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ATOMKS INTERNATIONAL A Division ef Merfh American Aviatlpn, Int.

TECHNICAL DATA RECORD

Bar* X

TDRNO l ^ ^ g

:5 9rSi

PAGE 1 OF I d

PPROVALS

W»»^

d f l ^ AUTHOR

J . Soanir

DEFT a GROUP NO

795-52 DATE 5 - 1 * ^

GO NO 7509 TITLE

S A NO 963 Iteaetor Taaaal Ballova

RECOMMENDED FOR OUTSIDE DISTRIBUTION

T W R

PROGRAM

KIFF

g3l8 SUBACCOUNT TITLE

Raaetor Struetvra

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SIGNATURE

DISTRIBUTION

USE AN ASTERISK ( 1 TO INDICATE THOSE

WHO ARE TO RECEIVE COMPLETE COPIES

•RtCf q i ^ £ . *W«T» WPMean *J . I . NriOMlatar *T» U<f9i-^.y, IWbtrg. *W»J« Saadara m*?. Balalar *yt F^fH *g. gola K!* Conaara

|l»g>». Ualt (IQ

*lllfr FTOdlgfe-r U f (750»J^963)

*TiB cwtrai r a t ^/f ^h^-lUi^'Vf^

STATEMENT OF PROBLEM

BataM.lali a tmlform baala for evaluating varloua ballova daalgaa l^ropoaad for tka HIFF Reactor Vaaaal.

ABSTRACT

Two typea of ballova eonflguz>«tloaa hare be«i prqpoaad for tlM HIPF Reactor Teaael. It haa been eatabllahed that the afely logical aaaner of ooovaring the two dealgaa la oa an evoal atraaa baala vhlch lapllea equal life for each ballova. Reaalta are preaented giving the rnlnlia auaber of eonvolutloaa required for each deaign.

The first several copies of page 1 of this TDK are for tear-off and retention. The TDK can then be passed on to others who wish to see it. Please do NOT remove the final copy of page 1.

367 002 •"40 O 5* RtV J eO

ATOMICS INTERNATIONAL A Divitkm of North American Aviolion, Inc.

Mfi rm »565

DATE ^-1-61

PAGE 2. OF- 18

I Paterwlnatlon of atreaaea Due to iaclal ttoveaant

VhMt the ratio of the naalanin to the alnlBna dlaaeter of an annular plate ag proaehea unity, Tlaoahenko (Ref. 1, Page 68) statea that the structure aaa be analysed vlth aufflelent aceixraey by eonalderlng an elemental atrip. Therefore, the analyala preaented in thla report haa been baaed oa atrip theory.

Configuration A

Conalder flrat the bellova design shovn in Fig. 5> A unit vidth atrip of each half convolution ean be treated as a guided cantelever beaa as shovn m rig. 1.

Defleeted Position

Figure 1

Using strain-energy relationships, the axial dlaplaeeBtent is.

c- - a ^ . r g - ^ ds O ^T J H ;5F

19M rotation at the free end is.

r C7 o e dx

(1)

(2)

yf m 0 for the ease being considered

3^7 003 730-V-45 (REV 5-59)

ATOMICS INTERNATIONAL A Division of North American Avioiion, Inc.

DATE hl=§^ PAfiF 3 r>r 18.

For convenience, the aoaent, M, la separated into three parta; i.e.,

M^ - M^ + FR (1- COS 9) 0 ^ © ^ -^

H^ - M^ + FR + Fx 0<; x<^ L

M/ - M^ + F(R + L) + FR sin 0-S -i "^

Considering BI to be a eonatant and substituting the above values of N into equationaCl) and (2) reaxilts in the follovingi

EI<5 + FR (1- COS 0 ) (1- COS 0) R^dO

•/ c. + FR + FSt)(R + x) dx

and

If/2

•i Tf/2

0 . j [• o

L

M^ + F(R + L) + FR sin |(R + L ) + R sin Rd|< (3)

M^ + FR (l-cos 0) RdO

• / '"o + FR + Fx) dx

<7772

/ M^ + F (R + L) + FR sin Rd^ («*)

Solving Sq. {k) for F, -2M

F - 2R + L (5)

004 730-V-45 (REV 5-59)

ATOMICS INTERNATIONAL A Division of Noriii American Aviation, inc.

NO. _ DATE. PAGE.

