ML20126B110

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Seismic Analysis of Reactor Internals
ML20126B110
Person / Time
Site: Dresden, Trojan  
Issue date: 12/24/1968
From: Leslie Liu, Watzel N
GENERAL ELECTRIC CO.
To:
Shared Package
ML17192A539 List:
References
257HA718, DAR-67, NUDOCS 8003050506
Download: ML20126B110 (39)
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_t ~ ' j t.. - ( APSIRACT a 'Ihc maximum seismic shears and mments of reactor internals for due to their respective Design Basis Earthquakes are II The The method used for these deteminations are detailed. determined. results for those components with significant seismic loads are given in the following table. Table I Significant Seismic Shears and Skrents - Dresden II . Plant Reactor Internals. Dresden II Vessel Skirt Moment 12,800.0 (Kip ft.) Vessel Skirt Shear 728.0 (Kips) Tcp Guide Shear 147.0 (Kips) Shroud Support Mcment 5,360.0 (Kip - ft.) Total Fuel Moment 700.0 l (Kip - ft.) Total CRD Housing $bment 880.0 (Kip - ft.) Stabilizer Force 202.0 J (rips) Notes: Dresden II Desi;:n Basis Ear-hquake: .10 g 1940 El centro Earthquake 1. N S Component. ( For maxi. mum credible earthquake values, mitiply the tabulated value 3. Dresden 2.00 W jQ~ 2 5 7 5,1 7 3 3 U $f $U bb [' ""'M w .e m y a i.-4 .$"r'.

4 GENER AL h ELECTRIC AToulC power EQUIPutNT oEPAf<rJENT 257M\\718 ENGINEERING 8'oRM SEIS5GC ANALYSIS OF REACTOR INTERNALS FOR THE DPESDEN II Pleur ~ I. IhTROCUCTION In order to insure safe operation and prevent unnecessary outate of nuclear boiler systems, seismic responses of reactor internals have to be determined. This report details the methods, approximations, and computer programs which It shculd he emphasized are being used at APED to obtain such respenses. that these metheds, approximaticns, and computer programs are constantly hhat is reported here is current as of the analysis date. being updated. In addition to the description of the method used for detemining seisnic response, the maxi:1mi seismic shears and moments for varicus reactor internal are also incidded in components of two plants (Dresden II this report. II. MATHE'MTICAL 50DELS OF THE NUCLEAR FIANTS_ plants were The nuclear steam supply system of the Presden II Several .. modeled with it: ped mass configuratiens as shcwn in Ficures 1 and 2. , features in each model carry the same asse: ptions in modeling, but each mdel j was generated by an individual author.* The assumptions associated with each ccmpenent of the Dresden II A discussion of each assumption is provided to indicate model are given here. the degree of confidence associated with the entire model. Vessel Head - The entire weight of the vessel head is it= ped at its center 1. The flexib'ility of the vessel head is detemined frem the of gravity. cunes given in Reference 1 where a tnmcated hemispherical head shcur. T below is analy:ed. rP n d. o ////////////// /// L. K. Liu, Dresden II Plant me - /h [ acy 0 1 o. 1 ~r n,...u, '"""g.:u,a -; v G r l ( J l'~~ r =

r GEN ER AL h ELECTRIC 4 s.. w C.. AToulC power EQUIPMENT oEPARTMENT 257M718 ENGINEERING FORM U" SEIS>fIC ANALYSIS OF REAGOR INTEFNALS FOR THE DRESDEN II PLANT The element flexibility matrix is defined by the following ratrix equation: ks bA E, A

4. 1P where:

'otation of the con of truncatad hemisphere {, lateral deflection thus: f is the rotaticn due to a unit moment M yy and f is the lateral deflection due to a unit moment M, etc. 12 The distance d is taken to be the distance frem top of the flange to the center gravity of the shell portion of the vessel head. Flanges - the weights of the two flanges are lumped at their center cf 2. The stiffness of the flanges compared to that of the vessel is gravity. such that the flanges move as a rigid body. Dryer - The dryer is relatively light and is supported off the vessel. 3. In no way does its respense affect those of the rest of the internals. For these reasons its weight is it ped to that of the vessel. Vessel - The vessel is divided into several sections. The weight within 4. The stiffness of these eachsectionisludpedatitscenterofgravity. vessel secticas are taken equal to those of beams having the same cross-A shear sectional moment of inertia and area as the cylindrical vessel. fem factor of 2 is used for detemining the deflection due to shear. Vessel skirt - The mass and stiffness catrices are treated in the same 5. manner as the' vessel. Bottomhead-Themassandstiffnessmatricesaretreated!thesamen 6. as the top head. l l_mgv O _a s C. 1 s am._,v.a I A--- E I 8h

