TECHNISCHE HOGESCHOOL utin , ;i : M VLlEGTUiGBOUWKUNDE CoA Note No. 110 Michiel d« Ruylerweg 10 - DELFT
THE COLLEGE OF AERONAUTICS
CRANFIELD
LINEAR BUCKLING OF AN AXIALLY REINFORCED
PRESSURISED CYLINDER
by
NOTE NO. 110 October, 1960.
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
Linear Buckling of an Axially Reinforced P r e s s u r i s e d Cylinder
b y
-D. S. Houghton, M.Sc.(Eng.), A . F . R . A e . S . , A. M. I. Mech. E. and
D. J . Johns, M.Sc.(Eng.), M . I . A . S .
SUMMARY
An analysis is presented using small deflection theory for the buckling of a p r e s s u r i s e d , axially reinforced cylinder, which is subjected to axial compression.
Various approximations to the analysis a r e discussed and Sonne results are presented which show the effects of internal p r e s s u r e and various s t r u c t u r a l p a r a m e t e r s on both panel buckling and overall buckling.
CONTENTS
Page Summary
Notation
Introduction 1 The basic equations 2
The axisymmetric buckling solution (n = 0) 4
A general buckling solution 6 4 . 1 . Panel buckling in a short cylinder
( i . e . e « R) under axial load ( ;^ = 1) 7 4 . 2 . Buckling in a long cylinder under
axial load 9 4 . 3 . Buckling in a long cylinder under
axial load and internal p r e s s u r e 10 The overall buckling of a heavily
stiffened cylinder (/3 » 1) 10
NOTATION s ( x , y , x y ) a b D D' E E I m M n ( x , y , x y ) N' y p R t t s u , v , w , a /5 e, , e^ , y V cr c r o s s s e c t i o n a l a r e a of s t r i n g e r s t r i n g e r pitch f l e x u r a l r i g i d i t y of skin effective f l e x u r a l r i g i d i t y of s k i n - s t r i n g e r combination Young's modulus of skin
Young's m o d u l u s of s t r i n g e r
m o m e n t of i n e r t i a of a s t r i n g e r about m e d i a n plane of skin length of c y l i n d e r
n u m b e r of half waves in an axial d i r e c t i o n r e s u l t a n t s t r e s s couples in c y l i n d e r n u m b e r of w a v e s in a c i r c u m f e r e n t i a l d i r e c t i o n s t r e s s r e s u l t a n t s in m e d i a n plane of c y l i n d e r N -pH y i n t e r n a l p r e s s u r e r a d i u s of c y l i n d e r t h i c k n e s s of skin equivalent t h i c k n e s s of s t r i n g e r s h e e t (= :-) d i s p l a c e m e n t s in x , 6 and z d i r e c t i o n s t^ c y l i n d e r p a r a m e t e r = 12R^ r a t i o of c y l i n d e r flexural r i g i d i t i e s = D m e d i a n plane s t r a i n s a r b i t r a r y c o n s t a n t s in e x p r e s s i o n s for d i s p l a c e m e n t s c y l i n d e r c o - o r d i n a t e E.in7r a x i a l r e i n f o r c e m e n t p a r a m e t e r = P o i s s o n ' s r a t i o s t r e s s p r e s s u r e p a r a m e t e r = E t s s E t (1 -v^) ( x , y , x y ) a x i a l f o r c e p a r a m e t e r = changes in c u r v a t u r e E t r - N (1 -vz) n E t
1
-1. Introduction
In the original investigations into the buckling of circular cylinders carried out by Timoshenko, Southwell, Flugge and others, a small deflection theory was used to establish the elastic deflection equations. Experiments however, show that, particularly for the case of a cylinder under an axial load, the classical deflection theory considerably over-estimates the critical s t r e s s .
