it-J^.
TECHNISCHE HOGBSCHOO VUEGTUIGBOUWKUNDE Ka&aalstMOt 10 - BfiLFT REPORT No. 612 7 JUNI 1952
THE COLLEGE OF A E R O N A U T I C S
CRANFIELD
THE THEORY OF GENERAL INSTABILITY OF
CYLINDRICAL SHELLS
by
J. R. M. RADOK, B.A. (Melbourne)
This Report must not be reproduced without the
^ïCHNTSCME HOGESCHOOL VLIEGTUIGBOUWKUNDE Kanaalstroat 10 — BELFT
2 7 JUN11952
R e p o r t Ho» 6^ Jvjies 1952 T H E G 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 DThe Theory of General Instability of Cylindrical Shells
by
-J.R.M. Radokf B.A., (Melbourne)» oOo
SUMMARY
Using a new approach to the theoretical study of thin-walled cylinders with discrete reinforcing members developed in this paper, the problem of general instability of such structures is solved with more than usual
generality. The principal stages are indicated which lead to the characteristic equation of the general problem in the form of a determinant of order three times the number of reinforcing members, i.e. stringers or rings. The less general problem of distributed stringers and discrete rings is solved completely and it is shown that for the case of one ring at the middle of the cylinder, buckling with axial symmetry, the characteristic equation can be reduced to a very simple closed form.
The method of solution, developed below, must not only be judged in its relation to the problem under
consideration. It will be found to be fundamental to the theory of plates and shells in the sense that most problems having an exact solution for the case of the homogeneous structure, can now likewise be solved in the presence of reinforcing members.
BHP
^ r . Radok is a member of the staff of the Structures Section of the Aeronautical Research Laboratories, Department of Supply, Australia, and is at present studying at the College. Acknowledgement is paid to A.R.L. for their agreement to publish this as a College Report.
LIST OF CONTENTS
Page
Notation 1 Introduction 3 1.1 Statement of the problem k
k
1.3 Assumptions 3 1.2 General discussion of the method of
solution
2. Deduction of the Basic Solutions 6 2»1 The general case of a circular cylinder 6
2.2 The simplified theory of a circular
cylinder with distributed stringers 9 2.3 Discussion of the properties of the ^^
basic solutions
3• The Characteristic Equation for the Simplified 12 Theory
3.1 Circular cylinder with several rings 12 3.2 Cylinder with one ring at centre. .^
Axially symmetrical buckling
k» Conclusions 15
5» Acknowledgement 15
References 16
1 -NOTATION a « b 9 c mn' nmi* mn Radius of cylinder Arbitrary coefficients d s f , g
n' n* *=n Arbitrary coefficients in basic solution
^ x ^ k , k
m* n
Skin thickness
Effective skin thickness for the case of distributed stringers. See (2.1.U) m, n u, V, w X, e ^i Length of cylinder Summation variables
Longitudinal, circumferential, radial displacements. Cylindrical coordinates Position of ring ^R A' = — Position of stringer
^R Area of cross section of ring E Young's modulus •R ^ Second moments of area of ring section
distributed stringers per unit length of circumference
a =
P =
12a
2 -\ _ mTta K» ^p See (3.2.1+)
. = I
P o i s s o n ' s r a t i o 2' N (1 - v^) ^ = " Eh^Ov-TT Magnitude of d i s c o n t i n u o u s change of f u n c t i o n a t x=x.
A—A. 1
A;^n ^®® (2.2.7)
K = k^ + v^d - k^)
N Direct stress resultant in x direction X
0(-^) Of order 1/A^
Q Radial shear stress resultant
5.. Kronecker'a delta
INTRODUCTION
The use of structures with thin skins and reinforcing stringers and ribs is due to two factors, the desire to save weight and the desire to save material. While the latter aim has o..*ly arisen during the last fov/ years, causing their use
in all types of construction, the former has been the guiding principle of aeronautical engineering ever since man made his first flight in a machine heavier than air.
