Buckling due to thermal stress of cylindrical shells subjected to axial temperature distributions

TECHN: GESCHOOL DELFT VLIHGÏUiGBOUWKUNDE Michiel de Ruyferweg 10 - DELFT

18aug.1981

CoA R E P O R T NO. 147

THE COLLEGE OF A E R O N A U T I C S

CRANFIELD

BUCKLING DUE TO THERMAL STRESS O F CYLINDRICAL

SHELLS S U B J E C T E D TO AXIAL T E M P E R A T U R E DISTRIBUTIONS

by

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

CORRIGENDA

Page 2, Equation 2.4 0 = Not Q

REPORT NO. 147 May, 1961.

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

C R A N F I E L D

Buckling due to T h e r m a l S t r e s s of Cylindrical Shells Subjected to Axial Tenxperature Distributions

b y

-D. J . J o h n s , M . S c . ( E n g . ) , M . I . A e . S .

D. S. Houghton, M . S c . ( E n g , ) , A. M . I . M e c h . E . , A . F . R . A e . S . J . P . H. Webber, D . C . A e .

SUMMARY

T h e r m a l s t r e s s distributions in uniform c i r c u l a r cylindrical shells due to axial t e m p e r a t u r e distributions a r e investigated. The discontinuity effect due to the p r e s e n c e of a cooler stiffening bulkhead is considered, and the possibility of t h e r m a l buckling of

the shell due to the circumferential discontinuity s t r e s s i s examined. The buckling

analysis i s based on Donnell's shell equation, and p a r t i c u l a r attention is given to shells having clamped edges.

An experimental investigation of this buckling problem i s d i s c u s s e d , and the r e s u l t s obtained a r e seen to agree reasonably well with theory.

Page Summary

Notation

Introduction 1 The Basic Displacenaent Equations 1

T h e r m a l Displacements in Unstiffened Shells 2 3 . 1 . Polynomial T e m p e r a t u r e Distributions 2 3 . 2 . Trigonometric T e m p e r a t u r e Distributions 3 T h e r m a l Displacements and S t r e s s e s in Plane Bulkheads 4

T h e r m a l Displacements and S t r e s s e s in Shells Stiffened

by Plane Bulkheads 5 5 . 1 . The General Discontinuity P r o b l e m 5

5 . 2 . Results for Simple Configurations 7 T h e r m a l Buckling of C i r c u l a r Cylindrical Shells 10

6 . 1 . Clamped Edge C i r c u l a r Cylinders 11 6 . 2 . Simply Supported C i r c u l a r Cylinders 13

Application of the Theory 13 7 . 1 . Uniform Shell T e m p e r a t u r e Rise 13

7. 2. Clamped Edge Cylinder with Non-Uniform Shell

T e m p e r a t u r e Rise 14 Experimental Investigation of a Clamped C i r c u l a r Cylinder 14

8 . 1 . Description of Specimen 14

8 . 2 . Instrumentation 15 8 . 3 . Heating Apparatus 15 8 . 4 . Description of E x p e r i m e n t s and Results 16

Conclusion 16 References 17 Appendix A - T r a n s i e n t T e m p e r a t u r e P r e d i c t i o n for

A r b i t r a r y Heating of a Shell-Bulkhead

Configuration 19 Appendix B - T h e r m a l S t r e s s e s for Assumed T e m p e r a t u r e

Distributions 25 Appendix C - T h e r m a l Buckling of Clamped Cylindrical

Contents (Continued)

Table 1 - General T h e r m a l Buckling Determinant

Table 2 - Symmetric Buckling Determinant Table 3 - Notation for Tables (1) and (2)

a Radius of shell

a , a Deflection coefficients in F o u r i e r s e r i e s representing radial displacement m p of shell

A ^a D

c c A , D j ( l + v^)

B 2a«Cj/kj (Appendix A)

c Heat capacity p e r unit volume (Appendix A) C Constants in equations (2. 2) and (5.14) t o (5.18)

m

D F l e x u r a l rigidity; dissipation function in Appendix A E Young's Modulus

i. . Radial t e m p e r a t u r e distribution, in bulkhead

F Constants in axiad t e m p e r a t u r e distribution in shell m

g Convective heat t r a n s f e r coefficient (Appendix A)

G Heat flow p e r unit length into the shell from the boundary layer at the Joint (Appendix A)

h Thickness

H Heat flow vector field (Appendix A) H. Joint t h e r m a l conductance (Appendix A)

J Additional heat input to shell fronx other s o u r c e s (Equation A. 20 and A. 22) (Appendix A)

k T h e r m a l conductivity (Appendix A) K Constants in equation (2. 3)

i T e m p e r a t u r e distribution p a r a m e t e r (Appendix A)

L Length of shell

Lo 'Buckled length' of shell P

Notation (Continued)

M , Bending moments per unit length about i^ and x shell aixes respectively M „ Bending moments p e r unit length in bulkhead about <j> and d i a m e t r a l axes n ^ (Appendix A); a number

cL

N , N, S t r e s s r e s u l t a n t s in x and é directions of shell respectively X 0

p Normal l a t e r a l p r e s s u r e in Donnell's equation; a dummy symbol; a number P P a r t i c u l a r solution of equation (2.1)

q Penetration depth of t e m p e r a t u r e r i s e in bulkhead (Appendix A) q., Q. Field and force p a r a m e t e r s in Appendix A

Q T r a n s v e r s e s h e a r force in shell and bulkhead p e r unit length r Radial co-ordinate in bulkhead

R Buckling factor = «T

S Coefficients in F o u r i e r s e r i e s representing circumferential s t r e s s distribution in shell

t Time (Appendix A)

t. ' T r a n s i t ' time (Appendix A) T Uniform shell t e m p e r a t u r e r i s e

c

T .. Critical uniform t e m p e r a t u r e r i s e to produce shell buckling T Axial t e m p e r a t u r e r i s e distribution in shell

T Value of T at ends of shell (x = - ö ^

T Radial t e m p e r a t u r e r i s e distribution in bulkhead T Adiabatic wall t e m p e r a t u r e (Appendix A)

V T h e r m a l potential (Appendix A) w Shell radial displacement

z Axial shell co-ordinate, measured from end, except for short shells (Section 3.1.6)

X Functions of y (equation 2.4) m

or Coefficient of t h e r m a l expansion P Shell p a r a m e t e r

rsd -v')i^

La«h« J

fi Stefan-Boltzmann constant ^ 2 6 Relative t h e r m a l r e s i s t a n c e (Appendix A) c Emissivity dw

6 -r— , angular co-ordinate for bulkhead

X Half wave length of buckling in circumferential direction V P o i s s o n ' s ratio

o S t r e s s , a Non-dimensional

T Elenaental volume (Appendix A)

^ Circumferential co-ordinate for shell

« - "^30 ^ '^3a _{3 / 3o 3a \}

o I T )

C \ CO /

^ Heat sink p a r a m e t e r (Appendix A) Suffices

t , a , 3 Member of configuration ( F i g s . 1 emd 2)

c Shell, used instead of i , g when shells a r e Identical I* Shell mid length

o- At X = 0, o r r = 0

E At edge of short shell • ~

-t A-tx^l, m e a s u r e d from end of shell

a At circiunference of bulkhead r Radial value in biükhead

Notation (Continued) Superfixes

T Due to T e m p e r a t u r e only

T Refers to differentiation with respect to time T' Refers to differentiation with respect to x

1. Introduction

Thin c i r c u l a r cylindrical shells stiffened at intervals along their length by bulkheads or frames a r e often used in engineering s t r u c t u r e s . When such shells are heated the thin shell tends to expand m o r e than the cooler stiffening m e m b e r s , and to resolve this discontinuity problem a system of selfequilibrating forces and moments must be i n t r o -duced at the shell stiffener joint, in o r d e r to nmake the deformations compatible. It m u s t be noted however that even in unstiffened shells, t h e r m a l s t r e s s e s can a r i s e due to non-uniform t e m p e r a t u r e distributions.

Considerable attention has been given in recent y e a r s to the problem of the buckling of cylindrical shells due to t h e r m a l s t r e s s , and in this paper the problems of t h e r m a l s t r e s s and t h e r m a l buckling a r e examined for shells subjected to non-uniform axial t e m p e r a t u r e distributions. In all the analyses it is assumed that m a t e r i a l p r o p e r t i e s a r e invariant with t e m p e r a t u r e , and a correlation is shown between the r e s u l t s of a t h e o r e t -ical t h e r m a l buckling analysis and an experimental investigation.