In 1934, Donnell first proposed the use of a non-linear theory to explain these discrepancies and this theory has been developed and used extensively in recent years for predicting the elastic buckling of cylindrical shells. Much of the recent work in this field is summarised by Nash and Thielemann .
In the last report use is made of the non-linear theory to analyse the effect of both internal p r e s s u r e and orthotropic properties on the post buckled behaviour of axially loaded cylindrical shells. The experimental investigations which are also carried out indicate good agreement with the theoretical r e s u l t s . However, the solution of the large deflection equations suggested by Thielemann, involves considerable computational difficulty, and it would seem reasonable in investigating the influence of axial stiffening and internal p r e s s u r e , to initially use small deflection theory in order to show more readily the effect of the shell p a r a m e t e r s on the critical s t r e s s .
3
An investigation has recently been undertaken by McKenzie , who used the small deflection theory in solving the problem of buckling of an axially reinforced cylinder subjected to axial end load and internal p r e s s u r e . McKenzie's solution i s , in fact, restricted to the problem of a cylinder having a large number of axial stiffeners. F u r t h e r m o r e , he is only concerned with the overall buckling of the stiffened shell after panel buckling has developed.
In this paper both panel buckling and overall buckling of the shell a r e considered, and the influence of the axial reinforcement is discussed,
- 2
2. T h e B a s i c E q u a t i o n s
The equations of e q u i l i b r i u m of an e l e m e n t of a stiffened cylinder can be obtained by suitably modifying the equations for an unstiffened s h e l l a s p r e s e n t e d by T i m o s h e n k o which a r e
aN • aN aM aM ' ^a aa ^ " a x Rae ^ a x " x 2
;,2 a^M a^M a^M a^M , ^ _ , v EN U ï + 2 ^ + R ?L+ _ _ Z 2EZ. + N ' y + <2.01)
^ ax^ 3 x 3 6 3x^ Rae^ ^^^^
( S^ w \
+ p w + — - = 0 . \ 36^ /
The sign convention i s the s a m e a s that given in Ref. 4, F i g . 231 . F o r a long s h e l l which i s stiffened in an axial d i r e c t i o n the s t r e s s r e s u l t a n t s and couples can be modified t o b e c o m e
N = E t e + - 5 1 _ . ( e ^.y^ ), N = -—-^i——( e + i^e )
X s s 1 1 - 1^2 •, 2 y 1 - y2 2 N = N = ^.)^ . , N' = N - pR , yx xy 2(1 +1^) ' y y (2.02) a n d M = - D (/9x + yx ) , M = - D( x + yx ), X '^ X y ' y y X - M = M :^ D ( l - y ) x yx x y x y ax ' (2.03) w h e r e e 1 ''x
au
" 9x •a' w
" 9 x ^ ' e 2 X = yav
R30 1 R^ w • R • y/ av aV\
I 3Ö '
ge^y
'
au
Rae
X = x y i _ / a v _!_ a f w \ R V^3x 3x36/ T h e s e equations a r e d e r i v e d on the a s s u m p t i o n that the axialr e i n f o r c e m e n t i s s y m m e t r i c a l l y d i s p o s e d about the skin c e n t r e l i n e . If t h i s a s s u m p t i o n i s not fulfilled, additional t e r m s m a y be n e c e s s a r y in the e x p r e s s i o n s given in equations (2.02) and (2.03).