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GENER AL $ ELECTRIC 257HA718 ATouic POWER EQUIPMENT oEPARTMENT 2, ENGINEERING FORM SEISMIC ANALYSIS OF REACTOR INTEP.NALS FOR 'UE DRESDEN I l Pedestal, Shield Wall, Building $ Foundation - The treatment is the same 7. as these given in Feferences 2 and 3. t Separators - The separators are mo' deled as beams for stiffness determ I 8. with masses lumped at the center of gravity. Standpipe Due to the bracing between standpipes, the stiffness is verf 9. As usual, the masses high and essentially rigid body motion results. are lumped at the center of gravity. Top Guide and Core Plate - These are treated as rigid bodies (for lateral 10. Net of motien) and their masses as included as part of the shroud mass. the fuel mass is considered to be part of the top guide and core pine mass. See 12 belcw. Shroud - The stiffness of the shroud is taken to be equal to that of a 9 11. The shear ferm factor is beam with the same mcment of inertia and area. 'n.e masses are luq>ed at the center of mass. J taken to be 2. The Dresden II plant This support is simulated by a hinge has the leg-type shroud support. This torsion spring rate is equal to the resisting and a torsion spring. Centributions to this moment per unit rotation of the shroud bottcm. resisting moment comes from: Axial deformation (tensien or compressien) of the legs in direct a. proportien to their distances from the retation axis. Bending of the legs due to structural ccmpatibility requiremnts. b. Bending stiffness of the shroud support plate assuming it to be an c. annular plate with a rigid center as show. M, N f Q / 90% of the stiffness centributien ecmes from (a). l m J lT o CON

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1 t. GENER AL h ELECTRIC AToulC power EculPMENT DEPARTMENT ENGINEERING form -l SEISVIC ANALYSIS OF FEACTOR INTEFNALS FOR 'I1E DRESDEN II PWT ) of these compenents are needed, they can be detemined after the system response has been found. III. HYDRODYNAMIC MASS l In order to prcperly account for the effect of the water enclosed in the pressure vessel en the dynamic characteristic of nuclear steam supply system (SSSS), hydrodynamic (or virtual) mass has to be included in the mass =trix. i ) The hydrodynamic mass ranifests itself as a dynamic coupling between the real ) masses and therefore should appear as off-diagonal as well as diagonal terms in the mass matrix. The hydrodynamic masses of the NSSS can be ideali:ed as thos'e between three cencentric cylinders. The inner cylinder representing the fuel, guide tube, or standpipes: The second and third cylinders representing f the shroud and pressure vessel. Based en this idealization,' the methed for generating the mass matrix to include hydrodynamic mass terms is derived in Appendix A. Tne method given in Appendix A is used to generate the mass matrices for both the Millstone :nd Dre:: den plants. IV. SYS~E4 FLEXI3ILITY Ah'D LCAD MATRIX Having derived the individual eierent stiffness or flexibility matrices as described in II, the overall system stiffness or flexibility matrix can be derived by either the matrix force method or the matrix displacement method. Both methods are described in Reference 5. The matrix force method has been chosen for this analysis due to the adaptibility of the computer program GDCP-I (See Beference O to this method. A brief description of the method, The fo11 ping closely the exposition given in Fmference 5, is given below. reader is referred to Feference 5 for the detailed logical proof of the validity of the method. 1. Derivatien of the systec flexibility matrix: If axia1' deformations are neglected for element a, the individual element flexibility ratrix can be represented a_L: F, = f f yy 12 f '21 22 I a i uE O j j......., n i i cows on eutt s su no. f& /5

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4 GENEBAL Q ELECTRIC Atomic power EoUIPMENT DEPARTMENT 257HA718 ENGINEERING FORM SEISMIC ANALYSIS OF REACTOR INTERNALS FOR 'DE DRESDE ~ He three lumped Puel - The fuel is modeled by three lumped masses. 12. masses are distributed such that the three largest periods are approx-imately equal to the theoretical periods of a prismatic beam with uni-This is accomplished by dividing the beam into formly distributed mass. The rem: :ning 8 segments, with 1/8 of the mass lumped at both supports. This is 3/4 of the mass equally distributed among the three masses. shown schematically in the figure below. E~ [ ^ E4 o~ts ~tM A o o~d:M .!L nqu ne 1/Bt at both supports are lu= ped with the mass of the top guide and core plate ento the shroud. The stiffness is taken to be that of a beam with mcment of inertia equal to that of the fuel channel alene. Guide tuke - Again the lunped masses are distributed such that the fund-13. ne amental period is the same as the theoretical fundamental period. effect of the axial load (due to the fuel weight) en the guide tube frequency is neglected. ne justification being that the axial load is less than 1% of the critical buckling load. Hence, the frequency chanr,e Be guide tube is treated as a beam for stiffness is extremely small. calculations. CRD and CRD housing - ne masses are distributed such that the fundamen 14. natural frequency is apprcximately the same as t.at determined frcm k The stiffness is taken to be that experiments reported in Reference 4. of the GD hcusing alene. Not included in the mathe.atical model are light compenents such as jet nis pump, in-core guide tube and housing, spargers and their supply headers The just-is done in order to reduce the complexity of the dynamic model. ification for this lies in the fact the response of these light ccrpenents do If the seistic respenses not materially influence the overall system respense. 88 j ... u, _L_ ~wuual -m'~, -.-w. ,.a