2. The Basic Displacement Equations

In the p r o b l e m s considered in this r e p o r t , the thin cylindrical shell i s , in general, subjected to the action of forces which a r e distributed s y m m e t r i c a l l y with respect to the axis of the shell. Because of this s y m m e t r y , considerable simplification to the theory r e s u l t s , and the r a d i a l displacement w i s given by the differential equation

dx

The derivation of this equation is given in Ti moshenko's 'Theory of P l a t e s and Shells' (Ref. 1). The general solution of equation (2.1) may be written a s either

w = e^^ f c ^ cos^x + C ^ s i n ^ x l + e ' ^ ^ jc^cos^x + C^sin^x 1 + P, ., ( 2 . 2 ) o r w = K sin^xsinh^x + K sin^xcosh^x + K cosjSxsinh^x

+ K^cos^xcosh^x + P/ ). (2-3) in which P . . is a p a r t i c u l a r solution of equation 2 . 1 , which depends on the form T. .,

and the constants C . . . C or K . . . K depend on the boundary conditions at the ends of the shell.

To apply the above r e s u l t s to edge loading problems we consider P . \ = 0- It i s assimaed that uniformly distributed bending moments M and s h e a r forces Q act at both ends of the shell, then the edge deflections and rotations a r e

w =

-° 2^

—

r^M

X + Q x i

2 -and

Q = t - J _ fajSM X +Q X 1 , (2.4)

°

2^'D

L ° ' ° ' J

_ c o s h 2 7 + c o s 27 ^ _ s i n h 2 7 - s i n 2 7 w h e r e X^ - g^j^j^27 + s i n 2 7 ' 2 ' s i n h 2 7 + s i n 2 7 ' X = c o s h 2 7 - COS27 ^^^ ^^^^^ 7 = ^ • 3 s i n h 2 7 + s m 2 7 ' 2 T h e s e r e s u l t s a p p l y d i r e c t l y t o a s h o r t s h e l l s y m n x e t r i c a l l y l o a d e d e . g. t h e s h e l l in F i g . 1 w h e r e t h e suffix s y m b o l ' E ' i s s u b s t i t u t e d f o r ' o ' .

T h e above f u n c t i o n s X, , X^ and X , a r e t a b u l a t e d in T a b l e 46 of Ref. 1, and it should b e noted t h a t t h e y a l l t e n d t o a v a l u e of u n i t y a s ^ L •• 00 , when t h e r e s u l t s a r e a p p l i c a b l e to a long s h e l l , e . g . s h e l l (2) in F i g . 2 . 3 . T h e r m a l D i s p l a c e m e n t s in Unstiffened S h e l l s 3 . 1 . P o l y n o m i a l T e m p e r a t u r e D i s t r i b u t i o n s 3 T h e p a r t i c u l a r s o l u t i o n of e q u a t i o n ( 2 . 1 ) i s given b y w. v = - a a T . v if T . \ = ?~) F X , a n d t h e g e n e r a l s o l u t i o n of e q u a t i o n ( 2 . 1 ) t h e n d e p e n d s on t h e b o u n d a r y c o n d i t i o n s t o b e s a t i s f i e d at t h e e n d s of t h e s h e l l . (a) F o r a long c i r c u l a r s h e l l t h e g e n e r a l s o l u t i o n of e q u a t i o n ( 2 . 1 ) f o r f r e e e d g e b o u n d a r y c o n d i t i o n s b e c o m e s ,, . . T T \ T " aa. . - - V [ e - ^ - [ c o s ^ x ( ^ ° . ^ ° ) - s i n ^ x ^ ] - 2T^^) ] , ( 3 . 1 ) w h e r e x i s m e a s u r e d f r o m t h e f r e e e d g e . At x = 0 t h e v a l u e s of t h e e d g e d i s p l a c e m e n t and r o t a t i o n d u e t o 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 T . . , a r e r . * T T I -w o and o r 2 T ' T " 2 1 ' -] f ( 3 . 2 ) 0 o d ' T , )

In the above analysis T* i s defined a s the value of at x = 0 in the shell. The , ° dx» other t e r m s , T , T ' and T " a r e s i m i l a r l y defined. The above r e s u l t s can be applied d i r e c t l y to shell (2) in Fig. 2 and s i m i l a r r e s u l t s a r e obtained for shell (1) when x i s negative.

(b) F o r a short c i r c u l a r shell (e. g. F i g . 1) the general solution to equation (2.1) i s taken in the form of equation ( 2 . 3 ) . If the t e m p e r a t u r e distribution is taken to be

symmetrical about the midpoint of the shell, x = 0, then the displacement mode for free edge boundary conditions is given by

w = K^sin^xsinh^x + K^cosjSxcosh^x - aa.T,y (3.3) where x is measured from the shell mid-length and T. ^ is assumed to be

T,, = è F Ixh.

(x) j ^ o m I I

K and K are found to be,

1 '/ 4 «

T T K = ora r—g- (sin7COsh7 + cos7sinh7) - —j- sin7Sinh7 | / rsinh27 + sin27 ] ,

^^^ r T " T ' -I r

K = aa — - (cos7Sinh7 - sin7Cosh7) - — - coS7COsh7 / sinh27 + sin27 1. The corresponding values of edge displacement and rotation a r e given by

T

w „ = K sin7Sinh7 + K coS7COsh7 - aaT_, ,

ill 1 4 ü.

and ,

T" r r n ' * ^ ' ^ F 1

6_, = - ^ K sin7Cosh7 + cos7Sinh7 + K coS7Sinh7 * ölh7Cosh7 1 - —s—

E L 1 L ^ J L d*T. J J ^ -^ In the above equations T* is defined a s the value of — at the free end of the

+ L d x ' shell when x = t — .

3 . 2 . Trigonometric T e m p e r a t u r e Distributions

F o r a generalised t e m p e r a t u r e distribution which t&tt be expressed in the form T. X = T F cos n'rx the solution to equation (2.1) cat! be written a s ,

(x) ^ n -JT"

w = K sin^xsinh^x + K^ sin^xcosh^x + KjCosjSxsinh^x + K^ cos^xcOöhjSx -£^C cos ^ ^ ,

g^j^ F ^ (3.5) where C = —=r , and x is measured from otie ertd of the shell.

n aD |^^^4 ^ (n^)4 j

F o r a short c i r c u l a r shell with free edge boundary conditions the constants K . . . K may be written a s

K = K = - R

2 3 n

and

4

-(sin27Cos27 + sinh27cosh27 ) + P (cosh27Sin27 + sinh27COs27) (sinh'' 27 - sin* 27)

K = R (cosh*27sin'27 + sinh'27Cos*27) - 2P (sinh27Sin27)

2 2

(sinh 27 - sin 27) where P = - L - > iH. (_i)»c V / H . Y ' '^n,

n = ^ I^KUJ ^-^'

-n-The values of edge displacements and rotation at x = O follow from equation (3. 5) a s

w^ = K - Yc ,

o A *-» n (3.6) and _ e = 2K Ö . o *

These generalised r e s u l t s will not be used or referred to in any of the subsequent analyses.

4. Thermal Displacements and S t r e s s e s in Plane Bulkheads

F o r the plane, constant-thickness bulkhead shown in F i g s . 1 and 3 it is assumed that there is a linear t e m p e r a t u r e gradient a c r o s s its thickness which is a function of radial position only i.e.AT =f^(r), and the mean radial temperature distribution is s i m i l a r l y assumed to be T^ = f ( r ) . In other words, if 1^, and 1^, a r e the a r b i t r a r y radial temperature distributions on opposite faces of the bulkhead, with linear variation between the faces

AT = T,, - T,^ = f^(r), and T + T

T = 2 = f (r) .

3 3

It can be shown that the edge deformations of the free bulkhead due to the assumed temperature distributions are

w*^ = - — / T . r . d r , a a ' -and ° (4.1) / T . r .

I

eT ^ 2a ^^ a ah '

The thermal s t r e s s distributions in the free bulkhead corresponding to the given axisymmetric temperature distributions can be easily deduced ( e . g . Ref. 2).

F o r the temperature distribution T , .a , r o ^ = oE I - . T

{7 I ^.••••'^- - ? ƒ T,.r.dr.l,

^ O o [• '^' •*" ? f ^' • r- d r + i , j T, . r . d r j . r and " " (4.2)

F o r the t e m p e r a t u r e distribution AT, the corresponding bulkhead moments a r e , r ,a r a P A T - -^ j AT. r . dr - i j AT. r . d r l , L r o a 0 -* M and r a (4.3) „ T aEh*

and the corresponding bending s t r e s s e s a r e easily obtained.