3
-Substitution of equations (2.02) and (2.03) into (2.01) y i e l d s
_2 a ' u , _ ( i + y ) a V _ aw „ , juR — — + R — ^ — ^:rrTa ~ ^^ TH ~ ^^4 R ax^ (1 + v) 3^u 3x30 ax axae + 1 -V .2 3=^ V + R (1 - y ) 2 / 9 x 2 a^ V -+ ax^ R^0 aV _ ae" af_v 2 a x ' afv_ 3x36 aw
ae
+ a ax a^v 3 ^ = 0 ae^ + a^w 2 / ae^ 2 a^'w + R „ 3u , 3v ^ " aT -^ a-F- - ^ 2Rf 3 V . „ , , ..X ^ a^v ^•\- 2(1 -1^) I^ , 30' ax^ 30 + yR 0 . .2 a ' v 30' Ox (2.04) ax''30 3x^30 •I- R V ax'» + a^w 1 2 n Q 2 ax^ao L J / , 3 V . , „ 2 a V 1 30 2 ax 0 .To solve the above equations the following m o d e s a r e a s s u m e d vdiich a r e a p p r o p r i a t e for a s i m p l y supported s h e l l a'^w 30^ and u = ^ cos n6 cos V = •n siii n0 sin w = ^ cos n0 sin R XX R Xx R (2.05) In t h e s e equations ^, 77 and g a r e a r b i t r a r y c o n s t a n t s ,
\ = —-g— and m and 2n r e f e r to the n u m b e r of axial and c i r c u m f e r e n t i a l
half w a v e s .
T h e s e m o d e s a r e identical with t h o s e a s s u m e d by Tim.oshenko and 3
McKenzie and include the p o s s i b i l i t y of a x i - s j m i m e t r i c r i n g shaped b u c k l e s and a s y m m e t r i c c h e s s b o a r d shaped b u c k l e s . However, in t h e i r p r e s e n t form equations 2.05 a r e inadequate to include the d i a m o n d - s h a p e d buckling m o d e which i s indicated in the e x p e r i m e n t s conducted by Thielenaann . Substitution of equations 2.05 into the e q u i l i b r i u m equations 2.04 y i e l d s the s t a b i l i t y d e t e r m i n a n t a l equation a s follows
4 -^^2 + y-j^jn 1 +y nX v\ . 7 ^ ' Xn 1 - y "(1+ v) ft ( y - ^ ) X 1
„ , (1 + 2o) + n^(l +c)-?i X^, n +an(n^+X^
^ / 2
, n -han' + 2 a ( l - y)nX^fyanX^ , 1 + o^SX +an''f
2an^X^ -<p {1 - n) - (f, X^
1 2 (2.06)
T h e solution of t h i s equation l e a d s to the c r i t i c a l value of the axial f o r c e p a r a m e t e r 9^2 for v a r i o u s v a l u e s of the p r e s s u r e p a r a m e t e r <P^ , and of the t e r m s d e s c r i b i n g the effects of the axial stiffeners u and /3 for any a s s u m e d v a l u e s of n and X. T h e r e i s of c o u r s e a m i n i m u m value of (p , and it i s the d e t e r m i n a t i o n of the v a l u e s of n and X c o r r e s p o n d i n g to t h i s m i n i m u m value which c o n s t i t u t e s one m a j o r p r o b l e m .
3 . The a x i s y m m e t r i c buckling solution (n = 0)
F o r t h i s c a s e the d e f o r m a t i o n m o d e s b e c o m e s i m p l y Xx u = ii cos R a n d w = ^ sin XX R (3.01)
and t h e s e m o d e s c o r r e s p o n d to the c a s e of r i n g shaped b u c k l e s . In the equation 2.04, it i s o b s e r v e d that the -gg t e r m s v a n i s h , and that v = 0, in which c a s e the equation b e c o m e s
d^ u dw MR a n d dx^ du d x V - <f) ^ - - T ^ - ^ / ^ d w , , _ 2 d w _ Ry 3 - - w - a R /9 — 7 - +5^ w - é B, — r " = 0 , dx dx 1 2 dx (3.02)
T h e s e can be combined to give dx^ 2 dx
_ • i > ^ [ l 0. (3.03)
F o r the c a s e of an unstiffened s h e l l ( i . e . /U = /5 = 1) and no i n t e r n a l p r e s s u r e ( i . e . 0 = 0 ) , equation 3.03 r e d u c e s to
This result c o r r e s p o n d s to t h e c l a s s i c a l equation for an unstiffened s h e l l .
