | |
| 3.
| |
| APPENDIX 2' r~'y /.Y - SAMPLE CALCULATION V 7
| |
| , SECTIONAL PROPERTIES OF RACK rl (
| |
| The cross section of the rack (see Fig. 1-1) consists of an orttoponal array of square cells of side -length c, separated by s t r f,o 2 of thickness t. Fig. A2-1 shows such an array with-7 cells in the x-direction and 9 cells in the y-direction.*
| |
| I.
| |
| In general there will be N x cells in the x-direction and
| |
| (
| |
| Ny cells in the y-direction.
| |
| A. Area The area of the (Nx + 1) " vertical" strips is (N x + 1)- t N yc and that of the Ny " horizontal" strips is (N
| |
| y
| |
| + 1) tN x c Therefore, the total area A is given by A=F (A2-1) j g et where (N x + 1) N y + (N y + 1) N, = 2N xy N +N x +N y F
| |
| A=
| |
| I B. Moment of Inertia (for distributed area):
| |
| a The moment of inertia (H.O.I.) about the x-anis of l'
| |
| typical " vertical" strip in Fig. (A2-1) is tb 3 t(Nyc)3 12 12 See comments regarding model on p. A2-9a. b A2-1 h-
| |
| | |
| O Since there 'are (Nx .+ 1)- such. vertical strips,' the (M.O.I.) of.all the vertical strips is therefore (N- + 1) N3 3 1A.S.-3)
| |
| TVERT' "~
| |
| c 12 i
| |
| The M
| |
| ..O.I. of horizontal strip i located at the height 3 Therefore, the M.O.I. of all y=yi is t(Nx c )yi .
| |
| horizontal strips.is given by S S 3
| |
| ( g/c)3 (A2-4)
| |
| T gog = 2t(Nx c) [ (y g) 2 = 2Ng tc i=1 i=1 where N is the total' number of horizontal strips in the upper
| |
| ~
| |
| half (yi>0) of the section.
| |
| l From Fig. A2-2, which shows the nodes on a typical vertical strip (for Ny either odd or even), it is apparent that for:
| |
| N odd Ny even y
| |
| y g = (1-1/2)c yi = ic S = (N y + 1)/2 5 = N /2 1
| |
| M= (N y - 1)/2 M= (Ny -2)/2 1
| |
| and that: l N
| |
| N= ,y g = ic (A2-5) f For N y even:
| |
| N +1 N 5 = Y ,y Y_ (A2-6)
| |
| For N odd: g = _2 2
| |
| l O
| |
| A2-2
| |
| | |
| ~ _
| |
| F l
| |
| p-(,)\ Therefore,'widm N y is even:
| |
| 2 5(N + 1) (25 + 1)
| |
| [ (2 g/c) 2 = li =-
| |
| i=1 in 1 6 (A2-7)
| |
| N y(Ny + 2)(N y+1) 24 Whereas for N y odd:
| |
| 1 N ; N N N N 2
| |
| { (1-1/2) =
| |
| [1 7t _
| |
| 7(g74)
| |
| { (y g /c)'''=
| |
| 1-1 = i=1 i=1 i=1 i 1 l 4
| |
| = N(9 + 1) (2$ + 1) ,
| |
| (5(5 + 1) +5 .
| |
| 6 2 4 l
| |
| k_ , Sj25 + 1? (25 - 1)
| |
| '12 (A2-8) i (N y + 1) ' N , _. 4- 2) N l
| |
| ~
| |
| 24 Comparing this result to Eq. (A2-7), we see that for N y, even ;
| |
| or odd:
| |
| 1 N + 1)(N y + 2)
| |
| [ (y g/c ) 2 , Ny(Ny _ (A2-9) ;
| |
| i = 1 24
| |
| ~ j Substituting tnis result in Eq. (A2-4) gives l N y(Ny, + 1)(N y + 2) 7 I " #3 l l HOR x 24 '
| |
| l l
| |
| AT-3
| |
| - -_.________-_-__a
| |
| | |
| O and the total M.O.I. about the x-axis.is given by
| |
| + 2)
| |
| + 1)(N 3
| |
| (N.* + 1) N Y3 + N*N Y(N Y Y
| |
| ]
| |
| = tc [ 12 I
| |
| x = IVERT + IHOR
| |
| .(A2-10)
| |
| =F IX ct NN 3 N +1 (N +-1)(N- + 2)
| |
| ^~
| |
| I where FIX " Ny 2
| |
| 12- Nx I
| |
| By interch'anging x and y we find (A2-12)-
| |
| I = F; ct3 y
| |
| NY N* 3 N Y
| |
| + 1
| |
| +
| |
| (N* + 1)(N* + 2) ]
| |
| (A2-13) where F IY = ( Ng 2
| |
| 12 N y
| |
| Numerical examplet Let N g =N y = 13, c = 6.363", t = 0.150"
| |
| , (13)(13)3 [ 13 + 1 + (14)(15) ] = 5521 p*,pIY 1
| |
| 12 13 -169 ,
| |
| I l
| |
| l = 213,300 in" (A2-13a)
| |
| Ix"Iy = 5521 (6.363)3 (0.150)
| |
| C. Moment of Inertia' for Lumped Area Model' (i) For an odd value of Nyt
| |
| ' Recalling that there are Nx-1 strips of type A, and l j
| |
| two strips of type B, and ref erring to Fig . A2-2, it will be seen f
| |
| that the M.O.I . of the lumped area: above the x-axis A2-4 l
| |
| i
| |
| | |
| /~~
| |
| t 2 N 2 k 2 1/2Tx" ("x~1) 5^+bI -) (- )+ ... ( l - 1) c 2) 2 2 2 2 N 2 Nc 2
| |
| +A
| |
| ' Y )
| |
| 2
| |
| }+2 {AT b-2 * - ) # *** ( l - 1) c 2
| |
| ' 2 2 2
| |
| Nc 2 (A2-14)
| |
| +A g (
| |
| 2 Y
| |
| )}
| |
| Upon* recalling (Eqs. 4-la, b, c) that A= 2ct, A = (3/2) et, Ag = ct (A2-15)
| |
| T and introducing the notation EE 1+3 2+ 5 2+ ... (Ny -2) 2 (A2-16)
| |
| ,O. It follows from Eq. A2-14 that
| |
| : ( ,/
| |
| 3N x +1 l I
| |
| x =
| |
| 2Nx+1 -
| |
| k + N 2
| |
| (A2-17) !
| |
| F IX i
| |
| 3 y
| |
| ct 7 4 To simplify the expression for k, we write Eq. (A2-16) in the form:
| |
| M 2 M 2 E= [ (2i-1) =
| |
| { (41 - 41 + 1) i=1 i=1 O# M M 2 . ( A2- 18 )
| |
| [i l E = 4 -
| |
| 4 [i +M iel i=1 l
| |
| where 1
| |
| 2 2 (2i max -II 'I IN-2I or M E i = (N - 1)/2 (A2-19) max y
| |
| | |
| We now make use of the algebraic formulas
| |
| * N M(M+1)(2M+1') (A2-20) g 2, i= 1 - 6 .
| |
| M(M + 1)- (A2-21) i i= 1- 2 which, when substituted into Eq. (A2-18), gives
| |
| ;,4 M(M+1)(2M+1) _.'4M(M+1) ,g 6 2 (N + 1) N - 1 (N -
| |
| : 1) (N Y- + 1) N Y Y + Y -
| |
| , Y - (N- - 1)
| |
| Y 2 2 6
| |
| - 2)/6 (A2-22)
| |
| = (N y -
| |
| 1)-(Ny) (N y (ii) For an even value of Ny:
| |
| 2 I
| |
| Ny -2 2 2= (Nx -1) {2ct [1 + 2 2, 31 , , ,. ( .. ) Jc 2
| |
| 2 2
| |
| N c y 2-
| |
| +3., eg ( )} + 2 {32 ct (1+2 3+ .
| |
| 3 2 2 l 2 2 N -2 Nc .
| |
| + ... (
| |
| Y
| |
| ) }c 2 + ct ( Y 2
| |
| )}
| |
| 2 l
| |
| Ny 2 I
| |
| x R 3
| |
| # (
| |
| F IX ct 3
| |
| "x ~ l ~~~
| |
| 4
| |
| ~~
| |
| 2 2 2
| |
| - N 2 y
| |
| +4 ( 33,(, ) } , ( g * , y ) {g4 , },_ Y g 2},3gg 2
| |
| Y l 24 2 Smithsonian Mathematical Formulae, Smithsonian Institution, O Washington (1957, p. 26)
| |
| A2-6
| |
| | |
| t f
| |
| , .m i
| |
| *2N X +1 3N* + 1 N 2 (A2-23)
| |
| E + l F = Y \
| |
| IX 4 2 l 2
| |
| N -2 {
| |
| - y )
| |
| (_
| |
| where h=1+2 2+ 32 ,,,,
| |
| 2 4
| |
| or, using Eq. (A2-20):
| |
| - 1)) b6 = NY(N -1)(N -2)
| |
| N -
| |
| 2 N Y Y (A2-24) l Y -.Y (N i k= 4[ 2 2 Y 6 f
| |
| Ny)
| |
| Note that Eqs. (A2-23) and (A2-24) (for this case of even are identical with Eqs. (A2-17) and (A2-22) (which are valid for f the case of odd Ny). \
| |
| For N g =N = 13, Eq. (A2-22)
| |
| Numerical Ex. ample:
| |
| t (N -1)N y(Ny -2) =
| |
| (12)(13)(11) --
| |
| = 286 gives k= 6 6 1 l
| |
| and Eq, (A2-17) gives
| |
| +1 2 = _27 (286) + 39 + 1 13 2= 5551 2N +1- 3N x x k+ N F = Y 4 .
| |
| IX 4 2 2
| |
| i l
| |
| Note that this value is within 1/2% of the value 5521 calculated l in part B of this Appendix for the distributed area model of the l cross-section.
| |
| The moments of inertia are given by 3
| |
| =I y =F IX ct= 5551 (6.363)3 (0.150) = 214,500 in 3
| |
| G I x
| |
| /g for c = 6.363" and t
| |
| = (0.150")
| |
| | |
| D. First Static Moments To uind . the first static. moment of -area above the x= axis the vertical strips shown in Fig. A2-2, we for either of introduce the following' notation:
| |
| k = index of a- nodal area starting with k=0 at the end of the strip and increasing to k=M M = the number of nodes above tiie x-axis in any vertical
| |
| . strip Ao = area concentrated at each of the interior nodes-a area concentrated at the ends of a vertical strip A,
| |
| Note that For Type A strips (as defined in Fig. 4-1):
| |
| Ao=A =
| |
| 2ct, A, = AT = 3ct/2, A,/Ao = 3/4 (A2-25)
| |
| T For Type B strips:
| |
| Ao = AT= 3ct/2, A e " A,L = et, A,/Ao = 2/3 (A2-26)
| |
| Also note that whether Ny is even or odd M= INT ( Y ) (A2-27) 2
| |
| 'With this notation, we see that the static moment about the x-axis of all areas in a strip from the outermost (k=o) up to and including node k is gives by Nc Nc Nc Nc Sk"Ae +A0 ( - c) + Ao( - 2c)+ ... A o( - kc) 2 2 2 2 Nc Nc Y (A e + kAo)
| |
| = Y (A e + kAo) - Aoc (1 + 2 + ...k) =
| |
| 2 2
| |
| IN + II
| |
| -Acck 2
| |
| A2-8
| |
| | |
| ~_
| |
| l O S k
| |
| =
| |
| A cc . [ (
| |
| Ae
| |
| +k ) N Y
| |
| (k'+.1) k) (A2-28) 2 Ao Using,Eqs. (A2-25).and A2-26) we find that For Type'A strips:
| |
| S k 3S kii = ct (( + k) Ny '- (k + 1)k]
| |
| ~
| |
| =ct 2 K A where KA( E +- ) N y - (k + 1) k (A2-30)
| |
| The max value of SkA occurs when k=M and is given by Max S kA * =ckK AM ^~
| |
| MA K 3 * ( + M) N y -
| |
| (M + 1) M (A2-32)
| |
| AM A For Type B Strips:
| |
| S k
| |
| 8 kB = ct [(
| |
| 2 + k) N y -
| |
| (k + 1) k}
| |
| ctK2 ^~
| |
| B
| |
| = N + k (N y - k - 1) (A2-34)
| |
| KB(k) y Max S kB E SMB = c t K BM 2 (A2-35)
| |
| ("} *
| |
| "y + " ("y - M - 1) .A2-36)
| |
| KBM " B O 1 I
| |
| A2-)
| |
| i
| |
| | |
| O A The preceding analyses are shown in detail for a honeycomb construction with all edges completely connected by welding. For.
| |
| assembly of the configuration does. not. permit a PWR unit, confirmation of weld integrity at all locations. Therefore, for conservatism, we neglect certain welds when computing rack inertia. Figure A2-1a shows a typical PWR cell configuration .
| |
| together with the idealized configuration used for computation of area and inertia properties of the rack cross section.
| |
| O 1
| |
| I A2-9a i I
| |
| e
| |
| | |
| I O
| |
| - c +-
| |
| 3 I=N t : = ,
| |
| P 1= 3 I
| |
| 'c , ..
| |
| !=2 9,
| |
| i= 1 Y -
| |
| C X
| |
| ' = ""
| |
| O .
| |
| t -= = -
| |
| 4 i
| |
| a - NxC r FIGURE A2-1 A2-10
| |
| | |
| 4 we O .
| |
| 1 A$ C crM9Tt0CT69 b$ bMhLYC6O yp r;~
| |
| - _ [ q a_
| |
| ~
| |
| I M T 1 h -9 4-d2,07
| |
| * U 7 p { a p gy - __.
| |
| : s. -
| |
| 3, . --. - u ) ,_
| |
| ( - K w l= -w R cHAopg(
| |
| >o eneumms cwv6criou L
| |
| V $ 9IC, A2-lq_ WO Ntt& ssacg ngs.eio a 0 A NUOT 88 CilFC}Lf f O
| |
| A2 10 a
| |
| | |
| O A 2 1 p t
| |
| = = =
| |
| i i i g D o o o 0 g ,
| |
| 0 b
| |
| y
| |
| ^
| |
| x=
| |
| - A a
| |
| S 2
| |
| 2 A
| |
| O .
| |
| E R
| |
| . U
| |
| - G I
| |
| F
| |
| ) ) )
| |
| 0 1
| |
| )
| |
| m.. .
| |
| = = .
| |
| =
| |
| k k . k, x
| |
| ( ( ( (
| |
| 2 5 2 1 h; m
| |
| = : = .
| |
| I I i 3
| |
| O 0 o o ;
| |
| 'J x
| |
| =
| |
| g
| |
| . i -
| |
| Y - e
| |
| ^ A O -
| |
| g-
| |
| | |
| gg s
| |
| )
| |
| - REFERENCES
| |
| : 1. S. Timoshenko, Strength of Materials, Part I, Van Nostrand, Princeton (1955).
| |
| Strength of Materials, Part II, Van
| |
| : 2. S. Timoshenko, -
| |
| Nostrand, Princeton (1956).
| |
| Goodier, Theory of Elasticity,
| |
| : 3. S. Timoshenko and J .N .
| |
| McGraw-Hill, N.Y. (1951).
| |
| and Shell Structures,
| |
| : 4. P. Kuhn, Stresses in Aircraft McGraw-Hill, N.Y. (1956).
| |
| : 5. T.H.G. Megson, Linear Analysis of Thin-Walled Structures, Halstead Press, Div. of J. Wiley, N.Y. (1974).
| |
| l 6. S. Timoshenko, "On the Correction for Shear of the of Equation for Transverse Vibrations Differential Vol.
| |
| Presmatic Barr", Philosophical Magazine (xli) Sec. 6, 41, (1921), 744-746.
| |
| Methods Elasticity and
| |
| ["
| |
| \-
| |
| : 7. K. Washizu, Variational Plasticity, Pergamon Press, Oxford (1960).
| |
| in "" ~~
| |
| : 8. B.A. Boley, "An Approximate Theory of Lateral Impact on Beams", J. Appl. Mechanics, Vol. 77 (1955) 69-76.
| |
| 1
| |
| : 9. N.J. Hoff, B. Klein and .P.A. Libby, " Numerical Procedures for the Calculation of the Stresses in Monoccques, (Part IV)", Nat. Advis. Comm. for Aeronautics, Washington, Tech.
| |
| Note No. 999 (1948).
| |
| : 10. Smithsonian Mathematical Formulae, Smithsonian Institution, Washington, D.C. (1957), p. 261.
| |
| : 11. B. Paul and M. 2 aid, " Normal Perforation of a Thin Plate by Truncated Projectiles," Journal of the Franklin Institute, Vol. 265, No. 4, April 1958, 317-335.
| |
| /^g ,
| |
| I
| |
| (_/ :
| |
| _ _ _ _ _ _ _N
| |
| | |
| ' ' ~ - ~ - - - _ _ _ _ _ _ _
| |
| o 1
| |
| 1 I
| |
| l O
| |
| II. DYNAMIC ANALYSIS AND STRUCTURAL INTEGRITY l 4
| |
| O
| |
| \
| |
| i 1
| |
| i i
| |
| l O
| |
| | |
| ~.
| |
| l O 1. INTRODUCTION
| |
| {
| |
| The purpose of this section is to develop governing equations, and the structural adequacy of the spent fuel rack to demonstrate l design under normal and accident loading conditions. The method of 1 analysis presented herein uses a time-history integration method similar to that previously used in the Licensing Reports on High' !
| |
| Density Fuel Racks for Fermi II (Docket No. 50-341), Quad Cities I I and II (Docket Nos. 50-254 and 50-265), Rancho Seco (Docket No.
| |
| 50-416), Oyster Creek 50-312), Grand Gulf Unit 1 ( Docke t No.
| |
| Summer (Docket No. 50-395). The (Docket No. 50-219), and V.C.
| |
| high density spent fuel racks are results show that the structurally adequate to resist the postulated stress combinations associated with level A, B, C, and D conditions as defined in References 1 and 2. !
| |
| 1.1 ANALYSIS OUTLINE l
| |
| The spent fuel storage racks are Seismic Category I equipment.
| |
| Thus, they are required to remain functional during and after an SSE (Safe Shutdown Earthquake) (Ref. 3). As noted previously, these f racks are neither anchored to the pool floor nor attached to the side walls. The individual rack modules are not interconnected.
| |
| Furthermore, a particular rack may be completely ' loaded with fuel assemblies (which corresponds to greatest rack inertia), or it may be completely empty. The coefficient of friction, u, between the supports and pool ffoor is another indeterminate factor. According 199 tests performed on to Rabinowicz (Ref. 4) the results of austenitic sta[nless steel plates submerged in water show a mean value of p to be 0.503 with a standard deviation of 0.125. The l upper and lower bounds (based on twice the standard deviation) are thus 0.753 and 0.253, respectively. Two separate analyses are performed for the rack assemblies with values of the coefficient of friction equal to 0.2 (lower limit) and 0.8 (upper limit),
| |
| respectively. The following analyses are performed for the geometrically limiting rack modules:
| |
| I e t-3 )
| |
| | |
| O Fully loaded rack.(all storage locations occupied),
| |
| u = 0.8; 0.2 (p= coefficient of friction)
| |
| Nearly empty rack p= 0.8, 0.2 O'
| |
| O Rack half full, p = 0.8 The method of analysis employed -is the time-history method . The pool slab acceleration data were developed by the Pacific Gas and Electric Company, San Francisco, California.
| |
| analysis .is to determine the The objective of the seismic deformation, rigid body motion, structural response (stresses, application of the three orthogonal etc.) due to simultaneous Recourse to approximate statistical summation excitations.
| |
| techniques such as -the " Square-Root-of-the-Sum-of-the-Squares" f
| |
| i is avoided; for nonlinear analysis, the only method (Ref. 5) practical method is simultaneous application.
| |
| Pool slab acceleration data are provided for three earthquakes:
| |
| Design Earthquake (DE), Double Design Earthquake (DDE), and Hosgri The specifications require design for Earthquake (HE). rack conditions including the DE and using the more severe of DDE or performed on a typical rack module show that HE. Studies displacements and stresses are more severe using the Hosgri seismic subsequent references herein refer to event; therefore, all on the rack structures. The Hosgri seismic imposing the HE time-history has,a peak "g" level that is much higher than the DDE condition. Figures 1.1-1.6 show the time-histories used for the analyses.
| |
| The seismic analysis is performed in three steps, namely:
| |
| dynamic model consisting of
| |
| : 1. Development of a nonlinear inertial mass elements and gap and friction elements.
