Unsteady thermal stresses in fuel elements under heat generation

NUCLEAR ENGINEERING AND DESIGN 30 (1974) 328--338. © NORTH-HOLLAND PUBLISHING COMPANY

U N S T E A D Y T H E R M A L S T R E S S E S IN F U E L E L E M E N T S U N D E R H E A T G E N E R A T I O N Y. TAKEUTI and N. NODA Department of Mechanical Engineering, Facultyof Engineering, University of Osaka Prefecture, Sakai, Japan

Received 8 April 1974

In designing some types of fuel elements, it is necessary to find the thermal stresses in a continuous medium perforated by cylindrical channels of circular cross-~etion. In order to set up the problem, assume that the strength and heat transfer calculation are based on the phee~omena that occur in a typical unit lattice cell, and that the heat generation is uniform over the,whole section: Therefore, the problems reduce to solve the ~emperamre and thermal stresses in a polygonal prism with a central circular hole under uniform heat generation and for insulated outer boundary. In our previous papers we obtained the steady tht;rmal stress distribution in such formed regions by use of the five elementary functions method. In starting up or shutting down of the reactor, however, the problem becomes one of the unsteady state. It is felt that no analysis has been given to the transient thermoelastic problem of the multiply-connected region. The problems considered here are concerned ~vith unsteady thermal stresses in a polygonal prism ~ t h a circular hole under a uniform heat generation. The solving process is divided into two basic steps. First, the heat conduction equation is solved by the technique of Laplace transform. Secondly, the plane thermoelastic problem is solved by use of the five elementary functions method. Throughout the treatment of both problems, in order to satisfy the boundary conditions, we use the point-matching technique in conjunction with the least square method.

1. Introduction As shown in fig. 1, a typical example of cell arr~mgement of fuel element in the reactor is reduced to solve the tern. perature and thermal stresses in a hexagonal pri.sm having a central circuilar hole with insulated outer boundary as shown in fig. 2. One of the present authors has recently solved the steady plane thermoelastic problem in a polygonal region with a circular hole by means of the five kinds of elementary stress functior~,s method [ 11 ]. More recently, as an extension of the five elementary functions method, the present authors have solved the transient thermoelastic problem in a polygonal prism with a circular hole under ~Loheat ge,nerration [2].

:, 0 , 0 r

oTo!

-,y."

,.y-"

'-y

I

!

):0:

,0,

,....'"..,,..,",. y .i--rl ~ • .

,'0.:0', I

!

Fig. t,,

' y,,,

...,

i

Example of celt arrang~ment,

*'

I

IT. Takeuti, N. Noda, Thermal stresses in fuel elements

k.21"

2)29

i

Fig. 2. Regularhexagonal polygon with a central circular hole. This paper is concerned with the uncoupled transient plane thermoelastic problem in a pollygonal prism with a conc.~ntric circular hole under constant heat generation. The analysis is developed by means of Laplace transforms and the point-matching method. In the first part a theoretical solution is given for the problem of transient heat condaction of a polygonal prism with a hole under heat generation accompanied by the thermal insulation on (:he outer boundary. In the second part the transient two-dimensional thermoelastic problem hag been solved for the same region with the help of the five elementary functions method. Numerical work has been carried out for the problem of a hexagonal prism with a cir('ular hole. The solutions obtained from this analysis may also be applied to the case of a thin disk insulated on both flat faces. From our results, it is noticed that there is an obvious difference between the problem of no heat generation [2] and the present one of heat generation. Comparing the hoop stress distributions in both problems, the'. tendency of the curve become inverse, that i's, the hoop stress on the inner boundary for the present problem becomes a large tensile stress.

2. Heat conduction problem Consider a prism, as shown in fig. 3, bounded externally by a regular n-sided polygon a,ld internally by a central circular hole. Let a be the inner radius of the circular hole and b be the outer boundary of the prism. For transient heat conduction, Q+AT=I 0~" X g at'

(1)

where 1 is the temperature rise; K is the thermal diffusivity; X is the thermal conductivity; 12 is the heat generation per unit area per unit time; and A is the Laplacian = 02]Sr 2 + r - l . O/Or + r -2 . 02fitO z. T h e boundary conditions for the hea~: conduction problem are

,r = 0

at the inner boundary r :: a,

i)r//b'. = 0

(I a)

at the outer boundary x "- b.

