Thermal stresses in thin cylindrical shells stiffened by plane bulkheads for arbitrary temperature distributions

CoA Note No. 83 ICHNiSCHE HOC-':'-^- VLIEGTUIGBOUV

Kanaalstraat 1 0

2 0 nov.tS58

THE COLLEGE OF AERONAUTICS

CRANFIELD

THERMAL STRESSES IN THIN CYLINDRICAL

SHELLS STIFFENED BY PLANE BULKHEADS

FOR ARBITRARY TEMPERATURE

DISTRIBUTIONS

by

NOTE WO. 83 J u l y . 1958

T H E C O L L E G E O P A E R O I T A U T I C S

G R A N F I E L D

Themial S t r e s s e s i n Thin C y l i n d r i c a l S h e l l s , Stiffened hy Plane Bulkheads, for A r b i t r a r y Temperature D i s t r i b u t i o n s

• b y -D . J . J o h n s , B . S c , M . I . A . S . SÜMARÏ A s t u d y h a s been made of t h e t h e r m a l s t r e s s e s r e u l t i n g n e a r t h e j o i n t of a c y l i n d e r a n d i n t e r n a l b u l k h e a d dxie t o a r b i t r a r y temperatijre d i s t r i b u t i o n s i n t h e c o n f i g u r a t i o n a n d t o t h e conseqijent c o m p a t a b i l i t y f o r c e s a n d m.oments a t t h e j o i n t . The method i s g e n e r a l enough t o p e r m i t t h e i n c l u s i o n of j o i n t t h e n n a l r e s i s t a n c e b u t c e r t a i n l i m i t a t i o n s a r e p l a c e d on t h e form of t h e a x i a l temrxirature d i s t r i b u t i o n i n t h e c y l i n d e r ,

An a p p r a x i m a t e method t o d e t e m i i n e t h e t r a n s i e n t t o n p e r a t u r e s f o r c o m p l e t e l y g e n e r a l h e a t i n g programmes i s a l s o p r o p o s e d .

2

-COOTEyiTS

1, I n t r od\x3t ion 2, Theory . .

3, Thermal Stresses for Arbitrary Temperature Distributions 4, Assumed Temperature Distributions

5, Possible Modifications to the Theory 6, Conclusions 7, References 8, NotaticQx Appendix 1 , P r e d i c t i o n of T r a n s i e n t Ternperature D i s t r i b u t i o n s i n Cylinder -Bullchea.d Configurations

"I • Introdacticn

The purpose of this note is to develop formulae for the thermal stress distributions in thin cylindrical shells and internal stiffening bulkheads Titoch arise as a conseqtience of the restraint forces at the

cylinder-bulkhead joint due to differential expansion of the tv/o components, The physical nature of the problem can easily be visualised and one obvio\is application occurs in the kinetic heating of any typical missile or aircraft fuselage.

Because the thin shell attains a higher tanpera.t\jre quicker than the thicker, internal bulkhead, differential expansion will result. A system of s elf-equilibrating forces and moments must therefore be applied to the tvro components at the joint in order to make the resultant deformations of shell and biilkhead compatible.

Przemieniecki (Ref.l) has already considered this problem and presented seme very useful resixLts. The approach in Ref. 1 vra.s to assume

a rather restricted flight programme i.e. constant height, instantaneous acceleration and then to derive exactly exparessions for the transient temperature distributions in the biilkhead diaphragm. The main assumptions made in the structural analysis were that biilkhead spacing was so large that restraint effects at any one joint did not influence conditions at other joints, and that the cylinder tcanperature was constant axially at any time,

The f onner asstimption regarding bullchead spacing should be valid for most practical configurations, since the discontinuity stresses introduced at the joints are extremely localised; and this assuniption will also be made in the present note. The latter assumption cannot be easily justified

since it is known that heavy internal members attached to the shell act as heat sinks thereby lowering the local shell temperature. To remedy this therefore the thermal stress problem will now be solved for arbitrary

temperature distributions in both shell and bulkhead. Suitable approximations to realistic temperature distributions will then be made yielding

-

4-I

2, Theory

Por the configuration shown in Pig, 1 it will be assumed that the temperatures in shell and bulkhead are constant thro\igh their thickness but vary axially and radially respectively.

