Thermal stresses in a box structure

CoA Note No. 8 ECHNISCHE HOC"^

VLIEGTUIGBOUW :Kanaalstraat 10

-2 0 noï.CT

THE COLLEGE OF AERONAUTICS

CRANFIELD

THERMAL STRESSES IN A BOX STRUCTURE

by

VLIEGTUIGBOUWKUNOE Kanaalstraat 10 - DEIFT NOL'E NO, a^ Ju,\7, 1958 T H E C O L L E G E O F A E R O N A U T I C S C R A N F I E L D

Thermal Stresses In A Box Structure

"by

-B. M. Lempriere, -B.Sc.(Eng,), M.Ae.E,

SUMMARY

This i n t e r i m note p r e s e n t s an a n a l y s i s , using an energy-method, of t h e thermal s t r e s s e s i n a f i n i t e length hox structijre r e s u l t i n g from uniform skin h e a t i n g . The s o l u t i o n depends upon an eighth order d i f f e r e n t i a l equation with constant c o e f f i c i e n t s . Numca?ical s o l u t i o n s are given for comparison with e x i s t i n g arid prf)jected experiments.

SÏMBOLS A, a

a

t

s

c,

Constants depending on temperature d i s t r i b u t i o n and m a t e r i a l p r o p e r t i e s .

Half width of I - s e c t i o n (= half web spacing)

Half depth of I-section (= half box depth)

Moduli, of E l a s t i c i t y 9P/2A

Stress Functions i» axial variable Length of box

Local and weighted average temperat\ares Ratio of width t o depth (= a/d)

Skin and web thickness

Non dimensional a x i a l co-ordinate (= ^ö.) Ncn dimensional t r a n s v e r s e co-ordinate (= y/d) Mon dimensional l a t e r a l co-ordinate (= z/a) Axial co-ordinate

Transverse co-ordinate see f i g . 2 L a t e r a l co-ordinate

Coefficient of l i n e a r expansion Poissons Ratio

D i r e c t s t r e s s Shear s t r e s s

-. >l _

I n t r o d u c t i o n .

The a n a l y s i s of t h e r m a l s t r e s s e s due t o k i n e t i c h e a t i n g i n multi-web box Tidng s t r u c t u r e s was f i r s t d i s c u s s e d by Hoff, who

c o n s i d e r e d a s t r u c t u r e of i n f i n i t e l e n g t h . T h i s reduced t h e problem t o a u n i a x i a l system, £ind t h u s l e d t o an e x t r e m e l y simple s o l u t i o n .

2

Fiorther unpublished Tvork has been done by Calkin who has generalised the infinite solution for unsymmetric cases, etc., and who also attempted an experimental investigeition. This however showed

no agreement with the analysis, the differences being put down to the simplicity of the theory, mainly in neglecting end effects in a short box.

The present analysis accounts for finite length of box, and could be extended to practical cases of wings with root fixings, etc.

Furtl er experimental work is under way, being directed at establishing means of measuring thermal stresses.

Analys is.

An element of t h e box beam of F i g . i i s i d e a l i s e d t o one I s e c t i o n .

¥ i t h uniform s k i n t e m p e r a t u r e s , t h e t e m p e r a t u r e d i s t r i b u t i o n a t any c r o s s - s e c t i o n ( F i g . 2) a f t e r a s h o r t t i m e may be approximated by : -T , . = -T s k m max T , = T . f l - ( y / d ) ^ + T ( y / d ) ^ web mm [^ J max^"^' ' The teirrperat'ure i s c o n s t a n t a x i a l l y ao t h a t t h e r e i s o n l y a x i a l s t r e s s due t o d i f f e r e n t i a l e x p a n s i o n . The t o t a l end l o a d t o c o m p l e t e l y s u p p r e s s e x p a n s i o n i s

P = - / EoTdA.

so that, assuming the properties are constant, an overall 'average' temperature which would require the same restraint load, may be defined as

2

-A thermal stress system due to the above temperature distribution, and having no resultant load is thus

0- = - E a ( T - T . ) X ^ max m m ^

v= -^ =

(y/d) 2 _ (-1 + 3^)

3(1 + * ) _

where "i is ratio of skin area to v;eb area (= 2 t a/t d ) . ^ s^ w '