J. GEN ER AL $ ELECTRIC 4 ATOMIC POWER EoVIPMENT oEPARTMENT ggg ENGINEERING FORM SEISMIC ANALYSIS OF REACTOR Ih*I~cRNALS FOR THE DRESDEN II et.pr The individual eleaent flexibility matrices is then formed into a diagonally partitioned flexibility matrix, Fv, Fa Fb py ~\\ Fs Forbeameierents,thisoperationisdonebytheprogramGE'(P-Iautcmatien11y where the eccent of inertia, effective area, length, Young's modulus, and Poisson's ratio is given. For non-beam elements, the individual cicnent flexibility matrix can be read in.- The next step is to rencve redundant supports temporarily from the system j In most nucicar plants, to reduce it to a statically determinate structure. the redundancies can be chosen to be the stabili:ers between the vessel an the shield wall, and between the shield wall and the building. Using the statically dete=inate system, the ith colen of a Matrix B,is obtained by finding the mcment and shear at every mass point when the structure is subjected to the external ferce Fg = 1 and all other external forces The complete matrix Bo is formed by varying i frem 1 to N, F -0 for fpi j is fermed in a where N is the nt=ber of mass points. Another matrix B1 similar manner, except that the redundant force, L takes the place of the is the moment and shear at external force, F Thus, the ith column of B1 all mass points when the statically determinate structure is subjected to X =1, and all other redundant forces Xpo for i/j. The complete B1 natrix tis forced when i is varied between 1 and n, where n is the degree of red-undancy. i Given the three matrices, B, F, and B, the system flexibility matrix, j n y 1 f, is given by: 0 U f=D -D10 1 10 og et "'O t n... - u. __-., -.. 2-n ~~~ pa a.p.g l

GENER AL @ ELECTRIC $ s# 257HA718 . ATOMIC POWER ECU PMENT oEP ARTMENT enanneenma ronu PL\\NT SEISMIC AN*AI.YSIS OF PDCIOR I.TERNALS FOR TIE DRESDEN II N where: T D oo = % g g T D10 = B F 4 y D =B Tv B 11 y y bot = transpose of Bo, etc. D -1 = inverse of D 11 yy The formation of the ratrix E, and the necessary matrix multiplication and inversion are easily acccmplished by using the program CDOP-I. The load matrix, the 1 column of which gives the moment and shear at each 0 mass point when the system is subjected to F =1 and g =o when jpi, is given i by: "1 B=B -B D D g y yy 10 It sheuld be noted that if the complete structure is statically determinate, \\ B - q. V. NATURAI. FREQCD,CIES AND 3CCE SHAPES Given the flexibility ratrix, f, the stiffness matrix, K, of the system can be found by a simple inversien. From the mass matrix (non-diagonal in this case) and the stiffness ratrix, the unda. ped natural frequencies and mode shapes can easily be determined frcm an eigenvalue reutine. These capabilities are again built-in in the GDCP I program. The modal matrix, 0, outputted by'GDCP-I is normali:ed such that: C' ? 0 = I T and O K0=('Ans) where M is the mass matrix l K is the stiffness matrix I is the identity matrix i [' X n s] is the diagenal ratrix of the eigenvalues, i.e., angular

frequency squared.

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4 GEN ER AL h ELECTRIC ATOMIC power EcVIPMENT oEPARTMENT 73 gig ENGINEERINo FORM SEISMIC A'lALYSIS OF RCAcr0R INTERNALS POR WE CRESCEN II.PLAur VI. EQUATIONS OF $0 TION The equations of motion in matrix form is as fo11cws: M (5 + Y) + C X + XX = 0 where: M = mass matrix, n x n (This includes the hydredynamic mass.) X = column vector of displacement relative to ground (n x 1) C = damping matrix (n x n) K = stiffness matrix (n x n) 'Y'.= column vector of ground accelerations (n x 1)