5. ThermalDisplacements and S t r e s s e s in Shells Stiffened by Plane Bulkheads 5 . 1 . The General Discontinuity Problem

In Sections (3) and (4), analyses have been presented enabling the t h e r m a l d i s -placements to be determined for unstiffened shells and unsupported plane bulkheads respectively. F o r a composite shell-bulkhead configuration it can be seen that the t h e r m a l displacements in the various m e m b e r s (e. g. shell (1) shell (2) and bulkhead (3) in Fig. 2) a r e not n e c e s s a r i l y equal. Therefore to produce compatibility of deformation of the various m e m b e r s , a self equilibrating system of moments and forces must be applied at the joints between the m e m b e r s . The s t r e s s e s in the various m e m b e r s caused by this system of moments and forces a r e known particularly a s 'discontinuity s t r e s s e s ' .

In this section the general discontinuity problem is considered i . e . Fig. 2, in which shell (1) and shell (2) a r e assumed to have d i s s i m i l a r axial t e m p e r a t u r e d i s t r i -butions. The two shells and the bulkhead a r e also assumed to be of d i s s i m i l a r materials. Because of this generality suffix notation is used with the r e s u l t s of sections 2, 3 and 4, when considering the various m e m b e r s simultaneously.

A major problem would be to determine exactly the correct t e m p e r a t u r e

distributions for a r b i t r a r y heat t r a n s f e r , to and from such a generalised s t r u c t u r e . However, although this aspect will not be dealt with in detail h e r e , an approximate analytical technique is presented and discussed in Appendix A for a particular a x i -s y m m e t r i c heating problemi.

F r o m the analysis of section (2), the values of edge displacements and rotations for the shells in Fig. 2 in t e r m s of the given edge moments and s h e a r s a r e

6 -w 10 e 10 W 20 e 20 _ « » 1

^K:

1 2 < 1

^^l

1 D, D 1 D»

r - i 3 M X + Q X 1

1 10 12 10 11 I

f- 2^ M X + Q X T

"^1 10 13 ^ 1 0 ^ *

r ^ M X +Q X 1 ,

1^2 20 22 ^ 2 0 21 1 ' ( 5 . 1 ) and , ( 5 . 2 )

r 2 ^ M X + Q X 1 ,

2 20 23 2 0 22 1 2 ^ D "^Z 2

w h e r e , f o r long s h e l l s t h e functions X a l l h a v e a v a l u e of unity. By u s i n g p l a t e t h e o r y f o r the bulkhead (Ref. 1) it c a n be shown t h a t

Q 3 a <^ -'-3 ) ^

* 3 a " • E , h , • ;

and M ( 5 . 3 )

0 - 3 a s a D j ( l + f j )

A s the s y s t e m of edge m o m e n t s and s h e a r s i s self e q u i l i b r a t i n g t h e n ,

Q + Q + Q = 0 , ( 5 . 4 ) ^ 1 0 ^ 2 0 ^ s a and M + M + M = 0 . ( 5 . 5 ) «0 20 s a F o r c o m p a t i b i l i t y of d e f o r m a t i o n s at the joint (x = 0, r = a) it i s r e q u i r e d t h a t , r r i r r i r r i 0 + e = e + e = e + e , (5.6) 10 10 20 20 3 a 3 a T T T and w + w = w + w ^ w + w , ( 5 . 7 ) 10 10 20 20 3 a 3 a E q u a t i o n s ( 5 . 4 ) to ( 5 . 7) m u s t a l l b e s a t i s f i e d if the s h e l l t o bulkhead c o n n e c t i o n s a r e a s s u m e d t o be fully c l a m p e d . If 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 apply, the e d g e t n o m e n t s a r e z e r o and t h e n only e q u a t i o n s ( 5 . 4 ) and ( 5 . 7) need to be s a t i s f i e d .

T T

T h e v a l u e s of 0 , w e t c . follow d i r e c t l y f r o m the a n a l y s e s of s e c t i o n s 3 . 1 ( a ) for long s h e l l s . F o r s h o r t s h e l l s suffix E i s s u b s t i t u t e d for suffix o in the above • q u a t i o n s , and the r e s u l t s of s e c t i o n 3.1(b) t h e n a p p l y .

Having d e t e r m i n e d the v a l u e s of t h e unknown edge m o m e n t s and s h e a r s in the f o r e -going a n a l y s e s , the r e s u l t a n t d e f o r m a t i o n m o d e s and the c o r r e s p o n d i n g t h e r m a l

5 . 2 . Results for Simple Configurations

5 . 2 . 1 . A long shell on a central bulkhead

In this section the configuration considered is that shown in Fig. 3a and the following assumptions a r e made

:-(a) Shells (1) + (2) a r e identical,

(b) the shell axial temperature distributions are symmetrical about x = 0, (c) there is no t e m p e r a t u r e difference through the bulkhead thickness, and

(d) the shell-bulkhead joint is clamped.

The above assumptions produce the following simplifications to the general analysis of section 5 . 1 . T T *^10 ~ ^ 2 0 T T 6 = - 0 , (5.8) 10 20

and e^ = e = 0 ,

s a 3a Hence we conclude that,

M = - M , _{10 s o}

M j ^ = 0 , (5.9)

and Q = Q = - Ö Q

^ 10 ^ 2 0 2 ^ 3 a

F r o m section 5.1 we obtain the r e s u l t s :

e = -0*^ = - ^ - r - r 2 ^ M - - ^ 1 , (5.10)

2 ^ \ L " 2 J '

Q r T T T «o 4^ r i T T -I

+ ^«D „ a 1 - ^ ^ + - ^ + - £ ? I . (5.11)

^ c [_ ^ 7^ 2^ J

and finally, substitution of equations 3. 2 and 5.1 into equation 5. 7 yields, a r T T T 4a /• T + - £ ° - - ^ - — ^ / L «^o 2^ 4^» J a» J o

Sa =

2« T + - ^ - - ^ ; T . r . d r c CO 2 6 4«s o» J „ 3

[

E h E h 6a . 2(1 - . , ) C C 3 3 (5.12)

NOTE Now that identical shells are considered the suffices 1,2 relating to structural, m a t e r i a l and temperature properties are replaced by a single suffix, c. Suffix notation is still n e c e s s a r y since shell and bulkhead p a r a m e t e r s a r e d i s s i m i l a r . The overall deformation mode of the shell is

w 2/3'D Q ^ (sin^x + cos/3x) + 6 D o a

_[-*(T-°^S?)

cos^x {-p - - ^ j J - a aT, . c (x) (5.13)

The values of the various s t r e s s couples and s t r e s s resultants in the shell a r e

e"^ r ( f'

M^ = - - ^ i j - Q^^ (sin^x + cos^x) - C, sin^x ( •

CO CO 3 26 ) (5.14) + cos6x ( CO

^C^')

M, N = X V M c X (5.15) (5.16) (5.17) and N, ix r f , T ' T ' s

- . 6a Q ^ (sin6x + cos6x) + C^ sin6x ( - | ^ + " ^ )

/ T ' T * V , ~ _(5.18)

where E a h c c c

6a and Q is given in equation 5.12. sa

Similar results were previously presented for this ex£unple by Johns (Ref. 3) but it should be noted that the value for N, given therein (equation 20) is in e r r o r . This present paper m a y b e deemed to supersede Ref. 3.

In conclusion it may be pointed out that the simple configuration considered in this example is that assumed in the heat conduction analyses of Appendix A and in the t h e r m a l s t r e s s calculations of Appendix B.

5 . 2 . 2 . A long shell on an end bulkhead

In a subsequent section reference is made to an experimental investigation

conducted on a long shell clamped at one end to a plane bulkhead. This configuration is shown in Fig. 3(b) and the assumptions made a r e identical to those in section 5 . 2 . 1 . By applying the method of analysis outlined in section 5 . 1 , the pertinent end r e s u l t s only will now be quoted for the clamped shell-bulkhead joint.

The total shell deflection mode is given by

W a a r r ,T" T v T T "^ 4 - [ e - ^ ^ [ c o s 6 x ( ^ o . - ^ ) - s i n 6 x • ƒ ] - 2T^,, ] 6x p -— I ^ ^ c o <^^"^^ " cos^x) + Q^^ cospx . (5.19)

-6x

where M CO

Q a a)3 fZTJ^ 2 T ' _ T '

t

2T 2T T T CO CO CO

^ 1 ^

•?"

J

26*D^ 2 |_ 6 j3* ^ ' and

[^c

' ^s<^' V ]

T ' A T* A T * / A + 2 A T A T A T / A . + 2 A \ , « f

^«cKo'*.-v-fT-^°-T -i?(-4—°)h<*.*V f- h-''"

Q » = — — —

[ Ê T h T

< ^ ' " V - F I <A,-H2A^)].

(5.20^

c c

In equation 5. 20 A, = D, (1 +y,), A » ^aD^. and the suffix symbol c again

relates to the cylindrical shell p a r a m e t e r s .