| |
| | |
| ~.
| |
| n C
| |
| : s. . , .v y
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| ~ '.'~~~~~.'&.'3.m.~same o
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| e.n .,. u N
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| O 2. Generation of the equations of motion and inertial coupling- I and solution of the equations using the " component element time integration scheme" (References 6 and 7). to determine f i
| |
| nodal forces and displaceme'nts. i i
| |
| : 3. Computation of the detailed stress field in the rack (at the critical location) and in the support legs using the nodal-forces calculated . in the previous step. These stresses are then checked against the design limits.
| |
| A' description of the dynamic model follows.
| |
| : 2. FUEL RACK - FUEL ASSEMBLY MODEL Since the racks are not anchored to the pool slab or attached to the pool walls or to each other, they can execute a wide variety of rigid body motions. For-example, the rack may slide on the pool floor (so-called " sliding condition"); one or more legs may momentarily lose contact with the liner (" tipping condition"); or the rack may experience a combination of sliding and tipping ,
| |
| conditions. The structural model should permit simulation of these kinematic events with inhereht built-in conservatism. Since the Diablo Canyon racks are equipped with girdle bars to dissipate energy by inter-rack impact, it is also necessary to model the l Similarly, the lift off l impact phenomena in a conservative manner. 1 of the support legs and subsequent impacts must be modelled using appropriate impact elements, and the Coulomb friction between the rack and the pool liner must be simulated by suitably piecewise linear springs.* These special attributes of the rack dynamics I strong emphasis on the modeling of the linear and require a springs, dampers, and stop elements. The model ;
| |
| nonlinear description in subsequent sections of this report show the detailed modeling technique used to simulate these effects, with emphasis placed on the nonlinearity of the rack seismic response.
| |
| m-u
| |
| | |
| n
| |
| ~
| |
| 2.1 Outline of Model
| |
| : a. The fuel rack structure is a folded metal plate assemblage welded to a baseplate and supported on four legs. The-rack structure itself is a very rigid structure. Dynamic analysis of typical multicell racks has shown that the motion of the structure is captured almost completely by the bahavior of a six degrees-of-freedom structure; there-fore, the movement of the rack cross-section.at any height is described in terms of the six degrees-of-freedom of the rack base.
| |
| : b. The seismic motion of a fuel rack is characterized by random rattling of fuel assemblies in their. individual storage locations. Assuming that all assemblies vibrate in phase obviously exaggerates the computed dynamic loading on the rack structure. This assumption, however, O greatly reddces the required degree-of-freedom modeling the fuel assemblies which are represented by two for lumped masses. One mass is located at.the top of the rack, while the second mass is at the rack base. The centroid of the fuel assembly mass can be located, relative to the j rack structure centroid at that level, so as to simulate a partially loaded rack.
| |
| l the reck-support interface is
| |
| : c. The local flexibility of ,
| |
| i modeled conservatively in the analysis. 1 I
| |
| I rock base support may slide or lift off the pool
| |
| : d. The floor.
| |
| : e. The pool floor has a specified time-history of seismic accelerations along the three orthogonal. directions.
| |
| : f. Fluid coupling between rack and assemblies, and between racks, is simulated by introducing ;
| |
| rack and adjacent J II-4
| |
| | |
| O appropriate inertial coupling into the system kinetic j
| |
| ener'gy. Inclusion of these effects uses the methods of References 9 and 10 for rack / assembly coupling and for coupling (see Sections 2.4 - 2.5 of this rack / rack I report).
| |
| impacts between rack and assemblies are
| |
| : g. Potential appropriate " compression only" gap accounted for by
| |
| (
| |
| elements between masses involved.
| |
| : h. Fluid damping between rack and assemblies, and between rack and adjacent rack, is conservatively neglected.
| |
| : i. The supports are modeled as " compression only" elements for dynamic analysis. The bottom of each support leg is attached to frictional springs. The cross-section inertial properties of the support legs are computed and used in the final computations to determine support leg stre'sses,
| |
| : j. The effect of sloshing can be shown to be negligible at the bottom of a pool and is hence neglected.
| |
| Inter-rack impact, if it occurs, is simulated by gap k.
| |
| elements at the top and bottom of the rack in the two horizontal directions at the corners of the rack. The is most conservative case of adjacent rack movement
| |
| ~ assumed; each' adjacent rack is assumed to move' completely l out of phase with the rack being analyzed. f 1
| |
| i
| |
| : 1. The form drag opposing the motion of the fuel assemblies j in the storage locations is neglected in the- results l l
| |
| reported herein.
| |
| ( m. The form drag opposing the motion of the fuel rack in the also conservatively neglected in the results j water is reported herein.
| |
| II-5 !
| |
| _--_-___-A
| |
| | |
| I
| |
| ~
| |
| i The rattling of the fuel assemblies inside the storage n.
| |
| locations causes the " gap".between the fuel assemblies and the cell wall to change from a maximum of twice the nominal gap to a theoretical zero gap.- Howev e r ', the fluid-coupling coefficients (Ref. 8) utilized are ~ based on linear vibration theory (Ref. 9). Studies in the literature show that inclusion of the nonlinear effect (viz. vibration amplitude of the same order of magnitude the gap) drastically lowers the equipment response as 1 (Ref. 10).
| |
| Six degrees-of- freedom Figure 2.1 shows a schematic of the model.
| |
| l are used to track the motion of the rack structure. Figures 2.2 and' 2.3, respectively, show the inter-rack impact springs and fuel assembly / storage cell impact springs.
| |
| The fuel assemblies are modelled by two lumped ' masses. The lower l mass is assumed to be attache'd to the baseplate and to move with the baseplate. The upper mass .is located at the top of the rack and is f ree to move in a horizontal plane relative to the rack.
| |
| Two degrees-of-freedom are used to track the motion of this mass.
| |
| 2.2 Model Description (8 DOF Model)
| |
| The absolute degrees-of-freedom associated with each of the mass locations i, i* are identified in Figure 2.1.
| |
| 4 S
| |
| | |
| j)
| |
| O Coupling Elements 2 Z.., ._ -
| |
| / ./ 'e
| |
| -T*
| |
| /, 2
| |
| =
| |
| As {
| |
| 1'
| |
| . 1 ( .)
| |
| 97 l 1, l -
| |
| ; i
| |
| % H a
| |
| l "I'
| |
| 1 _
| |
| AY =
| |
| L I
| |
| ( l t
| |
| I 3 , ,
| |
| S3 AX 1.
| |
| j , '! / y L- 4,
| |
| / d.v - '
| |
| i*'Xi- .)r N.*
| |
| / t 44 d '--1 _ . .
| |
| t-3- qs u Sp p o r t ,4 s,, -
| |
| i\ (3 S1 S2 + ; !
| |
| J
| |
| : r. 7/ '/) ,977 *~~'~ (
| |
| / ,
| |
| x n XB, YB - Loce.ticn of f u ?! ro : .
| |
| q, . group mass c.:ruroid - rc!:tti <e . ; ' .
| |
| centerlinc of fus.! rc.ck I
| |
| l i
| |
| t O .
| |
| .r....-.t e. ..' c. - 'm ., .1 Dynam;c ModeI
| |
| | |
| ./f
| |
| .. , A
| |
| .i, e
| |
| O .
| |
| Tyn. Ten imnect
| |
| ,.g Element -g
| |
| .E E '
| |
| fvA A v-l T
| |
| l w.
| |
| f s
| |
| f:
| |
| 1 Rack Structure f .
| |
| Typ. Bottom impact k E!ement s. g
| |
| / C Ww
| |
| . ps,ev
| |
| , y Ws / +h hfm. r f
| |
| 3 mr 1
| |
| i l
| |
| ( \
| |
| I
| |
| - FIGURE: .G.2 G.m Elcmento Tc Sin u'aie I Qa < inior- rcck Ir.:pacic i
| |
| o_-----_-__-- ___ - - _ _ _ _ _ _ _ - _ _
| |
| | |
| l l j
| |
| ,-~ .
| |
| \;
| |
| I 1
| |
| J Irnpact Springs y
| |
| hh l
| |
| 4 n,
| |
| {
| |
| ~
| |
| E- Mass
| |
| -ww-i 2* -0 ba T O a'
| |
| ~7% F!u:d
| |
| -/ dampers '
| |
| i
| |
| ,/e .
| |
| (001 USGil .ta t h 's .?. n a l u ';. ! ':. .
| |
| ' Frame 1
| |
| * \
| |
| . _ _ - - __.c_ ;;
| |
| FIGURE .2.3 Impact Springs And Fluid Damperu 1
| |
| .i j
| |
| | |
| W--,p I
| |
| f 1
| |
| \
| |
| l 1
| |
| Table 2.1 f ('] -
| |
| ]
| |
| C/ DEGREES OF FREEDOM 61splactmeat P.o ta t io n e s e u u a y g Location x y x (Node) pz p5 qw q3 Qe 1 p1 1* Point 1* is assumed fixed to base at X,Y,2=0 2
| |
| Point 2 is assumed att. ached to rigid rack at the top most point. 1 pe 2* py p = q (t).+.U (t) i i , i
| |
| ._ 1 1
| |
| U i(t) is the pool floor slab displ'acement seismic time-history.
| |
| Thus, there are eight degrees-of-f reedom ' i.n the system. Not shown V in Fig. 2.1 are the gap elements used to model t.he support legs and the impacts with adjacent racks.
| |
| An effect of some ' significance requiring careful modeling is the g so-called " fluid coupling effecc." if one body of mass (ml) vibrates ad j acettt to another body (mass m2), and both bodies are submerged in a frictionless fluid medium, then Newton's equations of motion for the two bodies have the form: 1 1
| |
| H (ml + Mll) X1 - M12 2 = applied forces on mass mi i
| |
| ~M21 X1 + !.rt2 + M22) X2 = applied forces on mass m2 accelerations of mass mi and m2i
| |
| 'X 1, E2 denote absolute l
| |
| respectively. !
| |
| and are fluid coupling coefficients g M11, til2, M21, M22
| |
| ,] shape of tne tvo bodies, their relative V which depend on the II-7
| |
| | |
| I 1 L
| |
| disposition, etc. Fritz (Ref. 6) gives data for Mj for various i
| |
| body shapes and arrangements. It is to be noted that the above equation indicates that the effect of the fluid is to add a certain amount of mass to the body (M 1 1 to body 1), and an external force which is proportional to the acceleration of the adjacent body (mass m2) . Thus, the acceleration of one body af fects the force field on another. This force is a strong function of the interbody gap, reaching large values for very small gaps. This inertial coupling is called fluid coupling. It has an important effect.in rack dynamics. The lateral motion of a fuel assembly inside the storage location will encounter this effect. So will the motion of a rack adjacent to another rack. These effects are included in the equations of motion. The fluid coupling is . between nodes 2 and 2*
| |
| in Figure 2.1. Furthe rmore , the rack equations contain coupling -
| |
| terms which model the effect of fluid in the gapc between adj' scent racks. The coupling terms modeling the effects of fluid flowing between adjacent racks are computed assuming that all adjacent are vibrating 180* out of phase from the rack being racks l
| |
| analyzed. Therefore, only one rack is considered surrounded by a hydrodynamic mass computed as if there were a plane of symmetry located in the middle of the gap region surrounding the entire rack.
| |
| Finally, fluid virtual mass is included in the vertical direction vibration equations of the rack; virtual inertia is also added to the governing equation corresponding to the rotational degree-of-f reedom, 9 6( t) .
| |
| 2.3 Development of Model Equations of Motion The rack is modelled as a rigid body governed by a general six-degree of freedom system. Figure 2.1 shows a schematic of the model. A lumped mass at height H and location X,Y represents a portion of the fuel mass that is assumed to rattle. The rattling lumped mass has two horizontal degrees of freedom. Its vertical motion is assumed to be the same as the rack motion at that location. The fuel mass not involved in rattling is assumed fixed to the rack base.
| |
| 11-8 .
| |
| | |
| 1
| |
| )
| |
| , t
| |
| /~' . 1 A certain portion of the fuel assembly mass is assumed to !
| |
| impact with the rack structure. The rattle and to possibly remaining portion of the fuel assembly mass is assumed fixed to the rack base, and is assumed to move with the rack base. We assume j that fluid virtual masses can be computed using Fritz's model (Ref. 6], and that all fuel mass moves in phase so that we consider the entire rack planform as a single quadrant. 4 i
| |
| l B. . Kinematics i
| |
| : 1. Velocity of Center of Base p " qi n 2
| |
| + 92 n2 + q3 n3 l
| |
| i
| |
| : 2. Velocity of Rack Mass Center P*_
| |
| t-%
| |
| N yP*
| |
| (91 + H- 4 5) n 1 + (q. g- H g) g2 + Q3 6 3
| |
| : 3. Velocity of Rack at Centroid of Rattling Fuel l tat top of rack) l V R= (q t + H q s - Y 4 6) ni+ (q2- H 94 + x 9 6) n 2
| |
| + (d 3 + y d4 -
| |
| x d s) n3
| |
| : 4. Velocity of Rattling Fuel Assembly Mass at x,y,H !
| |
| = - * -
| |
| * C x 9 5)n 3 V = q7n1+q gn2+ (9 3 + Y G4 II-9
| |
| | |
| C. Computation of System Kinematic Energy and Governing Equations
| |
| : 1. Fuel Rack and Rack Base and Feet 2T 1=MB +M + II xx +I xx I R
| |
| + (I yy +I yy ) d5 + (I +I z) 6 M'MR B represent the mass of the base and the rack, and R B I
| |
| xx t yxx , etc., represent the appropriate inertia properties -
| |
| with respect to point O.
| |
| : 2. Fluid Mass and Vibrating Lumped Masses We define the following quantites:
| |
| Y* W W *^* - - - W l
| |
| u, =
| |
| -s A W q ML) p m= - % a _ a cm a u y ,
| |
| 4 a _. _uu )
| |
| x;.~gL_o-agp A M &A 6 1L 4 pan ~+A1LJ%
| |
| wl=mofLA y 4 & w~_=l6 m w tL. M % w A 4 Ea J k O
| |
| : m. p + ga.s w. -
| |
| p a sa ZZ-1@ _ _ _ ____i l
| |
| | |
| meas
| |
| ^m$ A^
| |
| 1 L A_
| |
| u A J M. *
| |
| * 6:
| |
| -2 *
| |
| * t v . . t 2T 2 . * (9,2 + 9 , ) + a ., ( 7, 3 7J + % (y x 7 3 1
| |
| . . . . 1
| |
| + V 3 4~ f5 1
| |
| + <n b
| |
| +
| |
| 2dn ], (7t
| |
| +
| |
| Hfs ffs)
| |
| +
| |
| * n ( 9, + H .73 y 7, )2 )
| |
| + e., 9, +
| |
| 1 x,e 7, ( y - H7, t $ 7, ) -
| |
| Ah
| |
| , r , / / .- 1 L) m '
| |
| a: u ti,. m -
| |
| #0. - . E .D - ~ J ff.
| |
| 1I ,
| |
| L ,. 1. L . - ~ u A , f, Al- ) J k . ff, f ,9 ,,,JL: 1 o
| |
| .~
| |
| (-
| |
| ~~/<-
| |
| R
| |
| /
| |
| a, ) N. _> - > A A d A- -o ? w.? us VA. U 0 E.a0 1
| |
| j _
| |
| f
| |
| -.M L , ea L aif. J d .:JLX~J[
| |
| ~
| |
| . w w -,
| |
| ~
| |
| a 1 takN 3 uN$ t:" _ 2.s. 8 2
| |
| . . Eh __h .
| |
| ?1 :l % _s p-i A M J g dy == %
| |
| L L (L A & XL; LA e= L A R;7 l
| |
| WlaAty A K .~ L ) j Q y n, o u e aquKL/%fyJA } -~A .
| |
| l na
| |
| | |
| I y man ? mnAlk /fy_.pAuso h e W
| |
| /
| |
| ~
| |
| [ R M f_
| |
| tw lh f[
| |
| &w b d/ 1 Y f M'e _
| |
| 7
| |
| $*- a',, x bla
| |
| .. ,' d., e-s
| |
| - (U.n.
| |
| +
| |
| M i .) ,
| |
| * t's ,
| |
| * M, 4 h
| |
| The effects of the fluid adjacent to each rack outer wall \
| |
| {
| |
| are assumed to be represented by a virtual mass of fluid at the I level of the rack mass center. Denoting the fluid kinetic energy associated with this virtual mass as Tout and using the Fritz model to compute the virtual mass, we have i 2 Tout = B1 (di + 5) + B l$ (42 - S 44) 2 2
| |
| + 2B j (di + E ds) 0 i
| |
| 2 1
| |
| + 2B $ (d2 - E d4) 0 2 i
| |
| t 2 '
| |
| +B) 2 02+ B50 1 2 2 Bj i
| |
| the are fluidthe fluid coupling surrounding the rackcoefficients being studied.simulating the effect of
| |
| {
| |
| Ul (t), U 2( t ) ,\
| |
| are the input seismic displacements in the 1,2 directions, respectively. Forming the total kinetic energy T as the sum of rack, base, vibrating masses, internal fluid virtual mass, and external fluid virtual mass components yields !
| |
| the governing equations of dynamic motion in the form: ;
| |
| d ST
| |
| ( BT ) =0 i = 1,N dt 91 34 aqt l
| |
| p Since T = T(d), only the first term on the left hand side Q contributes.
| |
| l XV-12
| |
| | |
| Oy ^ % " ~ W'm A 4'%
| |
| ar.(u,+unq,+uy4,+a.zi,+<~t-aNi,+ui,gi.) n pg' .
| |
| + ml(q,y4.)+e,:(f,+yis)+elz0, l l
| |
| c a. . ua j ,
| |
| * u n e f, + v.1 j, + (ut- <.t)(pz - u py + #f.)
| |
| M'9*+=m; (4t + xi.) + e,? (it -B;jd+e,?o*z 3r =
| |
| +
| |
| *v. (p + f 9y - # fs )
| |
| . (Mo + Ms) 93 b
| |
| 2T , ( r + r: W "q (iz- fiv) - d 4 (it -g iv) vu " '
| |
| - G,,8 5e - d,s H f, - (M - 4t)H (fs .H hv + 4 hs) t
| |
| +~< .(93+g9r-17s)g i (If + r 'a u ) is + "T(fi + 14s) + e l (f, + 1 is)
| |
| 'f =
| |
| + 65 { 0, + x,, H 7, + (u,- e,e) H (f, + H f5 y fu )
| |
| - *l lis + yiv - 455 )f 2[ = ( ra. + Iz' ) f, - x,t g j, - (ut-<,tyy (p, a f, y p )
| |
| 4 x,e p + 4(u - a,e)(7, - H fa + $fs }
| |
| + ;
| |
| (1 + 4 [v (fu+ ri.) y (i,-y f.)]
| |
| _ - - - - - _ - - _ _ _ _ _ _ _ _ _ _ _ _ _ _ . 31-13
| |
| | |
| 1 1
| |
| /3 (J . . . .
| |
| 21 -
| |
| (m + e,n 9, + e,z ( 9, + u 7s g 7<. )
| |
| 3, 1 rr . ( m + a,Q cp + % ( 9, - u 7,, + 4 7. )
| |
| di, NL tobef4 fg M a. AeL o~e
| |
| &: A. f i i,1,3,7,8 (Le. 7; n 7; + 0; )
| |
| u-Le 9; - - 4 L/ W A & &
| |
| tl $>d M , TL,a % /a; &
| |
| klw.4 4 z. m c ht, s /A ,
| |
| p & & of'4. u.Jj' -
| |
| l M g + M e + H z - oc,z + ml + B,,' . 9, ,
| |
| + W,z 9,
| |
| : i. -
| |
| I I i= .
| |
| # ee y se
| |
| + L u qi + u - s + e., /2 ] u 9, -
| |
| y (u ,- e,z , m ) 7,,
| |
| l = 07, - [u, + na + e,,. + m - e,, + nl + e,, + e,; ] b,*
| |
| ,. .. *. I M g + Me +Mt ~ X,e
| |
| * M! + 0,i +
| |
| l1 W' t fa NRf*2 + L " W,t S hog,lQ l$ $ NL" Y ll Y /1 (,
| |
| n *
| |
| % =
| |
| Q -
| |
| [ u , e u, + ut + i.e e 8,,* + e,? ] o e l l E___ ___. _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ . _ _ _ 2Ro1A. _ _
| |
| | |
| ( .
| |
| G , .. . .. ..