The initi al condition is ¢=0

(Ib)

at t=O,

Applying the Laphtce transform to eq. (1), we have

(2)

At* (p/K)r*, =

OlD

where .

r* .

I e-Ptrdt .

.

(p =parameter to-Uansform).

Y. Takeud, N. Noda, Thermal s~!ressesin f,,~elelements

330

Y

Fig. 3. Regular n-sided polygon with a central circular hole.

Then the general solution of eq. (2) in plane polar coordinates can be represented as a~

¢, = Qg/~p2 + (Ao + BoO){ColoCqr) + DoKo(qr)} + 2 (C'mlm(.q r) + L)mKm(qr)) m=l

x

(Am cos mO +Bm sin mO),

(3)

where q2 = p/g; Ira, Km are modified Bessel functions; and ,ira,/Jm, Cm, Dm are coefficients of temperature function. By considering the symmetry of the problem, for an n-sided regular polygon, this solution reduces to 1"* -- ~g[~19 2 4" .~O./O(qV) 4" Bogo(qr)

÷ ~. {Anm[nm(ql,) 4"Br,mKnm(qV)~COg nmO.

(4)

m=!

By the use of boundary conditions around the circular hole, we obtain the following relationships:

Ko(qa) o/Zo(qa),

CS)

--

Thus~

Q"lo(qa)-- Io(qr),_

Bo {Io(qr)Ko(qa) Ko(qr)lo(qa)} ~ 'o(qa)

-

Barn

--,-,m = l

(6)

x {Inm(qr)Kn,a(qa)- gnmCqr)Inm(qa)} cos nmO.

The remaining unknown coefficients are now determined by satisfying the boundary conditions at the outer edge of the polygon.. Substituting eq. (6) into eq, (la) results in an infinite n~mber of equations fol: the inf'mite number of coefficients. However, since the boundary x =, b J,snot a coordinal~e line for the temperature function, an exact solution cannot be found. To obtain an approximate :;olution, we use the point-matching technique to satisfy the conditions at the selected points, that is, t~ cosOj = b (/= O, 1 . 2 , , . . , s) on the outer boundary, we can 0e~.ermine ~o and ~nm. Tl!e solutions obtained :satisfy the prescribed conditions in the interior and on the i~aer boundary of the body exactly, and those on the outer bclunda~: ~pproximately. Then, ~'* becomes J - " .... g

1 .J

.f j."

('7)

331

Y. Takeuti, N. Noda, Therrradstressesin fuel elements in which D &;notes a determinant of the order (s + 1) and is expressed as follows: = IApl,

q, z.~.

11, 21

(8)

3 , * • .t S '~" 1)

where

AN = q{~rn(i_li.l(qri_l)Kn(t,l)(qa)+In(i_l)(qa)Kn(l_l)_l(qrj_l) x {ln(l_o(qrl-t)Kn(i-t)(qa)

- ln(i-t)(qa)Kn(i-!)(qrj-l)}

} cos

n(i

,A,_'-. r, _~

- l)Oj_t cos 0 i - l -

(9)

cos {n(i - 1 ) + 1 }01.. I. m

Furthermore a new determinant of the order 0 + I) obtained by interchanging the (m + l)tL column in D by roll ll(qri_l) cos 0i- 1 is denoted as Dnm. From an inverse Laplace transform, we finally detezmine the solution for the temperature db~tribution

Qa2[

r = ~-

'2

1 - ~ " + 2Fo In

a ~ +

2

Fnm

--~

{(~")--(r)

-nrn } COS

nmO + 8

~1 exp (-h:/d°t?'t) (-~~:3

m=l x ~

-~

-

Ho~n{Jnm(r#(~)Ynm(al~a) Jnm(a#e)Ynm(r#¢)} cos

nmi~],

(I0}

m=O

in which p~ are ~thepositive roots of I Go(rol a, tt)...Gnm(ro, a, #)...Gns(ro, a, P)I . ! : = 0, I Go(r~, a, p) G~m(r~,a, p) Gns(rs,'a, IJ) I

(11)

where nm r/

Gnm(ri, a, p) --:la{Jnm(alz)Ynm-l(r/Iz) - Jnm-l(r/la)Ynm(al~)} cos nmOl cos 0i + m x (/nm(r/P)Ynm(eP) - Jnm(aP)Ynm(riP)} cos (nm + 1)0i,

(12)

and Jnm and Ynm are the Bessel functions. From a characteristic equation (11), it seems to have complex roots. However, if we consider the physical skuation of a characteristic equation in a partial differential equation of parabolic type in which temperature must be finite and an oscillating temp~, ='.~,.re distribution is not allowed under our given initial boundary conditions, it is sufficient to take the positive roots only. Moreover, the zero oflo(qa) becomes the removable singular points, and Fnm and H~n are ~iven by

liro~ns+1 •

ro

•

cos(ns-1)0o+

wh.,._ cOSO0,

I,rs

)0o

': ° '

!