If the a x i a l stiffening and the i n t e r n a l p r e s s u r e t e r m s a r e included in the a x i s y m m e t r i c buckling equation 3.03, the condition for minimumi
<p gives the r e s u l t 2 2a/9X - 2 • 1 - — -cj> ( 1 - - ) W h e r e 1 - - ^ - ^ ( 1 - 77) X^ = ' " < ! ' ' 0 , (3,05) u
T h e s e r e s u l t s could have been obtained d i r e c t l y from equation 2,06 with n = 0.
F r o m equation 3.05, the c r i t i c a l a x i a l s t r e s s b e c o m e s
N Et
/5 i L _ ^ (1 _ Ü)
x c R 3(1 - y^) (3.06) F o r no axial stiffening equation 3.06 r e d u c e s to
. E t r.
!>'.
'1
-R 3(1 v^) 1 +y
(3.07)
which can be c o m p a r e d with the c l a s s i c a l buckling solution for an u n p r e s s u r i s e d c y l i n d e r u n d e r a x i a l load given by T i m o s h e n k o a s
E t cr.
R 3(1 -y^
(3.08)
On e x a m i n a t i o n it i s found that the (ji t e r m in equation 3.07 i s v e r y s m a l l c o m p a r e d with unity, so that it can be concluded that the influence of i n t e r n a l p r e s s u r e on a x i s y m m e t r i c a l r i n g shaped buckling i s s m a l l , but i s slightly d e s t a b i l i s i n g . The fact that i n t e r n a l p r e s s u r e h a s negligible effect on r i n g shaped buckling i s o b s e r v e d by T h i e l e m a n n .
- 6 4 , A gv'^neral b u c k l i n g s o l u t i o n T h e b u c k l i n g d e t e r m i n a n t e q u a t i o n 2.06 c a n b e e x p a n d e d a n d e x p r e s s e d i n t h e f o r m w h e r e a n d C + C a = C 0 1 2 3^1 •kH^-v') . A^Z ( 4 . 0 1 ) = (n^ + X') - (2 + y ) (3 - y ) X^n' + 2 X ^ 1 -v^) -X^n''(7 + y ) + X' n M 3 + I') - 2n + n"* + ( ^ - 1) (y? - 1) 2X^ 1 - y 4/ ^ 2 , „ 2 \ 2 + (/? - l ) X ^ ( X ' ^ + n M 2 X^ ^ ^ 2 I „ 2 — X + n + ( ^ - 1)
(x"+
n^ r
( ^ r ^ ) X ^ + n' 1 - V - 2n + n -{3 - v) n^X' + (1 - y ) X" - C 3 = ( n ' - l ) ( X ' + n M ' + i > x ' + ( /J- l ) X " ( n ' - 1)(X^+ ^ n ) . ^•K\ 2n X^ { 1 -V ) + x{' + 2\' {-^ ^ 1 ) + n'In t h e s e e q u a t i o n s t h e s m a l l o r d e r t e r m s a^, ac/) , a.^ ,9^^, e t c . , h a v e b e e n n e g l e c t e d . ^ 2 1,2 It i s found t h a t f o r t h e c a s e of n o a x i a l s t i f f e n i n g t h e s e e q u a t i o n s d o not r e d u c e e x a c t l y t o t h o s e g i v e n b y T i m o s h e n k o * . T h e r e a s o n f o r t h i s i s t h a t T i m o s h e n k o ' s s o l u t i o n i s b a s e d on t h e e q u i l i b r i u m e q u a t i o n s d e r i v e d b y F l i i g g e w h i c h h a v e s m a l l d i f f e r e n c e s f r o m t h e e q u i l i b r i u m e q u a t i o n s 2 . 0 1 . T h e s e d i f f e r e n c e s a r e of l i t t l e c o n s e q u e n c e a n d s h o u l d not a f f e c t t h e s o l u t i o n f o r t h e c r i t i c a l a x i a l s t r e s s , T h e e v a l u a t i o n of t h e c r i t i c a l a x i a l s t r e s s , h o w e v e r , f r o m e q u a t i o n 4 . 0 1 , i s d i f f i c u l t a n d f u r t h e r s i m p l i f y i n g a s s u m p t i o n s w o u l d a p p e a r d e s i r a b l e t o o b t a i n a c l o s e d f o r m s o l u t i o n .