| |
| 3
| |
| ( Me + Ms < ><,)]..g & M e y 7v ~ Av 4 cf5 ' 9, 7 - (Hs + Ha + <) u, (rL rl, + m uy. e.' ny, + a ym-e..) + wl g' ) y, l
| |
| -n(mie+8,>Vt+m-x,)ft - u (g[m-*,,1) f. + g*l(j; y };)
| |
| e,, j', H 9 7, - y el di + ["% + (e.".e,?Jh + He ]H bh (r'+rf+m8Py*a:ayy n + 8 c(u,-e,,) + m; x') j',.
| |
| e H (my el/s + uz -e,e) f, - g n (nt-r,<)f - yml(ll + y };,)
| |
| +*1 H f *, = Qy, + 4 ml U, - [ "% + (ol+ s. )h +"t] H b* i 3
| |
| (v) -
| |
| I t,s + I,.2, + (m- a',s)(x *+2 3 ) + wl(x *g'), 9, - p x,e f.,
| |
| e
| |
| * g g,e 7,
| |
| - H (m-<,1)( y is + 4 f,) + (us - e,0(1f, - y y,)
| |
| > #~ (x p - y 7, ) =
| |
| 97, + (*:+ ut )(y o, x ot )
| |
| $ + W,1 , f Weg ,
| |
| + g, q & 3 - W,g b
| |
| ~ Q7, . - (m + % + x,< ) d,'
| |
| .. .. e. .. i
| |
| $ + Wel lg & W, L z
| |
| ~~
| |
| ,e fy + W,t f g
| |
| = 9 7,. -
| |
| ( ~ x,, r n ) 5, O) n-u
| |
| | |
| ~
| |
| '~
| |
| lO 1
| |
| IQ 2.4 Coupling Between Fuel Rack and Fuel Rod Group Figure 2.4 shows a typical cell of square cross section.
| |
| 1 l
| |
| We use the Fritz model for concentric cyliriders ' and define equivalent radii R 1 , R2 as R"
| |
| 2 //s ,
| |
| ; Rg =*
| |
| /6 For a unit of length H, we obtain, for the entire rack model M1 = w p, H R a = number of fuel cells f f a
| |
| containing fuel rods O') M2 = 5 0, H R n c n e = total number of cells If the gap g1 = (b-a)/2, then Fritz gives, for the hydrodynamic mass, the expression R
| |
| 'Hg ", y/(1 + 12 R /H )
| |
| Therefore, since we are assuming fluid coupling between rack and rod groups at a single location, we assign the following values to a t t, a 12, a2 2 for the rattling mass at the rack top as:
| |
| )
| |
| O J
| |
| | |
| a22 = $ (M 1 + M 2 + Mg )
| |
| all = $M H al2 = $ (M i+M) H can be taken as .375. The It is shown later that t non-rattling fuel mass fraction is assumed fixed to the rack base.
| |
| 2.5 Coupling Between Fuel Rack being Modelled and Adjacent Structures All. of the fluid inside the ' rack has been counted in the from the coupling analysis above. We now .must contributions consider the effect of the fluid between the rack being studied and For this simulation, we use a modified adjacent structures.
| |
| In version of case 13 of Fritz's reference to model the effect.
| |
| Fig. 2.5, the hydrodynamic mass term for this contribution can be j
| |
| written as i
| |
| ( '
| |
| }
| |
| MH" 12t gg g2 where Mi = water mass displaced by the rack as a solid body l
| |
| (M1 = Aw W H t) 1 l
| |
| = water density l ow l t,w = shown in Fig. 2.5 g 1,9 2 = Gaps as shown in Fig. 2. 5.
| |
| i O ,
| |
| g_g
| |
| | |
| 1 I
| |
| 'I O In Fritz's work, $= 1; it may be taken as $= 1 + W/H, to approximately account for the fact that fluid can escape around all four edges, rather than only two edges assumed in Fritz's work.
| |
| Therefore, in our model, we evaluate the constants Bij as B 1g=M;H B 12 = (M i+M} H Consider the rack configuration in planform view as shown in Figure 2.5. The values g 1, 92 for computation of fluid virtual mass are computed based on consideration of a single rack surrounded by or by fixed po61 walls. In other racks (moving out of phase) general, g i and g 2 for x-direction motion will be different from those in the y-direction motion.
| |
| i
| |
| - l l
| |
| i I
| |
| I i
| |
| j O
| |
| i 11-18 l I
| |
| i
| |
| | |
| I O
| |
| Blank page O .
| |
| i l
| |
| 1 I
| |
| l e
| |
| I; O l
| |
| )
| |
| 9 i
| |
| - XX-1@.
| |
| | |
| l
| |
| {
| |
| 2.6 guid Mass Effescts in Vertical Direction The added mass oa the base plate during motion in the vertical direction is modelled by case 6 of Fritz's reference.
| |
| 2 Mg = o g (AX) (AY) where K = K(AY/AX)
| |
| K [4 To simulate the effect of fluid inside the rack during vertical ;
| |
| motion, we add vertical mass to the rack equal to the mass of fluid l displaced by the metal volume of the rack (not including the base plate). ;
| |
| Effect of Fluid on Rotational Inertias Iij l
| |
| O. a. Base Plate: The total effective mass of the base plate of thickness t* is given by 9
| |
| 3
| |
| = metal mass density ,
| |
| p,= water mass density l
| |
| *
| |
| * 2 l
| |
| M e
| |
| = p e
| |
| AXAY t = o g AXAY t +K* p y AX AY 4
| |
| = [p s + P w ( )) AXAY t The above expression will define the effective base plate density p e which will be used to calculate contributions to the rotatienal inertias. Therefore, the j incremental contributions to tne rotational inertias due.to the-base plate are given as:
| |
| i II-20
| |
| & -____._____.__________w
| |
| | |
| o /
| |
| 4 n - - -
| |
| 't* AY31" g[k , P
| |
| . iAY t* AX )
| |
| ' . , p e:(AX 12 '
| |
| g e 12
| |
| * 6 -
| |
| iAXAY+Ay'3Ix}
| |
| ~
| |
| N 3h "#. (3 ,' fluid effect) 3
| |
| : b. Contributions from Fluid Inside Rack we assume that all of the fluid inside the rack gridwork moves with the rack insofar as contributions to rotational inertias are J Concerned.
| |
| O 2.7 Calculation of Rack Inertia Properties We now compute the inertia of the individual rack plates with respect to the mass center of the rack (assumed to be - alcng the vertical centerline at height H/2). Consider Fig. 2.6 which shows typical rack plates aligned along the x and y axss of the rack.
| |
| assume the origin of For the subsequent computations, we coordinates at the rack centroid P*.
| |
| The figure shows a typical panel aligned .along either the x axis or the y axis. If the rack has NX cells in the x direction l- and NY cells if the y direction, then the rack configuration has NX x axis and NY + 1 plates
| |
| + 1 plates perpendicular to the l
| |
| perpendicular to the y axis.
| |
| The total inertia of the rack is computed by adding up the
| |
| ' individual inertias of the panels after applying the parallel axis theorem.
| |
| 88-23 _ _ _ _ _ _ _ _ - _ _ _ -
| |
| | |
| }
| |
| O ,
| |
| / /
| |
| / ,.
| |
| //// - - / ;7 h
| |
| / ,
| |
| / /
| |
| I
| |
| ~a-=- b
| |
| ,/
| |
| /
| |
| U
| |
| '///., ,
| |
| /,
| |
| O a= out sid e dimension of f uel assembly b= inside dimension of f uel rack cell l
| |
| l l
| |
| l i
| |
| l FIGURE . 2.t - !
| |
| -I O ,
| |
| 1
| |
| | |
| 'l l
| |
| O .
| |
| = VIBRATION [/
| |
| /
| |
| /
| |
| /
| |
| / [ '
| |
| / /
| |
| ' /
| |
| /
| |
| /
| |
| p' W / \
| |
| I
| |
| , /
| |
| / /
| |
| / /
| |
| / /
| |
| * ' l . Y /
| |
| h l
| |
| ' /
| |
| W 91+ f ~92 *'' -
| |
| O ,
| |
| . l FIGURE 2.2 O
| |
| | |
| I I
| |
| I a
| |
| lz Ax A
| |
| /
| |
| /x Y
| |
| H / 's' \ - H C% / N
| |
| \
| |
| /
| |
| / \
| |
| (xJ,o) -
| |
| (o,yj) mx my l
| |
| l NX + 1 NY + 1 PLATES PLATES !
| |
| l FIGURE ' _i,6 -
| |
| O e
| |
| | |
| l l
| |
| O c t. of mx plates'about their local centers.
| |
| l m i I
| |
| xx
| |
| =2 (AY 2+H 2) 12 j
| |
| = 2 I H YY 12 m*
| |
| I = AY 2 12 1
| |
| tias of my plates about their local centers l l
| |
| = H2 /
| |
| I** 12 l
| |
| =
| |
| (AX2+H 2) 12 I gg
| |
| =y AX 2 l 12 - - - - - -- -
| |
| l efore
| |
| * NX+1 NY+1 m y k' m*
| |
| I xx
| |
| = [ [ AY 2+H 2] + y { H 2+m yk Yk }
| |
| j=1 12 k=1 12 NY+1 2
| |
| + [ mY yk k=1
| |
| | |
| O Similarly, ,
| |
| * NX+1 m NY+1 m I =
| |
| - [ -( l H 2,,
| |
| x 2
| |
| : 3) + [ l (AX2+H) 2 YY 12 12 or
| |
| * m NX+1 i
| |
| 'I P
| |
| =dH2 (NX+1) + g (NY+1)(AX2+H) 2
| |
| *k=1
| |
| { "x*k I YY 12 12 Finally,
| |
| * NX+1 m NY+1 m I,[ = [ (
| |
| * AY 2 , ,gx5 ) , y 12 2
| |
| ( ,,_y.
| |
| 12 A X2 + "y Yk I or
| |
| * m 2
| |
| m NX+1 ' 2
| |
| , J gy (NX+1) + E. . AX 2 (NY+1) +
| |
| I*P* 12 12 j=1[ m*x$
| |
| )
| |
| 1 NY+1 2
| |
| + [ m y
| |
| y k
| |
| k=1 The above formulations 'are internally computed in the algorithm l once NX, NY, AX, AY are given. As noted in the previous section, I
| |
| when the rack 'is simulated by a single rigid body, the fluid moments of inertia (for the rotational degrees of freedom) are ,
| |
| the fluid moves .with the rack. The :
| |
| computed assuming that 1 inertias are added to the i incremental fluid rotational contributions from the rack metal. i O
| |
| I RSLca9LS
| |
| | |
| 2.8 FORCE AND MOMENT RESULTANTS AT RACK BASE For considerations of rack structural integrity, we need to determine the resultant forces and moments . acting on the rack at-its attachment to the base. To obtain these values, we consider a free body of the rack base together with the feet. One foot is shown in Figure 2.7 with the reactions from the ground, and the reactions from the rack.
| |
| WTB is the dead weight of the. base plus the four feet, and the typical foot shown is assumed located at position xi, yi with respect to 0. The foot length is hi, i = 1,2,3,4.
| |
| i Force equilibrium equations for. the free body given previously are:
| |
| M B 1"Yx+ [F xi M
| |
| B 2=Vy+ [F yg M
| |
| B 3=N z
| |
| + Pgg - WTB where WTB = Mg B and q1, i = 1,2,3 are the transnational degrees of freedom at 0 the center of the base plate.
| |
| Therefore, the resultant forces Vx, V, y and N2 are determined, at any time t, by the equations V =M(B 1 + 1) -
| |
| F xg V
| |
| y
| |
| =M B
| |
| ('b 2 +b) 2
| |
| [F yi N g =Mq-B Fgg + M g (N 3 + 'U3) i 1
| |
| l X 3 "'NO - ___ _ - _ _ _ _ _ _ _ _ _ _
| |
| | |
| ~
| |
| i l
| |
| O .
| |
| o Nz Mz j Mx og r Vy Vx l %y O '
| |
| I t h; I "
| |
| l Fxi r F;y _
| |
| (mg R , mi x y
| |
| 1 Fzi l 1
| |
| FIGURE 23 7 l
| |
| O i
| |
| | |
| i 1
| |
| D i C/ In the- above relations, we have introduced .the rack transnational degrees of freedom ni(t), whe'_ce.
| |
| I q g(t) = ng (t) + U f(t) i = 1,2,3 and Ug (t) are the seismic motions.
| |
| kf we now write rotational equilibrium equations along the x, y directions, we obtain the following equations to determine Mx, My.
| |
| ** 4 B
| |
| M, = Ixx 94 - g,k ("xi + Y i F,g + h Fyg) ;
| |
| )
| |
| l 4
| |
| My " I'y ci s - (my g - xg g Fg-hFxt)
| |
| The dynamic analysis code is designed to compute the force and moment resultants at each instant of time. These values are thena used to compute the rack stress factors Rt (discussed in For structural analysis purposes, the separate section later).
| |
| logic of the computer code is set up so as to keep track of the l
| |
| 1 maximum value of each of the force and moment components.
| |
| 2.9 DAMPING reality, damping of the rack motion arises from material In hysteresis (material damping), relative intercomponent motion in (structural damping), and fluid drag effects (fluid l structures '
| |
| damping). In the analysis, a maximum of 4% structural damping is seismic the rack structure during HE imposed on elements of accordance with the FSAR and NRC simulations. This is in i
| |
| II-25 l
| |
| - - - .__ - --__ _ - - _ L
| |
| | |
| O guideline's (Ref. 11). Material damping is conservatively neglected. The dynamic model has the provision to incorporate fluid damping effects however, no fluid damping has only been used in this analysis to model the effects of dissipation by the fluid j
| |
| inside empty racks during a seismic event. We will discuss the l
| |
| damping model used in a subsequent se'ction.
| |
| Impact The fuel assembly node 2* may impact the corresponding structural mass node 2. To simulate this impact, four compression-only gap elements around the upper fuel assembly node are provided (see As noted previously, fluid dampers may also be Figure 2.3).
| |
| The compressive loads provided in parallel with the springs.
| |
| developed in these springs provide the necessary data to evaluate the integrity of the cell wall structure and stored array during the seismic event. Figure 2.2 shows the location of the eight any potential for inter-rack impact springs used to simulate impacts. Section 6.4 gives more details on these additional impact springs.
| |
| a
| |
| : 3. ASSEMBLY OF THE DYNAMIC MODEL !
| |
| l The cartesian coordinate system associated with the rack has the following nomenclature:
| |
| O x = Horizontal coordinate along the short direction of rack i rectangular platform ;
| |
| O y = Horizontal coordinate along the long direction of the rack rectangular platform O z = Vertically upward I
| |
| As described in the preceding section, the rack, along with the l base, sapports, and stored fuel assemblies, is modeled for the three-dimensional ( 3-D) motion simulation by an eight-general L-________-____ - _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ ___
| |
| | |
| I To simulate the impact and sliding degree-of-freedom model.
| |
| phenomena expected, 24 nonlinear gap elements and 16 nonlinear friction elements are used. Gap and friction elements, with their.
| |
| conne'ctivity and purpose, are presented in Table 3.1.
| |
| restricted to two dimensions (one If the simulation model is vertical motion, for example) for the horizontal ~ motion plus of purposes of model clarification _ og, then a descriptive model necessary gap and the ' simulated structure which includes all friction elements is shown in Figure 3.1.
| |
| The impacts between fuel assemblies and rack show up in the gap local stiffness KI, in Figure 3.1. In Table elements, having these elements are gap elements 7 and 8. The support leg 3.1, spring rates K 6 are modeled by elements 1 and 4 and 2 and 3 in Table 3.1 for the 2-D case. Note that the local compliancesliding of the floor may be included in K To simulate concrete 6 potential, friction elements 2 plus 8 and 4 plus 6 (Table 3.1) are shown in Figure 3.1. The friction of the support / liner interface is with a suitably large modeled by a piecewise linear spring stiffness Kg up to the limiting lateral load, pN , where N is the the interface between support and current compression load at the liner. At every time step .during the transient analysis, current value of N (either zero for liftoff condition, or a compressive finite value) is computed. Finally, the support rotational friction springs rlg reflect any rotational restraint rate is that may be offered by the foundation. This spring
| |
| : 4) and is calculated usin,g a ' modified Bousinesq equation (Ref.
| |
| the resistive moment of the support to included to simulate This counteract rotation of the rack leg in a vertical plane.
| |
| rotation spring is also nonlinear, with a zero spring constant value assigned after a certain limiting condition of slab moment ,
| |
| 1 loading is reached.
| |
| The nonlinearity of these springs (friction elements 9 plus 15 and 11 plus 13 in Table 3.1) reflects the edging limitation imposed i II-27
| |
| | |
| e "
| |
| 'l r
| |
| n l l
| |
| Tacle 3.1 I l.
| |
| ELEMENTS .,.
| |
| NUMBERING SYSTEM'FOR GAP ELEMENTS AND FRICTION 41
| |
| ['
| |
| I, Nonlinear Springs (Gap Elements) (24 total) n
| |
| . Description Node Location
| |
| ~
| |
| Number Support S1 2 compression only element 1
| |
| Support S2 2 compression only element 2
| |
| Support S3 2 compression only element 3
| |
| Support S4 Z compression only element i
| |
| 4 2,2*
| |
| X rack / fuel assembly impact element 5
| |
| 2,2* X rack / fuel assembly impact element 6
| |
| 2,2* Y rack / fuel assembly impact element 7 Y rack / fuel assembly impact element 8 2,2*
| |
| Top cross-section Inter-rack impact elements 9
| |
| of rack (corners) Inter-rack impact elements.
| |
| 10 Inter-rack impact elements 11 Inter-rack impact elements 12 Inter-rack impact elements 13 Inter-rack impact elements 14 Inter-rack impact elements l O 15 16 ,
| |
| Inter-rack impact elements Bottom cross-section Inter-rack impect elements 17 Inter-rack impact elements 18 of rack (corners) Inter-rack impact elements 19 Inter-rack impact elements 20 Inter-rack impact elements 21 Inter-rack impact elements l 22 Inter-rack impact elements 23 Inter-rack impact elements 24 II. Friction Elements (16 total) i Description Number Node Location Support S1 X direction support friction 1
| |
| 2 Support S1 Y direction friction X direction friction 3 Support S2 4 Support S2 Y direction friction 5 Support S3 X direction friction 6 Support S3 Y directic.i friction 7 Support S4 X direction friction 8 Support S4 , Y direction friction l 9 S1 X Slab moment Y Slab moment i
| |
| 10 S1 I 11 S2 X Slab moment l 12 S2 Y Slab moment :
| |
| X Slab moment 13 S3 14 S3 Y Slab moment '
| |
| i 15 S4 X Slab moment s 16 .54 Y Slab moment _
| |
| II-28 . . _ - _ _ - _ _ - _ _ _ L
| |
| | |
| (D i V ,
| |
| i on the base of the rack support legs. In this analysis, this effect is neglected; any support leg bending, induced by liner / baseplate by the leg acting as a beam friction forces, is resisted !
| |
| cantilevered from the rack baseplate. i l
| |
| i The spring rate K 6 modeling the effective compression stiffness !
| |
| of the structure in the vicinity of the support, is computed from I the equation:
| |
| 1
| |
| = . 1_ + .1_ +1 K
| |
| 3 Ki K2 K3 where:
| |
| l of the support leg treated as a K<= spring rate tension-compression member =
| |
| ESUPPORT ASUPPORT/h
| |
| ['- (h = length of support leg)
| |
| ~
| |
| j i
| |
| 2 spring rate of pool slab Kz= 1.05EcB/(l-v ) = local length of (Ec = Young's modulus of concrete, and B =
| |
| bearing surface)
| |
| K3= spring rate of folded plate cell structure above support leg (same form as K 2 with E chosen to reflect the local i stiffness of the honeycomb structure above the leg) l 3-D sintalation , all support elements (listed in Table For the model. Coupling between the two 3.1) are included in the horizontal seismic motions is provided both by the offset of the the rotation of the fuel , assembly group centroid which causes entire rack and by the possibility of liftoff of one or more support legs. The potential exists for the rack to be supported on one or more support legs or to liftoff completely during any instant of a complex 3-D seismic event. All of these potential I
| |
| events may be simulated during a 3-D motion and have been observed O in the results.
| |
| (.) l 11-29
| |
| __c
| |
| | |
| h l
| |
| l' O
| |
| V i '-
| |
| Table 3.la (Refer to Figure 3.1)
| |
| RACK 8x8 6xil 10x11
| |
| .127 x 10 66 (x direction)
| |
| K 1
| |
| .233 x 10 6 .744x10[(x)
| |
| .260 x 10 (y) .154 x 10 (y direction) 7(#/in) .