.--cos0s

•

/ \ns+l

,

cos(ns+

} i

cos(rim - 1)0o +

cos(rim + 1)0o . . .

)0s

Y. Takeuti, N. Noda, Thermal s fresses in .fuel elements

332

{ ( ~ ) ' - ' ~os (.s-.

~)0o+ (~)=+'~os (~s ~-~)0o)

=., COS(ns t\ a/

} cos (ns -t. 1)0,

l)Os +

Gdro,a, p=)...

(13)

Yl(Vop,Jcos 0o... Gin(to, a, P=)

Go(rs, a, I ~ . . . Jl(rsPa)'cos Hc~n =lEo(ro" a, l~a) . . . Gnm(r o,: a, I~) .... Gns(ro,. a, IJa)

Os... Gns(r~, a,#a)

, (14)

Go(re, a, IJ~) . . . Gain(re, a, ! ~ . . . Eas(ro, a, #a) c;o(v~, "a, ~ )

. . . Gam(r~; a, ~=) . . . g~(r~;

a,

~=)

where

Enm(r/, a, I~a) = #a[ri{Jnm(al~a) Ynm(ril~=) - Jnm(r/i~a) Ynn, fal~a) } + a {Ynm -l(r/l~a) Y~t,n-l(al~ r

- Jnm_i(apa)Ynm_l(rll~a)} ] cos nmO/cos 0 i -- nm I {Jnm-t(ril~a) Ynm(al~a) - Ynm(al~a) Yam-l(r/#=) } ri ~rJ "1

2rim + a(Jnm(r/,ua)Ynm-t(r/ga) - Jnm-l(aua)Ynm(r/P~)}- ~a {Jnm(r/l~a) Ynm(aP=) -Jam(a#=) Yam (ri#a) } ]

(15)

x cos (rim + 1)Op Thus temperature distribution is obtained entirely.

3. Unsteady thennoelastic problem Now the fundamental equation governing the two-dimensional thermal stress problems may be expressed in the known form, AAX = -kAT,

(16)

where X is the stress function; k is the material constant, i.e. k = atE for plane stress problems and k = ~tE/(l - o) for plane strain problems; c~t is the coefficient of thermal expansion; E is Young's modulus; and o is Poisson's ratio. In this paper, a two-dimensional thermoelas, tic analysis is being developed by the five elementary functions method frst introduced by one of the present authors [ 1]. This method looks somewhat complex superficially but has many advantages compared with problems of multiply-connected regions. The first advantage of this method is that it facilitates a physical meaning of the thermoelastic problems. Since five elementary functions are decided to satisfy each of the alloted boundary conditions, the composite stress function X perfectly satisfies all the boundary conditions as a whole. Referring to our pr~.vious work, it. can be easily verified what kind of temperature distributior~ is sufficient to ensure the presence of thermal stress for the given region and temperature conditions in a multiply-connected body. The second advantage of this method is its mathematical convenience which we shall explain a little later. ~Now, whet ',onnected, wesee that the stress function X may be expressed in terms of the system of.five _X~,Xo, Xtlt,.X2tandx3tso, that [ I]

2,3).

(17)

Y. Takeuti, N. Noda, Therma!stresses in f~Ielelements

333

However, pure thermal problems of zero traction on the boundary give ×o = 0. The functions on the ,.'it~y,ht-hand side of eq. (17) have to mttisfy the following equation: (i) Differential equations in the region, AAx~. =--kAl",

(i8) (19)

AAX~j = 0. (ii) Boundary conditions on the ruth boundary Cm (m = 0, 1),

(20) (21) (22)

(Xf)pm - (~vXT)pm --" 0,

(Xh I)pm = [(X/~)(6lh + ~ 2h) + ~ 3h]51m, (aoXhl)p m = [(Ptl)pm(~lil + ~ 2 h ) ] ~ l m ,

(h not summed) (h not summed)

whe,re oi = cos (xi, o); Pm i,'; an arbitrary point on the mth boundary Cm; 6i/is Kronecker's delta; and ~)i is a partial differentiation with respect to i. (iii) MicheU's conditions, 2zr

j" a,{zX(x, + C;,,Xh,) +

dO = o,

(23)

0 2~r

f (r sin 0. ar - cos 0 . a g){A(Xr + Chtx~i) + kr}r d0 = 0,

(24)

0 2n

(r cos 0" ar + sin 0 . as){A(Xr + C~ix~l) + kr}r d0 = 0.