The variation along the cylinder will be T = P(x) and in the bulkhead T,^ = f(r),

B

2,1, Thermal expansion effects in the cylinder

Consider the cylinder separated from the bulkhead and consider a small ring element, dx, separated from the rest of the cylinder. The radial expansion of the ring due to a teniperature change T is equal to O^Q^TQ»

It produces no stress in the ring,

Imagine now an external pressure p which is applied to the ring to restore it to its original diameter. The contraction due to p must equal the expansion due to T , Hence,

PR2 ^--. = ci RT E t c c o t and therefore p = a E T . ^ • (l) o C o K

Since T = P(x) , pressure p is also a function of x and produces a hoop stress in the ring. The remaining analysis follows that of Ref. 2 (p. 423) exactly. Since there is no actual applied pressure on the complete cylinder a pressure of -p must be applied to the now-undisturbed cylinder, which vd.ll cancel the hoop pressure due to p but introduce longitudinal "tiending

stresses as the sole equivalent of the axial tem.perat\jre distribution. The stresses produced by -p can be fcttnd if the deformation mode of the cylinder shell is known. The skin deflection, w, (positive inwards towards the axis) is given by the differential equation (Ref. 2 p.392 eq,230).

é*^^^ = "^ (.)

where ^ =

^(' " 'l^

' ^ V ^

S u b s t i t u t i n g from eq, (1) , nh- , E a T^t d w , /P4- c e o /-t\ + ^ ^ ^ = - - l ^ R • ( 3 ) dx The g e n e r a l s o l u t i o n of t h i s e q u a t i o n i s ,

'P^ K ° ° ^ ^ ^ + ^ 2 ^ ^ iSx J + y~^^ k , o o s i 9 x + K,sin/3x: J - K J ^ (4) w = -ge

where K , , , . . K . a r e unknown c o n s t a n t s depending on t h e boundary c o n d i t i o n s a t t h e ends of t h e c y l i n d e r s , and

E a t

K = - . ° ° - = a R . ' 5 " i ^ R °

It should be noted tha.t in assuming w = -K_T to be the particular integral of eq. (3) it is implicitly assumed that the function T = P(x) =

c 3 m E P X . I n o t h e r v/ords i f P i s a p o l y n o m i a l i n x i t cannot be of mr.0 m h i g h e r o r d e r t h a n a c u b i c . P o r a c y l i n d e r vd.th m^Dderately h i g h b u l k h e a d s p a c i n g t h e boimdary c o n d i t i o n s a t b u l k h e a d s d i s t a n t from x = 0 w i l l n o t i n f l u e n c e t h e c o n d i t i o n s e x i s t i n g a t x = 0. I n o t h e r words K = K„ = 0 and 1 W^x w = -ge ^ fK^GOs ^ + K, s i n /2x - K^T (5) L 3 4 _ i' °

To determine K,, K. it is convenient to assume for the moment that the cylinder has a free edge at x = 0 and therefore

i2 • .3

— ^ = -j = U at X = 0.

é -Prom eq. ( 5 ) dw dx = ^ ^ " ^ [^(K^ - K^)cos ^ - (K^+K^)sin /2x1 - K ^ ^ 2 ^ ^ • 4 / e ~ ' ^ r2K,oos/3x - 2 K ^ s i n ^ " | - K^ T* • ^ = ^ e " ^ r2(K^ + K^)cos/5bc + 2 ( K ^ - Yi^ais,^^ - K^T^ l ï h e n x = 0 d w ^ ^ ^ ^ /^ ^ 0 dx

v

5 oo ^ = (K, + K J /5^ - K. T " = 0 dx^ ^ 3 ' 5 co T h e r e f o r e , K, _{^2 ^co} / T" GO OO •5\ ^ "- ^2 (6) "ïühere T" i s v a l u e of T" a t x = O e t c , oo o T h e r e f o r e eq, ( 5 ) becomes cosiSx w = ^ Kr mil min rp// 0 0 CO \ - o . + —=• 1 - siiV?x and a t X = O. rp/Z m/* „ V » -I / o o g 0 0

^oc = S f ^ l ~ ^-7

['

_^^ - T CO dw dx _{0 0}

₅

"1 /2S" rjyUI T' / 0 0 0 0 \ T " H CO

]

]

K_T 5 o (7) (8)

2.2, Thermal Expansion Effects i n the Bull±ead

I f t h e temperature d i s t r i b u t i o n i n t h e bulkhead i s axi-syrametric and given by t h e function TL = f ( r ) i t can be shown t h a t the r a d i a l expansion of the bullchead a t t h e outer circumference i.s given by

^ ' T^ r dr (9)

^^ = - — ] ^B

o

and because it has been assumed that the temperature is constant through the thickness of the bvilkhead no curvature of the b'ulkhead m i l result, and hence no change in slope at the circumference.

2.3. Compatibility of Deformations

Having now considered the deformations in both shell(2.1) and bulkhead (2,2) due to thermal expansion it is necessary to determine the

system of internal eqLiilibrating forces and moments that will make the overall deformations of shell and bulkhead compatible.