  • *= second derivative with respect to time Removing the grcund acceleration vecter to the right side of equation (1),

the equatien reduces to the classical fom: MI+CX+KX= - F'r (2) In order to unceuple equation (2), we set: (3) X=0f Equation (2) then becomes: MOj'+Cfi+K/g= - hk (4) Pre-multiplying by the transpese of 0 and using the orthegenality conditions, we get: [ h% + Ol}+ [K0g = -5 (5) +[C0j+Oggc},, gM (6) T The abcVe procedure for uncoupling the equatien of metion by using the medal matrix of the undag ed system =ust require that daq ing in the system be small. I It will be further assumed that the da ping matrix C is such that C CO is a diagonal matrix. The elements of this diagenal matrix are the medal da. ping values eID m I:P 0 3........ ......, u .w_ ,gte a.av.sg ~" ~

G ENER AL h ELECTRIC ATOMIC power EQUIPMENT DEPARTMENT ENGINEERINo FORM PI.N A' flT6E SEISMIC MAf YP.IS OF P.DC OR INTED.HAI$ FOR 'rHE DRF2D; II Vith the akovo assunptions, ce,uation (6) ray he tvitten: (7), S rr (t) u ~ = - A 26 w q + u q g g qg g g i = 1,7 --- N darping ratio for tSe i ende exrressed as parcent oF criticci "bere S = g danping 1"S natural ancular frenuenciea h of t e systen u = j E node = -f HI nodal participation factor for the 1 S = g ground acceleration eine history U = T transpose of L'h rnde shape. a f = g (5x1) coluen vector 'ange clerants are all uniev I = u for dyaarie e-alysis nec TSc nodal danping values usually given in the nient design <pec, Ta le II

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1 b givenbelg'gg e i 5.0 Reinforced concrete meructure a. 1.0 b. Welded structural assenhlies 2.0 Bolted and riveted structural assenblins c. .5 d. Vital piping systen 1.0 Purr, fans, mechanical equipnent I e. 2.0 f.

  • cinforced or prestre<=ed concrete prienry centain-ment structures The resperac The accountable danping is, therefore, alvays less char 5* (3 g.05).

f frenuency or error associated with assur.ing danping not changing tSe resonant This general curve sets the mode shapes is shown in Figure 5 of Feference 7.(in* correnponds to 5* damping for error in the response at less than 10 percent the modeling of the two poner plants undertaken herein). i For conponents not covered in the Table I (fuel and control red drive), the dart rq for the CWD. The darpine factor for vnluca are assigned (7" for the fuel), and 1. the CRD was experir.entally daternined recently en be 3.5% (Ref. 4). Horve r, t' d a (10: vas not reported early enou?h for inclusion in the present analvste. Unvin- .j av ) j deternined the darping values of individual cerponerer, the I l? 'l I ie l'"*** gfd_s /J -a v-6fi c.~,.~.-<- , u._r.. ~~ ^%W%f s

\\ G E N E R Al. $ E1.E CT R10 .s AToulC power EoulPMENT oEPARTMENT 257HA718 ENGINEERING FORM J PLA.NT J SEISMIC A% LYSIS OF REACTOR INTEFNALS FOR THE DRESDEN II ,,,u. system modal damping value for each mode can be easily determined. The approach taken is to examine the mode shapes. If in any particular mode l cnly one component vibrated with the other components essentially stationary, j then the damping value for that component is taken as the system modal damping i If rcre than one component vibrates, than a modal damping value is value. A assigned based en the relative vibration arplitudes of these components. more refined energy balance technique of assigning modal darping is presently . being investigated and will be reported in RA-43. Presently, the procedure described abcve, selecting medal darping fica mode shape ar:plitude is considered within the accuracy of the model and with good engineering judgement, a quite j acceptable precedure. l VII. RESPC*,'SE CALCULATIGS The system of one degree of freedcm equaticas represented by eq. (7) subjected l f to the initial ccnditions: l 0 1 (0) l = q 0 q, (0) dete..ines the modal respense qt (:)Using eq. (7), the maxi um seismic respense (displacement, acceleraticn or load respense) can be determined by one of the two methods described below: Time History Method - In the time history methed, the ground acceleration 1. time histoty is divided into small tim increnents. For each of these (t), i, = 1...m, m f g, is small time increments, the modal respcase qt dete mined. The number of modes censidered, m, is selected in such that the respense cf a 1 (t), f = m,... Nis negligible cc mared to the Icwer ) i T i mode respenses. 1 Having found the modal respenses, qi (t),... L (t), the time history dis-l placement and acceleratica respcases of individual mass points becomes: X (t) = 0 (t) N (t) = 0 t) i l " ',.* O av C $ hE T 0 84 g as ( ( ? ,, f as se c. /[* l - ~ ~ " - -e----..