Great difficulties have been experienced in subjecting such structures to an exact theoretical analysis. A
considerable amount of work has been done on the stability problems of discretely reinforced rectangular plates, using energy methods. This work is described in some detail in Ref.1, and one of the principal exponents of this approach in England is H.L. Cox who has published a nimiber of papers on the subject (e.g. Ref.6). But this method leads always to an infinite system of simultaneous linear equations, and very often a great deal of ingenuity is required to reduce these to a closed expression.
To the author's knowledge, all attempts to solve stability and other problems in reinforced shell theory have been based on initial, very approximate, assiomptions, and even then have led to such involved calculations that it is often very difficult, even for the specialist, to follow the
reasoning and to grasp the physical meaning of the final-results (e.g. Ref. 2-k)*
The method of this report seems to avoid many of these disadvantages, and in some sense will be seen to be equivalent to the energy method, since results, obtained by it, probably could also be deduced in a roimd about manner by that method. Its strength lies with the fact that it has a simple physical interpretation. Its application is by no means restricted to the problem under consideration, nor, as a matter of fact, to problems of stability. The present paper deals only with
the application of the method to the problem of general
instability of shells. It is hoped that it v/ill justify the claims made here, and it is the intention of the author to support these statements in the near future by a more
comprehensive report, giving the solution of other important problems in the theory of plates and shells.
1.1 statement of the Problem
The problem of elastic instability is one of the most important of the theory of elasticity, and its nature is
explained in most relevant text books. For this reason, no attempt will be made here to describe it in detail. As in the later work, emphasis will be on the part played by the
reinforcing members.
Consider a stressed skin structure with ribs and
stringers firmly attached to .it in such a way that the elastic axes of the reinforcements lie in the middle plane of the skin. While this assumption is not essential, it will be made here
for the sake of simplicity, the modifications arising from its removal being obvious. It is known from the theory of
structures that, if such a structure deforms, certain of the
stress-resultants undergo discontinuous changes at the reinforcing members. From the point of view of the skin, these
discontinuous changes manifest themselves in local pressures and shears. On the other hand, from the point of viev/ of the stringer, they can be conceived as loading on the member causing its deformation. Vi/hile the reinforcing members are of finite width in practice, it is most convenient in theoretical work to Idealize them into lines. As a result of this step, the pressures and shears, mentioned above, become infinite, since the line of contact is devoid of area.
After these remarks, the problem under consideration may be formulated as follows: "To find the characteristic equation for the determination of critical loads of a cylinder with thin skln^ subject to localized infinite pressures and shears» which depend on the deformation of the shell and the stiffness of the reinforcing members.
1.2 General Discussion of the Method of Solution
A certain amount of hesitation may be shown at first at the idea of introducing infinite pressures. On the other hand, in many other branches of mathematical physics such localized infinities have been used for many years In the
form of line sources. The method of solution, to be used here, thus will be seen to be based on a concept which, to the
author's knowledge, has been little used in the theory of structures although it occurs in a disguised manner, for example, in Ref.5 in the solution of the problem of a plate loaded over a portion of its surface. But it has been arrived at in an arbitrary manner and the solution used there does
not by itself satisfy the differential equation.
What may be termed the source solution of the theory of thin plates, will here be shown to fully satisfy the
differential equation \inder loads, peculiar to the presence of discrete reinforcing members. Due to the indeterminateness of the pressures, as in the case of sources, for example, in aerodynamic theory, the source or basic solution will contain one arbitrary constant, or sets of arbitrary constants, v/hich therefore can be used to satisfy the "internal" boundary
conditions at the stringers or rings.
5
-In the case of stability problems these conditions lead to
a homogeneous system of linear equations in the above mentioned constants, and the condition for the existence of non-zero
values of these constants present:; the required characteristic equation.