To this end we assume that conditions in the cylinder on both sides of the bulkhead in Pig. 1 are identical. Hence if the bulkheads are spaced far enough apart, the resxiLts of Ref. 2 (p.393 eq.232) can be applied directly.

Therefore,

e w = —

•=— I/9M (sirv5x - cos/5x) - Q cog/9x I (10)

^3i) U 0 o J

expresses the deflection mode for the right-hand portion of Pig. 1 due to applied

along t] p.392).

applied bending moments, M , and shearing forces Q distributed uniforroly along the circumferential edge x = 0, (Por sign convention see Ref. 2

If the radial shear force reaction on the bulkhead in Pig. 1 is P, then by symmetry Q = - •^/2 and the value of /dw\ is given by,

dw' dx,

To c a l c u l a t e t h e moment M v/hich appears i n eq. ( n ) t h e boundary c o n d i t i o n f o r t h e s h e l l a t t h e j o i n t i s \ased v i z . t h a t t h e net slope a t x = 0 must be zero.

•- ö •* 2/3 T h e r e f o r e from e q s , (8) and ( 1 1 ) ,

1- [^H„-|]=. [1

2 T" CO ^ fp/// \ rp/ ^ ^ r p / rp// r p ^ — p I ' oSnir ° 0 0 ° ° ° ( i 2 ) (13) S u b s t i t u t i n g eq, (13) i n t o e q . (IO) g i v e s w = — 5 — (sin/9x -cosySx) - 7- 4. pLx^n 35^ D L *- ^ ^ •"DK. , _ r p / rp// rp/// -CO -CO -CO + — 2 + — 3 /3 yS'^ 2^^

}

p "-oosySx (14)

which added t o eq. ( 7 ) g i v e s t h e o v e r a l l d e f o r m a t i o n mode of t h e s h e l l , Hence,

-/Sx r

w = -^— ' 7- (sin/Sbc + cos/9ic) +/S^Kp, sin/Sx/ -— ^. — * 1 -cos/9x CO CO y9 2/9^/ 5 o T' CO

0

T'" V ^ c o \ 2^) . T h e r e f o r e a t x = 0 t h e s h e l l d e f o n n a t i c n i s (15) w _CO l^ D

! - ^ D % i 2T^o +

CO

P

rp*/ ^ CO 2^^ (16)

Similarly, for the bulkhead, the resultant radial displacement due to the applied force P and the thermal expansion is,

^ P(1 - Vg)R 2c^ ^ •• R R T_ r dr B (17) E q u a t i n g e q s . (16) and (17) t h e v a l u e of P c a n be d e t e r m i n e d , which on r e a r r a n g e m e n t y i e l d t h e r e s u l t P =

': i'-'i•%] -

^l\--(18) 2(1 - \ )

1

2,if. Thermal Stresses in Cylindrical Shells

Prom eqs, (14) and (15) it is now possible to determine the values of the various stress resultants and stress couples in the shell.

Eq. (15) is the sum of the tvro equations, eq. (7) and eq.(l4). The former represents the deformation mode of a. free ended cylinder and is a function of £Lxial temperature distribution only. The latter derives from the

compatability relationships between cylinder and bulkhead and is a function therefore of the external forces and moments applied to the end of the cylinder (x = O).

Since thermal expansion effects in the free cylinder do not introduce hoop stresses (see Sect, 2,1), the eq,(7) is not considered in calculating the circumferential stress i'esultant ^i> , Hence eq, (14) gives Nji directly. To determine the remaining stress resultants and stress

couples eq, (15) is used. Hence, N = 0 X (19) -ySx N56 = - - ^ /3 R Q^ = -M = X 2 3-/5x P ( s i n /2x + cos /Sx) + Yiy T ' _c /S rp// CO rp/ CO - ^ + — ^ + — ^ ) ( 3 i n /2x ~ Gds /&) /3 r 2/r

P cos/Jx + K J sin/Sxl j + cos/S'x ( — v j

r /^co ^GoA

P (sinySbc - cos/5x) - K^ sin/9x (—^ '—)

(20) + K, -" hr

-6

Co

JOS/Sx { —TT + M0 = y M e x where K^ (23) and P i s g i v e n b y eq, ( 1 8 )

*

10

-2 , 5 . Thermal S t r e s s e s i n C i r c u l a r Bulkhead.