F o r a simply supported edge joint equation 5.19 still applies with M^^ = 0 and the

expression for Q „ Is then,

T . _£2 _ ^ ° I . _L / T. r. dr

*=° 26' 26' J a'

JQ

'

Q _ ^ — . ( S . Z l ;

2a

c

The values of the various s t r e s s resultants and s t r e s s couples in the shell follow directly from equation 5.19. However, for the t h e r m a l buckling analyses in the following section, it is the circumferential s t r e s s resultant which is of particular i n t e r e s t .

10

This can be expressed in the form^ Eh

N , • _{(p a}

_{h x ) ^ «c ^"^(x)] • <5-22)}

where w is given by equation 5.19. It can be shown that N, is compressive near the shell-bulkhead joint in any heating problem, and it is this which constitutes the possibility of thermal buckling.

F o r the case of a uniform ahell temperature r i s e T , and a rigid bulkhead which does not expand, equation 5. 22 becomes

N

a, = -rr = - E a T e"^^ (sin6x + cos 6x) , (5.23)

9 h c c c

for a fully clamped shell-bulkhead joint, and for a simiply supported shell, N „

a.=-rf = -E a T e"*^^ cos/3x . (5.24)

^ h c c c "^ 6. Thermal Buckling of Circular Cylindrical Shells

The circumferential buckling of circular cylinders which a r e subjected to a uniform temperature r i s e is considered in this section. The analysis is similar to that of Hoff (Refs. 4, 5), who considered only the problem of simply supported cylinders, and of Zuk (Ref. 6), who considered clamped edge cylinders.

In Ref. 5, Hoff used an infinite s e r i e s to r e p r e s e n t the radial deformation and the compressive s t r e s s distribution for the simply supported shell. Donnell's shell

equations were used to obtain a direct solution to the problem, in the form of an infinite determinant which could be truncated to give a solution to any desired degree of

accuracy. F o r the clamped edge boundary conditions, a direct analysis as in the case of simple supports is not possible, and a solution to Donnell's equation r e q u i r e s the application of Galerkin's method.

It is evident from the form of equations (5. 23) and (5. 24) that the problem of circumferential buckling is due essentially to the effects of discontinuity, and the compressive circumferential s t r e s s e s caused by the restraint of the bulkhead decay rapidly a s distance from the bulkhead is increased. Thus local buckling near the joint may occur.

The thermal buckling of cylinders due only to circumferential temperature

variations has been analysed by Abir and Nardo (Ref. 7). Simply supported edges were considered, and the method of analysis was similar to that of Hoff (Ref. 5). The main conclusion reached was that the critical axial thermal s t r e s s under a variable

circumferential thermal s t r e s s distribution was close to the critical axial s t r e s s of the cylinder under axial compression.

6 . 1 . Clamped Edge Circular Cylinders

The method of solution is based on Donnell's uncoupled equilibrium equation for the radial displacement w of thin walled circular cylinders. An infinite F o u r i e r cosine Series is used to r e p r e s e n t the hoop s t r e s s e s which vary in the axial direction, and a r e the cause of buckling.

The unknown radial deflection w is expressed in t e r m s of a F o u r i e r Series which satisfies the boundary conditions at the ends of the cylinder, but not in general, the equilibrium equation. An approximate solution is then obtained by applying the Galerkin operation, and an infinite set of homogeneous linear equations involving unknown deflection coefficients is formulated.

The criterion for buckling is that the determinant formed by the mviltiplying factors

of the deflection coefficients shall vanish. The theory allows for both antisymmetric and symmetric buckle p a t t e r n s , but only the symmetric case is considered in the numerical calculations.

Donnell's uncoupled equilibrium equation may be written a s

a* ax* L ^ ax» x y ^ a x a y ; y \ a y » ; j "^ ' where the sign convention i s in accordance with Fig. 4.

F o r the case of an axi-sjrmmetric t e m p e r a t u r e r i s e , we take p = a = T = 0 ,

X xy

and the circumferential s t r e s s e s a r e represented by the infinite s e r i e s ,

m^O

where the positive a is taken to be c o m p r e s s i v e . dw

The boundary conditions w = — = O a t x = » 0 and x = L, a r e satisfied if we assume an expression for w of the form

w = s i n ^ ) a s i n ^ sin ^ , ('6.3) p L L or w 00

= I s i n O : ^ ap r c o s ( p - l ) ^ -cos(p + l ) ^ ] • (6.4)

p=l

12

-Substitution of equations (6. 2) and (6.4) into equation (6. 1) yields an equation which is not satisfied identically by any choice of the coefficients s^. At this stage in the analysis, recourse can be made to Galerkin's approximate method of solution, which yields an infinite set of linear homogenous equations in the unknown deflection coeffici-ents ap. F o r these equations to have a non-trivial solution, the infinite determinant of the coefficients ap must vanish and this condition is the criterion for buckling. The first four rows and columns of the determinant a r e given in Table 1. It is evident that this determinant may be divided into two separate determinants, one containing the odd deflection coefficients giving r i s e to symmetric buckling, and the other concerned with the even deflection coefficients giving an anti-symmetrical buckling mode. Only the symmetric buckling mode i s considered in the subsequent numerical analysis, and the appropriate 3 x 3 determinant is given in Table. 2.

The above criterion for buckling is used in a given problem by inserting the coefficients S , characterising the circumferential s t r e s s in equation (6.2), and then by carrying out first, second, third etc. , approximations to the problem, by

truncating the determinant. This leads to an equation in R, the degree of which depends on the o r d e r of the truncated determinant, and containing t e r m s in X. Of interest, is the value of X which makes R a minimum. The degree of accuracy depends upon the number of approximations made, and these should be continued until the overall minimum value of R is found. F o r the type of buckling considered in this paper, the length of the cylinder L may be taken a s the length near to each end over which com-pressive circumferential s t r e s s e s a r e present. This length is referred to a s the

buckled length, L Q . This concept has the advantage of requiring only a few coefficients, S , to r e p r e s e n t I h e s t r e s s distribution near the ends of the cylinder. The numerical calculations have shown that adequate accuracy is obtained by using a 3 x 3 determinant.

F o r a uniform temperature r i s e T in the clamped edge shell, the s t r e s s distribution is given by the equation 5. 23.

i . e . o = E a T e ^^ rsin6x + cos6x 1 _{y c c c L J} y

3""

It can be shown that o becomes zero when 6x is — , and then has a very small tensile

In ^

value until 6x ="2- • F o r all practical purposes therefore, the buckled length can be taken as

L^ = I f • <«-S)

Equation 5. 23 can therefore be represented in the form of equation 6. 2 a s rx

^6

E a T _

c c c r , , 2irx -\ to a\

o.. - 2 [- 1 + cos — j , (6.6) y

from which it can be inferred that S = S = i

The application of equation (6. 6), with a simple mode (p = 1) assumed in equation (6. 3) is outlined in Appendix C, and compared with s i m i l a r equations obtained by Zuk (Ref. 6) and Hoff (Ref. 5).

F o r a non-uniform axial s h e l l t e m p e r a t u r e r i s e , the method given previously applies directly, provided the s t r e s s distribution can be represented in the form of equation (6.2). Because of the minimisation p r o c e s s , involved in finding the critical t e m p e r a t u r e distribution, it naay be n e c e s s a r y to assume that the form of the axial temperature distribution r e m a i n s unchanged with varying temperature levels,

i . e . T. . = T_ F , . . (x) L (x)

The s t r e s s distributions for axial t e m p e r a t u r e variation can be determined from the analyses of sections 2 to 5. In this case it should be noted that the buckled length L^ will differ from that given in equation (6.5).

6 . 2 . Simply supported c i r c u l a r cylinders

The method of analysis for this case is much simpler than that for the clamped edge cylinder, because a direct solution is obtainable without r e c o u r s e to the Galerkin operation. The same expression i . e . equation 6 . 2 , is assumed for the circumferential s t r e s s , and the expression for w has the form

OO

"•y / • n^TX . „ „V

w = s i n - r ^ / a sin-7-—. (6.7) A J , m L

m = 1

This analysis and the resulting stability determinantal equation a r e given by Hoff (Refs. 4, 5).

7. Application of the Theory

7 . 1 . Uniform Shell T e m p e r a t u r e Rise

The r e s u l t s of computations using the buckled length of the shell L„, has enabled generalised curves to be drawn, which a r e illustrated in F i g . 5. These show the c r i t i c a l buckling factor R against the ratio of shell radius to thickness r a t i o . They compare the r e s u l t s for cylinders having simply supported or clamped edge conditions, and a r e independent of the cylinder length. It is seen that cylinders having simply supported edges a r e predicted to buckle at a factor which is 20 per cent lower than the corresponding clamped edge cylinder. This relatively close agreement is not s u r p r i s i n g , because although in the bulkhead-shell region clamped edges give a v e r y rigid support to the shell, the compressive s t r e s s e s act over a 50 per cent g r e a t e r length than for the simply supported shell. This will tend to make the clamped shell m o r e prone to buckling, and in fact, the two effects tend to counteract each othet";

14

-The governing buckling determinant for a clamped edge shell of radius to thickness ratio 7- = 452, is seen to converge to a satisfactory solution when only a third o r d e r determinant is considered (see Fig. 6).