| |
| 7 7
| |
| 7 .1 x 10 .1 x 10 K,(#/in) .1 x 10 .i 10
| |
| .2284 x 10 10 .1128 x 10 10 .1128 x 10 Kf (#/in) 7 7' 7 .515 x 10 .515 x 10 K6 (#/in) .534 x 10 O K R
| |
| (
| |
| ") .1284 x 10 10 0.
| |
| O.
| |
| rad 3.5 7.25 7.25 h (in.)
| |
| 165 165 H (in.) 165 1
| |
| Nominal gap values for impact springs between fuel assembly and cell wall are .151" on each side.
| |
| 2 Gap values used are the approximate distances between the wall and the girdle bar, or 50% of the spacing between girdle bars and adjacent racks. 'For a rack completelyFor surrounded the byrack 10x11 otherthe racks, gap the . gap for Kw springs is
| |
| .125".
| |
| spacings taken from the Joseph oat Layout drawing are .125" for all .l impact springs on walls adjacent to adjoining racks, and 4.13" and l 6" on the two sides of the rack that face on pool walls. For the H l (8x8) rack, the values for the rack-to-rack impact spring gaps are l
| |
| .125" on the two walls facing other racks, and 3" and 13.4" for the i two rack sides that face on pool walls. (Run ttl5 uses a gap l O_ reduced by 1.06" on the two rack sides that face on pool walls.)
| |
| II-29a e-_____---__ _ _ - _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ ___
| |
| | |
| m_ _ .
| |
| l' ,
| |
| < ? t h .) ,
| |
| " ..l
| |
| * a !
| |
| ' , e (f -
| |
| t.
| |
| /
| |
| DETAILED REMARKS.ON NON-LINEAR' SPRING RATES AND COUPLING o.,
| |
| 7' -
| |
| ,/.<3.1 COEFFICIENTS
| |
| /r 3.1.1 Impact Elements between rack and rattling ~ fuel' assemblies
| |
| /..
| |
| j ,<
| |
| ' A. Kinemati_c j
| |
| The n 1 and E 2 velocity components of a point on the I
| |
| rack at level z and location x,y from the rack i centerline are given as )
| |
| , . 'i
| |
| _R =
| |
| (qt + z q s - Y9 6) n i + (q 2 - 29 4 + Xq s) n 2 t.. -V
| |
| :[
| |
| J f.
| |
| Therefore , ,
| |
| _ R- ,, , , ,
| |
| V . n 1 = q i + zq 5 - Y9 6
| |
| } +H 2 " 4 2 - zd s + xq d;. I 9
| |
| /
| |
| The / velocity of the vibrating mass, . impacting the
| |
| / rack wall at the same location is (typically) g Q _M V .
| |
| n1=gg
| |
| . .M V
| |
| .n2*qg f
| |
| Therefore,.the impact spring relative velocities 8, 8 are e
| |
| 3 y i .t M R T" -E ) *E l = ig; - (il + Zd 5 - Yd 6) 5 x
| |
| = (E \
| |
| j ,
| |
| M R /
| |
| 't j S = (E '9 ) $$ 2' a dg . (d 2 - Zd 4 + xd6) y , ,
| |
| 1.
| |
| To simulate i Apay.ta r, betweed fuol mass and the rack
| |
| +
| |
| structure the compression only gap elements are used- j in pairs. Fig. 3.2'shows, the node at the top of the '
| |
| rack . " Knowledge of the kinematics enables.
| |
| computation of the coupling coefficients [Ref. 6]
| |
| 'f which enables us to asch rtaint the contribution of
| |
| " each impact elemeft to the generalized forces Qqi l in the governing equ'ations given. in Section 2.3-C.
| |
| P ,
| |
| .; }
| |
| e ;
| |
| . II-30 .l y _
| |
| .i
| |
| | |
| [
| |
| 1 l
| |
| l B. Computation of Fuel / Rack Impact Spring Rates
| |
| ( )
| |
| order to transfer impact forces, each fuel In assembly impacts on the thin side plate of tne fuel rack grid. The impact force is then transmitted of the grid cross section. To through the rest compute the impact spring constant, we consider the following analysis.
| |
| - The cross section of a rack having NX cells in the x direction and NY cells in the y direction is shown in Fig. 3.3a. We assume that all cells impact in phase, and that the unit is full.
| |
| Consider the strip segment " A" . We represent this segment as a beam of length L (either a or b) subject to a uniform load with rotations resisted by the vert sections between adjacent horizontal strips. (See Fig f\_)T 3.3b).
| |
| To see the ef f ect of the vertical members, we conside the deformed configuration shown in Fig. 3.4.
| |
| It is easy to show that the moment M acting on the vertical members when the rotation of the horizontal members is e is given as M=6EI g LX For ease in the subsequent solution, we replace these concentrated resisting moments by a distributed momen m = M/ly. Since Ex = ty = 1, we have 0
| |
| m= 0= K 90 ,
| |
| 2 1
| |
| ,/~N
| |
| (-)
| |
| Therefore, looking at equilibrium of the beam section with the uniform load q gives (say in the y direction
| |
| | |
| O Y
| |
| : v. O
| |
| & t Tr 9
| |
| , o
| |
| ~
| |
| R o
| |
| esA T _
| |
| sK sC a _
| |
| / e ss s
| |
| I
| |
| /
| |
| , uA rM O
| |
| 6
| |
| /
| |
| L5l/
| |
| - 7
| |
| # 2 G 3 N
| |
| I L
| |
| R E
| |
| T E
| |
| M R E
| |
| C Vo u
| |
| G Y / 6
| |
| , I F
| |
| C A
| |
| 1l.
| |
| (
| |
| )
| |
| O
| |
| | |
| 0 0 .
| |
| <( '
| |
| I t F
| |
| = o a 11 g
| |
| >- n =
| |
| Z >.
| |
| S m
| |
| I n o 0 X 4 Z
| |
| O !
| |
| i
| |
| | |
| ~.
| |
| M M-L .
| |
| T Ct:
| |
| W -
| |
| C 2 --e w
| |
| 2 O m --*
| |
| m -
| |
| p O
| |
| h -
| |
| h
| |
| -- 0 0
| |
| = m b
| |
| d _
| |
| O M
| |
| T 1!
| |
| .--e-l' l
| |
| I
| |
| | |
| - e egy 34 $a w e, 4
| |
| \
| |
| \
| |
| A
| |
| 'CWd'
| |
| \ \
| |
| \
| |
| t m
| |
| u .
| |
| O
| |
| | |
| L.
| |
| O' $
| |
| dy.
| |
| +m-V=0 (See Fig. 3.5)
| |
| $'+q=0
| |
| .dy 2
| |
| dM + dm- +q=0 or .
| |
| 2 dy dy Since M = -
| |
| EI d 2w/dy2 r m = Kg dw/dy we obtain the
| |
| - equation 2 6 d "w 2dw=q._._ ; . g2 = _Ke
| |
| =3 I -
| |
| S dy" dy 2. .EI EI I
| |
| The boundary conditions are w(y) = w(-y) : w ( b) =d"2 ( b) =0 2 dy 2 where we are obtaining the solution in the region -
| |
| 0$y$ b L = b,, for example-2 g, The solution of the differential equation is 2
| |
| NY + Cy + D w(y) = A sinh ( Sy) + 8 cosh (Sy) -
| |
| 2 2 8 EI i
| |
| By symmetry, A = C = 0, so that 2 2 w(y) = 8 cosh 8y + D - qy /28 EI
| |
| .l At y = $ , w(y) =0 i 2
| |
| 2 2 0 = B cosh Sb/2 + 0 - qb /8 8 gy.
| |
| l 1 I
| |
| II-32
| |
| _ _ _ - _ - _ _ - - - _ _ _ _ _ - _ _ _ _ _ __ --__ _ -_ _ -__ a
| |
| | |
| e -
| |
| D O
| |
| a g h ,
| |
| h d
| |
| " P.j V + aV l
| |
| p M+6M r 's '
| |
| ,- e2 e _ .d y M( o g V
| |
| i**'*
| |
| O 5 Iw .l l
| |
| L ._ __ __ __ __
| |
| t.y FIGURE 3.5 O
| |
| 1
| |
| | |
| At y = b/2 d 2w/dy 2= 0 2 2 0=BS cosh 8b/2 q/8 EI Solving, we obtain 2
| |
| (b 2 ,4y )
| |
| w(y) =B [ cosh By - cosh Sb/2 ] + _ 42 86 EI where 9
| |
| B=
| |
| 8"EI cosh ab/2 The maximum deflection is at y=0 and is given as gb 2 6 (cosh 6 $ -1 )
| |
| 6= 2 86 EI 2 qb 2 8 with B = 4; 4= 2 2 cosh Sb/2 8 ski Sb Therefore gb 2 (cosh - Bb/2 - 1) ]
| |
| 6 gg
| |
| =
| |
| [l-8 8 EI Since, if the total load from all cells is P, then qb =
| |
| P ; we can write NX P=K 6g MAX where
| |
| =
| |
| 8 NX EI ( Sb) 2 (3)
| |
| K
| |
| [1 - $x (cosh -1) ]
| |
| II-33
| |
| . . _ _ _ _ _ _ _ _ _ _ . _____-_m-__.___m
| |
| | |
| . (%, .
| |
| V Similarly, the spring constant Ky is~given as
| |
| ,8 NY EI ( Sa ) 2 . (g) g
| |
| [l' . $y (coch 8" - 1. ) ) .
| |
| As an example, suppose we consider' a rack having dimensions. a x b x H- (NOTES- These values are for
| |
| * illustration only. Specific. values aro for the PG&E 10xil' rack. We assume a ' width of' plate H/4 in the' calculations.
| |
| NX = 10, NY = 11, a = 108.125", b = 119.125"
| |
| ~
| |
| Ax = ty = 10 . 57 2 " ; H = 165.5" For KX, we, calculate KX = 12532#/in for 110 cells and in computer simulations we increase.this value by a factor of 10 for conservatism.
| |
| For KY, we similarly compute KY = 15400, and we increase this by a factor of 10 in the simulations.
| |
| O ,
| |
| 11-34
| |
| | |
| The above calculation looks . at ' the . rack .as a. unit.
| |
| .We between thould also look. at thel local . deformation vertical strips. Consider a pinned plate of length 1 and width w = H/4. Then, under~a uniform pressure p
| |
| * 3 5 p 1 t" y ,
| |
| t 6=
| |
| 384 EI
| |
| * 10.92 But p1W'=
| |
| (NX-NY)
| |
| So that 5 ~PL3 4,
| |
| 3 384 (NX-NY) E w t 10.92 Thus, we can define a local impact spring rate as 3
| |
| K =
| |
| 384 ,NC x E x w x t
| |
| (\ L 5 10.92 x A a For the unit considered in this numerical example, g = 10.G'x 10' 9/in Since- the two spring rates computed here act in. series, the overall' unit spring rate is K g Kg Kg IMPACT 1 + Kg/K g Kg+Kg where i can be either x or y.
| |
| As stated before, we arbitrarily increase the computed values by a factor of 10 to account for uncertainties in the analysis. Thus', the impact spring rates for the and O 10x11 rack are 154,000#/in (y direction).
| |
| 127,000#/in, ~(x direction),
| |
| II-35
| |
| | |
| Fuel Rack base Support i 3.1.2 The effect of the base supports at- each corner of the fuel rack . base are simulated by four compression only elements (stop elements) in order to permit lift off of any or all supports.. The stop element spring constant in
| |
| . compression is obtained from the following formula.
| |
| 1
| |
| _ =
| |
| 1
| |
| _ +
| |
| 1 #
| |
| 1 , 1 (5)
| |
| Ky K gp K PF
| |
| . 6. LR The spring constant Ky represents the elasticity of the rack support cross-section in direct compression.
| |
| EA g Ky =
| |
| h where As is the support metal ;ross section and h is
| |
| '.\ the length of the rack support.
| |
| The spring constant Kp L represents the local elasticity of the thick concrete floor under the rack. It is shown in Levy and Wilkinson, The Component Element Method in Dynamics, McGraw Hill, 1976, that for a square pad of side lengch B, sitting on a floor having Young's Modulus effect of local floor elasticity in direct Ec, the compression can be modelled by a spring with constant 2
| |
| K gp = 1.05 E c B/(l-v )
| |
| The e'ffect of local floor elasticity o'n rocking motion (bending of the support) is represented by a rotational ~
| |
| spring having spring rate KH*E c I K /B $
| |
| ( 1- V I O
| |
| II-36
| |
| | |
| r l
| |
| O a where K = 2 and I = B"/12 Therefore, for the given rack support pads.
| |
| KH" c B3 /6 ( 1- v2) is utilized in the following The spring constant KH manner. With the support leg rotation in each direction obtained from the dynamic analysis program at each time .f step, the base moment on each leg due to a rotation $ is
| |
| - determined from the relation M=K H
| |
| * applied ir. the x,y directions. Since the base moment cannot exceed the moment provided if edging occurs, these base rotational springs (2 on each support) are modelled as rotational frictional elements, with maximum allowable moment given by the current downward load times the edge distance B/2. If KH is set to zero, we are neglecting this effect in the sinalstioiis ,
| |
| l The spring constant KLR represents the local elasticity of the rack gridwork structure near the support attcchment points. To calculate this spring constant, we use" the formula for Kp L given above except that Ec is j repliced with E* of the gridwork and B is replaced' with B*, an assumed effective width over which local deformation of the rack gridwork to occur. For E*, we 'l assume O II-37 l
| |
| | |
| m * '
| |
| .(d E
| |
| =ESTEEL * ( Included Area of Single Cell l
| |
| The final constant Kpp represents the plate ' behavior of the floor during dynamic motion. Because of the plate like behavior of the pool floor slab, a rack in ' the center of the pool floor will feel- the effects of a finite Kpp. For a rack near the pool floor supports, Kpp + =,
| |
| We can estimate the magnitude , of Kpp for a rack near the center of the pool floor by calculating the central deflection of a rectangular plate loaded by a
| |
| ~
| |
| static uniform lateral pressure loading. For the PGSE conservative assumption that racks, we have made the Kpp = " .
| |
| The final values used' in the PG&E analyses is K = .515 x 10 7
| |
| lb/in.. The appropriate coupling I coefficients for the supports are obtained from the kinematic considerations shown in Fig. 3.6.
| |
| Let point s1 have location.31= x 1 E 1 + y i n 2 relative to node 1. Then the velocity of point s1 is i
| |
| . s1 = dl bl + d2 52 + d3 b 3 + (d4 bl + d5 b2+d6 b 3)
| |
| X (X 1 b l+y1 b)2 Therefore 951 = (d i - y 1 d 6) b l+ (k2 + X 1 k6) b 2+
| |
| 1 (d 3 + .Y 1 d4 - X 1 d 5)
| |
| Hence, the stop springs in the z direction have coupling coefficients 1, y1, -x1 to nodes 3, 4, 5 respectively.
| |
| (See discussion in Levy and Wilkinson.)
| |
| 11-38 l 1
| |
| l
| |
| | |
| O 95 il r =. _ _ -- - - y 42 O g5 n2
| |
| / 5
| |
| /
| |
| S1 _
| |
| /
| |
| /
| |
| /
| |
| X FIGURE 3.6 4
| |
| O
| |
| | |
| i t
| |
| 3 .1 ~. 3 Elements at z = -h to Simulate' Sliding of Supports l Associated with each' support stop element spring are 'two-orthogonal friction elements located in the plane l z = -h. 'The friction element spring constant is taken'as the spring' constant.of the support structure considered as a guided cantilever beam of length h under Ltip? load P. Thus, for a friction spring, we have, for example, the - illustration in Fig. 3.7.
| |
| 1 1 1 where 4 = 8.52 -
| |
| KI= h3 (1 + 4) AL 2 3
| |
| - 4.157 ( I
| |
| ) /2 AL 2 in The above formula is obtained from the solution Timoshenko, Strength of Materials, 3rd Edition, - McGraw Hill, p. 175. For numerical ~ stability,.we find it useful to _ arbitrarily increase this value by a factor 'of 100.
| |
| IU j
| |
| For the PG&E racks, the values used ' are .112 x 10 lbs/in.
| |
| The coupling coefficients are easily obtained by noting that the velocity of the lower point s* in the ' x-y plane is given as 9' = 981+ (q' u E t + q's 62 ) x (- h E 3) or (5-12)
| |
| ' 'q' g) d 2 +
| |
| 9 =
| |
| (dt - Yt d6 - h d5) 5 1+ (d 2 + X I d 6 + h .
| |
| O II-39 1
| |
| | |
| O Z
| |
| o .
| |
| BASE PLATE sq.
| |
| I h
| |
| h O e
| |
| ///////////
| |
| v FIGURE 3.7 l 1 '!
| |
| i n {
| |
| I O
| |
| | |
| l
| |
| <^
| |
| For example, for a friction spring in the n1 direction, 5 with coupling the coordinate coupled nodes are 1,- 6, coefficients 1, -y 1, -h, respectively.
| |
| 3.2 INITIAL DEFORMATION OF RACK At the start of a rack dynamic analysis, the initial loading in the support legs is adjusted to simulate the dead load of the rack. The total static load is PSTAT = weight of base + weight of fuel assembling +
| |
| weight of rack Given the spring constant of the support K, the number of l supports Ns, and the coordinates of the fuel assembly centroid XBR, YBR (which are assumed to coincide with the p point of application of PSTAT, we may write 1 Y N,
| |
| [Ft - PSTAT = 0 N,
| |
| { yi Ff - (PSTAT) x (YBR) =0 N,
| |
| [ xg Fi - (PSTAT) x (XBR) =0 q 5}
| |
| with Fg=-K (q 3 + yf qu- x g Solving for q3, q 4, q 5 permits us to establish the static ;
| |
| compression in each support leg. The initial load and displacement are used as the basis for determining the possibility of lift-off of a given support leg. i II-40 .
| |
| 1
| |
| - - - - - _ - - --._______.____N
| |
| | |
| g:=:rxa i
| |
| l 3.3 A REMARK ON COMPUTATION OF INPUT DATA A code has been written to compute all of the spring constants and coupling coefficients necessary for input into the transient analysis cc ,de . Inputs to this code are cell dimensions, support leg ' dimensions ,
| |
| dimensions, rack etc. The output from this code coritains all ?
| |
| properties, information that is necessary to create a data file for the dynamic analysis code used simulate a seismic event acting on'a I for the PGEE racks are typical fuel rack. Sample outputs contained in an appendix to this report.
| |
| l 3.4 EFFECTIVE RATTLING MASS FOR DYNAMIC ANALYSIS We have shown in previous sections that a complex model of a fuel rack cell structure and the internal vibrating masses can be We can choose to model the internal develoqed.
| |
| vibrating masses;4 fuel / enclosed containers) by discrete masses located in dif ferent rack quadrants and at different levels.
| |
| To simplify the modelling, we can choose to represent the vibrating internal masses by a single effective mass located at In this section. of the report, we ,
| |
| the top of the rack.
| |
| I demonstrate how the proper value for this mass can be obtained.
| |
| consider Fig. 3.8 which shows a typical slender fuel assembly inside of a fuel rack cell.
| |
| We assume that the actual mass is impacting the cell wall I
| |
| over the top le'ngth x with a relative velocity V. The total cell height is 1 and the density of the vibrating rod is p.
| |
| Our task is to demonstrate a replacement of this effect by a single mass impacting at the top of the cell. We choose the value for me so that the angular momentum of the two systems 7 are the same; the equated angular momentums are both computed g j about the base of the cell. i I
| |
| II-41
| |
| | |
| 6 8
| |
| a m.e a i d d
| |
| /
| |
| / l f
| |
| g /
| |
| / l
| |
| / I it /
| |
| L l
| |
| I 1
| |
| l om I 9 I I
| |
| I I
| |
| I I
| |
| l a y if 19 i
| |
| l 1
| |
| J FIGURE 3.8 O
| |
| w____._.___.m____.___ _ _ _ _ _ _ _ _ _ _ _ _ . _ . _ _ _ . . _ _ . _ . .
| |
| | |
| r~ '
| |
| From Fig. 3.8, we can write m V i= PAXV ( 1- x +1) e 2
| |
| + f '~
| |
| * p A V ( Y
| |
| ) y dy 0 L- x 1
| |
| section area of the vibrating body.