The s t r e s s d i s t r i b u t i e n i n a circiiLar bulkhead of imif orm

thiclaiess subjected to a r b i t r a r y axi-syrametrical d i s t r i b u t i o n of teniperatiire i s given by (Ref, 3 , P.3é6) ,R „ r % = % ^

\ = %^[

•^ o • R

\

ƒV^]

(24) (23)

vdiere cr and cr, are t h e r a d i a l and circumferential s t r e s s e s r e s p e c t i v e l y ^T 1?

due t o tempea?ature a l o n e . I n a d d i t i o n t o t h e s e s t r e s s e s t h e r e i s a uniform s t r e s s f i e l d throughout t h e bulkhead due to the t r a n s v e r s e shear i n t h e s h e l l , This uniform s t r e s s i s given by P and i s added t o b o t h equations (21+) and

d

3 , Thormal S t r e s s e s f o r A r b i t r a r y Temperature D i s t r i b u t i o n s

I n s e c t i o n 2 forraulae have been developed giving the thermal s t r e s s d i s t r i b u t i o n s i n a bulkliead and cylinder combination i n terms of a r b i t r a r y temperature d i s t r i b u t i o n s r a d i a l l y i n the bulkhead and a i a l l y i n t h e cylinder,

I t i s p o s s i b l e therefore t o use the forraulae with e i t h e r experimental or t h e o r e t i c a l temperature d i s t r i b u t i o n s . I n e i t h e r case the a x i a l temperature d i s t r i b u t i o n i n t h e c y l i n d e r should be expressed as a polynomral. i n x

up t o the t h i r d power, even i f the t r u e v a r i a t i o n i s of a higher order

(see Sect. 2 . 1 ) , To minimise any e r r o r s i n approximating t h e exact, a r b i t r a r y d i s t r i b u t i o n t o a polynomial of t h e form

T^ =p(x) = P^ + P^x -^ P^ x^ + P^x^, ' (26)

the choice of constants P , , , , , , P , shoiild be such as t o s a t i s f y conditions a t the j o i n t v/ith the t r u e taTiperature d i s t r i b u t i o n ,

i . e . P^

I n t h i s way no e r r o r s w i l l be introduced i n t o eqs. ( l 3 ) , (16) and (18) and i n Sect 2,h- t h e only e r r o r s introduced i n t o t h e various s t r e s s r e s u l t a n t s and s t r e s s couples w i l l be those i n t h e function T away from t h e j o i n t . Since t h e g r e a t e s t r e s t r a i n t thential s t r e s s e s are expected a t the j o i n t with a rapid diminution as x increa.ses, t h e e r r o r s should therefore be small, = T CO T ' 0 0 rp// 0 0 — T' 0 0 (27)

TECHNISCHE HOGESCHOOL VUEGTUIGBOUVVKUNDE K»naalitra<it 10 - DELFT

12

-4 . Assumed Temperature D i s t r i b u t i o n

I n a recent paper (Ref,4) Biot showed t h a t by assuming a r b i t r a r y , but r e a l i s t i c , temperature d i s t r i b u t i o n modes i n skin-web conbinations, f a i r l y simple analyses enabled t h e complete t a n p e r a t u r e time h i s t o r i e s of

such combinations t o be obtained. He assumed t h a t t h e temperature d i s t r i b u t i o n i n t h e web should be parabolic and the skin temperature v/ould be constant

except in t h e v i c i n i t y of the i n t e r n a l member which caused a l o c a l parabolic v a r i a t i o n ,

Vifithout any j u s t i f i c a t i o n such d i s t r i b u t i o n s w i l l be assumed here for t h e analogous case of a cylinder with a bulkhead,

There are two phases in t h e heating of t h e configuration shown i n P i g . 1. I n i t i a l l y t h e heat p e n e t r a t e s r a d i a l l y i n t o t h e biilkhead and t h e temperature a t t h e c e n t r e has not yet begun t o r i s e . The p e n e t r a t i o n depth at any time i s denoted by q and the corresponding assumed teniperature d i s t r i b u t i o n s are shcfwn i n Pig. 2. I t w i l l be noticed t h a t a teniperature drop i s considered over t h e cylinder bulkhead j o i n t ,

The corresponding temperature d i s t r i b u t i o n s during t h e second phase of h e a t i n g , i . e . a f t e r t h e temperature a t t h e centre has begun t o r i s e , are shown i n P i g , 3. This phase begins a f t e r a time t known as the " t r a n s i t " time,

Biot assumed t h a t f o r the f i r s t heating phase 1= q and he did not consider t h e second heating phase i n d e t a i l . I n t h i s a n a l y s i s the parameters 1, q w i l l r e t a i n t h e i r separate i d e n t i t i e s during both heating phases.

thios,

The temperature d i s t r i b u t i o n s shown i n P i g s . 2 and 3 can be expressed

J (28) ^o = ^ o " (^o - ^ l ) ^ ^ -f^^ f a r X < 1 = T for X > 1 o ^ 2

^B=^2 [ ^ 1 f o r r > R - q -^

^ "^ I- ^ -I f or t < t = 0 f or r <R - q J o 1 N / r \ 2 1 (29)

= ^3-(^2-V(f) ^-^o>*l J

The boimdary- condition at t h e bulkhead cylinder j o i n t , where a drop i n temperatxire of T - T» i s assumed, can be w r i t t e n ,

h ^ = ^^j(^i - ^ 2 ^ ' ^ = ^- (^°^

where H. i s the thermal conductance of t h e j o i n t and k^ i s t h e thermal conductivity of the bulkhead m a t e r i a l . Using eq, (29) the boundary condition g i v e s , (31) RH.