7
-4 . 1 . P a n e l buckling in a s h o r t c y l i n d e r (i. e. -& << R) under a x i a l load \P = 1)
F o r a r e i n f o r c e d cylinder with no i n t e r n a l p r e s s u r e equation 4.01 b e c o m e s
C, + C^a 4 ^Z (4.02)
The p r o b l e m i s to find the v a l u e s of X and n which c o r r e s p o n d to a m i n i m u m value of the axial loading p a r a m e t e r (f>^. If it i s a s s u m e d that /? = 1 while ii > 1, it is implied that even with heavy a x i a l
stiffening the value of n p r o d u c i n g t h i s c r i t i c a l yJ will c o r r e s p o n d t o , o r be a m u l t i p l e of, the n u m b e r of s t r i n g e r s , w h e r e each a c t s a s a nodal line and d o e s not bend.
It i s found that equation 4.02 r e d u c e s to the following form XM ^-v^ + a ( n ' + X")" F ^. nX (4.03) X^ F nX w h e r e F 2 / . 2 . 2 n nX
(ii^+x')^+ ( ^- 1)X (x'=+ f^J
In equation 4.03 low o r d e r t e r m s in X^have been neglected which i s justifiable for the c a s e of the s h o r t c y l i n d e r , provided that I « R and I » VRt.
3 y differentiating <p^ with r e s p e c t to both X and n, and equating to z e r o , one obtains the two e q u a t i o n s ,
F {^-v.^)
nx
and Fnx
a( X"- n M a(n^ +XM XI 2 X^ n^ + X^ +{^ - DCkU 2 . n ) ( t f +X^) + ( ju 1) nX (4.04) (4.05)When n i s given by the p a r t i c u l a r s t r i n g e r s p a c i n g , equation 4.04 y i e l d s the value of X c o r r e s p o n d i n g to the c r i t i c a l value of 0 .
Solution of t h i s equation i s not a t t e m p t e d h e r e . If however the s t r i n g e r s p a c i n g i s i n i t i a l l y unknown, then the s a t i s f a c t i o n of e q u a t i o n s 4.04 and 4.05 s i m u l t a n e o u s l y , r e s u l t s in a l o w e r c r i t i c a l value of yi, .
8
-F r o m equations 4.04 and 4.05 the s i m p l e r e l a t i o n s h i p i s obtained between n and X which i s
n = v\^ (4.06)
It i s i n t e r e s t i n g to note that t h i s r e l a t i o n s h i p i s indeper.dent of the a x i a l r e i n f o r c e m e n t p a r a m e t e r /i . Substitution of equation 4.06 into equation 4.05 gives
X^ n^ 1 - y^ ^ a ( l + y)^
.. J (1
-v')
N a d + v)^ and (4.07) ^ 2 o c c u r s when (n^ +X^)^ X" 1 -v' ~ N aAgain t h i s r e s u l t i s independent of ju and c o r r e s p o n d s with Tinnoshenko's r e s u l t for an a x i a l l y loaded s h o r t c y l i n d e r which s u g g e s t s that a m i n i m u m
(4.08)
It i s p e r h a p s worth noting that by a s s u m i n g axial stiffening, i. e. /J > 1, one obtains explicit v a l u e s for X and n c o r r e s p o n d i n g to a m i n i m u m value of ^> even though t h e s e v a l u e s a r e independent of i" .