| |
| where A- is the cross (
| |
| .j Performing the integration and simplifying yields m* = m
| |
| '( 1 -
| |
| ) + "( 1 - 1.- )
| |
| 1 21 3 1 where m = pA 1 i
| |
| Typical effective mass valut. are x/ A m,/m 0 .333
| |
| .25 40625
| |
| .5 .45833 1.0 .500 l
| |
| we are already making the assumption that all l
| |
| Since cells in a given quadrant move in phase when we replace the effect by a single mass, we assume that we need only consider impact over the top 1/8 of the cell. Thus, when we choose to model the , impacting system by a single effective mass, we will The remaining mass fraction of .625 is use me /m = . 3 7,5 .
| |
| assumed to be located at the base and undergoes all necessary base motions but does not impact. A series of runs (not reported herein for the 10x11 and 6x11 racks) were carried d out for mass fractions m e/m = .4, and attached base mass
| |
| - .6.
| |
| II-42 1
| |
| | |
| O' This was done to assess the effect of increased rack flexibility by increasing the portion of fuel assembly mass that is assumed to rattle. The results of this study are reported in Appendix IV (Supplementary Studies). It is shown there tnat an increase in the rack flexibility can be accounted for by increasing the rattling mass by 2.5% of 2
| |
| the total mass. Changes in the rattling mass fraction have only a small effect on the results. On all runs carried a
| |
| out, subsequent to the studies on increased flexibility, rattling mass fraction of 0.4 was used. It is noted that all conclusions concerning rack integrity are unchanged by small perturbations in the fuel assembly rattling mass fraction.
| |
| l II-42a
| |
| | |
| l I
| |
| l 36 ew! mL & n 246; M .
| |
| is ,
| |
| A %. M - c h ,4p , .
| |
| - _ n & u.i a ,
| |
| A 6 wAA % ,fo kl L 7;-
| |
| & % lC] M& 4 & .1 ,n \
| |
| nss a p , _Ls a n, O bb e
| |
| =
| |
| - cp A 7-p-p
| |
| &AMka u c = 23 e w, &
| |
| : n. uz - % - ~ g& as & M
| |
| : 2. s - c ) , t% w . , a
| |
| % Cue S&
| |
| ~ c. H z a L J w - L ,' M ;
| |
| & A .
| |
| $ w .nl J c. i . ry, c = . e x m a ( m - n.) l O = 2 s v (uc - u, ) * ~ % ' \
| |
| II-43
| |
| | |
| O s Rz=h*/g = /- dam fmcaff
| |
| $f As = a'/ff
| |
| * W f o'a Nc -
| |
| , v ede- lk n$ d NF,4 - i''t hk kig$ uk He. nHl,Az.'Nc A4, :
| |
| p H f, A,' NF)
| |
| Om A a= y qu A , s.. na4 lA. T & ,iu p-v-1 C= 7, st/17 N f, ( A 1.* Aic Aj ' NfA) f%. Ya fG + 6 & , M = /. 45 , 4 x= 6.0 9 '' Ai = Y. 79
| |
| = , ocoo 9 s 9 # * */a V
| |
| , F<r. a. M d N c4 0o 8 , a>e & l$< Aa.<A. (NFA=//a)
| |
| & $ ~so ' h g 0 (DM = t/ )
| |
| aan ;
| |
| OD
| |
| "# *^ 'a %
| |
| C = 599 C = N3 G II-44
| |
| | |
| i l
| |
| k
| |
| - 1 i
| |
| % c l'-
| |
| A,1y W y & !'
| |
| auj& w & Amee ~&") w fle
| |
| %q 6 we. % 66 wwas m p au~o q 4 4 ~4 4 A,4
| |
| ~ n 4 can a.
| |
| i i
| |
| O) l II-45
| |
| | |
| i I
| |
| i I
| |
| : 4. TIME INTEGRATION OF THE EQUATIONS OF MOTION f' i
| |
| j 4.1 Time-history Analysis Using 8 DOF Rack Model
| |
| !(
| |
| Having assembled the structural model, the dynamic equations 'of motion corresponding to each degree-of-freedom are written by-using of Section 2.3. The system
| |
| 'Lagrange's equation as shown in equations can be represented in matrix notation as:
| |
| 3
| |
| [M ] {I[} = (0 ] + {G }
| |
| function of nodal is a non-linear where the vector [0 ] the coupling on displacements and velocities, and {G } depends Premultiplying the above inertia and the ground acceleration. f
| |
| 'l equations by (M]-1 renders the resulting equation uncoupled in N mass.
| |
| O We have: {2[} = (M ]-1 (O ] + (M ]-1 {G }
| |
| run verify numerical simulations to .
| |
| noted earlier, in the As structural integrity during a seismic event, all elements of the ,
| |
| This will 'rovide p
| |
| fuel assemblies are assumed to move in phase. l maximum impact force level, and induce additional conservatism in the time-history analysis.
| |
| This equation set is mass uncoupled, displacement coupled, and is ideally suited for numerical solution using the central difference for this The computer program " DYN AHIS"
| |
| * is utilized scheme. ,
| |
| l
| |
| ~
| |
| purpose, i
| |
| 4 This code has been (Docket previouslyNo. utilized 50-341), in Quad licensing of Isimilar Cities and II racks for Fermi II (Docket Nos. 50-254 and 265), Rancho Seco (Docket No. 50-312),
| |
| ! Summer (Docket No.
| |
| Oyster Creek (Docket No. 50-219), and : V ,C. j 50-395). .
| |
| u f s
| |
| II-46
| |
| _______-_a
| |
| | |
| i
| |
| 's stresses in various portions of the structure are computed from known element forces at each instant of time.
| |
| Dynamic analysis of typicp1 multicell tacks has shown that the captured osast completely by the motion' of the structure j is behavior of a six-degree-o'f-f reedom structure; therefore, in this rack cross-section at any analydis model, the movement'
| |
| , .7 7 of the height is described in' terms of the rack base degrees-of-freedom (qti t ) , . . .q 6( t ) . The remaining degrees-of-freedom are associated with horizon
| |
| * 31 movements of the fuel assembly masses. In this a single lumped mass is used to represent fuel dynamic model, assembly movement. Therefore, the final dynamic model consists of rack plus two additional nass six decrees-of-freedom for the The single lumped mass, degrees-of-freedom for the one fuel mass.
| |
| used to represent the fuel assembly vibrating mass, is located at the top of the rack to obtain the maximum moment during impacts.
| |
| The ef f ective mass of .a dingle impacting body is chosen so as to yield.the same angular 1 momentum at the base of the-rack as would be obtained by the continuous rod-l'ike fuel assembly structure. 'The remaining portion of the fuel assembly mass is assumed to move with the rack base. Thus, the totality of fuel mass is included in the simulation.
| |
| : 5. DESIGN CRI'rERI A 5.1 Structural Acceptance Criteria ,
| |
| There are two sets of criteria to be satisfied by the rack modules:
| |
| : a. Kinematic Criterion
| |
| ' to ensure that the rack is a This , criterion seeks physically stable structure. Diablo Canyon racks are certain inter-rack impact at designed to sustain designated locations in the rack modules. Therefore, i
| |
| i
| |
| '- physical stabili$y of the rack is considered along with the localized inter-rack impacts. Localized permanent deformation of the module is permissible, so long as the minimum flux-trap gap in the active fuel region is nct violated. ,
| |
| II-47 l
| |
| | |
| IR )) .,
| |
| : b. Stress Limits limits of the ASME Code, Section III, The stress Subsection NF, 1983 Edition are used since this code provides the most appropriate and consistent set of loading limits for various stress types and various The following loading combinations are conditions.
| |
| applicable (Ref. 1).
| |
| Loading Combination Stress Limit D Level A service limits D +To D +To+E D ^ Ta + E Level B service limits D + T o + Pg D + Ta + E' Level D service limits
| |
| ['w-J' D + Fd The functional capability of the fuel racks should be demonstrated where:
| |
| D = Dead weight-induced stresses (including fuel assembly weight, F
| |
| = Force caused by the accidental drop of the d
| |
| heaviest load from the maximum possible height Pg = Upward force on the racks caused by postulated stuck fuel assembly E = Design Earthquake E' = Hosgri Earthquake T c = Dif ferential temperature induced loads (normal or upset condition) .
| |
| T* = Differential temperature induced loads (abnormal design conditions)
| |
| (-
| |
| U II-48
| |
| | |
| y e
| |
| Y
| |
| +
| |
| 0 .
| |
| and To cause' loc Q thermal stresses to be The, cogi tions J Ta produced.- The worst situation will be htitained when an isoluted
| |
| - storage incation has a fuel assembly whicfi 'is generating heat at the maximum postulated rate. The surrounding storage locations are assumed to contain no fuel. Tne heated water makes unobstructed
| |
| & J lI contact uith t's inside :of the st'orago .wa.1.ls, thereby producing the difference' be ty,ee n .'the adjacent
| |
| ~
| |
| maximu%,7 possi ble temperature .
| |
| Jl .
| |
| t.o the-ceI ls . // ,The secondary stresses thus producedi are' Iimited t
| |
| ' oody of the rack; that in, the support legs do not experience / he
| |
| /
| |
| uecondary (thermal) s t re r,se s .
| |
| S.2' Material Properties y
| |
| 'I '
| |
| i T'he data en the physical propeytbes o f' the rack and' support-materials, obtained ' f rom the ASMEl Boiler & Pressure Vessel Code, are listed in Section . III, appendices, and supplier's catalog, ,
| |
| N Tables 5.1 and 5.2. Since the maximum pool' bulk temperature (except for t'rA full core discharge case) 'is 140*-147'F, 150* is used a s' the reference desig.t temperature ,for evaluation of material properties.
| |
| l Table 5.2 RACK MATERIAL DATA Young's Yield Ultimate Modulus Strength Strength psi)
| |
| E #
| |
| y 98 u Property ,
| |
| 27.9 x 10 6 23150 63100 l '
| |
| Value i Table Table Table Section III I-2.2 .;I-3.2 Reference I- tr. 0 0
| |
| (s' a e ' 11-49 C_
| |
| | |
| I
| |
| [* Table 5.2 )
| |
| I 1
| |
| SUPPORT MATERIAL DATA Young's Yield Ultimate Strength Strength Material gdulus 94,350 psi 1 ASTM 479-S21800 27.9 x 10 6 44,900 psi (top part of support) i 6 101,040 psi 145,000 psi 2 SA564-630 (hardened 27.9 x 10 at 1075'F)
| |
| (bottom part of support)
| |
| I-2.2 I-3.1 Section III Reference I-6.0 5.3 Stress Limits for various Conditions the The fo . lowing stress limits are derived from the guidelines of ASME Code, Section III, Subsection NF, in conjunction with the material properties data of the preceding section.
| |
| 5.3.1 Normal and Upset Conditions (Level A or Level B)
| |
| : a. Allowable stress in tension on a net section j
| |
| =Ft= 0.6 S y or F = (0.6) (23,150) = 13,890 psi (rack material) t F = is equivalent to primary membrane stresses t
| |
| F,= (.6)(44,900)= 26,940 psi for upper part of I support feet)
| |
| - (.6) (101040) = 60,624 psi for lower part of support feet i
| |
| : b. On the gross section, allowable stress in shear is:
| |
| b4 (23,15G) = 9,260 psi (main rack body)
| |
| F = (.4)(44,900) = 17,960 psi (for upper part of e
| |
| support feet)
| |
| = (.4) (101040) = 40416 psi II-50
| |
| | |
| Allowable stress in compression, Fa:
| |
| X( ) c.
| |
| )y
| |
| [1 - ($1)#
| |
| _2C c F* = 5 3
| |
| ( - )+' -[3 (kl) 8C c) ~ b(~k1 I c 3) 3 r r t
| |
| where:
| |
| 1/ 2 Ce = [ )
| |
| 8y k t/r for the main rack body is based on the fulli height and cross section of the honeycomb region. Substituting numbers, we obtain, for.both support leg and honeycomb region:
| |
| Fa = 13,890 psi (main rack body)
| |
| Fa = 26,940 psi (support legs - upper part)
| |
| =-60,624 psi (support legs - lower part)
| |
| : d. Maximum allowable bending stress at the outermost fiber due to flexure about one plane of symmetry:
| |
| Pb = 0.60 Sy = 13,890 psi (rack body) l Fb = 26,940 psi (support legs - upper part)
| |
| = 60,624 psi (support legs - lower part) i
| |
| : e. Combined flexure and compression: l C,xfbx C,yfby fa , 4 y F, DF g bx DF y by where:
| |
| f a
| |
| = Direct compressive stress in the section ,
| |
| f = Maximum flexural stress along y-axis bx ;
| |
| \
| |
| 1 11-51 u__ .
| |
| | |
| f
| |
| = Maximum flexural stress'along y-axis
| |
| ( by C ,x = Cmy = 0.85 f
| |
| a D x. =1-7,ex f
| |
| a
| |
| .D =1-y F'ey where:
| |
| 2
| |
| : p. ,
| |
| 12w E ex,ey 2 23 .( k1 * 'Y ' )
| |
| bx,y and the subscripts x,y reflect the particular bending plane of interest.
| |
| : f. Combined flexure and compression (or tension):
| |
| I f
| |
| a , bx , by < 1.0 0.6 S y F F bx by The above requirement should. be met for both~ the direct tension or compression case.
| |
| 5.3.2 Level D Service Limits F-1370 (Section III, Appendix F), states that the limits for the Level D condition are the minimum- of .l.2 (Sy/Ft) or times corresponding limits for Level A (0.75u/Ft) the condition. Since 1.2 Sy is less than 0.7 Su for- the rack upper part of the support feet, the material and for the multiplying factor is 2.0. For the lower part of the support feet, the multiplying factor is 1.67.
| |
| II-52
| |
| | |
| ( Instead of tabulating the results of these six.different stresses as dimensioned values, they are presented in a dimensionless form.
| |
| These so-called stress factors are defined as the ratio of the actual. developed stress to its specified limiting value. With this definition, the limiting value of each stress factor is 1.0 for DE and 2.0 or 1.67 for the HE condition.
| |
| 6.0 RESULTS l
| |
| 6.1 Remarks Results are given here for a 10x11 module (the largest module), and for a 6xil configuration (which is a module with the largest aspect ratio). ,
| |
| A complete synopsis of the analysis of~the 6xll module subject to the Hosgri earthquake motions is presented in a summary table O- labelled as Table 6.1. Table 6.1 gives the maximum values of
| |
| = 1,2,3,4,5,6). The values given in the stress factors Ri (i tables are the maximum values in time and space (all sections of the rack). Table 6.2 gives typical results for a 10xll rack. The stress factors are defined as:
| |
| R1
| |
| = Ratio of direct tensile or compressive stress on a net section to its allowable value (note support feet only support compression)
| |
| R 2= Ratio of gross shear on a net section to its allowable value R3 =
| |
| Ratio 'of maximum bending stress due to bending about the x-axis to its allowable value for the section i
| |
| II-53 1
| |
| L--- _ - _ - _ - _ _ _ _ _ _ _ - _ _ _ _ _- _ _ _ _ _ _ _ _ _ _ _ . _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
| |
| | |
| O R4 = Ratio of maximum bending stress due to bending about the y-axis to its allowable value.
| |
| R 5 = Combined flexure and compressive factor (as defined in 5.3.1e above)
| |
| R6= Combined flexure and tension (or compression) factor (as defined in 5.3.1f above)
| |
| As stated before, the allowable value of Rt (i =1,2,3,4,5,6) is 1 for the DE condition and 2 or 1.67 for the HE earthquake.
| |
| The results given in the tables are for the Hosgri esrthquake and have been given in the licensing document. The maximum stress factors (R i) are below the limiting value for the HE condition for all sections. It is noted that the critical load factors reported for the support feet are all for th'e upper segment of the foot and are to be compared with the limiting value of 2.0.
| |
| Tables 6.1 - 6.4 also present results which are used to show lh that significant margins of safety exist against local deformation of the fuel storage cell due to rattling impact of fuel assemblies and against local overstress of impact bars due to inter-rack impact.
| |
| The tabular results shown assume that the rack metal thicknesses are based on a total metal thickness of 0.08 inch in the cell structure area (Region I configuration). The racks with cell wall metal thicknesses = 0.09 inch will be slightly heavier but will not significantly affect the safety margins since the rack weight is a small fraction of the total dead weight of a fully loaded unit.
| |
| O 11-54
| |
| | |
| 6.2 Comparison of Computer Results and Tabulated Results ld*
| |
| in Tab'les 6.1-6.3 Tables 6.1-6.2 were performed using The runs summarized in incorrect values for yield stress. Namely, for the rack section For the upper section of 25000 psi-was'used instead of 23150 psi.
| |
| For the the support feet 55000 psi was used instead of 44900 psi.
| |
| lower section of the support feet 125000 psi was used instead of 101040 psi. Therefore, all tabulated results for R i factors in amplified _ by the following the computer outputs must first be factors 25000 = .08 kACK = 23150 UPPER SUPPORT = = 1.225 44900 p
| |
| N.
| |
| .232 LOWER SUPPORT = ,
| |
| The results in Tables 6.3-6.4 do not need this correction since the proper allowable stress values were input into the program.
| |
| The results for R 1 factors in the computer runs also need to be corrected for an increase in the support foot cross section area in the upper section. The computer outputs for the 10x11 and 6 x11 racks were all carried out using an annular cross section for the upper portion of the support leg of 5.25" 10 and 8" 00. The actual cross section dimensions of the section as installed at Diablo Canyon have dimensions 6" 10 and 9" 00 for the upper component of the support leg.
| |
| O -
| |
| * Entire section, including tables, replaced.
| |
| II-55
| |
| | |
| ~
| |
| l i
| |
| O A comparison of section properties is given below:
| |
| Assumed for' Computations As Installed 28.62 35.34 A (in 2) 163.0 258.0 I' . ( i n ")
| |
| the support leg, therefore, 'all Ri For the- upper section cf factors in the computer outputs must be attenuated by either the-area factors
| |
| " *0I BA* 35.34 f
| |
| or by the inertia factors
| |
| '63 = .63 s
| |
| Qk I= 258 1
| |
| BA, while R 3, Rq are Therefore, R 1, R2 are attenuated by attenuated by Bl. The factors R 5, R6 involve both direct tension or compression plus bending so that all we can conclude in general I is that the appropriate attenution factors for R S, R 6 be in the range
| |
| .63 < attenuation factor < .81 Since R 5, R6 are usually dominated by bending, we will generally ,
| |
| l find that the ..co r r ect ion is closer to .63.the Where results are true attentuatior, critical for structural integrity checks, factor will be used. To. cross check the computer results with the summary tables, we provide the following summary:
| |
| O II-56 i I
| |
| 1
| |
| - - _ _ _ _ _ _ = _ - -
| |
| | |
| Tables 6.'1, 6.2
| |
| -Multiply all rack Ri factors by ARACK-Multiply all support leg factors by (AUPPER SUPPORT)*(BA or B)I or-recalculate R 6 as R6= (R1
| |
| * BA + (R 3+ R g) *B) AupPER I SUPPORT Table 6.3 reactions multiplied by appropriate B factors 1 Support leg only.
| |
| i Table 6.4 No modif'ications needed.
| |
| The following example should provide the appropriate clarification.
| |
| Consider the 6x11 rack, run ee01. From the computer output, we have Ri= .084. Table 6.1 has the corrected value R i= .084 x 1.08 =
| |
| .091. For the upper support leg section, the raw computer output gives R t = .177. The corrected value (including both amplification due to incorrect yield stress and attenuation due to as-built leg area) is Ri= .177 x 1.225 x .81 = .175 The factor R3 =
| |
| ~
| |
| 575 from the raw computer output; the corrected factor is R3 = .575 x 1.225 x .63 = .4438 t
| |
| II-57 1 l
| |
| | |
| t' actos R 6 from the output is'.864. The corrected value is (for T he section 8).
| |
| (R 3 + Rg) x .63) =-
| |
| R6 _= 1.225 (R 1 x .81 +
| |
| 1.225 (.144 x .81 + .850 x .63) = .799 The following summary tables show corrected results for all runs.
| |
| 6.3 Tabular Results the licensing report; va lues T ab les 6.1, 6.2 are reported in inc luded herein re f lec t the above considerations on yield . stress and area changes. ' T ab le s 6. 3 , 6.4 present additional results not issued in the licensing report. Note that since values for R 5 are ,
| |
| O always less than values for R 6, we do not report R 5 values in the I
| |
| summary. tab les.
| |
| 4 f
| |
| l II-57a
| |
| | |
| a eh H .f B )
| |
| ks 6 7 4 5 9 6 6 9 t et w ct 2 3 6 5 1 8 7 6 ol r ar 4 1 2 0 0 1 0 0 1 f b o t api Ro R . . . . . .
| |
| ewpm p
| |
| rp l oui ou bl sl fS al warl 2 5 3 9 4 8 o ee ir 5 4 7 9 6 8 wv Ro f 3 8 4 1 4 9 6 0 0 1 0 d 2 la2l l 0. ol s R . . . . . . e .
| |
| (
| |
| rd 0 eE on .thH t a 2 et c 4 8 3 0 5 7 9 5 a 1 0 2 3 3 4 5 3 ( e 0 f ;6 F 2 1 2 0 0 0 0 1 R . . . . . . e s d stl1 1
| |
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| |
| 1 0 1 0 0 0 1 0 1 asdu r aq Y R . . . . . . . .
| |
| roe T rol r K of C
| |
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| |
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| |
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| |
| )
| |
| e b et x1 x 1 x af cm S l e 5 x 9 x 0 f b T ( wkk 65 1 7 5 deh a U t cc 57 6 7 9 darc T U s eaa . . . aooi S d bRR 01 0 3 3 l
| |
| ol mh E a R o me L
| |
| * euh s F l 5
| |
| * 5 " rmt r ey 05 0 " 0 ai O t xet c neul 10 1 0 1 navc Y a e gF b x1 x 1 x mma R p ea/m 0x 1 x 0 A m wrl e 60 9 6 0 u hf M I t ol s 33 9 6 3 l r M et es .. . . . oatd U BSCA 15 0 1 2 cl ea S ieo hmf l t ci
| |
| / c ast r kk a s s s s e r ccp e e e e nrpo eo aam Y f Ys l
| |
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| |
| ea f k q8 q2 q2 t r q8 tr ff oa r . . . a . a oo t a i i i l y i l y m r= r= r= ut r= .t tt rsa e g g g mp g mp eean R sP sF sP i m sP im s s p oO oO oO se( oO se(
| |
| HC HC HC HC rrrti O 1 2 3 4 eeea pwpc p o p U l u
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| |
| * um
| |
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| a B
| |
| s kt 7 6 cr '
| |
| 8 3 ao 4 0 4 Rp R . .
| |
| p ru oS f 7 4 r 6 6 3 1 3 io Rf R -
| |
| sd rn t
| |
| oa 7 4 9 0 c 2 0 2 a R . .
| |
| F 1 d 1 a x o 9 5 6 L 6 4 1
| |
| 0 1 R
| |
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| |
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| |
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| |
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| |
| b et
| |
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| |
| (
| |
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| |
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| |
| BRR
| |
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| |
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| |
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| |
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| |
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| |
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| |
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| |
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| |
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| |
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| |
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| |
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| |
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| |
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| |
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| |
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| s .