(25)

0

The remaining three constants C~t in eq. (17) are to be determined so as to satisfy the three integral relat:ions of eqs (23), (24) and (25). After simple considerations on the symmetry of the figure and temperature distribution of the body, and the Michell's conditions, we find that C'll and C'12 in eq. (17) must be taken as zero.• On the other hand, an example of the case of having a term C~1 was given in our previous work [3]. As the functions Xhl have the same values at a given re[ion of same the figures irrespective of the thermoelasticity or isothermal elasticity. Therefore, the mathematical procedure is not as troublesome as one would expect, because if we calcttlate tht=.: exact values as much as possible on Xhm and Xr, we can use them on any kind of new problem such as unsteady or steady thermoelasticity and i,,;othermal elasticity. Then, in the present problem, we have X - X, + C~lx31.

,1126)

Now, it is convenient to split the solution of eq. (18) into a biharmonic function and a particular solution which will be denoted as X¢o and X~, respectively. Then, each function satisfies: AAX¢o = 0,

(27)

AX~p = - k r .

(28)

From eqs (10), (27) and (28), we find that the acceptable solutions for ×To and ×~p are given by

{

-

X~o = Ao + Bo rz + Co In r +Do rz In r + Y. (Ann rnm + Brimr-rim + Cnmrnm+2 + Dnmr'nm+2) cos nm[~

}

,

(29)

m=l

(,/+ + 8 ~_"

+Q,,[l

Xfp=~

a=l .

.

.

.....

(a ,O*Jo(al .

"

~ Hoan{Jnm(ri~)Ynm(aljo)-Jnm(alla)Ynm(rbla) } c o s n m 0 ) m=O

]. (30)

334

K Takeuti, IV. Noda,

Thermal stresses in fuel elements

Thus we may find X~, and two kinds of the unknown constants in eq. (29) are determined by the boundary conditions of e q . ( 2 0 ) at r = a. Then we have the following relations between unknown coefficients:

Ao--T

-

kQa4{ eo = T

=

I l

,,-~.~-~.'o(..----Z + (½-~n")c° +,½~Oo, 1~

- ~ ~ +~

=

2H.o e_xP(-KP~t) / ½a-2Co_ ½(2 in a + 1)Do, ~(~)'So(~o~ J kQ a4

Anm =

Bn m

t~=l

n m + I a2Cn m I_La_2(nm_l)Dn m + ~ nm T nm X

a2(nm+l)Cn/_ n~m

I--L nm

2a-nm ~ nm

Of)

Uoen exp (-gp~t)

~p~'J~

~-.~=

kQa 4 ~__m ~_. Hoen exp (-~/~t) a2Dnm - X " n m ~="=i n(alza)4Jo(al~) "

I

-

.

nm

After substitution of these relations into eqs (29) and (30), the X¢ can be rewritten in terms of only four kinds of unknown coefficients. 4

kQa 4

X~ = S

+

+I nm + 1

m- I

-n m-

}

I :,"2

nm

1 nm

2

2

+ r -nm

1-

+

m

I

r

+In

r2

+2 anm+2Cnm

!

r

1

( )} r2

Co+ ~ln--a +-2- 1 - ~ l -- n-m ~

nm

-In mnm-

a2Do

1

-rim

[r~'-nm+2] p ( - K ~d1c~) 2~')( - L:-J j a-nm+,DnmJ] cosnmO+- - ~ - k Q a 4 ' l ~(~/2/e2xcz)4Jo(a/ 0 {l(/'2a'2--I ) +Jo(rlaa)Yo(a~a)

-J~(a~Ja)Y~(r~a)~ +~=~~n-~{(r)nm-(r)-nm}+J-(~'v)Ynm(a~L)Ynm(~)Jnm(a~)~ Similarly as before, for the determination of unknown constants Co, Do, Cnm, D u n the point-matching technique is used again here in order to satisfy the outer boundary conditions at a selected finite set of points (Po)i (i = 0, 1, • . . , s) a't the outer boundaqr. Thus we have to solve the following simultaneous equations:

m= 1 +

tittl

tim

t~') rim+2} anm+2cn m _ Ilz-l--m(~)nmllm-l(~)-nm(~)-nm+21a-nm+2Dnm]COStimOl +. . . . nm