= ^^1 /i'^^è) + ^ 3 / ( ^ 1 ^ ^ ^ f o r t > t .

^ ^ '3 / I \ b / Equation (31) can be generalised t o give

T T , S

T . = — ^ + - ^ , (32)

(.^ I) (1 +e)

^ n '

where n = _£ and Ö i s t h e non-dimensional parameter known as the r e l a t i v e R /' 2k^\

thermal r e s i s t a n c e (= rr-rrr] , and i n the f i r s t phase n < 1, T, = 0; whilst

V ^ j V ^ in the second phase, n = 1, T^ ;^ 0.

It is seen that two extreme values of the parameter 6 can be considered, corresponding to,

(a) Zero joint thermal resistance or 6 = 0

(b) Infinite joint thermal resistance or Ö = a ,

Using the foregoing formulae it can be shown that the values of the parameters necessary to determine P (eq.(18)) and hence solve the therma.l stress problem are

:-ƒ ?^r dr = (^^^^ R (T2 + T3) {53) o T = T^ CO 1

T':_ = 1 (T^ - T^)

CO (34) ^^o = 'l.^^o-^l) T'' _ 0 CO

1 4

-Hence t h e t h e r m a l s t r e s s problem can be s o l v e d i f t h e v a l u e s of t h e t e n p e r a t i j r e s T , , . . , T , , and t h e p a r a m e t e r q a r e known a s f u n c t i o n s of t i m e . B i c t ' s analysis-^has b e e n dev el ope d f o r t h i s p u r p o s e f o r t h e a n a l o g o u s c a s e of t h e skin-web c o m b i n t a t i o n (Ref. 5) SÏI<3. w i l l be s i m i l a r l y d ev e l ope d h e r e i n Appendix 1 .

5 . P o s s i b l e M o d i f i c a t i o n s t o The Theory

5.1» AssTJimption. of Simply Supported Edge C o n d i t i o n s on t h e S h e l l a t t h e J o i n t

I n t h e a n a l y s i s of s e c t i o n 2 . 3 i t was assijmed t h a t clamped edge c o n d i t i o n s e x i s t e d a t t h e c y l i n d e r - biolkhead j o i n t . Such c o n d i t i o n s coiold r e s u l t from t h e method of a t t a c h m e n t a n d / o r t h e f a c t t h a t t h e c y l i n d e r i s c o n t i n u o u s over t h e j o i n t . I f hov/ever t h e c y l i n d e r i s d i s c o n t i n u o u s a t t h e j o i n t and depending on t h e method of a t t a c h m e n t , s i m p l y - s u p p o r t e d edge c o n d i t i o n s may p e r t a i n ,

The e f f e c t of t h i s m o d i f i c a t i o n on t h e a n a l y s i s of S e c t . 2 . 3 i s t o n e g l e c t t h e c o m p a t a b i l i t y of s l o p e s a t t h e j o i n t a n d t o remove t h e d i s t r i b u t e d moment M . Hence e q , ( l ) becomes

^-/5bc w = "• ^ D Q cos /2x, (3^) Using t h e f a c t t h a t Q = - -p t h e a d d i t i o n of e q s . (8) a n d (33) g i v e s t h e f o l l o w i n g e q u a t i o n f o r t h e r e s u l t a n t s h e l l d e f o r m a t i o n , a t x = 0 1 oo

A>

-P 2 On e q u a t i n g t h e s h e l l d e f o r m a t i o n s and bulkhead d e f o r m a t i o n s a t x = 0 t h e follov7ing v a l u e of P i s o b t a i n e d , 2 a T -- CO mil CO ml CO 2^ 2^-- — ^ / T^r d r R^ b ^

r 2

_^ 2(1 V ) n B^ E t U o \ ' (37) The v a l u e s of t h e s t r e s s - c o x i p l e s , s t r e s s r e s u l t a n t s i n t h e s h e l l and s t r e s s e s i n t h e b u l k h e a d f o l l o w from eq'uations (37) aii'i (35) i n t h e same manner a s i n S e c t s , 2 , 4 and 2 . 5 .