GEN ER AL $ ELECTRIC I. Atomic PCwER EculPMENT 02PARTMENT g g 71g i PLANT ]" SEISMIC ANALYSIS OF REACTOR Ih*TERNALS FOR TIE DRESDEN II where: / \\ e ' ' 0, h 31 3 (g) 'l '(t) = j 6 0 f(*) O O N1 Nn) The tire history of the load respense becomes L (t) = BK X (t) where B is the previcusly defined load matrix K is the stiffness matrix. The quantities of interest are usually the maxi.m values of the displacement, acceleration, and load responds. These quantities are easily determined by finding the maximum values of X (t), 'X (t), and L (t). Of course, these maxi =um values need not occur at the same t. 2. Restonse Soectrm N thod_ In the respc.se spectrum method, the maxwa modal respcase for each natbral frequency of interest is found frca respense spectra curves for the particular earthquake record under censideraticn. Response spectrum curves for most strong motion earthquakes which have been recorded are given in Reference 8. The response spectrum cur /es in Reference 8 have not been smoothed. It is a more comon practice to use smoothed curves such as those given in Reference 9.

Response

spectrum curves are essentially plots of the maxirum responses of single-degrees-of-freedcm systems described by eq. (7) with S t = 1. 0 against various natural periods or frequencies. Having fousd the maximum modal responses, qi, i = 1,... m, the maxim. th I physical displacement for the i mode is giten by: xi = Siy ei 4 sh. l 0. o .......s. CCutousutt? _h

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~.. e, G EN ER A t h. E LE CT RI C ATOMIC PCwER EQUIPMENT DEPARTMENr,337 t p,7 3 p ENGINEERING FORM ' P T / "~ SEIf"It" A"ALYSIS OF F/.C~0' I!**EP"M.S FM T'!" DPf 5'E" II TIT 6C e 7 l 'h e r e .! .t = 11 li i = d 121 "21 ~.- ,j "ni / "!. As pointed out in reference 10, the -cy.inun nhyqical rc.enonse for each re.ss is then usually taken to be the nn,ttere rnet of the sunn of the squares of each of the maxinum responses for each rede, ie, 1/2 T. (i I i(,

w:

j i i - 1,..., finilarly, the ra::f r.un Ioad response f er t te i's i rMn is fourA frnn BEX ,Lg g = where ? ~ L L = i li 21 "bere n is t'c mtirber of rons in the B ma':rfx. e Lei ~ of the sums Themax1=un1oadresponseisagaintakentobethese!uareroot of the squares of each of the raximun responses for each rede, ie, m 1/2 F + L .(L,)=ax ij = J l ,j= 1 i = 1,... n l -~ v. l r.cv 0 -o. 257 t').718 ....o.. l' _. s ~ ~ o. L co - o~s-ar* R/p ;;-D J'.4 Y l .m. m - -w ,amm.m..mm w.m e ageP"'M k 9F" 4*' O ** *

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ATOMIC POWER ECUIPMENT DEPARTMENT 2 57!'A719 ENGINEERING FCRM ftTLE II PL A N T' SEIS"IC A!!ALYSIS OF P.EACTOR I';Tr.P2:ALS 10P. TFE DP.ESDE': ',oth the tiec history nethod. of analysts and t c renpense spectriin t ethod of b analysis can be perfor=ed by CEM/P-I. VIII MASS Al.D ELASTIC PR0prp;7tg cy tP" FSSS The mass and elastic properties of the !!SS.c for the Dresden II plant are given in Appendices 9 and C, respectively. IX DE.ct*LTS OF TIT A':ALYSIS ' 1. Drenden II Plant s The shear and noment diagrars for the design hasis cartbouake (. log loan F.1 Centro, N-S conporent) are shot.-. in Figures 3 en 10 T' e s hear and cocent values shot.n for the separatnes, standpipes, fuel an? guide tubes ' ~ are total values. They should be divided by the accronristo nunber to obtain individual elenent values. ~here are 210 scoaretors and standpipes, 724 fuel bundles, 177 guide tubes and centrol red drives. In ell c'c figure =, vessel "o" is used as refererce fer the elevatinns, t i DISCL'SSION OF TUE RF.St*LTS Of all components in the nuclear boiler, the design of the folle+ing iteca may be significantly influenced by seis=le loads: s1irt, stabilizer, top guide, fuel, shroud support, CSP housing. ~herefore, t'e lead.s on these co.penents are tabulated in Table I. ~hese loads shoult. 5e used to determine uhetkar nr l:p n s .....,.u. 5 7 F t; $ e f.[y /2, A c/-b V\\ (eo~,e:s-es, s-~o. t .... c.n