3»l-6l

OF. 18

After integration, Bq. 3 reduces to,

,2 ECS

+ F

TT^' + RL (1 + -21 ) + |-

i^ [ 1 + R^ (3 + TT) + H.2 a + -f^ ) + n (6)

Substituting Sq. (5) into Eq. (6),

^l!)'-l!)'*-f 1!)* "o"-

Therefore,

R ^ T (7)

M^ - - Kl5 C, (8)

Vhere 1 R ^ 1 1

1 ^m^-nr^^iiYh I * 0.5 1

-? s -^ 1-57(11 -2(f) .0.785 (I) > 0.0834

Since -Uiis analysis is based on strip theory, the aaxiinuiB bending stress in the bellova due to an axial diaplacement is.

^

^ •

N t o w

(9«)

note that in Sq. (9a) the minus algn has been dropped since it sinq ly represents vhleh side of the bellovs is in tension or eceq>re8sion. For a bellova vlth N convolutions subject to a displacement d, the peak bending stress is,

Std » ^ ' ¥r (9b)

307 QQ^ 730-V-45 (REV 5-59)

ATOMICS INTERNATIONAL A Division of North Amtrican Aviation, Inc.

NO. TBR k^f nATF Z'l-6l

PARF 5 np 18

The T rlTwiH atress in the bellova i s a combination of bending and diz'eet stresses and la equal t o .

or

^

^3^

Etd

Etd W

h*

^1

t

1 1 * ^ ' M2k + L)

(10)

Iq. (10) demonstrates that the direct stress eomponent is usually very small (i.e. t la much amaller than R and L) and can be diaregarded.

Configuration B

The bellovs design shovn in Fig. 6 can be analyzed in a maimer similar to that used for configuration A. Consider the convolution fixed at the end of the flat portion aa shovn in Fig. 2, equations (3) and (It-) are reduced to.

irfz m S' j M^ + ra (1- cos e) (1- cos 0) R^dO

and

I 7r/2

(M + FR + Fx)(R + x) dx o

(11)

M -I- FR (1- cos 0) RdO

h r (M^ + FR + Fx) dx (12)

Solution of tto equations (ll) and (12) result in.

M F

fa + 1

:4-)i!i Hfhi

,"07 006 730-V-45 (REV 5-59)

NO MB ^5^5

ATOMICS INTERNATIONAL A Division of North American Aviation, Inc.

DATE 3*1*^^ PAGE 6 0 F__ i8 ,

^ •

(?MfH 7 i f ^ ^ lW- f ^ l lMs

X . - * -

Figure 2

The equation relating stress to displacement is in a fora similar to Eq. (9b) except that C, is replaced by C i

Etd -^ " ur °2r

2 Vhere 1

75 ^2 -

(13)

0.571 (fj + 1 + 0-5 -g -- u; - — — - g .

0.231 (f) + 0.785 (f) • (|) + 0.521 (I) + 0.083^

orn 007

730-V-45 (REV 5-59)

ATOMICS INTERNATIONAL A Division of Norfh American Aviation, Inc.

wn

DATE-PAGE-

•HM k56'i 3-1-61

7 OF 19

Since it haa been d«Bonstrated that the direct atress eomponent is relatively small, Eq. (13) ean be used directly to obtain the maximuB dlsplaceooent stress in 'ttie bellovs.

II Determination of Stresses Due to Pressurisation

Configqration A the

As im/previous analyala, a unit vidth strip is considered as a guided eantelever beam as shovn in Fig. 3*

p (Pressure Load)

> > t t J I L_i e

x-^

Figure 3

Since the horizontal components of the pressure loading res\ilt in hoop tension, or eos^ression ;these forces do not affect the analyeis of the elemental strip. From symmetry.

P(R + |) (llf)

Using strain energy relations and noting that for eraQ>atibility the rotation at the free end is zero, the follovlng relationship is obtained:

r I M ds (15)

008 730-V-45 (REV 5-59)

NO. ——ZISLiSf i i .

ATOMICS INTERNATIONAL A Division of North American Aviation, Inc.

DATE. PAGE, J OF 18

As vaa dana prevloualy tha moment, N, la aeparated Into three parts; I . e . ,

2 Mg - FR (l-eosO) - M - ^ (l-coaO)^ 0 ^ 0 .<! - ^

W| - F (R + x) - M - 1 (R + x)^ 0 - c X .<. L

M/ - [ F - P ( R + I . ) ] Rsln^ - £ | - sin^^ + F(R + L)

. M - 1 (R + L)2 0 <. ^ ^ -f:

Substituting the above values of M into Sq. (15)i and uaing the relatlonahlp given in Sq. {ik), reaults in the follovlng!