It is encouraging to note the agreement between a numerical example worked out by Hoff (Ref. 5) for a simply supported cylinder, and the general curve in Fig. 5. Hoff considered a cylinder with a = 10 in. , h = 0.3331 in. and L =3.14 in. , and the buckled length concept was not used. By using a determinant of order five, he found the critical buckling factor R to be equal to 0.0173, and this value agrees with that

calculated from the general curve. It should be emphasised that the length of cylinder which was chosen in this ntunerical example, being rather short, is not representative of shell like s t r u c t u r e s found in m i s s i l e or aircraft design. If a l a r g e r cylinder had been considered (say L > 10 i n . ) , a l a r g e r number of s t r e s s coefficients would have been required to represent accurately the s t r e s s distribution in the vicinity of the bulkhead. As a consequence of t h i s , determinants of much higher o r d e r would have had to be evaluated to give satisfactory convergence. The use of the buckled length in numerical calculations therefore, is seen to simplify the numerical work considerably. 7 . 2 . Clamped Edge Cylinder with Non-Uniform Shell Temperature Rise

In section 6 . 1 , it was suggested that in o r d e r to predict buckling of the cylinder with a varying axial t e m p e r a t u r e distribution, it would be convenient to assume a distribution of the form

T = T- F . . , X L ( x )

where only T^ is a function of t i m e .

This approach has been used in a theoretical analysis of the shell which was used in the experimental investigation, in which F . > was found to be adequately represented by '"'^

F , , = (0.467 + 1.167X - 0.667x*) , (x)

with X measured from the bulkhead.

The critical value of T is predicted as T . = 324 C, which should be compared

L I j C r i t

with a value of T ,^ for a uniform t e m p e r a t u r e r i s e T ,^ = 230 C.

crit crit 8. Experin:iental Investigation of a Claimped Circular Cylinder

8 . 1 . Description of Specimen

A cylinder of 14 in. diameter was constructed from stainless steel shim, having a thickness in the range 0.0025 - 0.003 in. This gave a mean value for the — ratio of 2540,

h and a theoretical uniform shell buckling t e m p e r a t u r e r i s e of 230 C. A value of

E = 27 X 10 lb/in , and a yield s t r e s s of 50,000 lb/in* were obtained from coupon t e s t s , carried out on the shell m a t e r i a l . It was found that a satisfactory axial lap joint could be made by spot welding.

In o r d e r to minimise steep t e m p e r a t u r e gradients near the bulkhead, at one end of the cylinder, the bulkhead was made from 0.5 in. thick Sindanyo asbestos board. This board was stiffened on either side by 12 s . w . g . mild steel d i s c s , which were of s m a l l e r diameter than the bulkhead.

A steel strip clamping ring which was insulated from the shell, was used to clamp the shell to the bulkhead, and the complete installation is shown diagrammatically in Fig. 7.

8.2. Instrumentation

Temperature m e a s u r e m e n t s were made using ten 40 s . w . g . chromel-alumel thermocouples connected to a calibrated 12 channel recording unit (New Electronics Products Ltd. , Type 1050, ultra violet t r a c e r e c o r d e r ) .

No attempt was made to m e a s u r e direct s t r a i n s within the specimen, for the following r e a s o n s .

(a) Under transient heating conditions, this would have required the use of t e m p e r a t u r e compensating gauges involving additional recording equipment.

(b) The type of discontinuity s t r e s s i s such that only mean s t r a i n gauge readings could be m e a s u r e d .

Three dial gauges were used and observed visually in o r d e r to detect the shell displacements in the region of the bulkhead.

8 . 3 . Heating Apparatus

Twenty four infra red quartz tubular h e a t e r s were spaced equally inside a c i r c u l a r polished aluminium reflector, to form the heating s o u r c e . These lamps were capable of giving a normal operating output of 1 kw at 240 v, and can be over run to give 2 kw. A voltage control box in the power supply c i r c u i t enabled the heat flux to be varied. The specimen was supported horizontally by four r o d s a s shown in Fig. 8.

8.4. Description of Experiments and Results

The specimen was made with a standard stainless steel finish» and It was found from the initial s e r i e s of experiments that t e m p e r a t u r e g r a d i e n t s exióted in the shell n e a r the bulkhead. These gradients were alleviated however, by an application of black paint over a narrow band of the shell, which improved the absorptivity in the region.

16

-The specimen was observed to buckle elastically at a galvanometer reading which corresponded to a maximum shell temperature r i s e of about 300 C. On cutting the power supply the buckles disappeared, and the cylinder returned to its original shape. Some difficulty was experienced in estimating the buckling point, and altogether seven runs were carried out at voltage inputs ranging between 175 and 225 volts.

The r e s u l t s were analysed to give a mean t e m p e r a t u r e distribution which is shown in F i g . 9. A photograph of the buckled pattern over the upper portion of the shell is shown in F i g . 10.

Despite precautions taken, t e m p e r a t u r e gradients were still found in the shell near the bulkhead (see Fig. 9). Because of t h i s , an analysis was carried out in the manner described in section 6. 2 and the critical theoretical temperature distribution obtained. This is compared with the experimental value in Fig. 11. The measured distribution is slightly lower than the theoretical value, but the comparison is considered acceptable.

The circumferential t e m p e r a t u r e distribution was found to be non-uniform (see F i g . 9), but the variation was small over the upper portion, and it was considered that this should have negligible effect on the m e a s u r e m e n t s and observations in this region.

The thermocouples placed on the steel d i s c s which were used to stiffen the bulkhead, indicated a very small t e m p e r a t u r e r i s e , and in the analyses which were carried out, expansion and flexibility effects in the bulkhead were neglected.

It should be emphasised that in the theoretical analysis, the mean shell thickness was taken to be 0.00275 in. whereas in fact the m.aximum and minimum, values varied between 0.0025 in. and 0.003 in. F o r the case of a uniform temperature r i s e , these thicknesses if constant would correspond to critical t e m p e r a t u r e s of 209°C and 249''C. It would not be unreasonable to suppose therefore that this range of ± 20°C should apply to the theoretical r e s u l t s obtained.

9. Conclusion

A method of analysis has been developed to determine the thernaal s t r e s s distributions due to axial t e m p e r a t u r e distributions in thin circular shells.

The discontinuity problem caused by the presence of stiffening bulkheads has also been considered and s e v e r a l specimen calculations have been made.

The circumferential s t r e s s e s induced near the shell-bulkhead joint may produce buckling, and a general solution has been derived for cylinders subjected to a uniform shell t e m p e r a t u r e r i s e . The critical buckling t e m p e r a t u r e for clamped edges, was found to be about 20 per cent higher than for simply supported edges. The use of the 'buckled length' concept was shown to greatly simplify the numerical calculations.

Elastic buckling caused only by t h e r m a l hoop s t r e s s e s was observed in a test specimen, and a buckling analysis, which was based on the form of the measured axial

temperature distribution gave reasonable agreement with the experimental r e s u l t s . It i s concluded that this type of elastic instability is unlikely to occur in shell s t r u c t u r e s normally found in airframe construction.

10. References 1. 2. Timoshenko, S. Timoshenko, S. Johns, D . J . Hoff, N.J. Hoff, N.J,

Theory of P l a t e s and Shells.

McGraw Hill Book Co. 1st Edition, 1940. Theory of Elasticity.

McGraw Hill Book Co. 1st Edition, 1934. Thermal S t r e s s e s in Thin Cylindrical Shells, Stiffened by Plane Bulkheads for A r b i t r a r y T e m p e r a t u r e Distributions.

College of Aeronautics Note No. 83, July 1958. Buckling of Thin Cylindrical Shells Under Hoop S t r e s s e s Varying in Axial Direction.

Jnl. App. Mech. vol.24. No. 3 , pp 405-412, Sept. 1957.

Buckling at High T e m p e r a t u r e .

J n l . Royal Aero. Soc. v o l . 6 1 . No. 11 pp 756-774 Nov. 1957. 6. 7. Zuk, W. Abir, D. , Nardo, S.V.

T h e r m a l Buckling of Clamped Cylindrical Shells. J n l . Aero. Sci. vol.24. No. 5 p . 3 5 9 , May 1957. T h e r m a l Buckling of Circular Cylindrical Shells Under Circumferential T e m p e r a t u r e Gradients. Jnl. Aero/Space Sci. Dec. 1959.