There is then a constant stress in the skin and a parabolic stress distribution in the web. For a finite box with free ends a correction system must be superimposed on this having equal and opposite edge stresses, i.e.

x = o , I : o - = E a ( T - T . ) ' X ^ max m m '

= o

Considering the web and skins separately as plane stress systems and treating only half the skin as there will be a shear discontinuity at the junction, the boundary conditions for all the edges of the two elements may be written as :

¥eb: X = 0 , I : V = * 1 5 cr^ = A T = o cr = 0 y 2 (^ ^ 3^ ^ 3(1 + *

T =

'^X-^)

.4

3

-where ( i ) the t r a n s v e r s e co-ordinate y i s replaced by v = y/d ( i i ) the l a t e r a l '• z " " " w = z/a ( i i i ) the web a x i a l s t r e s s i s p a r a b o l i c , with /i=Ea (T -T • )

(iv) the skin i s assuiaed t o have no bending r i g i d i t y so t h a t i t cannot r e s i s t normal loads from the v/eb.

(v) the shear s t r e s s e s at the j u n c t i o n are t o be considered l a t e r .

(vi) the box beam i s considered t o be symmetric so t h a t t h e r e i s no shear between adjacent I s e c t i o n s .

( v i i ) the l a t e r a l skin s t r e s s i s taken t o be zero a t the edge of t h e I - s e o t i o n (a minimum i s implied in v i ) .

Nüfce; The two halves of the skin element are i d e n t i c a l so the l a t e r a l s t r e s s i s continuous a t the web l i n e and a f i n a l boundary

condition on t h i s s t r e s s i s t h e zero value at the outer edge of the opposite half.

The simplest a x i a l s t r e s s system which s a t i s f i e s the boundary conditions if and v a r i e s a x i a l l y i s

web : o-^ = P..(x) + •v^2(^) skin : cr = F;,(x)

The equilibrium equations for plane stress with no body forces (the thermal 'body forces' are not to be considi'red in this correction stress system) are satisfied by taking the other components of stress as web skin ^ = -vdp;,

^y = i^^^;'

^z = h\^'^

where dashes denote differentiation with respect to x, and where the functions of integration P, to F-, are in x, and are chosen in a convenient form.

if

-Satisfying cM the boundary coy.ditions if (except the shears at t h e junction) r e q u i r e s F / o ) = P ^ ( Z ) = - | [ 1 ^ ^ - ; p ; ( o ) = P ; ( l ) = 0 P2(o) = P2( I ) = A ; P^ (o) =: P^ ( I ) = o ^3(0) = F 3 ( l ) = ^ 2 ^ ^ , p ^ (^^ ^ P ^ ( l ) = 0 F' (o) = P ' ( J )= 0; Ff = 0 .*. Ff = c o n s t . = o ^ M- 4- k-^ 5 k-^ = 2 k-^ 1 2 k-^ F'g (o) = F^ ( l ) = o ; P^ = - P ^ ^7 = 2 ^ 3 ^ 6 = 2 ^ 3 . . . . 7 The s t r e s s e s now become :

vreb : cr^ = P^ + ^r'^^ T; = -vdPjj - -iv^dP^ o-y = - 1 ^ (1 - V2)P:; - -f- (^ - ^V^ skin : cr = P , X 3 -r = a(l-w)p^3 2

"^z =|-(i-w)V^ . . . . 8

As t h e r e are t h r e e unknown functions, two conditions r e l a t i n g t h e s e may be imposed a t t h e junction. These are eqviilibrium of shear and conrpatibility of a x i a l s t r a i n . Noting t h a t t h e r e are two halves of skin, t h e f i r s t of these conditions r e q u i r e s t h a t

; aP

s

3 - - i v ( ^ ; *-fV

This becomes on integration

P^ + 4 ? ^ = - * F , + Const. 0 ....9 i 3 2 3

5

-Por the second condition, the value of Young's Modulus for the T/eb and skin may be taken t o be the same, so t h a t an equation of a x i a l

s t r a i n r e q u i r e s t h a t

Vp, n

1^ + j?2 - J^3 2 ^3 10

Solving equations 9 and 10 gives

P^ = - ^(l+3i/')F3 + -J; va.^'^ + ^

Fg = 1(1+^)^3--^ i^aS^3 - | c 11

Prom the f i r s t t h r e e conditions of equations 7 these lead t o the following conditions:

P^(o) = P ^ ( l ) =: 2 A / 3(1+1^)

F^ (o) =

F^(e)

=

P^Co)

=

r^) =

= P; (^)

=

«

C = 0 . . . 1 2 The s t r e s s system may nov/ be v/ritten in terms of the »ne

unkncwn F , , which w i l l be abbreviated t o F:

web: 0-^ = - ^ ! (1+3^) - 3(l+^)v^ j ^ + 4 i^a^(l-3v^)F" r = ^ j (l+3^)v - (l + ^)v^ P " ~ 4 i^a^<i(v-v^)ï"' a-y = ^ ^ [ ( l + 3 ^ ) (l-v2) - 1 ( 1 + ^ ) (1-v^) - l y a ^ d ^ r (i-v2) - 1 ( 1 - V ^ ) I F ' " skin: cr = F X r = a(l-w)F' 0- = I (1-7/) V . . . 1 3 z ^

The problem is nov/ reduced to finding the function P for this system., satisfying the conditions 12. The internal energy may be found and minimised to yield a differential equation in F

5

-Since energy is quadratic in the stresses, the highest order term will be a fourth derivative squared, which <?n minimisation will lead to an eighth order term. Thus the equation vvill require 8 conditions for a solution, and these are given in equations 12. There will be no need to establish boundary conditions in the minimisation process, so that the boundary terms will not be evaluated.

The strain energy is given by

U

1_

]

14-the integration being taken over 14-the whole area cf web and skins. These may be considered over the ranges o<x< I , and o<v<1, for half the v/eb; and the skin over the ranges o<x<l and o<w<1 for half of one skin. Then the former must be doubled and the latter quadrupled.

The system is finally non-dimensionalised by substituting for the axial co-ordinate from

so that

X = ud

^ x

15

au etc.

The non-dimensional parameter r = a/d is introduced also. The expression for the energy becomes

l/d 2EU. w 2N - _{p" p} / ( "1 0 0 = / du -(^+-J^|^+6^ )P - r ^ vr^ (3+8^) - |g^ (2+18^+51/) - -:^ vV(2+9i^)P"'P + |(l+y) r ^ (2+18* + 5U^) + r^^l P'P' - ^ % H ^ vr2(2+9^)P'' P' 105

+

[y^

(1+1U +3l/) +

y-Hv^-^f) -

:f^

^^r^

(2+9*) J P'F"

" [3fe^^^^2+11*)-j^,.^r^ I P«"P"

+ % T ^ ^ r V F* + 7 ^

j^^J^u»^,,

105 030 ... .16

7

-This energy expression can be minimised by t h e v a r i a t i o n a l calc\iLtis, r e s u l t i n g in i n t e g r a l s of v a r i a t i o n s of d e r i v a t i v e s o A t y p i c a l term i s l/d • V d 6 f p ' p " da = f ( p ' 6 p ' " + P'' 5 P ' ) du o o I n t e g r a t i n g by p a r t s t h i s becomes p - . y d r^/d b ' 5 s" - p " 6p' + 2P'* 5 p j ^ 2 j P"6 Pdu ° 0 •y;h.cre ^ i s an a r b i t r a r y v a r i a t i o n i n t h e function P,

Nov/ a l l t h e boiondary tcnns w i l l be zero, a s every term contains one of the f i r s t t h r e e d e r i v a t i v e s or i t s v a r i a t i o n , and these a r e zero from the bcAjndary conditions 12, Then since Sp i s a r b i t r a r y , a l l the i n t e g r a l terms may be c o l l e c t e d together and equated t o zero.

For t h e p a r t i c u l a r case of a box 7d.th square c o l l s , having the skin thickness the same a s , and twice, t h a t of t h e web, r e s p e c t i v e l y

( i . e . r = 1, * = 2 and 4 r e s p . ) t h i s r e s u l t s i n t h e follov/ing d i f f e r e n t i a l equations f o r P in terms of u:

* = 2: p " ^ ' " - . I 6 6 P ^ ' + 8 0 6 5 P ' " - 4 7 , 0 0 a p " + 546,000? = 0

* = 4: F^ - 313P'^ + 2 7 , 0 0 a P " " - 209,00CP«« + l75,00CiP = 0 . . . . 1 7 The solutions of these equations s a t i s f y i n g the boundary ocriditions of equations 12 w i l l y i e l d t h e c o r r e c t i o n strest^es, from equation 13, t o be imposed on the i n f i n i t e system of equation 3.