\ -r/z 2 1 0 « r [pR (R + |)(l-coa 0) - M - 2|» (LCOS O)^

o ••

L , + M P (R + 1 ) (R + x) - M . 1 (R + x)2

o

RdO

dx

7r/2 2 + r . 2 | i sin ,< - ^ sin^^ + P (R + |)(R + L)

. M . 1 (R + hf\ M4 (16)

Solving Eq. (l6) for M ,

„ ^,2 n r III •' 2 l i r ? li:) * 12 ^ *M M » p L J-KT

3 2 ^ 0.785 111 + 1»07 Ir + 0.5 (1) + 0.083'*

K, «» L 1 3 . 1 ' * ( | N l

Then,

orp 009

730-V-45 (REV 5-59)

K.n TDR'»565

ATOMICS INTERNATIONAL l^ll l'^'^^^^ .Q . . . . . M ^ . u A A • .• I PAGE Z OF ±2. A Oivition or North American Aviation, Inc.

The maxtmmn bending atraaa in the bellova la.

*o

^ • ^

^ - ^ K^ (18) t

TliM MAvlwnn n-t THvaA 4n ^Mm >)*11 ova la -feh* aiim nf -th* tMndlncr and dip«a-fc

eoagponants.

^mx-^h*ih («)

Vhere

h - ^ [(I * l] The direct stress is relatively small for a typical bellova design (i.e. t<< l) and ean usually be neglected. Therefore, for engineering asyplications, Eq. (l8) can be uaed to obtain the maximum stress in the bellovs.

Configuration B

Conaldering the convolution to be fixed at one end of the flat portion as ahovn in Fig. 'i-, Bq. (l6) la reduced to

n^ r 0 - f FR (1-008 0) - M^ - pR^ (l-cos 9f

J^ I 2 J RdO

L

+ J F (R + x) - M^ - 1 (R + x)^ dx (20) 0 L -•

orp 010 730-V-45 (REV 5-59)

NO. TDR k^^

ATOMICS INTERNATIONAL A Division of North American Aviation, Inc

DATE 3 - l *6 l PAGE XO OF lfl_

V

p (Pressure Load)

V / f » ^ » t ^ t I /

rr\H e

K - * -

Figure 4

To flM the force F, the relationship given in Sq. (l) is used. Substituting the values of MQ and M^ on Page 8 into Eq. (l), and noting that the deflection, C , is zero results in,

^/2

^ . 0 - j [FR(1-COS 0) - M^ - 2|> (l-cos 0) J (1-eos O) R dO

o

L + I [F (R + x) - M^ - I (R + xf] (R + x) dx (21)

Eliminating M from equations (20) and (21) and solving for F,

F = pL ,'¥'-§4lf*

I ^..2 ^' -A l!l * f If I

% / _ . 3 _ . 2 ~T. : ^ ¥^ ^(ihfe

?07 Oil

730-V-45(REV 5-59)

ATOMICS INTERNATIONAL A Division of North American Aviation, Inc.

Kin TEB »565 DATE_Jhi=fiL PAGE_J1__0F. 18

Subatltutlng the above value for F into Eq. (20) and aolving for M ,

The maximom stress at the apex of the bellovs i s ,

'% ^^w:^ ^2^1 2 (22a)

Where:

^ Kg - r 0.017(1) + 0.104(1) + 0.275(f) + 0.399(f) + 0.279(f) + 0.179(f)'

+ o.o525(f)+o.00694 l | ^0 .23 l j f ) +0.785(f) +(f) +0.524(f)+0.083ia.57(f|+i

O.ioolfl + 0.452(f) + 0.804(f) + 0.691(f) + 0.279(f j+ 0. * 2 " i|. 3 2

0.23llf) + 0.785(f) +(f) + 0.524(f) + 0.0834

04l6

As vaa demonstrated before, the direct stress eon^onent in Eq. (22a) can be neglected and the maximum stress can be adequately represented by,

^ W-!l 2 t

The [

maximum moment at the

"r-^o

Consequently

or

w -

^rWa -

*1 , the

f l .

fl

root

(R + L)2 -

of the bellovs ia.