P r z e m i e n i e c k i , J . S . Transient T e m p e r a t u r e Distributions and Thermal S t r e s s e s in Fuselage Shells with Bulkheads or Franaes.

J n l . Royal Aero. Soc. v o l . 6 0 , No. 12, pp 799-804, Dec. 1956.

Blot, M.A. New Methods in Heat Flow Analysis with

Application to Flight Structures.

Jnl. Aero. Sci. vol.24. No. 12 pp 857-873, Dec. 1957.

10. Carslaw, H.S. , J a e g e r , J . C.

Conduction of Heat in Solids,

References (Continued) - 18 1 1 . 1 2 . Ingersoll, A. , Zobel. O. Webber, J . P . H . , Houghton, D . S . Heat Conduction.

McGraw Hill Book Co. . p. 176, 1948.

Thermal Buckling of a F r e e Circular P l a t e . College of Aeronautics Note No. 105, Aug. 1960.

APPENDIX A

Transient Temperature Prediction for A r b i t r a r y Heating of a Shell-Bulkhead Configuration

A. 1. In general the t e m p e r a t u r e of a structure depends upon the heat being applied to the s t r u c t u r e . F o r aircraft s t r u c t u r e s the heat source may be the propulsion system, radiation from the sun or friction of the moving a i r on the external surface of the s t r u c t u r e , etc. This latter t e r m , which is commonly known as kinetic heating, con-stitutes the major effect on the basic s t r u c t u r e s of high speed aircraft and is convective heating. All components of the h e a t t r a n s f e r problem a r e involved however in d e t e r -mining the t e m p e r a t u r e distribution in the s t r u c t u r e , e . g . conduction, convection, radiation and heat storage.

The solution of the general h e a t - t r a n s f e r problem for typical s t r u c t u r e s and a r b i t r a r y heating p r o g r a m m e s is outside the scope of this paper. Because kinetic heating i s , however, such an important problem this appendix will deal briefly with an approximate method of predicting transient t e m p e r a t u r e distributions in cylinder-bulk-head configurations for a r b i t r a r y kinetic heating p r o g r a m m e s .

In Ref. 8 Przemieniecki analysed exactly the transient t e m p e r a t u r e distributions in a cylinder-bulkhead configuration for the case of convective heating of the cylinder, for constant values of the heat t r a n s f e r coefficient, stagnation t e m p e r a t u r e and axial

t e m p e r a t u r e distribution in the cylindrical shell. Since these assumptions do not permit a completely general study the analysis of Ref. 8 will not be adopted h e r e , nor will it be extended to nmake it m o r e general since the r e s u l t s of such a study would be tedious to obtain, and inconvenient to incorporate in the main analysis of this paper.

Recourse will therefore be made to an approximate method of heat flow analysis based on an extension of Ref, 9. The basic method of Ref. 9 will not be discussed in detail here and the miain extension introduced will be in allowing the shell temperature T to be a completely a r b i t r a r y function of t i m e .

A. 2. The variational principle derived in Ref. 9 yields n equations for the field p a r a m e t e r s q of the heat flow problem

These a r e the analogue of the Lagrangian equations for a mechanical dissipative system with a potential energy, V, and dissipation function D, and have been derived for an isotropic medium for which V, 'the t h e r m a l potential' is defined a s ,

20

-and D, the 'dissipation function' is defined as

v2

T

The generalised force Q, will be referred to as the 'thermal force' and is defined a s ,

^'f^^d)

In the above equations H is defined as a heat flow vector field and, if cT, is the heat content distribution, energy conservation i s expressed by the relation

cT = - div H . (5) The technique of Ref. 9 was to assume temperature distributions of parabolic form

and hence to solve equation 1 to obtain the required result. This has been done here and the end results a r e now quoted without the intermediate analyses.

A. 3. Assumed Temperature Distributions

There a r e assumed to be two phases in the heating of the configuration shown in Fig. (3a). Initially the heat penetrates radially into the bulkhead and the temperature at the centre of the bulkhead has not begun to noticeably r i s e . The penetration depth at any time is denoted by q and the corresponding assumed temperature distributions a r e shown in Fig. (2). It will be noted that a temperature drop is considered over the shell-bulkhead joint to allow for joint t h e r m a l r e s i s t a n c e .

The corresponding temperature distributions during the second phase of heating, i. e. after the temperature at the centre of the bulkhead has begun to r i s e , are shown in Fig. (3). This phase begins after a time t known a s the 'transit t i m e ' .

Blot assumed that for the first heating phase * = q and he did not consider the second heating phase in great detail. In this analysis the p a r a m e t e r s I, q Will, Initially

retain their separate identities during both p h a s e s .

The temperature distributions shown in F i g s . 12 and 13 can be expressed thus

1 (6)

T = T ^ - (T , - T ) (1 - r ) for X < e

C C * C * CO *

= T - for X > «

[^-^]

2 T = T I ° " '^ " ^ I for r > a - q

.-' H ^ .1 1 fort<t^

= O f o r r < a - q J . „ .

= T + (T - T ) f-Y for t > t^ .

30 sa 30 \ a y t T h e b o u n d a r y c o n d i t i o n s at the b u l k h e a d - s h e l l joint, w h e r e a d r o p in t e m p e r a t u r e of ( T - T ) i s a s s u m e d , can b e w r i t t e n CO s a k ^ ' = H. ( T - T ) at r = a , (8) 8Ï^ 3 CO s a w h e r e H. i s t h e t h e r m a l c o n d u c t a n c e of t h e joint and k j i s t h e t h e r m a l c o n d u c t i v i t y of the b u l k h e a d m a t e r i a l . U s i n g e q u a t i o n (7) t h e joint b o u n d a r y condition g i v e s

2k T = T / f 1 + S - - 1 f o r t < t . s a c o / v H . q y t J

/(-fe)-„/(-if)

2k , . aH, (9)

^coC-Wj)'-\o [''•2ït] '^^'>\

w h i c h m a y b e g e n e r a l i s e d a s , T T .6

T =/ _E£.s^ + 4 1 _ _ , (10)

s a

'(-1)^ (-^. •

w h e r e n = -^ and 6 i s the n o n - d i m e n s i o n a l p a r a m e t e r known a s the r e l a t i v e t h e r m a l

^ / ^ ^ 3 \ r e s i s t a n c e ( =» —— ] , and in t h e f i r s t h e a t i n g p h a s e , j ^ ^ n < 1 T = 0 , so and in t h e s e c o n d h e a t i n g p h a s e , n = 1 T =^ 0 . so

It i s s e e n t h a t two e x t r e m e v a l u e s of the p a r a m e t e r 6 c a n ' b e 'comsidered, c o r r e s -ponding t o ,

(a) Z e r o joint t h e r m a l r e s i s t a n c e o r 6 = 0 (b) Infinite joint t h e r m a l r e s i s t a n c e o r ó = oe

22

-Using the foregoing formula it can be shown that the values of the temperature p a r a m e t e r s introduced in Sections 3 and 4, n e c e s s a r y to solve the thermal s t r e s s problem are : / T r d r H ^ ^ l i ^ ) a* (T + T ) , (11) J s \ 12 / so sa o T CO T' CO T" CO T ' CO = 3 = 3 \ ^ c o •

l^^c*

4 ( T ,

-0 . — / T ) CO T ) CO (12)

Hence the t h e r m a l s t r e s s problems of Sections 3 , 4, 5 can be solved, using the approximations of this appendix, if the values of the t e m p e r a t u r e s T , T . , T , T and the p a r a m e t e r q a r e known as functions of t i m e .

A. 4. F i r s t Heating Phase in Bulkhead

The p a r a m e t e r q can be expressed a s a function of the temperature T as

foUows (see Fig. 12). . ' ^ 5n* T (6 - n)

nn T (13 - 3n) + ^ = "^^Sa (7 + n) , (13) 4 g

-2a* c

where n = ^ , B = —r , and where c , k a r e the specific heat and thermal conductivity of the bulkhead m a t e r i a l respectively. The above equation can be solved for q using the boundary condition, q = n = O a t t = 0. Hence it is possible to find the value of t ^ t , which makes n = 1.

Equation 13 has not in fact been solved and it is proposed to use the result quoted in Ref. 2 (p. 370), and also in Refs. 10, 11, given by exact analysis for T constant with respect to t i m e ,

a»c

t^ = 0.025 - — (14) i

Biot showed in Ref. 9, that q was proportional to (t)^ hence, if it assumed that

6 . 3 4 . 1 - | - , (15) s

Although t h i s d o e s not follow from equation 13 it i s considered to be accurate enough for m o s t practical problem in which T > 0 for 0 < t < t .