Finally, one short remark will be made with regard to the mathematical character of these source solutions. It will ba shown later on, that these solutions are in actual fact ^
Cümbinations of the complementary functions and the particular integrals of the non-homogeneous differential equations, which follow from the ordinary stability equations after introduction of the earlier stated type of loading. The solutions, v/hich are in the form of Fourier series, contain a "singular" part in that in the present problem their first or third derivatives with respect to the coordinate at right angles to the stringer or rib are discontinuous. The "singular" part, which is of the form used by S. Timoshenko in Ref.5 is easily separated from
the basic solution, and the remaining part can then be shovm j to be "regular". It is for this reason that the term basic
solution has been preferred to that of source solution, because the latter term is normally used for solutions which are purely singular»
1«3 Assumptions
Some of the assumptions to be stated here have been referred to earlier, others will be introduced during the actual analysis. Nevertheless it will be worthv/hlle to state them here in full. They
are;-Thin shell theory is applicable.
The elastic axes of stringers and ribs lie in the middle plane of the skin.
Stringers and ribs have no torsional stiffness and may be idealized into lines.
The free edges are simply supported.
When dealing with the simplified shell theory, based on distributed stringers, the follovang additional assumptions will be required:
The stringers are so closely spaced that each
circumferential wave in the buckled state contains several stringers.
The stringers add effective area to the skin only as far as the longitudinal direct stress flexural stiffness and radial shear stress are concerned. The skin lacks torsional, flexural and shear
stiffness and the circiimferential displacements are small as compared with those in radial direction. The rings have no stiffness for deformation out of
their plane.
(The last four conditions are discussed in detail in Ref.3)
/ As far A.I A. 2 A.3 A.U A.5 A. 6 A. 7 A. 8
TECre^nSCHE flCXaKCHOOL VLIEGTUIGBOUWKUNDE Kosaaalstiacrt 10 - MUI 6
-As far as A.1+ is concerned, this condition will be seen to be automatically satisfied. However, it will be suggested here that the edge conditions may be varied in a manner which is very close to practical conditions, and v/hich it would be very difficiilt to achieve without the help of the basic solutions. In fact, by choosing the length of the theoretical cylinder somewhat larger and by Introducing very stiff rings in the vicinity of one or both ends of the theoretical cylinder in such a v/ay that either the distance between the stiff rings or between one simply supported end and the stiff ring is equal to the length of the actual
cylinder, one could solve the problem for the case of clamped ends. The choice of the stiffness of the artificially
Introduced rings would thus give the means of varying the degree of clamping. Hov/ever, it is not proposed to
investigate this point further in this report, and A.ii- will hold throughout the subsequent work.
DEDUCTION- OF THE BASIC SOLUTIONS
In order to save space, the relevant differential equations and boundary conditions v/ill be stated here v/ithout deduction. In the case of the general problem they may be found in section 8i+ of Ref.1, v/hile for the case of the simplified problem they are essentially contained in Ref.3» As far as possible, the notation used agrees with that of Ref.1.
2»1 The General Case of a Circular Cylinder