The general solution of t h e s e equations i s

m^u -«i.u m„u -m„u m, u P = C.e ' + 0„e + 0 , e '^ + C, e + (Cr-cos m,u + C^sin m,u)e

1 ^ 3 k- 0 D ^ J

V

-m, u 4 + ( C ^ o s m_u + OnSin m_u)e with t h e i n d i c e s having t h e values t a b u l a t e d below:

.18 If 2 4 "^1 1.252 .976 m^ 2.227 2.744 " ^ 1.395 1.439 \ 9.045 12.416

For a s e m i - i n f i n i t e box ( i . e . with one f r e e end a t x = 0 and one a t X = 00 ) a l l p o s i t i v e order exponentials must vanish, so t h a t only four c o e f f i c i e n t s remain, t o be determined from the boundcjry conditions for u = 0.

8

-These coefficients for the two cases are :

If' 2

4

°2

3.054

^4

°6

°8 .

]

]

Y/here A i s defined i n equations 4 note i i i , and r e p r e s e n t s the tonperatxare difference and psroperties, wlriilst t h e odd c o e f f i c i e n t s are zero.

For a short free-ended box, t h e system i s symnetric, and t h e origin of axes may be toJ-con a t the c e n t r e , with t h e boundexy conditions expressed a t i •& / 2 , The general s o l u t i o n 18 may be v/ritten i n terms of even functions only, with four c o e f f i c i e n t s determined a t e i t h e r edge,

The s o l u t i o n s a.re:

P = C'cosh m u + C'cosh m u + C'cosh m^^u.cos m,u + C/^sinh m^^u, s i n m,u . . . . 20

where the values of the m are as hetere, and where the C' for different

box sizes are as below:

1^ ^/d C' * C« G« 0 ' 1 2 3 4 2 2 +1.222 - . 2 9 7 +3.59x10"^ +1.40x10""^ 4 + . 4 8 7 5 -.0498 +1.57x10"^ -8.65x10"^ <s.x,222A 7 + .0791 -.00186 +9.33x10"^ 5 _^-,^ ^ 3 ^ Q-1 5 J 4 2 + .9827 -.0675 +7.737x10""^ +2.097x10""^ 4 + . 4897 - . 00685 - . 7595x1 o""" ^ +6.041 xi o"^ ^

The functions of equations I 8 and 20, v/ith t h e c o e f f i c i e n t s 19 and 21 are p l o t t e d i n F i g . 3.

• 9

-Now t h e d i r e c t web s t r e s s e s of equ'^.tion 13 added t o the i n f i n i t e syston of equation 3 , for t h e two cases of ^ = 2 and 4, niay be v/ritten i n the follov/ing forms, t o give t h e a c t u a l s t r e s s e s i n t h e short box;

o" = A X .777(l-f) + .0i66f" I - J ( i - f ) + .05Qf" v^ ( ^ = 2 ) cr y = A rj . 8 6 6 ( i - f ) + ,010 f" j - j ( i - f ) + .030f" v^ h (v = 4) = AV(.306f" - ,0042f'V)_(,38if" -.0083f»^)T+(.083f" - . 0 0 4 2 f ' ^ v ^ / (*

= A{(.235f" -.0025f^) -(.433f" -.005af'v)v^+(.083f" -.0025f'^v^} (\f

where f = —r- and = -—-- f o r f = 2 and 4 r e s p e c t i v e l y . The rcanammg s t r e s s e s a r e not c o n s i d e r e d h e r e , b u t m a y b e obtained, from e q u a t i o n 1 3 .

F o r p u r p o s e s of comparison w i t h e x p e r i m e n t , t h e d i f f e r e n c e between a x i a l a n d l a t e r a l Y/eb s t r a i n f o r t h e c a s e of * = 4pWill he c a l c u l a t e d a s t h i s c o r r e s p o n d s t o t h e r e s u l t s o b t a i n e d by C a l k i n . He assumed t h a t t h e l a t e r a l s t r e s s e s v/ere n e g l i g i b l e , s o t h a t two s t r a i n gauges p l a c e d

p e a r p e n d i c u l a r l y on t h e v/eb v/ould g i v e t h e a x i a l s t r e s s o n l y , t h e gauges b e i n g temx3era,ture compensated. The a p p a r e n t a x i a l s t r e s s v/ould t h e n b e talc en a s :