+ 1 -1

- Jo(al~oJYo(Ctll~

x Hoen cos n i n o n , /

+

Hao

:~2

,=

I'[ l

-

+~ ~=l

~m-~

(apa)lo(apa)

~

-1

+ Jo(rip~)Yo(apa)

+ J n m ( r ~ p . ) r m ( a P a ) - Ynm(rlP~)]nm(aP¢O

1 (33)

335

K Ta:ceuti, N. Noda, Thermal stresses in fuel elements

a-lc°sOt'C°+2riln~ac°sOl'D°-a a

(am+l) --

nm . .1

ri

m-I

xcos(nm+ [(~).-1

Ir'\nm+l

1)0t+Ia )

cos (nm-

+

x {nm cos (nm+

l)Oi....O~m-

l)Oi

XcosO i+ ~. Ham m - ,

"'

] 2cosn,nO, cosO,} anm+lCnm cos (rim + l)Oi +

rt nm-I

(~)-nm+l

1)

Oi}a-nm+lDnm

C O S 0 i +oL=, ~ 2"e(x~p~( -(' ~ " ~)e t )

[\a]

"

(rla)-nm-I

1)

2 cos nmOi x cos

-~

= --

{nm cos ( n m - 1)0i+

-'tlm-I

cos(nm-.l)Oi+ ~1

(H.o

"ri - alaa{S,(rlao)Yo(alaa)-;lo(a,~)r,(rla¢)}

nm 1

cos (rim - l)Oi +

cos (nm + 1)Oi + a~..{J,m-l(riV.~)Ynm(aa~)

a

-- gnm_t(rilae~Ynm(ald~) } cos nmO i cos 8 i --

x cos (nm-1)o,])

!

nm r~ - {Jnm(ri ld~) gnm(ald~)

-- gnm(rill~)Jnm(ald~)}

(34)

Applying these equations for the points (Po)i(i = 0, 2 , . . . , s , . . . 6 and 6 > s) at the outer bounda ry, and solving these: equations, we can determine the remaining coefficients. Thus Xr is now dete:cmined entirely. W.: now consider X31 in the same way. The following function is applicable to this problem: r

X3t = A3o + Bao

lnr

+ C3o In -a + D3o

+ D3nm ( r ) - n m + 2 }cos nmO.

£

-- +

a

(r) nm

{ A 3rim

(r) -nm + B3nm

(a)nrn+2 + C3nm

m= 1

(3s)

For this function, the boundary conditions are at

r = a,

at

x = b,

aXm/ar= O, X31 = 8X31/ax = O.

Xal=

1,

(36) (36a)

And we also can determine the unknown coefficients Am, B~,..., A3n m, B3nm, • • • in the above equation by a point-matching technique. Therefore .~helast function X3t may be calculated in a similar way. Substituting eqs (10), (29), (30) and (35) into eqs (23), (24) and (25), we obtain = -(Do + Qak4FolSX)/D

(37)

Then we can determine the form of stress function from eq. (26). For convenience, in accordance with the equivalent coefficients in Xr and Xa~, the next notations are to be used: D t C~ _- Co + C:~7~11, C*nm = Cnm + C~mC~ll, D~ = Do + D~A~f31, Dr~n = Dnm + 3nmC3l, (38)

Y,. Takeutt iV. iVOdal.Thermnl.stmssesin fuel elements . :

336

:-_ :

. . • ': : : . . : ' . .

Thus the approximatesolution of a stress •function for this problemmay be expressed entirely, and the, thermal stress components.corresponHng to tlgs,stre~ function are gg,en. Here,we list the solutions in their final forms: "

"a2

"

,I

2

+ (rim+ I) ~a I

,a/...,) anmC*~n'(nm'l) ,(a)" :'[3 r ' x cos nmO + kOa' ~ ii1-~-~-

.