The most general stability equations for the case of a thin walled homogeneous cylinder in compression are:
P 2 ? è'u "1 + V d V V dw . 1 - V 3~u p^ . 2 •*• 2 aaxde " a ax 2 2..2 " ^ + V d u a(l - v) 8 V d V ÖW dxde •*• 2 - 2 "^ „,„2 " aae * ÖX ao0 r, + T + a —7,— + a(1 - v) — T ! - a0 — r = 0 \2AA)
aae" aae^ byC'b<d ÖX •' ax
^ a w au av w - a0 —- r + V r— + —rr - — ^ 2 dx aae a ax ia v /„ \ a V 3 aSy d\r rt a Y; \ _. - a' r + (2 - v)a — ^ — + a^ —r- + + 2a —-—- r ] = 0
'aae^ b7i bQ ax^ aae^ by. h^
7
-These equations represent the conditions of equilibrium in the longitudinal, circumferential and radial directions
respectively at each point of the shell. As mentioned in the Introduction, these equations will be solved for the case, when infinite pressures and shears act either along a circle at a station x. (case of a ring) or along a generator,
specified by e. (case of a stringer). This loading condition will be produced by introducing on the right hand sides of these equations the follov/ing sets of loading functions:
7 I , . g . s i n ^ k cos 2 ^ , ^ , . £2. c ™ / ne d . c o s ^ sin-n i 2 mTOCi s m ntTOC n=0 m=1 . mTOC. f . s m - r - s m sm—— (.2.1.2) n i 2 -.-^ •:: '
for the case of a ring, f$r* V / • i . . . — VXKX. n e . n=0 m=1 ^ . cos-r— sln-TT^ s m r r -m i n0 > 2 , •'- / - , imoc , no -— d . sm—TT" k cos-r-n=0 m=1 mi n n0. •r~- V" n o . „ .C. / . . f^ s m - — s m - ^ s m ~ ( 2 . 1 . 3 ) n=0 m=1 f o r t h e case of a s t r i n g e r , where j TOC. X % cos — + J^ k
mToc. cos-; OTt
cos e. X m = oos 7^: n0. 1 - m nac k = cos-n 1-n 2 m >y 2 n >y 2 (2.1.4)
have been chosen i n such a way t h a t the cosine and s i n s e r i e s used i n (2,1 2) and (2 1 3) are completely equivalent (See also section 2.3)
I t i s e a s i l y v e r i f i e d t h a t t h e s e l o a d i n g f u n c t i o n s h a v e t h e r e q u i r e d p r o p e r t i e s , s i n c e t h e r e l e v a n t p a r t s o f t h e F o u r i e r s e r i e s c o n v e r g e t o z e r o i n a c o n v e n t i o n a l (Ate]) S e n s e f o r a l l v a l u e s o f x o r e i n t h e r a n g e 0 i x ^ f o r 0 < 6 .< 27i e x c e p t when x = x . , 6 = 6 . , t h e s e r i e s b e c o m i n g i n f i n i t e t h e r e . I n o r d e r t o s o l v e t h e d i f f e r e n t i a l e q u a t i o n s ( 2 . 1 . 1 ) w i t h t h e l o a d i n g s y s t e m s ( 2 . 1 . 2 ) o r ( 2 . 1 . 3 ) , s u b s t i t u t e f o r t h e d i s p l a c e m e n t s t h e f o l l o w i n g s e r i e s : / u = . . .
- 8
u=>
I
a V w n=0 ID:,T
n=0 = / -n=0 in=1 m=1 (i.... ra=1 mn n n inn s m ne o OS s i n ne no cos irtKX s m ( 2 . 1 . 5 ) s m nTcx €which a f t e r some i n t e r m e d i a t e c a l c u l a t i o n s reduce t h e d i f f e r e n t i a l e q u a t i o n s t o t h e f o l l o w i n g system of e q u a t i o n s f o r t h e d e t e r m i n a t i o n of t h e a r b i t r a r y c o n s t a n t s of ( 2 , 1 . 5 ) \ 2 ^ 1-v 2 1 + V \ — 2 — ^ ^ 1 + V y. 1-V V 2 . 2 / . \ v 2 2 \ 2 2 liA - ^ > + ^ t S a ( l - v ) A S a+x'^- X^jZl vX [ii1 + a |i +(2 - v)>i j I V \ (\^ 2x t mn cm mn =: A (X^)
-„. 1 /
mn mn ran Sni d . n i f . n i k n •amu siiï—— .^^•^t \ne.
d . k n i n n 0 . f . s i n - ~ nx 2for the case of a ring
for the case of a stringer
(2.1.6)
The determinant of this system of equations vanishes for critical loads, i.e. values of 0, which correspond to the buckling of the homogeneous cylinder, since
A (>,^) = 0
is the characteristic equation of the homogeneous problem» Hence (2.1.6) will normally have finite solutions for the a^^^, b , c . The solutions (2.1.5) with the appropriate values
mn' mn \ ^/
of the constant coefficients are the basic solutions of this problem.