A. 5. Second Heating P h a s e in Bulkhead

In t h i s p h a s e , for t > t , if the g e n e r a l i s e d co-ordinate i s T the t e m p e r a t u r e at the bulkhead c e n t r e , the following differential equation i s obtained using the t e m p e r a t u r e distribution of F i g . 1 3 ,

b

3T

B 1 ^ T „ + ' * 42T - 12T (16)

3 a so 'so 2

A. 6. F i r s t Heating P h a s e in the Cylinder

In this a n a l y s i s the l o c a l h e a t - s i n k effect of the bulkhead i s considered and the t e m p e r a t u r e distributions of F i g . 12 are a s s u m e d except that I i s a s s u m e d equal to q. The subsequent a n a l y s i s y i e l d s the r e s u l t

G = h c T + i- h , c , T (1 - 7-) , (17) CO c c c o 2 ' 3 3 a 4

w h e r e G i s the heat p e r unit length t o have flowed through the boundary l a y e r into the skin at the joint station x "^ 0.

If G =« h c T 1^ ,

CO c C CO

h c T (4 - n)

C C CO

A. 7. Second Heating P h a s e in the Cylinder

G - h c T + | h c ( T + T ) . CO c c CO 8 3 s s a ao

C C CO

A. 8. Shell T e m p e r a t u r e

F o r the s h e l l away from the joint (x > t) the t e m p e r a t u r e T , o b e y s the following differential equation

K^'c^ol = 8 ( T 3 - T ^ ^ ) - e 6 3 T ^ + J^^ . (20)

24

-where e is emissivity,

6 is Stefan-BoltzmEmn constant, s

g is convective heat t r a n s f e r coefficient, T is adiabatic wall t e m p e r a t u r e .

8

and J is additional heat input from other s o u r c e s .

cl

The corresponding value of G , is h c T . and therefore an equation of the form of equation (2) will define the temperature variation of T provided the p a r a m e t e r h c is factored by tp . This may be generalised from equations (18) and (19) by

h . c (T + T ) (4 - n)

* • ' -

• \ w

T "

•

<">

C C CO

A . 9 . P r o c e d u r e

The procedure n e c e s s a r y to determine the transient temperature distribution in the shell and bulkhead is a s follows

(a) Using equation (20) determine T as a function of t i m e .

c Zf

(b) Using equation (13) or (14) determine q a s a fimction of tinae.

(c) F o r the two distinct phases of heating, the t e m p e r a t u r e s T , T , T a r e

CO 3 a 30

determined from the following three equations solved simiultaneously,

h c d i T - g ( T - T ) - e / 3 T * + J , (22) C C ^ CO ^ S CO "^S CO CO / 11 • 3T \ B V " 2 ^ " "^ —^r 42T - 12T _{\ 2 / s a so} T T 6 T :, CO so ' a A + i \ (1 + Ó) (23) so T T 6 CO so (24)

where )p is given by equation (21) and depends on the value of n. F o r a completely a r b i t r a r y flight programme it should be e a s i e r to solve the above equations than to solve the exact equations of heat flow in the shell and bulkhead.

APPENDIX B

Thermal Stresses for Assumed Temperature Distributions

To Illustrate the thermal stress formulae of Sections 4 and 5, applicable to long shells stiffened by bulkheads, the temperature distributions considered in Appendix A will also be assumed here.

B. 1. Unsupported Plane Bulkheads

In Appendix A the assumed radial temperature distribution for the bulkhead was parabolic and of the form

T = T „ + ( T ^ - T ) ( r ) ' . t > t , . (1) 3 30 3 a so a t

Therefore, the thermal stress distributions of equation 4.2 are given by T

aE(T - T ) ^ 4 \^-^I^ r "'T

3 a 30

and

aE(T - T ) ' 4 P -3<a> r '^e

3 a 30 L 1

(2)

(3)

These non-dimensionalised stresses are illustrated in Fig. 14. By inspection it can be seen that tf the temperature distribution AT is of the form of equation (1), the results of equations (2), (3) and Fig. 14 can be applied also to the stress distributions of equations 4 . 3 .

It is worth noting that for the unsupported bulkhead the stress distribution OQ i s compressive for — > .577 and consequently thermal buckling is a possibility, (Ref. 12). However, the presence of the shell in the combined shell-bulkhead configuration has two stabilising effects

(a) circumferential edge of bulkhead is supported and number of buckles depends probably on the number of shell-bulkhead attachment points.

(b) the reaction force between shell and bulkhead introduces tensile forces into the latter.

26

B . 2 . ' Infinitely'Long Shell with Central Bulkhead (Fig. 3a)

The analysis of Section 5. 2 can be applied to this problem. If the temperature distributions considered in Appendix A a r e applied to the analysis, two significant effects can be investigated.

(a) The relative importance of shell and bulkhead radial flexibility on the force Q acting on the bulkhead.

(b) The influence of variations in the a r b i t r a r y temperature distributions T , T Because it is one of the most significant parajneters in the analyses, Q will be considered for these investigations

(a) For the clamped shell-bulkhead joint, analysis gives

s a Q. 3 a 2a c T CO 6a E h _{c c} T' T ' 1 CO CO

2^ 4 6 ' J

4a 3 a ' r 2E h (1 - v-)-l , . c c ' . ' • ^ ^ 6a a

j

T^r dr

o (4) where stiffness. 2E h c c E h 3 3 ( 1 - V^

6a is the ratio of shell to bulkhead radial

F r o m the definition of 6 one obtains i

6a a 1.285 ( r— 1 and hence R becomes

R _1.09 E h c c E h 3 3

ff

(5)

Since for most shells h / a « 1/10 and in general E h > E h it can be inferred C 3 3 c c

that the effect of radial bulkhead flexibility is small. In the analysis that follows R will be assumed to be z e r o .

(b) Using the temperature distributions of Appendix A, for t > t for both shell

(6) (7)

and bulkhead ( i . e . equations (11),

' ^ » ^''^^ CO * ct (12)) one obtains, = J (T + T ^) 4 so s a - T ) . CO

Hence

r , T , 1 a , T

T + T V so sa \ T ) CO ' . (8) 3 a ^ a ' 1 / E h a T C C C CO

The dependence of the non-dimensional s t r e s s Q on the various parameters of sa _,

equation (8) can now be investigated. The parameters ct and ^34 determine the T

CO

magnitude of the heat sink effect of the bulkhead and the distance over which it acts;

T T the parameter = is a measure of the joint thermal resistance as i s also, =— ;

CO a CO

the parameter — i s extremely important since if it i s greater than unity, c

thermal s t r e s s e s can be minimised in the transient heating stage, whilst at thermal equillbrluca large thermal s t r e s s e s can persist.

In Fig. 15 the quantity Q i s plotted as a function of =— and ^l for values of

a . T t ' ^ T CO

the product x =• - ^ ( ~ ^ ^ ~ T — ^ ^ )equal to 0.25, 1.0 and 2.0.

C ^ CO '

Hence one can infer the effect of individually varying any of the parameters in equation (8). It i s worth stating that T + T corresponds to a uniform

- ^ ^ r - ^ =2.0

CO T o

temperature throughout the configuration, hence, if -= * 1.0 and —^ =« 1,0, Q - 0.

Other similar results can easily be determined, e . g . if in the case above — » 0.5,

— c then a value of Q > 1.0 i s attained.

28 -APPENDIX C

T h e r m a l Buckling of Clamped Cylindrical Shells (first o r d e r solution)

The analysis is for a cylindrical shell which i s assumed to be unrestrained longitud-inally, fully r e s t r a i n e d laterally at the ends, and subjected to a uniform t e m p e r a t u r e r i s e .

The method of solution i s based on Galerkin's method in conjunction with Donnell's shell equation. Donnell's shell equation may be written a s

It la assumed that the circumferential s t r e s s a may be approximated by the function

[l.cos?p]

E a T

a * C C C I , , A l l JL I . - y

y ^ I 1 + cos -f- I . (2)

F o r a first o r d e r solution, the radial deformation may be assumed to be

w . a , s i n ^ r sin«^"j. ^ sin ^ [ " l - c o B ^ l . (3)

This mode satisfies the boundary conditions of z e r o slope and deflection at the ends

of the shell, a^ is the generalised co-ordinate, and X is the half wavelength of buckling In the y direction.

Substitution of (2) and (3) Into (1) yields eventually

T "-'f.

Eh

a'

0 .

The left hand side of the above equation i s not identically equal to z e r o , and to obtain an approximate solution to the equation, the Galerkin operation can be applied.