cr = ( e - e ) E / ( I + v) app ^ X y

Hov/ever t h i s would r e a l l y become * Of = cr - cr app X y F o r t h e c a s e c o r r e s p o n d i n g t o C a l k i n ' s t e s t s ( i . e . f = k., -& / d = 4 , ^ = 1J £^d. t a k i n g v = 0 . 3 ) t h e p r e s e n t t h e o r y g i v e s cr = ( . 3 5 0 - .350v^ - .034v'^) E a(T - T . ) 23 V a l u e s of t h i s s t r e s s a r e c a l c u l a t e d u s i n g C a l k i n ' s measured t e m p e r a t u r e d i s t r i b u t i o n s , (assuming t h e s e t o b e p a r a t o l i c ) and w i t h t h e same m a t e r i a l p r o p e r t i e s , namely E C( = 303 p . s . i . / C. These a r e p l o t t e d i n P i g . 4 t o g e t h e r w i t h t h e e a r l i e r e x p e r i m e n t a l r e s i i l t s and i n f i n i t e t h e o r y c a l c u l a t i o n s , and w i t h t h e c o r r e c t e d a x i a l s t r e s s of e q u a t i o n 2 3 .

10

-Conclusions

1, According t o t h e present theory for short boxes, t h e a x i a l s t r e s s e s are about h a l f those p r e d i c t e d by the i n f i n i t e theory, fcr the box examined. 2, A short i n i t i a l period occurs before t h e heat flow frcm the skin p e n e t r a t e s t o t h e c e n t r e of the v/eb, so t h a t in t h i s period t h e web

tei!iperat\:!re d i s t r i b u t i o n i s not a s assumed, namely p a r a b o l i c . Hence the p r e s e n t theory cannot be r e l i a b l e then.

3 . The experimental r e s u l t s ajid t h e calciolated values do not agree v/ell having d i f f e r e n t forms.

4 . The weakness of the present theory probably l i e s in t h e f a c t t h a t i t i s based on a simplified temperatiore d i s t r i b u t i o n . IThile t h e web parabolic approximation ma.y be j u s t i f i e d in some c a s e s , t h e skin temperature v/ould always have a s i g n i f i c a n t drop a t the v/eb, and the maxinicci web

temperatxjre v/ould, be dilTerent frcm t h e skin temi^erature, due t o thermal r e s i s t a n c e a t t h e j o i n t . The theory i s a l s o only a second approximation t o t h e t r u e s t r e s s d i s t r i b u t i o n , even i f the i d e a l i s a t i o n s are j u s t i f i e d . However a more r e f i n e d s o l u t i o n would be impractical and probably unnecessary. 5. I n a long box, the i n f i n i t e theory would be acc\irate for p o s i t i o n s more than about 2-g- depths ("^/d - 5) from a free end.

6, For boxes of tjrpical p r o p o r t i o n s , t h e s t r e s s d i s t r i b u t i o n i s only s l i g h t l y a f f e c t e d by t h e r a t i o of skin area t o v/eb a r e a .

References;

1, Hoff, N . J . S t r u c t u r a l problems of future a i r c r a f t ; Third Anglo-American Conference, 1951. 2, Calkin, P , An a n a l y t i c and experimental study of some

problems of thermal s t r e s s e s ; College of Aeronautics Diploma T h e s i s , 1956.

FIG. I MULTICELL BOX STRUCTURE.

FIG 2 IDEALISED I-SECTION WITH TYPICAL TEMPERATURE DISTRIBUTION.

1 0 ' 4 .2 n

]

V

1

jV

1

f

1

• o x PAOPORTIONS : » - 4 , e/d = 4 , r = (

AT 15 SECS

Z

AXIAL STRESS (PRESENT THY.)

I. - ( I N F I N I T E T H Y . )

CALC. STRAIN DIFFERENCE e x P L .

AT IIO SECS. STRESS C P . S J )

-AT 4 O SECS

.COO

AT I70 SECS.

FIG. 4 . WEB AXIAL THERMAL STRESSES ACROSS CENTRE LINE OF SHORT BOX

3 0>_ (A a _. O 1 O rr! 0 --n - ( jS m r. O n c 3: C /L m O T Z en O X m I O O m O X O O