GENER AL $ ELECTRIC s. ATOMIC power EQUIPMENT DEPARTMENT .!57P.A71P tuoiNesmiso ro"" a SCIS* TIC A:'ALY. TIS OF P.EACTOP. I I::Tt*J.\\Le TM. THE DVSDF?f II PI MT TITLE = i not the corronents are edequately designed. The ratio of the naxirun credible earthquat<c load co the design cartFqun! ^ lead is conrenly tehen to be equal to the ratio of the t-axinun ground acceleration of the na:cinun credible eert!*qunho to that of t e dc.11:n h earthcuahe. Thin is correct only if the structure remained elastic and herce, linear. Fe+ever, due to the difficulties of nonlinear analysis of highly complex structures, no attenpt has been nade to deternine the loads uhen the structure goes into the inelastic ranae. Therefore, the loads for the naxinun credibic eertF"uahe cre ciatermit'ect 2*: if the structures ra.nained elastic. } In this report, only those results undr: t' c tine history nethod ' ave ' ee-reported. Also, only those results corrnspont:irg to the earthquehes specified for the Dresden II plcnt (Il Centre) are giv2n. The results for other earcheuakes 2.nt th.e.results usine a the response spectrum nethod will be reportad in a future DA?. =... .sne g

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_... 7 l GEHER Alh ELECTRIC Atomic poDER EQUIPMENT DEPARTMENT 2571'A 71R - ENGINEERING F'oRM F E^# eEIc"It A'!AtfrIS OF ftEACTon I I?"TE"*:ALS FO". ?"P. ""PSN?' IT i P EFT ~tE' CES C. P. Steels, ""onsyr etric !?efernatien of Done-Ehaned $ bells of 1. ?. evolution", Journal of.a.polied "echanics, June, 1062. Eartheua!:e Analvsis of the Seactor Pressure t'essel, Dresden II and TIT, 2. Feb. 24, 1^4 ':uclear Plant, by John A. Blu e and Assceiates, Tnd neers, rentrol P.ed Drive "otising Smith, " Vibration Testing of a Preduction S. ti. '4 "C SP.12 5. Installed in the 30" Vessel in Building; G", TP-A315, 257FA600, APED internal docunent. 9 E.. C. Pestel, F. A. Leckie, '"tatrix "ethods in F.las to-Mechanics," 5. a and 10 ?!c-Grati uill Dooh Co., Inc., ?!ett forh, 1063. Chapters V. '4. !!et:el, "0C0?-I, A General Matrix Operation Progran", SAR-36, j 6. I i !!ovenher, 1967. >!. Curtin, "Vihr$ tion Analysis of Piscrete Mass Systens", Peport h'o. 7. 59GL 75, General riectric Con.pany, Schenectady, ?!. Y., " arch,1950 l G. 11. Fousner, et al, "Spectrun Analyais of Strone '!otion Earthquakes", S. Sulletin of the Feisnological Society of Ar. erica, Volune I.3, ?!un'er 2, April, 1953. 0. ":*uclear "cactor and Earthquakes", TID 70?A, August, 1063, USAEC, 01 vision of Technical Infornation. R. W. Clough, " Earthquake Analysis by Se.sponse Spectrun Superposition," 10 Bulletin of the Seisnological Society of Anerica,, 'lolume 52, Nenher 3, July, 1M2. e f h em, I i g.. d /N/ 9Ev o. O d Q J f. ) asseOva68 av CONT @M, MEET TM 48 0. ' \\

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s i ' GENER AL h ELECTRIC ATOMIC r' #ER EQUIPM2NT DEPARTMENT j 257PA713 ENGINEERING FORM i v e t t. c SEIF"IC A! ALYSIS OF REACTOT! I I!CER?!ALS FOR TFE "*f'SDC: II Pl Aer l I APPT.?IDIX A j i i HTDi'0DY"AMIC }fASS i The basic formulae used in this section Perc first derived by John Corr l and Ed Miss. The formulae for hydrodynanic nass derived by then '.' ore in terms of two concentric cylinders. Because the nuclear boiler, for the purposes of hydrodynanic nass analysis, can be thought of as consisting of nany concentric cylinders, a slight modification is necessary. The 4 nodel chosen for the present analysis is she"n in the figure belo". The method for deriving the complete nass natrix (Nc:4 w real nass and the hydrodynanic nass) is essentially as follo"s: j