Comparing e q s , (37) f^^d ( l 8 ) i t can be s e e n t h a t t h e former

e q u a t i o n g i v e s t h e lower vaJ-ue of P and hence t h e lower t h e r m a l s t r e s s e s , 5 , 2 . Use of C i r c u l a r Prames (See Ref. 1)

I f circxolar frames a r e used. a.s i n t e r n a l s t i f f e n i n g members i n s t e a d of c i r c u l a r b u l k h e a d s t h e s o l u t i o n i s o b t a i n e d b y ocsuparing r a d i a l d i s -p l a c e m e n t s of t h e j o i n t between t h e frame and t h e s h e l l . Assuming A-p, t o be t h e c r o s s - s e c t i o n a l a r e a of t h e frame w i t h a mean frame t e m p e r a t u r e of T and a mean r a d i u s of R^ t h e n t h e frame r a d i a l d i s p l a c e m e n t a t t h e j o i r i t i s

^ = - k «n, T^ + P

= - [ ^ ^ ^p *^4/'^^]-

R ; / ^ ^ E , . (38)

Equating eqs, (38) and (l6) yields the result rp / rp /// /? R 2 " ^ ,2 S i n c e , i n g e n e r a l , 2 R^ „/ s

^ f i ! ^ (o.. e,.(,8))

a n d , cc T «0= «-nT^JV, ^> -, ^ -o • ^ ^n ^ ' c CO * P F P , t h e val\je cf P i n eq, (39) R

should be s m a l l and s e v e r e t h e r m a l s t r e s s e s s h o u l d not r e s u l t frcm frame s t i f f e n i n g ,

5 . 3 . Use cf a C y l i n d r i c a l S h e l l S t i f f e n e d b y L o n g i t i J d i n a l S t i f f e n e r s (See Ref . 1 )

The p r e c e d i n g a n a l y s e s v/hich have been dev el o pe d f o r t h i n i s o t r o p i c c y l i n d r i c a l s h e l l s i s a l s o apjDlicable t o s h e l l s Tri.th l o n g i t u d i n a l s t i f f e n e r s . The main e f f e c t i s t o n e g l e c t f l e x u r a l r i g i d i t y i n t h e c i r c u m f e r e n t i a l

d i r e c t i o n and t o l e t D = E IT^D vdaere I i s t h e moment of i n e r t i a of t h e s t r i n g e r - s k i n c o m b i n a t i o n and b i s t h e s t r i n g e r p i t c h . With t h i s m o d i f i c a . t i o n t h e p r e v i o u s a n a l y s e s a r e i d e n t i c a l except t h a t M0 = 0, and P i s r e p l a c e d b y y , vAiere y*= b t

4 IR c

16

-6, Conclusions

The general problem of the thermal stresses resulting in a cylinder bulkhead configaration due to arbitrary tanperature distributions has been considered and suitable formulae derived.. The method is restricted in that the axial tanperature variation in the shell must be a polynomial of positive ordner n less than or equal to tliree; but it is general enough to permit the inclusion of joint resistance. An approach has been sviggested for minimising errors inc\mred in approximating any arbitrary temperature distribution to satisfay the above restriction.

The method ideally awaits the development of a concise theory to predict the transient temperatinre distributions in both cylinder and bulkhead. One approximate method based on the assumption of parabolic temperature distributions is presented in the Appendix giving formulae sufficient to determine the complete transient teniperature distributions. The accuracy of this method has not been assessed by comparison with either experimental or more precise theoretical results. It is considered that the results of the Appendix can be safely applied in project design studies at least.

7, References 1, Przemieniecki, J,S, 2, 3. 4. 5. 6. 7. Timoshenko, S. Timoshenko, S. B i o t , M.A. J o h n s , D . J . C a r s l a w , H . S , , J a e g e r , J , C , I n g e r s o l l , A, Z o b e l , 0 , T r a n s i e n t T a n p e r a t u r e D i s t r i b u t i o n s and Thermal S t r e s s e s i n F u s e l a g e S h e l l s v/ith Bxolkheads o r Prames,

J o u r , Royal Aero. Soc. Deconber I 9 5 é . Theory of P l a t e s and S h e l l s .

McGraw-Hill Book C o . , P i r s t E d i t i o n ( l 9 4 0 ) . Theory of E l a s t i c i t y .

McGraw-Hill Book Co. P i r s t E d i t i o n ( 1 9 3 4 ) . New Methods i n Heat Plov/ A n a l y s i s v/ith A p p l i c a t i o n t o P l i g h t S t r u c t u r e s . J o u r . I n s t . Aero. S c i . V o l , 2 4 , December 1957. PP.857-873.