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| ucf a ;
| |
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| |
| Rp ao R . . . . . .
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| |
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| |
| 2 8 1 7 1 8 et oS 6 6 el r f 4 2 7 9 3 3 8 9
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| |
| . b a p io R . . . .
| |
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| t . 0w c R . . . . .
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| |
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| |
| b : s E tl P krel Y cov T apeE
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| |
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| |
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| |
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| |
| 6 7 att a L t cc 77 7 f ci T U s eaa ;
| |
| 2 ar S d BRR 1 6 7 df E a a R o
| |
| * odee L
| |
| * 5 5 5 l ar F
| |
| O t l
| |
| ey 5 5 00 0 0 0 el o
| |
| c neul 11 1 1 1 t a xx x x x n r Y
| |
| R p egFb ea/m 40 6 7 7 e l am A m wrl e 93 2 9 2 m l u a wmt M
| |
| M I t ol s et es 48 4
| |
| 4 2
| |
| 1 2
| |
| . e.w v
| |
| on oi rxe U BSCA 28 2 mid a S -
| |
| r hmar t 9a c
| |
| / c s s w ar kka s s e e xa 7. o eat l
| |
| ccp e e Y Y m2t aam Y Y nie RRI i m
| |
| - i k sst c -
| |
| e
| |
| '0 l O l Oa l O l ueo l rp s E2e E 2 e k E8.eu .u E r 8. eu .u aap r i f i f ir f i f v a r= r= re= r -
| |
| m g l d g- l d gn - ld g ld f e e sPl a sFl a srFl a s l a oue R oOao oOuo ooOuo o 0uo .l HCFl H Fl HCCFl H Fl t a ev s
| |
| f 2 ro O
| |
| 0 1 1 e ptr 1 2 n n pea 0 0 r r U s n . 0 0 o o uo a a c c RN a a a a -
| |
| s7mO -
| |
| ; f
| |
| | |
| a 2 9 1 9 Bs 5 4 1 9 q 1 0 t 0 3 kr R . . .
| |
| co ap Rp
| |
| .u rS 3 6 0 1 o 9 3 7 2 f r 3 0 3 1 1 o R . . . .
| |
| if R
| |
| d -
| |
| sn 5 4 3 7 ra 2 5 6 2 5 0 0 o 0 1 t R .
| |
| c a
| |
| F
| |
| _ l l
| |
| d 7 3 3 8 x a g 3 5 6 9 0 o 0 1 0 1 1 L R . .
| |
| _ E P
| |
| _ Y
| |
| ) T
| |
| _ d e K u C
| |
| _ n A
| |
| _ i R
| |
| _ t "
| |
| _ n 0 "
| |
| o - ) no 1 x
| |
| c b et 0 .
| |
| ( l e 1
| |
| _O 2
| |
| S T
| |
| L
| |
| (
| |
| s wkk tcc eaa 4
| |
| 9 s
| |
| e 6
| |
| x
| |
| _ 6 U d BRR 4 i 4
| |
| _ S a l
| |
| _ e E o b "
| |
| _ l R L l m k
| |
| _ b ey " e O c
| |
| _ a F t neul 0 s l a T O c e gF b 1 s x r a ea/m x a 7 y
| |
| _ Y p wrl e 6 1
| |
| _ R m tol s 3 l 3 t p
| |
| A I etes . e .
| |
| M BSCA 2 u 2 m M f e
| |
| _ U
| |
| _ S t gk s c nc e
| |
| /
| |
| kka s i a s t ccp e nr e a
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| |
| ay Y l u
| |
| RRI tt m k np k i ome
| |
| _ c c se
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| _ a cek a k r a r ,a ssu su
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| _ s r8 l eQ r2l Q
| |
| _ k r
| |
| : o. l t o.l ei
| |
| _ i eai i
| |
| - a r= cl r r= cr
| |
| _ m e ug e -g e tF 1 ms tF1 s R nO 1i o nO1 o l iC ( sI l iC( I O 3 0
| |
| 4 0
| |
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| |
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| |
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| |
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| |
| e 0 6 0 0 6 4 1 0 2 1 t kr 4 1 3 0 0 0 0 1 3 0 1 e cr R . . . . . e f
| |
| ao Rc t n :
| |
| 2 5 r ru 7 5 9 7 5 7 3 1 4 4 o f
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| o( 3 1 6 2 .
| |
| 1 0 0 1 6 7 1 1 2 4 1 3 0 1 p s R . . . . . p it u Rr s o 2 6 9 2 d sp 9 0 4 7 5 5 rp 2 8 0 0 2 5 5 4 1 6 n ou 2 1 2 0 0 0 0 0 1 0 0 a tS R . . . .
| |
| e d c s e
| |
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| |
| a t Fo b a f r d 0 7 7 3 7 8 7 4 6 4 ad 7 0 0 2 5 4 6 8 1 4 k o K on 1 0 2 0 0 0 1 0 1 0 0 c p C La R . . . a r A r o R c
| |
| _ h n
| |
| _ t i 1
| |
| 1 o
| |
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| |
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| |
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| |
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| |
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| |
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| |
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| |
| - S b et 1 eT I e 3 . x x . t L ( wkk 2 O 5 5 0 e n bU t cc 8 0 6 r e
| |
| _ aS s eaa a m TE d BRR 8 1 3 e
| |
| _ R a s v o n o L L l - " 4 o m A ey 0 " 0 " i neul 0 " 0 t l N t 1 1 a
| |
| O c e gF b x 1 x 0 1 i
| |
| _ I a ea/m 6 x 3 1 x d i T p wrl e 2 0 4 x 3 n x
| |
| - 3 6 3 o a
| |
| _ I m t ol s 2 4 c
| |
| _ D I et es . . 0 .
| |
| D BSCA 2 1 2 1 e A t 7 E h c) D t t k l ao
| |
| _ / c c l ps S S O r o kka Sa aml O E E N o t ccp N Y Y f ER wia g aam Y( s RRI - n l - - m - r i e m m e m o d ru e e s e t n ef t s s s s k c o
| |
| _ n' s s s a s c a p
| |
| _ r1 e a a a a f s s o1 i
| |
| ~
| |
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| |
| lt et lt d r r - bF e . P e . F u. F e . rF a r a mO us O us O f s O us eO o o m 1 k eC f e C f e C e C f e nC l c
| |
| _ e xc s i ' 0i i r s
| |
| R 0a s8 1 l 2 1l 8 1 l o8 l 1 r a .
| |
| 1W 2 1 . 1 b . 1 b . 1 b c . l a
| |
| s o
| |
| _ l
| |
| _ t a y h g
| |
| _O l
| |
| 3) a a
| |
| b a
| |
| 3) b a
| |
| 3) b 3) a 3) a t
| |
| e r
| |
| e n
| |
| _ n. 1 E 3E 1 E 3E 3E t E
| |
| _ uo cll cD cl l cD cD o RN a( a( a( a( a( N t
| |
| - U$ l fLl
| |
| | |
| a B -
| |
| 7 4 6 5 ks 9 5 9 0 ct s 3 7 2 4 ar R . . . .
| |
| Ro p
| |
| rp 3 0 7 7 ou 4 7 6 1 2 8 6 0 1 fS R . . .
| |
| ir Ro f
| |
| s 1 7 2 6 rd 3 8 3 9 on R 3 2 5 2 2 t a . . .
| |
| c a
| |
| F-7 1 6 1 d 1 3 3 5 a 2 1 2 0 0 o R . . .
| |
| L K
| |
| C 6 7 A 9 4 3 3 R 0 2 8 5 1 . 0 1 H R .
| |
| 4S "
| |
| .T O 6L ) no 0 l x
| |
| U O b et 1 eS l e x 5 l E ( wkk 5 4 bR t cc 1 0 a s eaa .
| |
| TF d BRR 7 9 O a o
| |
| Y L l R ey A t neul 0 0 M c e gF b 1 1 M a ea/m x x U p wrl e 5 1 "
| |
| S m tol s 4 5 5 I et es . . 0 BSCA 5 4 9 2
| |
| t
| |
| / c =
| |
| kka S S ccp E E t aam Y Y n e
| |
| RRI m
| |
| e c
| |
| a .
| |
| l
| |
| * p s 8 2 s k . . i r i i d a r=
| |
| g r=
| |
| g .
| |
| m e sF sF x R looC oO a I
| |
| l l C M O 4 5
| |
| : n. 0 l uo t t RN t t Uhw
| |
| | |
| 13 '
| |
| Rack to wall impact:
| |
| and Modules located on the outer periphery of the rack array, proximate to the pool walls (to within less than 4") may impact the wall under certain conditions of fuel assembly loading and Coulomb and the liner.
| |
| coefficient of friction between the rack feet Calculations show that the maximum impact load corresponding to a nearly empty 10x11 rack is 38550 lbs and the duration of impact is less than .01 sec. Using a factor of two and rounding out numbers, a constant impact load of 80,000 lb acting for a period of .01 sec (total impulse = 800 lb-sec) is a realistic upper bound estimate of the dynamic loading on the pool wall. Whereas the peak impact load may exceed the surmised limit of 80,000 lb for some rack geometries the net impulse of 800 lbs-see is and conditions of loading, expected to define an upper bound.
| |
| Multi-Rack Collisions:
| |
| The dynamic model of the rack employed in our analysis seeks to maximize the impact load between the racks, or the rack and the wall. For this purpose, the racks are assumed to move 180*
| |
| out-of-phase, resulting in a reference plane of symmetry. This model seeks to impose an order in rack motions, which, of course, is a very slim possibility scenario. The unlikelihood of occurrence of such an organized motion becomes evident when the following j l
| |
| facts are considered:
| |
| (a) Rack module aspect ratio, and location of support feet within vary from rack to rack leading to the rack ' modules, non-identical dynamic behavicr. ;
| |
| (b) The number of fuel assemblies stored in different racks at any point in time will be different, until the pool begins to fill up. Even at that time, the fuel assemblies are of several different groups of manufacture, resulting in non-identical structural stiffnesses.
| |
| II-62b
| |
| | |
| -g (O (c). The assemblies may stand up straight in a storage cell, .'or north, east, west or they may lean against a cell wall -
| |
| The initial.' orientation of an assembly. greatly south.
| |
| In other words, an influences its dynamic impact locus.
| |
| orches'. rated (out-of-phase, or in-phase) motion of a. rack requires that all fuel assemblies in all racks be disposed in a certain prescribed order!
| |
| is an orthotropic, non-homogenous (d) Since the pool slab continuum, its local compliance also varies from one support leg to another. This too results in inducing dissimilar vibration patterns in different racks. . .
| |
| It is clear.that given such a wide scatter in the input data, an '
| |
| The racks-would-undergo organized motion is entirely improbable.
| |
| random collisions of far less severity than postulated in the opposed phase model utilized here. However, this model has the twin virtues of simplicity and conservatism.
| |
| ]
| |
| l l
| |
| II-62c
| |
| | |
| Table 6.5 presents results for maximum ' vertical and horizontal Note that- maximum shear loads do- not loads on the liner.
| |
| necessarily occur at the same time as maximum vertical loads.
| |
| Table 6.5 I
| |
| FLOOR LOADS MAX. VERTICAL MAX. VERTICAL LOAD (SUM OF FOUR FEET) MAX. SHEAR LOAD LOAD (1 FOOT) (1bs)
| |
| :RUN (lbs) (lbs) 5 5-5 6.74'x 10 1.825 x 10 acorn 10 2.95 x 10 5
| |
| aa001 2.32 x 10 5 6.77 x 10 5 1.467 x 10 5
| |
| ee01 1.68 x 10 5 2.975 x 10 5 1.067 x-10 5
| |
| 5 .961 x 10
| |
| ~tt04 1.71 x 10 5 ~2.062 x 10 l
| |
| O The results reported as acorn 10 are for the single -10xil corner '
| |
| rack (rack M). These results were not reported in the Licensing not complete at the time of document since the analysis was submittal. The results reported as aa001 are for a hypothetical 10x11 rack surrounded'by racks on all sides. The largest interior rack is 10x10; therefore, the loads for module aaOOl' can be reduced by .91. i l
| |
| Table 6.6 summarizes deflection and floor loads for all of the j runs. Wall impact loads are also given.
| |
| 1 II-62d
| |
| , O I
| |
| | |
| ~
| |
| )
| |
| l n
| |
| Table 6.6-
| |
| | |
| ==SUMMARY==
| |
| VALUES FOR ALL RUNS Total Max.
| |
| Max.-Y Impact Load Top Rack / Wall Max. X Impact Load Disp. Disp.' Load on One Run (in.) (in.) on Floor Foot (#) .(#)
| |
| ttl5 1.997 2.136 1.832x10 5 101700 63280 (8x8) 5 No wall impact tt04 1.849 2.905 2.062x10 5 1.712x10 (8x8)
| |
| .8425 4.167 4.39x10 5 2.395x10 5 38550 -l I ac13aal (10xil) ac33ab .0076 .0389 5.104x10" 2.209x10" No wall impact (10x11)
| |
| .3428 3.975 5.651x10 5 2.84x10 5 No wall impact
| |
| /] ac13ab (d (10xil) 5 5 ac33b .1263 .1662 3.721x10 1.862x0 No wall impact (10xil) ac33aa .0922 .4088 9.297x10" 5.163x10" No wall impact <
| |
| (10xil) 5 5 aa004 .160 .167 3.847x10 1.879x10 Interior rack l (10x11)
| |
| .156 .168 2.305x10 5 t,gixto 5 Interior rack aa003 (10xil) 5 5 aa001 .176 .196 6.772x10 2.323x10 Interior rack (10xil) 5 5 aa002 .193 -
| |
| .184 2.829x0 1.191x10 Interior rack !
| |
| (10x11) 5 I ee01 .19 .15 2.98x10 5 1.675x10 Rack assumed I to be (6xil) completely surrounded
| |
| () n a n
| |
| .177 1.736x10 5 7,ggxio4 ee02 .191 (6xll) ,
| |
| II-63 L ___ __ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ . _ _
| |
| | |
| q 1
| |
| l C .
| |
| Table 6.6 (continued)
| |
| Total- Max.
| |
| Max. Y Impact Load Top Rack / Wall Max. X Impact Load Disp. Disp. ' Load on One
| |
| - (in.) on Floor Foot (#) (#)
| |
| Run (in.)
| |
| 5 n . a ee03 .173 .159 4.055x10 5 1.59x10 (6xil) 5 a n n ee04 .193 .147 2.29x10 5 1.444x10 (6x11) 5 a a n ee05 .1674 .1746 2.428x10 5 1.653x10 (6x11) acorn 12 2.79 1.37 2.76x10 5
| |
| 1.263x10 5 No wall impact g j
| |
| f' (10xil)
| |
| .478 6.753x10 5 2. 95 5 x10 ,5 No wall' impact acorn 10 .3402 (10xll) 4 1
| |
| l i O
| |
| L _ ____ _ _ ___ _ ___________________ _____ ___ _ _ _ _ _ _
| |
| II-63a
| |
| | |
| 6.4 Impact Loadings 6.4.1 Impact Loading Between Fuel Assembly and Cell Wall The local stress in a cell wall may be estimated from the loads obtained from the various ' dynamic analyses. In peak impact this section, we estimate values for the worst case load per -cell A survey of - the between the fuel assemblies and the cell wall.
| |
| loads actually obtained dynamic analyses shows that the peak impact are less than'the limiting values' calculated here.
| |
| We assume impact occurs over a rectangular clamped section whose width is equal to the inside dimension of the rack cell and whose length is equal to 1/8 of the rack height. We replace the impact load P by two equal line loadings acting on the plate. The spacing of the line loadings is equal to the outside width of the fuel assembly.
| |
| We assume the impact load is transmitted to the cell wall l
| |
| only at the edges of the fuel assembly since any deflection of the I cell wall during the impact tends to shift the load transfer points out to the edges of the fuel assembly.
| |
| l l
| |
| l For the PG&E configuration, t = 8.85", a = 8.47" and L =
| |
| 20.6". (see Fig. 6-1). Since L/1 > 2.0, we may compute the stress based on cylindrical bending in the short span of the plate.
| |
| j Consider a beam of length 1 and width L, loadcd by line load P/2.
| |
| (See Fig. 6-2).
| |
| The dotted lines show the limit deflection shape of the beam. A rate of work balance yields O ;
| |
| i l
| |
| l II-64 A i
| |
| | |
| I O sc l
| |
| b N
| |
| % [ DgkN/d
| |
| %[fsS((% '
| |
| Nr %. N' L n A l
| |
| FIGURE 6.1 O
| |
| pJ f
| |
| ) i f 1 r / $
| |
| --+
| |
| y ._ _ _ _ __ _ -
| |
| Wc 7 ,S .
| |
| . , .a '
| |
| a 1
| |
| FIGURE 6.2 I
| |
| i O
| |
| l
| |
| | |
| c-1.
| |
| O -.
| |
| a-Lt, .
| |
| S .6 =2 ( Y )(d) 2 c 4
| |
| or P =' o y bt 2 c
| |
| The total impact' load in the unit is Q = PxNC where NC = number,of cells in the unit. Therefore, based on a safety margin of . 1. 0 against excessive plastic deformation, the limiting value of Q is
| |
| . O = abt 2 NC c
| |
| For-where oy is the yield stress of . the cell wall material.
| |
| typical units 2C = 1-a= .38" ; ay = 23150 psi; t = .08" 20.6 QLIMIT = 23150 x x (.08) 2 NC
| |
| .19
| |
| = 16063 NC Allowing for a safety margin of 2, we have the following results:
| |
| i NC OLIMIT *}
| |
| ^
| |
| 110 883483.
| |
| 66 530090.
| |
| II-65
| |
| | |
| O .
| |
| From the results of the dynamic. analysis, we find that the actual impact loadings do not exceed 249,400# for NC=110, and. h do not exceed 136,000# for NC=66. Thus, bending of the cell wall under the impact loads is not a safety problem.