+~

exp ( T g / J ~ ' ) / H I ==, 2(atta)'Jo(ag=) '< ao

1

-(nm'2)(r)-nm}a-nmD'l

+ (al~(~)2{jo(a~la)Yo(rl~=)_

,

_a~-: . {Jo(aga)y,(rlaa)-J,(rga, yo(ala~}]+ sm~=H , [ I {(nm - l) (r)nm=~ - ( "

(,

"

+ m(nm + 1)-~ -

z {Jnm(rtaJYnm(al~J - Ynm(rl~J&m(alaJ

}

Jo(rgL~)Yo(ala=)} -

+ l) (~)-nm-~- l

a2l~ar {Jnm-l(rlaa)Ynm(agJ

(39)

-Ynm-q(rla=)Jnm(ala~} cosnmO>i,

(a.) -~.-!

on. =

{

-(rim+It -(rim-l)

a-2C~J+21nrD~+ •'

a

+

(a)

'"--' j +(nm-2)~a)

a=l

x anmC*m +(nm - 1)

+(nm + 1)

k Q a ' l ~ 6 r2 +~ +-3?-I ~- 1

2exp(-s;~a~t)

(a#a)4jo(a#=)

a-nmD~n ] cos nmO

- (nm + 2) 2

a2/~

o -n + r - {Jo(alaa)Yl(rg.) - Jt(rl~=)Yo(ag=)) +

re=IXH(~m

m=l

x - -~ (rim - 1) -

- (rim + 1)

+ ..... r {Jnm-l(rg.)rnm(at~a) - Ynm-t(rlaa)Jnm(ata.)}

nm(nm + 1) a-~ {Jnm(rla=)Ynm(ag~ - Jnm(aIJa)Ynm(rg=). . ~ . .

~ cos nm

, . + ,> , - . ,> •

x

m = t

"

(40)

,.

..,.+,, .

+ n m(.,~ \ a ] - ),-m~ } i:

.!],i, nmO+~ ~ m ~ i

(a~ua)'l/°(a#=)

{41y

:

Y. 7"akeuti,N. Noda, Thermal stressesin fuel elements

337

4, Numerical resullts The foregoing solution will be illustr~tted numerically by solving the problem of transient thermal stress distribution on the hexagonal :region with a centred circular hole under constant heat generation specified by the following values: n=6,

b/a=2.

For simplicity and generality, the temperature and stress distribution in the cylinder or disk can r ow be, ,ondimensionalized by defining the follc,wing dimensionless variables: T=

XTt~ 2,

p =r/a,

td=Kt/a 2 and 0~0= Xooo/kQa2,

~p = Xo,,/kQa ~.

I.,1"01 Fig. 4. Temperature distributlon for time t d = 1.0.

05.

N..

0.0

%.

-0.5-

4'

Fig. 5. Stress distribution of O~ for time td = O.I. 0~5"

O0 ~ ~

i

-,5 t

,~1\J

.

Fig. 6. Stressdistribution of o"Ofo,~time td = 0.5.

Y. Takeuti, N. Noda, Thermal stresses in [uel elements

338

o,s

Fig. 7. Stress distribution of 0"0 for time t d = 1.0,

18:0.60.4- ~

td.i.0

oz \ ~ 7 / ~

'd'O~

• " ~-~~_~Z~../ta=O.I

Fig. 8. Variation in ~ and p~p along the 0 = 0.

In order to decide the unknown constants in ~temperatured stress functions by means of the te,:hnique of p•3intmatching, taking four points for the temperature problem and ten points for the stress problem on the outer bound. ary and the least square method for the stress problem is used. Variations of the temperature and hoop stress are shown in figs 4 - 8 for various times. Regarding the accuracy of the point-matching method, we see that the values of oxx and Oxy on the outer bound. are x = b, which are calculated from err, ooo and or0, are nearly equal to zero. Therefore, the results were justified by the boundary conditions of zero tractions. In fig. 8, it is obviously seen that the tendency of hoop stress distribution between the present problem and no heat source problem [2] is opposite.

References [ 1 ] Y. Takeuti and T. Sekiya, Approximate solution o!~thermal stress problems in plane elasticity, Prec. 8th Japart Nat. CongL Appl, Mech. (1958) 119; and Y. Takeuti and T. Se]~tya, Thermal stressesin a polygonal cylinder with a central cbcular hole under internal heat generation, ZAMM 48 (1968) ;~37, ~2] Y. T~keuti and N. Noda, •Transient themmelastie problem ina polygonal cylinder witha circular hole, ASIa[E, E40 (4)(1913) 935; [3 I Y, Takeuti, Thermalstresses in heat-generating multt-boresquare cncireular region, NucL Eng. Des. 14 (19'10) 201. ,-

,,

, , .