Approxirmte Pcarmulae f o r Thermal S t r e s s A n a l y s e s . J o u r . I n s t . A e r o , S c i . V o l . 2 5 A u g , 1958, p p . 524-525. Conduction of Heat i n S o l i d s . Oxford U n i v e r s i t y P r e s s 1947, P . 1 7 4 . Heat Conduction.

18 -8, Natation ^ Prame c r o s s - s e c t i o n a l a r e a b Stringer p i t c h i n s t i f f e n e d s h e l l 2 B Heat Parameter = ^ / ^ » 0 Specific h e a t / u n i t volume d Bulkhead thickness D S h e l l f l e x u r a l r i g i d i t y E Youngs modulus

f / \ Radial temperature d i s t r i b u t i o n function i n bulkheadi P/ \ , P Axial temperatiire d i s t r i b u t i o n functions i n s h e l l

G. Heat flow per u n i t l e n g t h i n t o t h e s h e l l , from t h e boundary l a y e r , a t t h e j o i n t (eq. A,5)

h Convective heat t r a n s f e r c o e f f i c i e n t H. j o i n t thermal conductance

1 Moment of i n e r t i a of the s k i n - s t r i n g e r conbination k thermal c o n d u c t i v i t y

K Parameters d>.efined i n Text (eq, ( 4 ) ; eqs,(20-(22)) 1 Temperature d i s t r i b u t i o n parameter

m Power of x i n polynomial expression for P/ \

MfM/i Bending moments per u n i t length about /6 and x axes r e s p e c t i v e l y

n (= % )

N ,N/ S t r e s s - r e s u l t a n t s in x and / d i r e c t i o n s r e s p e c t i v e l y p h y p o t h e t i c a l pressxoro (eq, ( l ) )

P Radial r e a c t i o n between t h e s h e l l and bulkhead or frame q P e n e t r a t i o n depth of temperature r i s e i n bulkhead

r Radial co-ordinate in bulkhead R Radius of cylinder

R^ Mean radius of Prame

t Shell thickness; t =time; t^=transit time ' o ' 1

T Temperature r i s e above the i n i t i a l value T Adiabatic wall teniperature

w Radial displacement

X Co-ordinate measured along shell axis a Coefficient of expansion ^

P Shell parameter =

3(1- "1?^

P Stefan-Boltzman Radiation constant s

V Poissons ratio

6 Relative thermal resistance (= 57-=')

3

<f> Angle denoting position on shell penphery

/ • dQg(T^T ) ( 4 - n ) \ f Heat sink parameter due to bulkhead ( = 1 + 'w •. m '^ J

\ c 1 / (T ,0 Radial and tangential stresses on the b\ilkhead respectively

e Emmissivity Sirffioes c Cylinder B Bulkhead P Prame 00 at X = 0 in cylinder

0,1,2,3 refer to Temperatures in configuration (Pigs. 2,3) T refers to differentiation with respect to time T' refers to differentiation with respect to x

20

-APPENDIX 1.

P r e d i c t i o n of Transient Temperature D i s t r i b u t i o n s in Cylinder-Biglldicad Configurations

In Ref, 1 Przemieniecki analj^ed exactly the t r a n s i e n t temperat'ore d i s t r i b u t i o n s i n a cylinder-bulkhead configuration for t h e case of

convective heating of the cylinder f a r constant values of the heat t r a n s f e r c o e f f i c i e n t , stagnation temperature and a x i a l temperature d i s t r i b u t i o n i n t h e cylinder. Since these assumptions do not allow of a completely general study t h e a n a l y s i s of Ref.. 1 w i l l not be adopted h e r e , nor w i l l i t be

extended t o make i t more general since the r e s u l t s of such a study would be tedious t o obtain and inconvenient to incorporate into t h e main a n a l y s i s of t h i s note,

Reooijrse w i l l t h e r e f o r e be had t o an approximate method of heat flow analjrsis based on an extension of Ref. 4. The b a s i c method of Ref, 4 v i l l not be discussed here and t h e main extension introduced w i l l be i n allowing the s h e l l temperature T t o be a completely a r b i t r a r y function of time. This extension has already been applied t o t h e analogous skin-web problem of an a i r c r a f t ving i n Ref. 5.

The r e s u l t s of t h e a n a l y s i s vrere as follows. A , 1 , F i r s t Heating Phase i n Bulkhead

The parameter q can be expressed as a function of t h e temperature Tp as follov/s (see P i g , 2 ) ,

2

^ 5n T (^ - ^) ^2 .

lA 1^(13 - 3^) + 1 = 21 B^ (7 + n) A . 1 , a 2 R \

vdiere n = ^ , B = —r-—— , and where C kL are s p e c i f i c heat and thermal conductivity bulkhead m a t e r i a l r e s p e c t i v e l y

The above equation can be solved for q using the boundary condition t h a t q = n = 0 a t t = 0 ,

n

Hence i t i s p o s s i b l e t o find the value of t = t , v/hich malces n = 1. Equation A.I has not i n fact been solved and i t i s proposed t o use t h e r e s u l t quoted by Timoshenko, (Ref, 3 p.370) f o r t h e problem with Tp= c o n s t , , by exact analjrsis which gave

2

*1 = . 0 2 5 R _ ^ ^^ t^ =.0125 B. A,2.

h

Biot showed i n Ref. 4 t h a t o a t ^ therefore i f i t i s assumed , '^ - '~ o

that q = 6.34 y p , A.3.