| |
| We.need also examine . shear punching of the cell wall. Let Ty be the yield stress In simple . shear. Then the limiting load Q in shear is Q ' = 2NC T t (a + L) ay/2, and assume a safety margin of 2.0. Then Let Ty a a
| |
| QIMIT L
| |
| = 2E ( Y
| |
| ) t (a + L)/2.0 2
| |
| (D
| |
| (,) = NC (11575)(.08)(8.47 + 20.6) = 26919 NC-We see that Q LIMIT < O L iMIT so that bending limit loads govern the impact.
| |
| 6.4.2 Impact Loading Between Adjacent Racks The magnitude of the seismic events is such tha't inter-rack impacts may occur during certain seismic events, at either (or both) the top of the racks or at the base plate location. To preclude a safety problem arising in these locations because of an inter-rack i'mpact, the rack has been strengthened by a band b of metal area A around the top of the rack.
| |
| v II-66
| |
| | |
| .c f
| |
| i Referring to the rack layout in the ' pool, it is seen that most inter-rack impacts, if they occur will'be at the corners, 3",
| |
| since the rack edges are -lined up. Reinforcing bars 1.0625" x made of steel with a. yield stress of 55000 psi are welded to the top of each rack side wall to form a rectangular frame at the top of each rack. The top frame is above theqactive : fuel height; the base plate serves as an absorber of rack to rack impacts at ,
| |
| s the bottom of the rack.
| |
| We show first that corner -impact loads can be adequately absorbed by the edge bars.. The limiting axial' load in the corner bar is set so that the yield stress in the bar is achieved unde:
| |
| direct bar compression. Therefore, with oy = 55000.
| |
| PLIMIT " 'y (T W) where T = 1.0625", W = 3.0"
| |
| '\,
| |
| = 175312.5#
| |
| O This is greater than the maximum value reported in Tables 6.1-6.4, thus, corner impacts can be. tolerated with the addition of the high yield strength material framing.
| |
| We now examine the case where one rack impacts in the middle This may happen since two of the racks are
| |
| ~
| |
| of an adjacent rack.
| |
| not lined up with adjacent rack structures. Consider a rack wall (at the top) with reinforcing bar as shown in' Fig. 6-3.
| |
| s We also show the rack shell structure behind the reinforcing y bar. For discu,ssion purposes, we have shown 11 cells falong the side. The construction is such that there are 20 " cell wall" over the total side length L. For i plates, each .08" thick non-corner impacts, we can treat the rack cellular structure as an
| |
| ,\
| |
| O II-67 I
| |
| | |
| t 1
| |
| I i, .. ,
| |
| f '(h I
| |
| . t
| |
| 'y , ;e>
| |
| - 4.c s n.
| |
| . y t
| |
| E
| |
| (
| |
| To P Vs s w w
| |
| O >
| |
| t
| |
| / 29 A P-Q # Q Fi--,-4
| |
| ' p l f' .
| |
| )
| |
| 7
| |
| ~ -.
| |
| n p,, ,
| |
| (! I lql.
| |
| I y l i ,
| |
| - ' esu, . "
| |
| I i) _ - - - - - '
| |
| i W L z . i.-. w i o t
| |
| FIGURE 6.3 1
| |
| s
| |
| | |
| m The l equivalent elastic foundation which helps support the bar.
| |
| (a) spring constant of this foundation can be estimated from the l formula !
| |
| * I
| |
| =
| |
| 4Etw K
| |
| e L A where t = cell wall thickness , w
| |
| = span between cell centers,
| |
| * j 55" (max.) and w is f L* = half length of adjacent side walls =
| |
| I the active length of cell wall that is assumed to *help support the
| |
| = 7.85" (i.e. a bar. For purposes of this analysis, we assume w Using the given i width equaj. to the inside free span of the cell) .
| |
| dimensions 6
| |
| K
| |
| =
| |
| 4 x 30 x 10 x .08 x 7.85 = .12571 x lo s #/in 2 55 x 10.9 l
| |
| To estimate the ef fect of a non-corner impact, we assume that local j l
| |
| fuel area, occurs so that an l (j defo'rmation above the active impacting load of 1000#/in is spread over the entire length of the !
| |
| bar. If we take L = 120",
| |
| then the loaded configuration is as l shown in Fig. 6-4. Note that 1000#/in x 120" is greater than any i
| |
| /
| |
| impact load reported for rack-to-rack or rack-to-wall impacts in Tables 6.1-6.4.
| |
| 1 the foundation load directly supports the Away from the ends, applied pressure.
| |
| Therefore, the actual foundation over a 10.94" l
| |
| span must be capable' of supporting 10940. From the typical rack cross section, we see that over the cell center-center distance of
| |
| .08" thick, 10.94", there are two " elastic foundation" members, having a 7.85" free span. In order to estimate the carrying capacity of the foundation, we need to compute the buckling load of 1
| |
| (b II-68
| |
| | |
| q 1
| |
| 1 l
| |
| l l
| |
| **
| |
| * eee tooofyg ~
| |
| L e Lll L L j j j'j
| |
| ~ -0 -1/l(//$///,,,,.,]--...-- .
| |
| _____________ "" > 6e . .
| |
| O 6 FIGURE 6.4 b
| |
| \
| |
| ; i i_-____________________,_____._________________________
| |
| L'_ _ _ _ __ . - _ _ .
| |
| | |
| - these members. We assume that the depth of the member (for the purposes of carrying load) is equal to the characteristic dimension 7.85". Thus, we examine the buckling load for the plate structure shown in Fig. 6-5. The length used is based on the span between lateral supporting members. This lateral support is provided by gap channels.
| |
| ld From Theory of Elastic Stability, S.P. Timoshenko, we can show that K v 2
| |
| Et 2 g ,2 Et 2
| |
| = =
| |
| o cr b 2 12(1-v2) 10.92b 2 where K = K(a/b). If the sides of length b are simply supprted (instead of clamped) K = 1.7. In reality, the supports on all three supported edges are clamped. Recalling the relationship between the buckling loads of a clamped beam versus a pin ended our purposes as K = 4 x 1.7 = 6'.8. i beam, we estimate K for e Therefore, 2
| |
| .08 o =
| |
| 6.8 x 9.86 x 30 x 10 6 x( ) = 19130 psi 10.92 7.85 ,
| |
| i Over a 7.85" span, the total load that is supportable is PTOTAL
| |
| = o cr t b = 12014#. Since there are two such column like members in each 10.94" cell to cell cente.line, the foundation can support 24028# which is equivalent to a 2196#/in. distibuted pressure due to any impact load on the external bar.
| |
| The bending, stress induced at the end of the bar due to a distributed load q (#/in) is o= 34 2 2 d
| |
| WT 3 where 48" = K*/EI = .0112392. ;
| |
| Therefore, under a loading.of 2196#/in along the bar, the maximum
| |
| ( bar strass due to bending is o= 31739 psi < o = "5000 psi y
| |
| II-69
| |
| | |
| 1 O
| |
| i i
| |
| )
| |
| 1
| |
| -_g free
| |
| -- a f e Q --* . I 4 $ h 7,y #
| |
| O
| |
| -, e-W C L
| |
| ff g (.f.ff'
| |
| [
| |
| u- - 4 7. VS m
| |
| .. n os . -
| |
| FIGURE 6.5 i
| |
| O l
| |
| l h________.___'______.________..___.___._m _ __ _ __ _.____J
| |
| | |
| 7 for a bar with dimensions w = 3.0", T = 1.0625".
| |
| All local V
| |
| deformations, are restricted to the area above the active fuel region during impacts of the top bar. Impacts on the bottom bar I
| |
| are supported by the 5/8" base plate, acting as the foundation, so that the rack cell walls carry :.3 ttle of the impact load.
| |
| 6.5 Weld Stresses The critical weld locations under seismic loading are at the bottom of the rack (at the base plate connection) and in the welds on the support legs. The results from the dynamic analyses using the simulation codes are surveyed and the maximum loading used to For the rack welds, the qualify the welds in these locations.
| |
| allowable weld stress is taken as the ASME Code allowable value of 18520 psi. For the support legs, made of high strength material, the allowable weld stress is taken as 80% of the yield strength (for HE conditions). We look at the welds considering a maximum V weld stress = 35200 psi. The welds at the rack base are .09" fillet welds. The shear stress T is the weld throat, induced by a normal stress o in the rack is given as i= ct/1.414h (h = weld size)
| |
| {t = skin thickness)
| |
| The above result assumes a continuous weld at the critical location ar.d a uniform o along the weld length. For the weld between the rack bars and the cell walls, we have t= .08" , h y .08 .09" Examination of the structural acceptance factors from the transient analysis yields a maximum value for R6 at the rack base = 0.453 o= 6292.2 psi. Hence, the (8x8 rack). Since R6 = c/13890, then weld shear stress at the rack base, doubled to account for skip n)
| |
| (
| |
| v welding is II-70
| |
| | |
| ( 6292.2x2 x .08 T = = 8900 psi 1.414 .08 The equivalent fillet weld in the upper support leg is a 1.25" weld. We estimate the " skin thickness" in that location as 1.5". Using a maximum load factor of 1.44 (10xil rack) in that location and noting that the allowable stress for load factor R6 in the support is 26940 psi, yields
| |
| ,, 1.44 x 26940 x 1.5 = 32922 psi 1.414 1.25 which is less than 35,200 psi. Note that the above condition occurs for an HE event.
| |
| The stress e, which must be supported by the weld, is calculated using the R 6 factor from the computer runs. The weld stress formula is taken from " Welding Design" by C.H. Jennings, ASME Transactions, O Vol. 58, p. 507, 1936, and is also found in the Heat Exchange Institute's Standards for Steam Surface Condensers, 8th Edition.
| |
| The configuration is shown in Figure W attached.
| |
| The maximum shear stress at the rack base is given by T = R2 X 9260. The maximum value cf R 2 reported for the rack base is R 2"
| |
| .156 (acorn 10, Table 6.2). Therefore, the shear stress is r gy = 1445 psi This shear stress must also be resisted by the weld between the gap channels and the tubes. This weld is continuous for the first 1.5" above the base plate and is then a skip weld (l" on 3-1/2" a
| |
| spacing). The value of 1445 psi is for the HE and assumes continuous plate with thickness t = .08". The weld in question is a .08" fillet weld. Therefore, the actual longitudinal shear stress in the tube-to-channel welds is 1
| |
| t = 1445 x x = 7151.3 psi 1 .707 II-71 _ _ _ _ _ _ _
| |
| | |
| O N.J t
| |
| a.. t > V*T Y2 N
| |
| +.- MA
| |
| ,n .
| |
| _ _.._____.....\ 4 . . . _ . . .
| |
| . . . . _ _ N. . .
| |
| N W . .. . ._. ...__ .
| |
| % y_g y. n FIGURE W !
| |
| WELD CONFIGURATION FOR' FILLET WELD DESIGN STRESS i i
| |
| (
| |
| N . . . . . . . . . . . . _ .
| |
| l l
| |
| (,
| |
| II-71a
| |
| | |
| q For HE conditions, the allowable shear stress in these welds is 2 x
| |
| -Q 9260 = 18520 psi.
| |
| =
| |
| For the DE condition, the maximum reported value for R2 is R 2
| |
| .052 (run ac33b, Table 6.3). Therefore, the weld stress in the channel-to-tube weld is x = 2384 psi i = .052 x 9260 x 1 .707 For DE conditions, the allowable shear stress in these welds is 9260 psi.
| |
| Near the bottom of the rack where this shear stress dominates, there are no other stresses on the weld which need be considered.
| |
| O Stresses in the char.nel-to-cell welds may also develop due to assembly impact with the cell wall near the top of the rack.
| |
| fuel This will occur if fuel assemblies in adjacent tubes are moving out of phase with one another so that impact loads in two adjacent cells are in opposite directions which would tend to separate the channel f rom the tube at the weld. The maximum impact load per cell is 2300 under HE conditions over the top 20" of the tube. Over a 20" length of tube, we have 5.71" of channel-to-tube fillet weld.
| |
| Therefore, the There will be two channels to resist this impact.
| |
| weld shear stress due to this load is
| |
| ~
| |
| 2300 = 3561 psi j T =
| |
| I 7
| |
| (.08" x .707) x 5.71" x2
| |
| \
| |
| If only one channel resists this load, the shear stress will be 7122 psi.
| |
| l II-71b !
| |
| I
| |
| - _ _ -___a
| |
| | |
| 1 O 6.6 Local Buckling of Fuel' Cell Walls The allowable local buckling stresses in the fuel cell walls are obtained by using classical plate buckling analysis. The following formula for the critical stress has been used.
| |
| 2
| |
| - 8w Et ( _
| |
| ).
| |
| a#C 12 b 2 (1, y 2) where E = 27.9x10 6
| |
| psi,.t = .08, b = 8.5". The factor' 6 is suggested in the reference to be 4.0 for a long panel loaded as shown in Fig. 6-1.
| |
| 1 For the given data i
| |
| 'cr = 8935 psi O It should be noted that this calculation is based - on the applied stress being uniform along the entire length of the cell wall.
| |
| i l
| |
| Strength of Materials,, S.P. Timoshenko, 3rd Edition, 1956, Part II, pp. 194-197.
| |
| II-72
| |
| | |
| 5 O. In the actual fuel rack, the compressive stress "comes from i
| |
| consideration of overall bending of the rack structure during a seismic event and as cuch is negligible at the rack top and maximum at. the rack bottom. It is conservative to apply Eq. (6-1) to the rack cell wall if we compare acr with the maximum compressive stress anywhere in the cell wall.
| |
| The output of the dynamic analysis program DYNAHIS provides the time history of critical stresses at various levels in the rack and ,
| |
| in the rack support feet. In particular, an output is provided for the maximum direct plus total bending stress in the outermost cell wall at the bottom of the rack. This translates into a maximum compressive stress in the cell wall at some critical time in the seismic event. The output is given in terms of the stress factor R 5 or R 6. As defined, the stress factor is the ratio of the' actual stress to the allowable value. (allowable = 13890 psi based on yield and/or overall column buckling).
| |
| Therefore, the maximum compressive stress in the cell wall is given as a = R 6 x 13890 psi
| |
| = .453, at the root. of the 8x8 rack, For a maximum load factor R 6 we obtain e = 6292 psi Thus, the factor of safety at the root of the rack against panel
| |
| ^
| |
| buckling is in excess of 1.4.
| |
| O II-73
| |
| | |
| i l
| |
| 73 b In the above, note that results are obtained for HE conditions. The calculation is based on a plate simply supported on four edges. In the actual structure, the boundary condition is probably more nearly clamped on at least two opposite edges. This would increase l the factor 8 to a value near 7.0 in accordance with Timoshenko's analysis. Thus, the actual safety factor is on the order of 3.0.
| |
| 6.7 Analysis of Welded Joints in Rack due to Thermal Load l I
| |
| Welded joints are examined under the loading conditions arising from thermal effects due to an isolated hot shell. Under the postulated load condition, the weld stresses are found to be below the allowable value of 24000 psi in shear that is given in Table NF329.1-1 of ASME Section III, Division 1, Subsection NF, 1980.
| |
| A thermal gradient between cells will develop when an isolated storage location contains a fuel assembly emitting maximum postulated heat, while the surrounding locations are empty. We can obtain an estimate of weld stresses along the length of an isolated hot cell by assuming a beam strip uniformly heated by 48'Ft, and restrained from growth along one long edge. The detailed computations are presented in an appendix.
| |
| Using shear beam theory, and subjecting the strip to a 1 uniform temperature rise aT = 48'F, we can calculate an {
| |
| I estimate of the maximum value of the average shear stress in the ' strip. The strip is subjected to the following boundary conditions.
| |
| I i
| |
| t i 48* is upper bound on the local temperature rise. l l
| |
| I1-74 ;
| |
| _. _ ___-_a
| |
| | |
| O a. Displacement Ox(x,y) = 0 at x = 0, all y and at y =
| |
| w/2, all x.
| |
| Average force N x, acting on the cross section Hxt
| |
| = 0 b.
| |
| at x =L, all y.
| |
| The final result for wall shear stress, maximum at x =L, is found to be given as
| |
| ,M AX , E a A T
| |
| ,933 where E = 27.8 x 10 6
| |
| psi, a = 9.5 x 10-6 in/in *F and AT = 48 ' F .
| |
| Therefore, we obtain an estimate of maximum weld shear stress in an isolated hot cell, due to thermal gradients, as T = 12424 psi HAX 6.8 Factor of Safety Against Overturning With regard to the safety margin for overturning for the limitin'g rack module, the OT position paper, by reference to the Standard Review Plan, stipulates that a factor of safsty of 1.1 over che ground motion (Hosgri Earthquake for Diablo extreme condition Canyon racks) be provided against rack overturning. Towards this end, the rack module with the worst aspect ratio (6 x 11 module) was run with 1.1 times the Hosgri Seismic excitation. A coefficient of friction of u = 0.8 was used to produce the maximum overturning condition. The x-axis is parallel to the short side and the y-axis is parallel to the long side. The x,y origin is at Two cases of eccentric loading are the rack centerline.
| |
| O . New section added.
| |
| 11 75
| |
| | |
| p 1 considered. .Both cases have 50% of the cells fi lled with fuel
| |
| - assemb lies . Case 1_ loads all fuel.in the positive x ha lf of the rack; Case 2 loads all fue l in tne _ negative y half of the rack.
| |
| The following table summarizes _ the results of the two analyses.
| |
| Rotation in x-z plane ,
| |
| Rotation in y-z p' lane, decrees decrees va lue Va lue Factor for Factor for of Load Ca lcu lated Incipient of Ca lcu lated Incipient- ~ Safety safety Maximum Tippinc Case Maximum Tippina 33.4- 1.15 35.94 31.2 1 0.58 19.36 21.14 27.4 1.26 33.38 26.5 2 0.77, These results show that a large margin of safety against kinematic instability exists in all rack modules.
| |
| 6.9 Definition of Terms Used in Part II of the Seismic Report S1, S2, S3, S4 Support designations Pi Absolute degree-of-f reedom number i qi Relative degree-of-freedom number i u Coefficient of friction Ui Pool floor slab displacement time ,
| |
| history ' in the 'i-th direction f x,y coordinates horizontal direction.
| |
| z coordinate vertica l direction, KI Impact spring between fue 1 assemblies and ce ll Kg Linear component of friction spr.ing K6- Axial spring of support leg !
| |
| locations ,
| |
| O II-76
| |
| | |
| N Compression load in a support foot Rotational spring provided by the KR pool slab Subscript i ' When used with U or.X' indicates direction (i = 1 x-direction, :i = 2 y-direc tion , 1 = 3 z-direction)
| |
| G O
| |
| M O
| |
| I - E 8-7 7 - _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ - _ _ -
| |
| | |
| i REFERENCES TO SECTION 6
| |
| [1} USNRC Standard Review Plan, NUREG-0800 ( .19 81 ) .
| |
| [2] ASME Boiler & Pressure Vessel Code, Section III, Subsection NF (1983).
| |
| ' Guide 1.29, " Seismic. ~ Design
| |
| [3] USNRC Regulatory C classification," , Rev. 3, 1978.
| |
| [4] " Friction Coef ficients of Waster Lubricated Stainless ' Steels for a Spent Fue l Rack F ac i lity , " Prof. Ernest Rabinowicz,
| |
| ' MIT, a report for Boston Edison Company, 1976.
| |
| [5] USNRC Regulatory Guide l'. 9 2 , " Combining Moda 1 Responses ~and Rev.. 1, Spatia 1 Components in Seismic Response Ana lysis ,"
| |
| l February 1976.
| |
| l
| |
| [6} "The Component Element Method in Dynamics with Application to l
| |
| l Earthquake and Vehic le Engineering," S. Levy and J.P.D.
| |
| Wilkinson, McGraw shill, 1976.
| |
| [7] " Dynamics of Structures ," R.W. C loug h and J . Penzien, McGraw Hill (1975).
| |
| [8] " Mechanical Design of Heat Exchangers and Pressu;e So le r , Arcturus Vessel Components," Chapter 16, K.P. Singh and A.I.
| |
| Pub lishers , Inc., 1984.
| |
| [9] R.J. Fritz, "The Ef fects of Liquids on the Dynamic Motions of Immersed So lids , " Journal of Engineering for Industry, l
| |
| ' Trans. of the ASME, February 1972, pp. 167-172. ;
| |
| l Coupling a C lose ly Spaced Two-Body System
| |
| [10} " Dynamic in
| |
| ' Vibrating in Liquid Medium: The Case of Fue l Racks," K.P.