Writing the left hand side a s the e r r o r function ^, the first o r d e r solution i s given

when

X L

0. (5) / f * 8 i n ^ f l - cos ~ 1 d x d y

By p e r f o r m i n g t h i s operation the following r e s u l t i s obtained

By further simplifying equation (6), the c r i t i c a l buckling t e m p e r a t u r e i s obtained a s

which h a s to be m i n i m i s e d with r e s p e c t to X.

If the a n a l y s i s by Zuk (Ref. 6) i s p u r s u e d , the equation for T , b e c o m e s identical to equation (7), except for a factor on the right hand side of equation (7) which i s equal to

[

1

X * X « 7-128 (j-) - 16 (^)

No explanation can be given for t h i s d i s c r e p a n c y , and since Z u k ' s r e s u l t s do not a g r e e with the e x p e r i m e n t a l r e s u l t s , it i s concluded that the a n a l y s i s by Zuk i s i n c o r r e c t .

Equation (7) c o m p a r e s well with the f i r s t t e r m of the buckling d e t e r m i n a n t d e r i v e d by Hoff (Ref. 4) for s i m p l y supported s h e l l s . By taking S = S » 1 , t h i s f i r s t t e r m b e c o m e s _ , , , . * 2

AMML-^ÈL

aT ,, - "•" ^ ^^ U L i . - - + l±Li (8) c r i t

As L •• 00 equation (7) and (8) give r e s p e c t i v e l y

Tf'h* a T . = 0.544 —j-~ (clamped ends) a T .^ = 0.363 -75— ( s i m p l y supported e n d s ) . Z u k ' s r e s u l t gives a T , = 0.0777 >* (clamped e n d s ) . c r i t "•

T h i s l a r g e d i s c r e p a n c y for t h i s c a s e t e n d s to confirm that Z u k ' s a n a l y s i s i s in e r r o r . I", d e r i v i n g the above r e s u l t s , the t r u e length of s h e l l w a s used. If however the buckled lf!:^.g-ih. concept i s introduced, then L should be r e p l a c e by Lj3 ,

X '

IT

< i . 2

TABLE 1

GENERAL THERMAL BUCKLING DETERMINANT

(a.) i I k <2A + A ' ) + K {2B R L 1 1 ' * -2C (2S - S ) - C ' (2S„ + 1 o a 1 0 +Cj(2S^) - C ( S , - S , ) - C ' (S + S ) +C (2S ) 2 1 • 4 « + a 1 1 S ) 4 - 4 FK A' + K B ' -C(2S 1 R L 1 1 « 1 Ï s J - c ; ( s . + s.) + c ; ( 2 s ^ + +C,<S. + S, ) -C^(2S^ -C^'(Sj+S^) +C^ (S, + S,) + C.(2S,)

V

)1

(a,) -2C (S - S ) - C ' (S + S ) 1 1 3 1 1 S ^C (S + S ) » 1 ' ! - - - _ _. 1 4 rK(A + A' )+K(B + B' R L_ 1 1 a 2 9 2 ' -C (2S - S ) - C ' (2S + S ) 2 o 4 2 0 4 - C ( S + S ) - C M S + S ) 3 1 » 3 1 T • *C', ( S , + Sj) + C j ( S j + S J - ;_{R L 1 2 2 a j} P K A ' + K B ' l +C' (2S + S ) +C (S + 5 ) a o * 4 4 *

)]

(a,)

I^- S [ M , * ^ ^ . ]

-2C (S - S ) - C ' (S + S ) 1 1 a 4 1 a « 1 +C (2S + S ) 3 o 4 ' - c , ( s , - s . ) . c ; ( s + S , ) 1 +C (S + S ) 1 4 1 S U) , - 2 C ( S - S ) - C M S + S ) 1 1 3 » 1 3 7 ' +S(S .S,)

' - 5[^^"^^^

- C ( S - S ) -C'<S + S ) 2 2 < 2 2 • -C (2S + S ) 1 4 o 6 1 4 pK (A + A' ) + K (B + BOl ' -C (S + S ) - c'(S + S ) R | _ . 1 3 3 2 3 s j » 1 » ' 1 » 1 1 , - C ( 2 S ^ + S ; - c ; ( 2 S ^ + S,) '+C^(S^ + S J + C , < S , + S,) L « _ _ _ , ' -C (S + S ) - C ' (S + S ) 1 4 1 1 4 1 9 '+C^(S^ +S^)+C^(S +S^) 1 1 J _ _ _ _ _ _ _ _ _ +C' (S + S ) +C <S + S ) 1 1 3 7 9 1 7 ' i l [^<A4^A>K<B^.BJ] | - C / 2 S ^ + s ; - C M 2 S „ + S ^ J , +C' <S + S ) +C <S + S ) 1 2 2 • 4 2 1 1 f -- A —

SYMMETRIC BUCKLING DETERMINANT | [ K , ( 2 A , . A ; ) + K (2B + B ' ) 2 1 1 -> - . 7 2 6 0 ^ + .051Cj - 5 [ ^ ^ ' ^ ^ . ^ 1 ^ - .051C, - . 4 2 3 0 , .365C3 - .04020^ 1

- 4 fK A + K B '

R L 1 3 2 3 . - .828C + .413C 1 1 ' 1

1 5[^i<^-^^3>

,

+ K P ,

+

B;)

• - . 4 1 3 0 ^ - .3540^

' - k

fi^i

^

"^y^

' + .354C - .412C 1 1 1 , - .09920^ + .423Cj

- 4 IK A + K B 1

1 R |_ 1 » 2 s 1 - . 4 2 3 0 , + .365Cg

I 5[^<^^^^>

1 + K^(B^ + B^' ) - . 3 6 5 C , - , 3 6 1 C ,

TABLE 3

NOTATION FOR TABLES (1) and (2)

K

K

[(^

[(^

e

-J

- ) *

-)• ^ i^r

-)•

^

i.]-2Dg'X*

Eh

2X*

»

m

03a O,

j / y

Wj ° . <>3a

FIG. I. SHORT CIRCULAR SHELL WITH EDGE BULKHEADS.

1 é

Tj = f j W .

I

P

l i ^ Z 's.W"

m « 0

FIG. 3a CONFIGURATK)N OF EXAMPLE S.2.1

SHELL I.

«lO ^ 3 . °JO

FI6. 2. LONG ORCULAR SHELLS WITH BULKHEAD.

GCOMCTRY AND O3-OR0INATES

MEMBPANE STRESSES AND MOMENT BESULTANTS

F)G.4.GENB)AL MEMBRANE STRESSES & MOMENT RESUCIANT5; OOOROMAES k GEOMETRY OF CYUNDRKZAL SHELL.

o 10 M I-*

=

1 1 1

u

U

L

p

i- [-\ %.s. eoats—-1 eoats—-1 eoats—-1

1

III ^ N V ^ — " 1 I I I 1 1 M

-1

^CLAMPED EDGES

-J

\ H

^ X TEST J X.X^SPECIMEIjT'l I I I ! soo ooo RADIUS/

"'A

FIG.5. BUCKLING OF CIRCULAR CYLINDERS DUE TO UNIFORM SHELL TEMPERATURE RISE

« O O 5 0 0 K

?

4 0 0 3 0 0 O m \ • -«52 y9 . 3 - 9 / I N A^ ' 2 2 - 4 RAOS. / 9 L ^ . 2 - 3 8 RAOS. i

s

Sod APPBOX T,^,.|,30CPC

K/

^ _f \ - ^ Ird. APPROX c l * t « l 2 4 0 ° C IsL APPROX. > ^ "^ i /

y

/

j /

l O O •2 -3 -4 -5 • * - 7 HALF WAVE-LENGTH OF BUCKLMC C>J IN INCHES

FIG.6: CRITK:AL B U C K L I N G TEMPERATURE FOR A CLAMPED CYLINDER

SUBJECTED TO CONSTANT SHELL TEMPERATURE RISE

0 - 5 THIC :LL 0OO27S THICK) 2-12 SWG M/S PLATES BOLTED TO aNDANYO SECTION A - A T/C I i 2 ON OPPOSITE SIDES OF •UUCHEW ON M/S PLATES

FI6.7. EXPatMENTAL INVESTKiATtON CIRCULAR CYUNER - GEOMETRY AND THERMOCOUPLE

BUCKLING OF A CIRCULAR CYLINDER - TEST SPECIMEN AND LOADING RIG SPECIMEN NO. 1

24 I n f r a - r e d Quartz Heating L a m p s Test Specimen / Cold Junction Reflector Rods to React End Load (4) / 3 Dial Gauges — Clamping Ring Thermocouple Leads FIG. 8.

P H O T O G R A P H SHOWING D I S C O N T I N U I T Y B U C K L E S IN S H E L L O F S P E C I M E N N O . 2 B u l k h e a d -..Shel _Reflc-'t:ting Sliini F I G . 1 0 .