1.
  • Obtain the kinetic energy function in terns of the lateral velocities f

of the cylinders. This kinetic energy function @cu2d include t'te i hinetic energy of the real masses as well as the kinetic energy of the J fluid nasses. % e kinetic energy of the finid nass is determined by assuming incenpressible and non-viscous flow. I 2. Obtain the potential enerCy function in terna of the lateral coordinates and the real spring constants. l ? l 3. Use the Euler-Lagrange equations to obtain the equations of motion. I 4 Arrange coefficients of the terms involving the second tin.e derivations ' John Corr, "Some !!ev Results on Virtual " ass Ef f ects for reactor Internal Vibration Analysis", January 3, 104, "eno to T. Trocki, i I Component Engineering. Ed Kiss, ""1bration in Fluids", " arch 20, 1967. g i l 0." 0 j j....~. 2ggg J n nui t, 3 co=, o ,m u t s= w. ...u.4 / dM I, 3 ~ '^^ ( .- i 1 1 ' ~ ~ ~ ~ ~ ' ~ ~

i GENER AL h ELECTRIC ATOMIC power ECUIPMENT DEPARTMENT ENGINEERING FORM l$71{A718 l tiv6s SEIS? TIC Apf.fSIS OF PTACTOR I IrlTANM,5 Fot! *UE DP.ESDEM II PLArdT l of the lateral coordinates into the e. ass ratrix (") such that the e. atrix product of M and the colurn vector representing the acceleration of the cylinders produces the inercia forces. The above nrocedures t'ere used to oktvin tSc results sunr.arized below. 4 e 9 4 6 9 e 0 . = =cv g ~o. 257FA715 amenova or cowt ow satte .- sw we. b l' [b M"[ Y*0 [ l

i EU G!1. M CTRIC C0, ,,t (, N,cle w E m ryy C M slen ENGH9EE11364 CALCtJLATION 5HEIT _ Mad 12 / W DArs SHOP ORDSR NO. L e SHEST I CF Ibst &~ Ya SA*& N Y t E C" V/ b 'l N#35 E1 ECT (*) g. e _..........- (,3.. q) / ..( .g. / / M-Ansv/,s fn.d.:, f /,....... O) N W Nr l y / s / / / \\ Q d ovve/ p adsoldr. f m/n.. / f' Yi X1 ~ ..~ g*n y, n- _. of e a ed c7 ider n re/o he . nfe ouke-boor l4*/s / mesf c,7/is/*r w his.A n coesidered a< al.ulafe co orda./e. ~ E<ch cyliab ee a conulerel.4 mova /<Ar*lj na a. l riy1/ hady C.,/ a s Amy e l-re-s /4 an youAd ~ ~~ 7 . -.. Aa n. ne new H ede u /ani.,,w,a %1 l e f cM 4' s le Adv /e7Yee e + /f*ech eto,/tt 7.- _*' ?* d ,.m 1 -e,, o6m: 1.. J

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1 m......_ G H E A!. f.!ISTilt0 C0. ~ 7 ( '1 Hnleer Ensar DivisIn SNGlHEERINO CALCULAffCN SHt!7 OAfr SHOP OROGA NO ~ SY SHEET 2 OF $ULJECT ' hens . o f -f/sy/ u snowb.s, $ = f( __.... We e/ _f/wdl mess 5 a.nov44 s' I h5eI-f/nl gess h *nn/n, d ' M. t ree/ m n J.,L eA,. ii e-6, i - Mc.. ~~ M*_p de +f e ss o[ ae* /cr / 8 9/s acte 1 1 88E' <de'e. for wd ah./ /e-p% .k; ' A fY I; (R,.,-i7,. ) {i' 1/1/:t ep[ebe*,s-fre~Aehee, n:emJes s s'-> Ei ) Zh Carr as s ~ ....I?g. p //g 17Rl h* me 14-k e.r,e. Ne er Te elemen/a oS Nie mass 4 a M*+M,.y+ 4.,v + /fy, + 2 Q, /7;; i /1;V,, + //,.p .. ff. s'+1 =.~ g \\ $s,* f.f., -.' kg*, [a $ t' s 4'fa. Anno /os C is a/we/ys af:r</caemjeei.) n.'& I - V. -4* t 4. i .n>7:;Ay33 l L.. u i 1J.LL' * ."_.1.".i:, M g . _..... ~ - 4. j Sf.e 8%e. ATM U H --.. .___...._q_ 9 "01-17" 9 g.* 9 9s9 c:a=-.a

MkP.r.K AL EJ.LV i mie vse g* - 4 (*-' Nseletr Energy Of rision ENCINCRINO CALCULATION SPEET DATF $ W ORDER NO-SY $HEET J OF SUBJECT ...._..-..z I .M X + c. x rit x = 17? x, 7~4a co/en r vee.lor on hfe re7 7' da.s.Yalue.s of m 4 &$ $ CL/nrkcl evtf4 Me !? test inebr~ix* e b k s) Are&y' n yeou /,y$*t. f fbe'