2
F o r the unstiffened s h e l l h o w e v e r , the only r e s u l t which i s obtainable i s in t h e f o r m of equation 4 . 0 8 .
Substitution of equations 4.07 into 4.03 gives the r e l a t i o n s h i p 0 = 2 a ( l - y ^ V
2 N
T h i s r e s u l t a g r e e s e x a c t l y with the c l a s s i c a l solution for an unstiffened c y l i n d e r which shows that panel buckling in a r e i n f o r c e d shell o c c u r s at the s a m e value of r e s u l t a n t force a s in an unreinforced s h e l l , provided that equations 4.07 a r e s a t i s f i e d .
4 . 2 . Buckling in a long c y l i n d e r u n d e r a x i a l load
F o r a long cylinder it i s r e a s o n a b l e to a s s u m e that X b e c o m e s s m a l l in the g e n e r a l buckling solution, equation 4 . 0 1 , so that high o r d e r t e r m s in X m a y be n e g l e c t e d . Hence if /5 i s s m a l l / 1
X^ n^ (n^ + 1) ^2 •"- ^^^ ^ """^^ + '^ P + ^^^''^ " ^'"'^ O + ^) - 2n* + n^ + ?^^n^(3+^')1 + a ( / 3 - l)n^X^+ a(/^ - 1) 2X^ if , 2 ,,2 (4.09)
F o r the c a a e when n = 1 then X"
<f>
(1 -y") 1 + ^ s '^s ^ a ( ^ - 1) Et ( 1 -v") .. and the axial f o r c e / i n . b e c o m e sN 2 X"
E t . E t ,lÜA<^Jli2
If E = E , the c r i t i c a l s t r e s s b e c o m e s s cr = If E R 2 I' 1 + a t t + t (/g - 1) (1 - y ^ ) JT h i s c o r r e s p o n d s to E u l e r ' s f o r m u l a for a stiffened shell buckling a s a s t r u t .
If n > 1, the value of X which m a k e s <{> a m i n i m u m in equation 4.09 i s found from the r e l a t i o n s h i p ^
g n^ (n^ - 1)^
(U -y") + a ( / 9 - Dn'*
(4.10)
When /? and ii a r e both unity, t h i s equation r e d u c e s to the form given by T i m o s h e n k o f r o m a m u c h simplified v e r s i o n of equation 4.09.
Substitution of 4.10 into 4.09 gives
+
2(n^ - 1) i « n^ + 1 a. 2 , n + 1 4rf ( M - y ' ) + c t ( p- l)n^ (7 + y ) n^ + (3 + !^ ) (4.11)10
-T i m o s h e n k o h a s justified the o m i s s i o n of the t e r m s
4n^ - (7 + y)n^ + (3 + y) and by m a k i n g a s i m i l a r a s s u m p t i o n , equation 4.11 r e d u c e s to that of T i m o s h e n k o for the unstiffened s h e l l .
It i s s e e n by i n s p e c t i o n that ^5 i s a m i n i m u m when n = 2 and i s
2
i 2 .2 '
2 0 a(n -y^) + a M / 9 - 1)16 . (4.12)
] •
4 . 3 . Buckling in a long c y l i n d e r under axial load and i n t e r n a l p r e s s u r e T h e a s s u m p t i o n s and a n a l y s i s will be s i m i l a r to Section 4 , 2 , and equation 4.01 r e d u c e s to b e c o m e i d e n t i c a l to equation 4.09, except that in t h i s c a s e an additional p r e s s u r e dependent quantity a p p e a r s on the right hand s i d e . T h i s quantity can be shown to be
</> (n - l)n^ .
1
If n = 1, t h i s t e r m d i s a p p e a r s and the obvious r e s u l t i s obtained, that the i n t e r n a l p r e s s u r e h a s no effect on E u l e r buckling.
F o r n > 1 the value of X which m a k e s </) a m i n i m u m i s found from equations 4.10 with the following additional t e r m in the n u m e r a t o r
4 2 (/> n (n - 1) .