^l ^ " • •

I n g e r s o l l * s ^ 4 x a c t r e s u l t i s obtained for t = t v^en q = R.

Although t h i s does not follow from eq. A.I i t i s considered to be accurate enough f o r most p r a c t i c a l problems.

A. 2. Second Heating Phase i n Biilkhead

I n t h i s phase, i f t h e generalised co-ordinate i s taken as T , , t h e temperature a t t h e bulkhead c e n t r e , t h e follovving d i f f e r e n t i a l equation i s obtained, using t h e temperature d i s t r i b u t i o n of P i g . 3 .

3 * •

B ( ! f 3 + - ~ ) = 4 2 T 2 - 1 2 T 3 A.4

A , 3 . P i r s t Heating Phase in The Cylinder

I n t h i s a n a l y s i s , the l o c a l h e a t - s i n k effect of t h e bulkhead i s considered and the temperature d i s t r i b u t i o n s of P i g . 2 a r e assumed except t h a t 1 i s assumed equal t o q, t h e p e n e t r a t i o n depth i n t o t h e bulkhead. The subsequent a n a l y s i s y i e l d s t h e r e s u l t

G

_{•1 = * ° o ^ i + ^ ^ ° B ^ 2 ( ^ - t ^ ^ - 5 .}

where G, i s the heat p e r u n i t length t o have flov/ed through the boundary l a y e r i n t o t h e skin a t t h e j o i n t s t a t i o n .

I f G, = t c T^ If-, 1 c 1 »

c 1

-1.4. Second jjeating Phase i n t h e Cylinder The resxiLt i s obtained.

S = * ° c ' ^ 1 - ^ i ^% (^2 ^ ^ 3 ^

3 d Q^(T + T ) A.7. .or 12^ - 1 + — ^ — ^

ï^ - 1 + 8 t c T^ c 1

22

-A.5. Shell Temperatures

Por t h e s h e l l av/ay from the j o i n t (x > l ) t h e temperature T obeys the d i f f e r e n t i a l equation

t o T = h ( T - T ) - e i ^ T ^ A.8. C O " S O' S O

Tdiere e i s emissivity

/? i s Stefan-Boltzman Constant s

h is convective heat transfer coefficient T i s Stagnation temperature.

s

The corresponding value of H i s t c T .

•^ '^ o C O

Therefore, an equation of t h e form of eq. A.8 vd.ll define the temperature v a r i a t i o n of T a l s o , provided t h e parameter t o i s factored by f , which may be generalised as

d G^ (T + T ) ( 4 - n)

' ='^ 8 t c T,^ • ^ • 9

-c 1 A. 6. Procedure

The procedure necessary t o determine t h e t r a n s i e n t teniperature d i s t r i b u t i o n i n t h e s h e l l and bulkhead i s as follov/s :

(a) Using eq. A,8 determine T as a function of time 0

(b) Using eq. A. 3 or A,1 determine q as a function of time (c) Por the tT,TO d i s t i n c t phases of heating t h e temperatures T T^T,

are determined from t h e following t h r e e equations solved simultaneously, •t c ^f^ ^ h (T - T,) - e y? T^ A, 10 C 1 ^ 3 1 ' S 1 B (T3 + 3T2) = 42 T^ - 12 T3 A.4 • ~2~

^2 = A _ 4._!3_!_

(1+ t ) (1 4. e) (32) where f i s given by eq. A. 9 and depends on t h e value of n ,

Por a completely a r b i t r a r y f l i g h t programme i t should be e a s i e r t o solve t h e equations above ( p o s s i b l y by numerical i n t e g r a t i o n ) than t o solve the exact equations of heat flow i n the s h e l l and bulkhead,

FIG I. SHELL - BULKHEAD CONFIGURATION

FIG. 2 . TEMPERATURE DISTRIBUTIONS DURING FIRST HEATING PHASE . ^3 r« BULKHEAD ^ 0 T , '1

y

°

fc.r

n

y^— / ^ SHELL i • - » :

n

FIG. 3. TEMPERATURE DISTRIBUTIONS DURING SECOND HEATING PHASE.