9 -2 . -2 The S i m p l i f i e d T h e o r y o f a C i r c u l a r C y l i n d e r w i t h D i s t r i b u t e d S t r i n g e r s . U s i n g A . 5 t o A . 7 o f s e c t i o n 1.3» t h e d i f f e r e n t i a l e q u a t i o n s t a k e i n t h i s c a s e t h e f o l l o w i n g f o r m : 2 2 V ^ ^ 1 + V a v . 2 •*• 2a dxdQ ex 1 - V a u V dw -2 , ^ -2 " a ax ~ a ae 1 + V a u (i - v) a V aw _a axao dx a a e aae^ = 0 (2.2.1) n I ax2 ^^ ^^0 ^ öx^
The two s t r e s s r e s u l t a n t s , a f f e c t e d by A,6 and used i n deducing ( 2 . 2 . 1 ) a r e now g i v e n by N. . 2 f- ax a ^aehS i^ au V fèv "^ T^ OU V / O V \ f ^ „ _ o w
a \
'J * x x ^ 3 1 - v ex (2.2.2) where 2 , ^ ^ x ^ ^ - ^ ) K = k + v'^(l - k ) , p = -X 2v a h (2.2.3) The e q t i a t i o n s ( 2 . 2 . 1 ) a r e the c o n d i t i o n s of e q u i l i b r i v m f a r an o r t h o t r o p i c s h e l l . Using A. 8, the l o a d i n g system f a r t h e case of one r i n g becomes0, I Z a.
mTOC. n=0 irusi 'ni s i r cosne sin"t
mTix. y a. . sxn "•' sxn no sxn—— n'=ol5;=1 ^ C- <^ (2.2.4)and p r o c e e d i n g i n t h e saLie manner a s i n t h e g e n e r a l case one a r r i v e s a t t h e system of e q u a t i o n s T^,2 1-v 2 K > + - y - n 1+v 2 n X V X 1+v \ -TT" n A 1 - v v 2 2 n -V \ n +pX^-0>-2 r X a mn b ^ nn c^ nn
j-— 1
0 3 . mTOC. d . sxn i n i - ^ mTOC. f . s i r n i < ... (2.2.5) / Again .10
-_, Again, the vanishing of the determinant of this system gives the characteristic equation for the homogeneous orthotropic cylinder. Assuming the critical loads of the cylinder with a ring to be different from those of the homogeneous shell, one finds from (2.2.5)
, nroc, ^ '-^An _ 1-v ,' 2 .21 v l + f . - r r - n -v> f >• ( . mTtx. nrnx. ^^'^ ^ ^ • > n . n-KX. m x . ^^^ ^
^f /l^(K>^+n^)+(K-v)xVil
>.nn
where (2.2.6) A , ^ =Rp>2- j 2 j | t o ( A K)^)+ n^A^^^v)] + iliix^CK - v2)j>2 ( . . 2 . 7 )Thus, s i m i l a r l y to (2.1.5) one has i n t h i s case the basic s o l u t i o n s
n;=0 m=1 ' ^ni '^d ^ ^ni ' V ' ^ ^ " ^ T ^^"^ "" ® ° ° ^ T '
V = Z . .-/- ! d . p , + f . p_j s i n 3. cos n 6 s i : ^ (2.2.8)
n=0 m=1 • " ^ '^ ^ ^ C'
w = ./.. Z . l d Y + f Y^ i s i n — ^ s i r r ^ S gin n 6 ,=0m=:1 - n i ' d ^ n i ' f
^-These solutions will be used in section 3« to obtain the characteristic equation for the critical loads. Hov/ever, the \mderstanding of the work of that section v/ill be greatly helped by the short discussion of the properties of the basic
solutions, which is given in the next section.