Hence it can be concluded that che effect of i n t e r n a l p r e s s u r e i s to r e d u c e the axial wavelength of the b u c k l e . A s i m i l a r r e s u l t h a s b e e n shown by T h i e l e m a n n .
5. T h e o v e r a l l buckling of a heavily stiffened cylinder (/?>> 1) 3
R e c e n t l y McKenzie h a s obtained a solution for the p r o b l e m of the
h e a v i l y r e i n f o r c e d cylinder subjected to an a x i a l load and i n t e r n a l p r e s s u r e . T h i s solution w a s for the c a s e of a cylinder having p r e m a t u r e buckling of the p a n e l s so that s o m e of the skin t e r m s could be n e g l e c t e d in the e q u i l i b r i u m e q u a t i o n s .
An o r d e r of magnitude a n a l y s i s , using M c K e n z i e ' s shell p a r a m e t e r s , s u g g e s t s that the full d e t e r m i n a n t a l equation 2. 06 can be reduced to the following
1 1
-.x=+ \
1 + i A . 2 ; ^ " 'yX y 2 n• V 2 y ^^
. (1 -f)X^ H-n^ nQ^'^l.
n , 1 + o/ïX"^ + an'* + 2ccn^ X^ + 9i n^-^ X^ 1 2 . (5.01) The solution b e c o m e s 2 ^^ ^ ^ " ' ' ' ! V + ^ T2 +(«^X^ + ctrf + 2 a n " X ^ ) , ^z (5.02) 1 - yEquation 5.02 w a s solved g r a p h i c a l l y using the p r e v i o u s l y a s s u m e d s h e l l p a r a m e t e r s , from Ref. 3.
_2
The m i n i m u m value of ^i i s 1,37 x 10 ', and c o r r e s p o n d s to v a l u e s of X = 4 and n = 7.
The solution of the full d e t e r m i n a n t a l equation 2.06 gave _2
i> = 1,33 X 10 which c o m p a r e s favourably with the a p p r o x i m a t e
solution above. T h e value for <p obtained by McKenzie w a s
2
_2
(p = ,607 X 10 (Exact solution b a s e d on full d e t e r m i n a n t ) 2
-2
o r ^^ - ,702 X 10 (Solution b a s e d on a simplified equation), It would s e e m that the p r i n c i p a l difference between t h i s solution and that offered by McKenzie i s in the definition of the a x i a i f o r c e / i n . N . The value used in t h i s a n a l y s i s , i . e . equation 2.02, w a s
N = E t e + ^^ (e +ve ) , (5.03)
X S S I 1 _ y 2 1 2
and the c o r r e s p o n d i n g value used by McKenzie w a s
N = E t ( e + ïï ^—^) , (5.04)
12
-where H is the distance between the centroid of the stringer cross section and the median plane of the skin. It is seen that equation 5.04 neglects the skin t e r m s in 5.03 but includes an additional curvature
correction t e r m . However, this t e r m vanishes when the stringer material is symmetrically disposed as assumed in Section 2. This
curvature t e r m in McKenzie's analysis gives a large nega-^ive number replacing the square-bracketed t e r m s in equation 5.01 and it can be shown that it is these t e r m s which mainly cause the difference in results between equation 5,02 and the corresponding result of Ref. 3,
6, References
1. Nash, W.A, Recent advances in the buckling of
thin shells,
Applied Mechanics Review, Vol, 13, No, 3, March 1960, pp 161-164.
2. Thielemann, W , F , New developments in the nonlinear
theories of the buckling of thin cylindrical shells.
D.V. L. Internal Report.
3. McKenzie, K.I. The buckling of a pressurised stiffened
cylinder under axial load.
R . A . E . Report STRUCTURES 247, 1959.
4. Timoshenko, S, Theory of Elastic stability.