| |
| Singh and A.I. Soler, 3rd International Conference on Nuclear !
| |
| Power Safety, Keswick, England, May 1982.
| |
| [11] USNRC Regu latory Guide 1.61, " Damping Va lue s for Seismic Design . of Nuc lear Power P lants ," 1973.
| |
| [12} "F low Induced Vibration ," R.D. B levins , VonNos trant. (1977 ) .
| |
| Foss, Rona ld Press, !
| |
| [13] " F lu id Mechanics," M.C. Potter and J.F. 1
| |
| : p. 459 (1975).
| |
| 0 II-78 l
| |
| | |
| 0 APPENDIX I PRELIMINARY COMPUTATIONS FOR DYNAMICS DATA FILES O
| |
| I O
| |
| | |
| Preliminary Calculations for PreDyna t n
| |
| Configuration of Cell - Assume a ,,
| |
| fr TS ,,
| |
| *~~~ V. F $ se-- 9,4 t4 --e 7: 'l 1. f
| |
| . ,, g
| |
| . #1 . .. . . .
| |
| ' /t 4 '
| |
| V 1500VaS).
| |
| f of a y gg 4 0,31 L e kL Overall Rack Configuration 10x11 1
| |
| *) j I ,/ No8.7# -Jieers
| |
| $7.,It1 I - . . . . .
| |
| I D , y,& y5
| |
| . ,p X3 = ll''1 6 _ j,3 ,,,,,yz 6
| |
| .; t- ,
| |
| Il9 l0 (xs,y$) y% :. H9. I 21 /.Sxfo,S7s s VL FCV3 L
| |
| /
| |
| Assumed Support Details Fixed 10" Sc 160 A = 34.02 in 2 I = 399. in" Adjustable Upper
| |
| * 2 A=3 (8 2 -
| |
| 5.25 2)=28.618 in '
| |
| f / I= .049 ( 8 "-5. 2 5 ") = 163 in" !
| |
| 7' I t The manual data preparation presented in this appendix is the raw material used by the analyst. The purpose is archival documentation, not detailed exposition of any principles.
| |
| * Actual upper support.used in construction.
| |
| l Al-1 _ _ . _ _ _ _ _ _ _ _ - _ _ _ _ _ _ _ _ _ _ _ _ _ _
| |
| | |
| 1
| |
| <^x V, Middle -
| |
| 2 A = 37.7 in y
| |
| u I = 188.16 in" Lower (solid)
| |
| A = 12.566 in 2 I = 12.54 in"
| |
| _9' t Use configuration for upper and lower in Predyna.
| |
| (^g
| |
| %.)
| |
| 2 A = 2 (92 -6.5 2) = 30.43 in l f
| |
| i 1
| |
| 9" w >
| |
| I= .049(9"-6.0 4) = 258 in" k Therefore, some early analyses are way too conservative. May need to correct R valu,es1
| |
| ( l 1 i l
| |
| Al-2 l
| |
| | |
| oO ' Calculation of Correct : Gap. Spacing . (Dynahis runs (undocumented) on previous jobs indicate results not too sensitive to gap size!)
| |
| Max. Opening = 8.85 + .015 = 8.865 inch-Min. Opening. 8.58 + .01 = 8.59 inch Average = .8.865 +.8.59 = 8.727' inch 2
| |
| Gap = 8.727 - 8.426 = .302" 2
| |
| Use this in-lieu of .426"ti lo l
| |
| .i O- A1-3
| |
| | |
| Check of Inertia Properties in FRI 5
| |
| Program computes Ixx = 1.4579 x 10 5
| |
| Iyy = 1.369 x 10 Izz = 9.639 x 10" l FAH - FAhl l FA N/~ FAM4 , ,,
| |
| --l <y ~g, - l-FANI
| |
| ~~
| |
| j
| |
| , ^
| |
| i n . --
| |
| .p.....
| |
| Y ftA = 3rY q -
| |
| /08 ,, .
| |
| I Ig v.09 */or ^ ,
| |
| )
| |
| (
| |
| k ,
| |
| , \ ~ .- .t I( n o? x /20
| |
| /20''
| |
| if Steel 1 p (329.6) + p g (12960)
| |
| I oc = .0007505 x 4025 = 1.905 x 10 i
| |
| l i
| |
| l O Al-4
| |
| | |
| O .
| |
| -Approximate Hand Calculation 2
| |
| -5 # sec Total mass = 1.905' x 10 x 165 x 108 x 120 = 40.74 Ln Total wt - 15742#
| |
| z+z 3 y)
| |
| I xx 2
| |
| := f f f - (y 2,x) pdxdydz = px ( 7-.3 12
| |
| +y) pdxdydz = " (x 2 , y 2) = 88487 I
| |
| zz
| |
| =
| |
| fff (x 12 O
| |
| 4 l
| |
| l l
| |
| l 0 i Al-5 i
| |
| I
| |
| - = -- - _ _ . - - . _ _ _ . _ _ _ - _ _ _ _
| |
| | |
| 1 O N 'Y - C l RC. L E b ITEM S WED F, R -
| |
| IN D V uA1413 DSTh Ft c. E3 '
| |
| trun ppredyn2 PG +g
| |
| /*%
| |
| pacific gas and electric 10x11 rack 2/95'
| |
| '*UT A TITLE FOR JOB INPUT A,B,L,NX,NY,NT. h [ 19.125, . ,h,h,0 INPUT C,T,NC:C1,T1 10.572,.04,110,8.85,.08
| |
| ^
| |
| INPUT ASSEMBLY METAL AREA, WEIGHT 39.,1550..
| |
| pacific gas and electri' 10x11 rack 2/95 A,B,L,NX,NY= 108.125 119.125 169.000 10 11 C.T,NC 10.5720 .400000E-01 110 C1,T1= 8.85000 . 8000 0_0 E-01 EFFECTIVE WALL THICKNESS = T.146969J-
| |
| ~
| |
| 8/A= 1.10173 INPUT BA1,K11,K2Qi,K2P1 1.0,.1406,.208,.208 INPUT BA2,K12,K2Q2,K2P2 1.2,.166,.219,.235 Ki,K2Q,K2P= .153520 .213595 .221734 FA,FIX,FIY= 241.000 3650.08 2210.00 AM,IX,!Y,J= 374.456 383787. 321378.
| |
| FUEL ASSEMBLY DENSITY = @460212.
| |
| 3616E-03)
| |
| Ov SHEAR,Y-Z DENDING SPRINGS U EKAM,EKBM,1/KY= 5 33.2500 24.2500 2.23994
| |
| .dX,KDY,PS!Y= .308260E+12 .421733E+08 48.1369
| |
| -X SHEAR,X-Z BENDING SPRINGS M,EKAM,EKBM,1./KX= 4 27.5000 20.0000 2.20073 KBY,KDX,PSIX= .257069E+12 .427328E+08 39.4405 RACK EXTENSION AND TORSIONAL SPRING RATES
| |
| 'KDZ,KRZ= .627048E+08 .206899E+11 COEFFICIENTS FOR EGELAST FILE CFXR,CFYR,CFZR= .801162E-02 .801162E .267054E-02
| |
| -C= "., M R.CTX.CTY= 129424r-03 .140966E-03 .249594E-03. .259104E-READ IN SUPPORT NLP18ER-IF ZERO, END
| |
| * INPUT NSUP,XSUP,YSUP 1,38.2045,43.7045 q l
| |
| RESULTS FOR SUPPORT NSUP= 1.
| |
| XSUP= 38.2045~ YSUP= 43.7045 W >MO * *' N o f M READ:IN TOP AREA, INERTIA, WIDTH, HEIGHT,NS 28.618,163.,8.,4. 1 T 01 SUPPORT i
| |
| CFX=CFY= .786218E-01 CFZ= .349430E-01 X=CMY= .24ssear-gi .
| |
| "AD BOTTOM AREA INERTI A , WIDTH, HEIGHT ,NS 12.566,12.54,8.,2.25,1 8 ,T OM OF SUPPORT ,- )
| |
| CFX=CFY= .179055 CFZ= .795798E-01 CMX=CMY= .318979
| |
| _ _ _ _ _ _ _ m _. ___
| |
| | |
| READ IN FLRc0 IF NO MOM. SUPPUKI AI e s t.
| |
| NOW READ IN AS,EIS,FLR 28.618,1G3.,0.
| |
| READ IN FLOOR ELASTICITY CONST.
| |
| IF KFLR=0. THEN ASSUME RIGID FLOOR 0.
| |
| / .515037E+07 s
| |
| d ?,KLF,KLR,KS=, .887629E+08 .221175E+08 .726313E+07 ,
| |
| d= .000000E+00 SUPPORT SPRING RATE KFRIC= .112755E+
| |
| MISC. CALCS. I WT.0F SUPPORT = 50.6181 READ IN SUPPORT NUMBER-!F ZERO, END INPUT NSUP,XSUP,YSUP 0,0.,0.
| |
| FWRX= (8_
| |
| READ IN t.,A FOR CHANNELLED FUEL 0.,0.
| |
| CHANNELLED FUEL SPRING = .000000E+00 KX,KY= C12664of 5382.8 FOR 110 ASSEMBLIES INPUT NOMINAL CELL CLF.armrst. .426 CELL-ASSEMBLY ',NITI AL GAP VALUES
| |
| .213000 .213000 ;
| |
| .266250 .159750
| |
| .319500 .106500
| |
| .372750 .532500E-01 l .426000 .596046E-07 GEOMETRY VALUE FOR ACCIDENT CALCS.
| |
| , CSAl= .179949E-01 0 0F PROGRAM V 6 9.t f f C.A T ro O %Y HAub CoM para ppo y fo LLO Ws i
| |
| l
| |
| )
| |
| k O ,
| |
| l l
| |
| | |
| January, 1985 INPUT DATA FOR DYNAHIS-
| |
| )
| |
| PRE-DYNAHIS Hand Calculation 10xil Rack (see PG&E documentation report) carried out for Original runs of 10xil Rack to verify'PREDYNA' B = 119.125 L = 169 NX = 10 NY = 11
| |
| : l. A = 108.125 EL4 = 42.25
| |
| : 2. 'C = 10.572 T= .04 NC = 110 C1 = 8.85 T1 = .08 Tc = 2 x .04 + .08 x 8.85/10. 572 = .1469"
| |
| : 3. AA = 39, WT = 1550#
| |
| B/A = 1.101 Final Interpolated values K1 = .154 K2O = .2135 K2P = .2217 (50% between B/A = 1.0 and 1.2)
| |
| FA = 2 x IOxll + 10+11.=.241 3
| |
| FIX = 10xil x [( 13x12 ) , 31 ] = 2650 2 10 12 11 l
| |
| 3 FIY = 11x10 [ lix12 2
| |
| ,12 ) = 2210 12 10 11 AM = 241 x 10.572 x .1469 = 374 l-L EIX = 2650 x 10.572 3 x .1469 = 460196 EIY = 2210 x 10.572 3 x .146969 = 383786 O
| |
| EJ = .154 x 108.125 3 x 119.125 x .146969/10.572 = 321377 A
| |
| | |
| Bending, Torsion, Ext. Spring Calcs Not Needed for Rigid Rack Stress Coefficients - Rack CFXR = 3/374 = .008021 = CFYR CFZR = 2.674 x.10-3
| |
| ~4 CMX = .5 x 119.125/460196 = 1.294x10 CMY = .5 x 108.125 /383786 = 1.4087x10~4 CTX = .146969 x .2217 x 10.572 2
| |
| x 100 x 11 = 4005.88 CTX = 1/CTX = 2.496 x 10-4
| |
| -4 CTY = 2.496 x 10 -4 x .2217 = 2.594 x 10
| |
| .2133 Stress Coefficients - Supports l
| |
| l AS = 28.618 I = 163 WIDTH = 8 HEIGHT = 4 (TOP) l CFXS = 2.25/28.618 = .0786 CMXS,= .5 x 8/163 = .0245 O ,
| |
| l l Al-7
| |
| | |
| NS = 1 Support Springs A1 = 28.618 H = 4.00 A2 = 12.566 H = 2.25 A = 28.618 EK1 =
| |
| * *
| |
| * 1
| |
| = 202.47 x 10 6 4
| |
| * *1 *1 *5 = 158.05 x 10 6 EK2 =
| |
| 2.25 s
| |
| 6 EKF = 202.5 x 158 x lo 88.75 x 10 360.52 WS = 1 x 8 = 8 O . x 4 x 14697 x 10 6 .787 x 10 6
| |
| ESTR =
| |
| 2 x 10.572, 6
| |
| ELKF = 1.05 x 28.3 x 10 ,1.27 x 8 = 22.12 x 10 6 15 .91 x 8 7
| |
| EKLR = 1.05 x .787 x 10 6 = .7265 x 10
| |
| .91 BB = 0 1 1 1 1 , , ,
| |
| 6 6 6 EKS 88.75 x 10 22.12 x 10 7.265 x 10 lh = (.01127 + .04521 + .13765) 10 6
| |
| = .19413 x 10 6
| |
| EKS = 5.1512 x 10 6
| |
| -- ~a
| |
| | |
| O Check of Friction Springs -
| |
| Not critical - just make sure a suf ficiently large value is used!
| |
| Check of Calculation of KX, KY C = 10.572 2 =
| |
| 62 = 6/C .05368 8 = .23170 Cosh sb/2 = 137741 1/ cosh =0 0 =
| |
| 0 = .0103 PX =
| |
| 773.06
| |
| (.2317 x 120)
| |
| O PY =
| |
| 8 = .01275
| |
| (.2317) x 108.125) 3 6
| |
| EKK (KX) =
| |
| 8 x 10 x 28.3 x 10 x( .146969) x 169 = 16.4494 10.92 119.425 4 2
| |
| EF3C = 16. 4494 x ( . 2317 x 19.125 ) = 12532 OK For 110 assemblies filled Increase these by factor of 10 before inputting into program for solution stability.
| |
| O Al-9
| |
| | |
| run ppredyn2 INPUT A TITLE FOR JOB pge 6x11 rack INPUT A,B,L,NX,NY,NT 64.125,119.13,165.5,6,11,0 INPUT C,T,NC,C1,T1 10.572,.04,66,8.85,.08 INPUT ASSEMBLY METAL AREA, WEIGHT 39.,1550.
| |
| pge 6x11 rack A,B,L,NX,NY= 64.1250 119.130 165.500 6 11 C,T,NC 10.5720 .400000E-01 66 C1,T1= 8.85000 .800000E-01 EFFECTIVE WALL THICKNESS = .146969 BiA= 1.85778 INPUT BA1,K11,K201,K2P1 1.5,.196,.231,.269 INPUT BA2,K12,K202,K2P2 2.0,.229,.246,.3097 K20,K2P= .219613 .241733 .298123
| |
| ,FIX,FIY= 149.000 1634.42 524.000 AM,IX,IY,J= 231.510 283832. 90997.5 95902.9 FUEL ASSEMBLY DENSITY = .621487E-03 Y SHEAR,Y-Z BENDING SPRINGS M,EKAM,EKBM,1/KY= 5 33.2500 24.2500 2.19372 KBX,KDY,PSIY= .194138E+12 .271964E+08 49.0383 X SHEAR,X-Z BENDING SPRINGS M,EKAM,EKBM,1./KX= 2 10.5000 7.50000 2.10984 KBY,KDX,PSIX= .622412E+11 .270644E+08 15.1207 RACK EXTENSION AND TORSIONAL SPRING RATES KDZ,KRZ= .395876E+08 .630467E+10 COEFFICIENTS FOR EGELSST FILE CFXR,CFYR,CFZR= .129584E-01 .129584E-01 .431946E-02 CMXR,CMYR,CTX,CTY= .209860E-03 .352345E-03 .515666E-03 .635956E-03 READ IN SUPPORT NUMBER-IF ZERO, END INPUT NSUP,XSUP,YSUP 1, 0, , 0.
| |
| RESULTS FOR SUPPORT NSUP= 1
| |
| .000000E+00 YSUP= .000000E+00 S"SUP=
| |
| QD IN TOP AREA, INERTIA, WIDTH, HEIGHT,NS 28.618,163.,8.,4.,1 idP OF SUPPORT CFX=CFY= .786218E-01 CFZ= .349430E-01 CMX=CMY= .245399E-01 READ BOTTOM AREA, INERTIA, WIDTH, HEIGHT,NS 12.566,12.54,8.,2.25,1
| |
| | |
| CFX=CFY= .179056 CFZ= .796/9dt-01 CMX=CMY= .318979 ADDITIONAL CALCS. FOR DYNAHIS SUPPORT SPRING RATES f5ADSUPPORTAREA,INERTIATOBEUSEDFORSPRINGS TAD IN FLR=0 IF NO MOM. SUPPORT AT BASE 4 READ IN AS,EIS,FLR 28.618,163.,0.
| |
| READ IN FLOOR ELASTICITY CONST.
| |
| IF KFLR=0. THEN ASSUME RIGID FLOOR O.
| |
| KF,KLF,KLR,KS= .887629E+08 .221175E+08 .726313E+07 .515037E+07 KH= .000000E+00 SUPPORT FRICTION SPRING RATE KFRIC= .112755E+09 MISC. CALCS.
| |
| WT.0F SUPPORT = 50.6181 READ IN SUPPORT NUMBER-IF ZERO, END INPUT NSUP,XSUP,YSUP 0,0.,0.
| |
| FWRX= 4833.70 READ IN E,I FOR CHANNELLED FUEL 0.,0.
| |
| CHANNELLED FUEL SPRING RATE = .000000E+00 KX,KY= 7440.87 26018.8 FOR 66 ASSEMBLIES INPUT NOMINAL CELL CLEARANCE .302 CELL-ASSEMBLY INITIAL GAP VALUES
| |
| .151000 9 151000
| |
| .188750 .113250
| |
| ?26500 .755000E-01
| |
| .264250 .377500E-01
| |
| .302000 .000000E+00 GEOMETRY VALUE FOR ACCIDENT CALCS.
| |
| C8A1= .281168E-01 END OF PROGRAM O
| |
| | |
| ]
| |
| l 4
| |
| DESIGN ANALYSIS OF H-RACK l
| |
| i l
| |
| This rack is only partially loaded and is not surrounded by other racks. The rack is analyzed as a rack with 24 cells filled i The rack cross section is square. The and located of f-center.
| |
| rack differs from other racks in the pool in the construction of its support feet. Four pipes with a total height = 3.5" are used for the feet. Preliminary calculations atte attached showing details of const: :cting appropriate data input for DYNAHIS.
| |
| .8, The following runs have been carried out: HE quake (COF
| |
| .2); DE quake (COF .8, .2). See Table 6.4 of the main report.
| |
| Tae results show that all margins are below the critical values for rack integrity under both DE and HE earthquakes for the complete range of coefficients of friction.
| |
| )
| |
| i j
| |
| f O
| |
| /M-M
| |
| | |
| February 5, 190'i O .
| |
| Configuration for Rack H 4 fixed feet made of 24x160 pipe with 1" thick 14" diameter b.ra plate. The total height f rom the rack baseplate of each foot is 3.5".
| |
| * Wt = 1 x 542. 09 = l ? '.: . 5 #
| |
| 4 lo3(1
| |
| /
| |
| j 1 OD = 12" ID = 9.376" R ,=12" I
| |
| (t d_, A=2 (OD2 _yp2 ) = 44.05.in 2-4-
| |
| O I= .049 (12" - 9.376") = 638. 7 in "
| |
| tesuu 3 ''
| |
| f?/ /
| |
| [ h I
| |
| // /- 1
| |
| ~ " 1's , ,
| |
| ,, //,/, I
| |
| &a, .
| |
| /
| |
| .///
| |
| tt. 25
| |
| /If /].E
| |
| %'///j/
| |
| Wy ,
| |
| t.,, ,,.
| |
| m" !
| |
| (/ /' /t'/J .
| |
| ' ~
| |
| l l
| |
| -'p -
| |
| r O .
| |
| u.e i
| |
| i
| |
| - m,
| |
| ___mm_______ _ _ _ _ _ _ _ _ . . _ - _ _ _ _ _ - . _ _ _ _ _ -
| |
| | |
| e O
| |
| 9 y > e p j g ** 4 e p arg ,* d'a=
| |
| * 7 w*** *"
| |
| ?'
| |
| + . .
| |
| * * ~ r*
| |
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| i 1 .
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| l l
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| APPENDIX II I
| |
| OUTPUTS FROM DYNAMICS RUNS l (UNCORRECTED)
| |
| I i l
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| O l
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