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l GENER At h ELECTRIC A70MIC po#ER ECUIPM!NT DEPARTMENT ENGINEERlHG FORM 237pA718 1 l 7876C SEIS?!IC Ah'ALYSIS OF P.EACTOR 1 I!!TER'!AI,5 TOR TifE DRCSDCI II PLAw APPO' DIX R " ass and Stiffness Pronerties of the Dresden II ??SSS The cass n chors and elenent stiffness or flexibility nunbers refer to che nunbers sho m in Figure 1. I. " ass Properties a. Seal ?fass (10'T slugs)

  • s s ?!u4 e r-Naam 1

2.67M 2 8.:2o81 3 7.74571 4 5.03105 5 4.0o637 6 3.54637 7 4.09037 8 4.00037 9 .83850 10 5.40690 s 11 2.45341 12 1.7nnn7 13 3.26096 V 14 1.52173 nu 0 ~o. ......,46. ' 't f& /99V4l l l eee e,i?13 2571L' 4R _ s ~ ~o. U s~ee-a o. u. m.-<w-o w --. - + <.

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a GEN ER At h ELECTRIC ^"# 2571't 718 ENGINEERING FORM rer6: S. TIS"IC A!:ALYSIS OF P.F)CTOR I 1 Cr""ALS FO' T"r. PR95DE'? TI PLAtJr 3.66450 15 16 .90062 4.22360 17 4.22360 1.9

4. 22.T60 19

.52795 20 21 .52795 1.72670 22 1.72770 23 i 1.80124 24 h. Real " ass plus hydrodynanic " ass " atrix Ter:s tith only one : ass nun *rer indicate dia;;onal eierents; tems t.ith cro nass nunkers indicate off-diagonal tems. I 1 Mas s ? uf-ber Maas (IF slucs) 1 2.47 1 n.23 2 7.70 3 10.33 4 21.73 5 s 21.12 6 l AEV g e. l 257PA713 l' ^ /Sh /e2-J '/-// l co~v:~swcet s-=c. "' " ^^ - = = =,

1 s' GEN ER AL h ELECTRIC 257HA718 Escissemimo romu l r,16: SEISMIC A!!ALYSIS OF REACTOR I ItCE"sVALS FOR TFF DD.T.SDEN II P LAyr 21.7S 7 25.19 8 0 n.n4 10 in."4 1 11 2.67 12 1.73 i i l 13 1.73 i 14 15.fo 15 17.71 16 17.96 17 4. II S l a. 4.45 1" 4.45 20 .945 21 .96F 22 1.73 23 1.73 24 1.80 4-11 .921 4-12 -1.43 5-13 -14.09 6-14 -14.09 7-15 -14.co 10-21 -?.74 .sv n ~0. .......w. 2'71',\\713 So ^ dia /.;1-AJ-4[ l

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4 GENER AL $ ELECTRIC 4 ATOMIC 70wf R EQUIPufNT CEPARTMENT 2571tA71R ENGINEERING romM TT'E DF,E!DE" II PL AruT-ff f f. E SIIS'rIC A"ALYSIS OF P.EACT0". I I:""T.R"/.LS Fn8 ^ .514 13-17 .514 14-18 .514 15-19 .01 16-20 .81 16-21 Thereferc, there are terms in reverse order The mass natrix is symnetric. as these listed above in off-diagonal positions. II. Elenent Stiff.ess Pronerties ~he element The element nunber corresponds to these shew in Fipare 1. essel Top 1'ead (clerent No.1) is v flexibility eatrix for the 1 (3.2? (10 ') (F.ip-ft)- ~ f = 3 f = - (2.704) (10-8) (tip)~1 f,1 = 13 (5.259) (10~7) (ft/ Kip) f = 22 4 The ele =ent is given en page Definitier.s of fu, f;1, f,, f22 y flexibility estrix of the botten vessel head (ele =ent No.12) is [ .c, o ] w S. 257nA718 31 -.- -o. 2 & /A.;L66.# l ee, .. c. w 1 -w-mre y F d' W - E TF J

.~ GEN ER AL $ ELECTRIC ATCulC POWER EQUIPMENT DEPARTMENT 2571!A718 EHoiNEEmiNo reau 70" ?Fr'nerer" IT P L arm r f e t h,8 !!ic'IC M!Al?'TS O' ??MTo" T It*??"'!ALC ( (1 1) (10 ) $1p-ft) f = g1 (2.30) (10~) (1:1p)~1 21 (12 f = (2.55) (10~7) (ft/r:1p) f = 22 Ele =ent 10 represents the shroud sunport tihich is redeled as a torsion ~1 ) rad /in lb spring 'rith a flexibility constant of (2.M) (10 other elenents are treaded as bene..; trith the eff nctive recent of inertia All shear area, length, Young's '!odulus, and Poisson's ratio given in the table en the f olleving page. l' ) i 5 RK 4 ( -o O 9 5711A719 l co-,o...c., _ L',_... 4

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