P (R + L)

maximum stress at

S * ^

[•^2*1

(R +

(R +

^f-

i.f-

the

• ^

'^2

root

^2

(RH

of the bellovs is,

(R +

^L)l

L)

-^Zl 012

(22b)

(23)

(24)

730-V-45(REV 5-59)

ATOMICS INTERNATIONAL A Division of North American Aviation, Inc.

Ill Evaluaticm of HMPF Reactor Veaael Bellova

Three bellova configurations vere propoaed for the HHPF Reactor Vessel. Two vere of the type ahovn in Fig. 5 vlth the follovlng dimenaions:

No. 1: L « 2.187 in., R « 0.688 in., t - 0.093 in.

No. 2: L - 3'625 In., R > 0.688 in., t - 0.093 in.

The third configuration vms of the type shovn in Fig. 6 vlth the follovlng dimensions:

No. 3i L • 2.125 in., R » 0.375 in., t « O.050 in.

Itie maximum stress In each type of bellovs is in the form

^ " ^ d * ^

or

Etd « 6p „

The principle bellovs parameters, C and K, az*e plotted in Fig. 7 for convenience*

During normal reactor operation, the thermal expansion of the reactor vessel and supports vlll be ppiTOxlmately 2-3/4 in. at the bellovs. Consequently, it was decided to cold-spring the bellovs 2-3/4 in. so that the displacement stress during normal operations vould be approxizaate-ly zero, ^le deaign preaaure at normal operating eondltiona ia 5 pal.

Baaed on information given in Ref. 2, the limiting stz^ss for the cold-spring vas 90,000 pal (i.e. tvlce the yield atrength). The atreas due to preaaure vaa limited to 18,750 pal ihiloh is I-1/2 times the ASME Boiler Code Allovable for Type 304 SS at 1000 F. Table I aummarlsea tha deaign information for the various bellovs. Table II Indlcatea the atraaa levela and the nuaber of convolutions required. It can be aeen la Table II that the bellova are atreaaed to approximately their limit for diaplacement atreaaea ( T" ) And a large margin exlats for px>essure atreaaea, ( )>

3C7 013 730-V-45 (REV 5-59)

Mr> TDR 4565

DATE y^'6). PAGE _12 OF la

ATOMICS INTERNATIONAL A Division of Norffi American Aviation, Inc.

NO se&ls^L DATE hl=SL

PAGE__li_-OF. 18

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730-V-45 (REV 5-59) 307 014

Mn TDR 4565

ATOMICS INTERNATIONAL A Division of North American Aviation, Inc.

HATF V l - 6 1 PAfiF 14 OF 18

REFSKBHCBS

1. Tlmoahanko, S., "Theory of Platea and Sheila," Mcarav-Hlll Book Co., Nev York 1940.

2. Fenley, A. B., TDR 4826, "Phyaical Propertlea of Material from Zallea Brotbara Teat Bellova."

^07 015

730-V-45 (REV 5-59)

K

Mr> TBR 4565

ATOMICS INTERNATIONAL PA I F 15 " ^ OF I T A Division of North American Aviation, Inc.

NGNBNCLATURS:

d

E

F

I

L

N

N

P R

t

U

ds

x>0, 4 C^, Cg,

Kg, Pj ,

s ^

Vi d

^P

^

^2

Total Bellovs Displacement (in.)

Young's Modulus (psi)

Force (lb) 1

Section Moment of Inertia (in. )

Length of Straight Portion of Bellovs (in.)

Bending Moment (in-lb)

Number of Convolutions in Bellovs

Pressiire (psi)

Mean Radius of Bellovs Convolution (in.)

Thickness of Bellovs Material

Strain Energy (in-lb)

Incremental length

Coordinates

Bellov's Parameters

Displacement (in.)

Rotation (radians)

Direct Stress (psi)

Displacement Stress (psi)

Pressure Stress (psi)

,07 0 ^ 730-V-45 (REV 5 -59)

NO. _JBBJ t5 f i 5 .

ATOMICS INTERNATIONAL A Division of Norfh American Aviation, Inc.

DATE. PAGE.

3-1-61 16 ,0F. 18

L

Configuration A

Roll Formed Expanaion Joint

Figure 5

?07 017

730-V-45 (REV 5-59)

NO TUB k9^i

ATOMICS INTERNATIONAL A Oivilion of Norfh American Aviation, Inc.

DATE 3rlr6l PAGE 17 OF 18

Configuration B

Welded Expanaion Joint

Figure 6

'^07 018

730-V-45 (REV 5-59)

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