11
-2.3 Discussion of the Properties of the Basic Solutions
It is obvious from the form of the solutions, deduced above, that they satisfy the assTimption of simply supported edges. So there remains only to shov/ that these solutions are complete, that their relevant derivatives are
discontinuous and that they are otherwise "regular" It is seen from (2.2.6) and (2.2.7) that
^nn =
Q ( V A ^ ), c ^ = OdA^) (2.3.1)
whilePA
Thus the first and third derivatives of v and w with respect
1 X -éT^-Z 0 !i X i X t o X a r e d i s c o n t i n u o u s a t x ^ , s i n c e A = m-Ks. and C".7
1
1 ._ X mTOC » ( 2 . 3 « 3 ) ~ sxn-' ,' cos—r- = for ^ -^ ^/ n -^ -£. n=1 ' ^ *- 1 / . X\ ^. /?If A denotes the magnitude of the discontinuities, it is seen that
"(ax/ - 1-v V f * - \^ 3' - p ^ni .^3 (2.3.^; x=x. ^- ex x=x. ^ 2u
X X
The expressions (2.3o2) show, that, as far as the differential equations (2.2o1) are concerned, the remainders of the basic solutions are regular in x. Similar arguments can be applied to the other derivatives of u, v, v/ which occur in the differential equations, and it can be shown in this manner that the discontinuities, indicated above, are the only ones of sufficiently low order to be of interest here. Further, since the basic solutions contain regular as well as singular parts they will be complete, and by a suitable uniqueness theorem they are the only solutions, as they satisfy the boundary conditions. This latter point will become yet more obvious in the next section dealing with the deduction of the characteristic equation and the "internal" boundary conditions.
12 -3 . T H E C H A R A C T E R I S T I C E Q U A T I O N FOR T H E S I M P L I F I E D T H E O R Y It h a s b e e n s h o w n i n s e c t i o n 2.3 that t h e basic s o l u t i o n s ( 2 . 2 . 8 ) p o s s e s s a n a l y t i c a l p r o p e r t i e s , w h i c h w i l l b e r e q u i r e d i n t h e p r o c e s s o f s a t i s f y i n g t h e " i n t e r n a l " b o u n d a r y c o n d i t i o n s . A s i n d i c a t e d i n t h e I n t r o d u c t i o n , these c o n d i t i o n s l e a d t o a h o m o g e n e o u s s y s t e m o f equations f o r t h e d e t e r m i n a t i o n o f t h e d_. a n d f .. Since there are only two sets o f such
n i n i c o n s t a n t s f o r e a c h r i n g , it w i l l b e e x p e c t e d that t h e c h a r a c t e r i s t i c d e t e r m i n a n t i n t h e p r e s e n t case w i l l b e o f o r d e r 2 g , w h e r e g is t h e n u m b e r o f r i n g s . 3.1 C i r c u l a r C y l i n d e r w i t h S e v e r a l Rings T h e r e l e v a n t b o \ m d a r y c o n d i t i o n s a t t h e r i n g s a r e d i s c u s s e d i n d e t a i l i n R e f s . 3 a n d k» T h e y a r e :
f^i
, a^3^-06
few i-^ ^] -
,, I'avKli^
u,- ^i j = -
_ A( ^X±\T1J d^^^^'^'^^
x=oc. '" x=x. (i, I = 1, 2, ..., g)
r_, a^ (dw ] ,, a 1'av il h A<^3U dv)
x=x xaX w h e r e t h e left h a n d s i d e s a r e t h e elastic f o r c e s i n the r i n g
w h i c h h a v e to b e i n e q u i l i b r i u m w i t h t h e e x t e r n a l l o a d i n g , f r o m t h e p o i n t o f viev/ o f t h e r i n g , r e p r e s e n t e d b y t h e t e r m s o n t h e right h a n d s i d e s . I f there i s m o r e t h a n o n e r i n g , t h e a p p r o p r i a t e e x p r e s s i o n s f o r the d i s p l a c e m e n t s m a y b e o b t a i n e d b y a d d i n g s e v e r a l o f t h e b a s i c s o l u t i o n s ( 2 . 2 . 8 ) . S u b s t i t u t i n g t h e c o m p o s i t e s o l u t i o n s , t h u s o b t a i n e d into (3.1.1) a n d u s i n g
(2.3.U) one finds t h e f o l l o w i n g d e t e r m i n a n t a l c o n d i t i o n f o r the e x i s t e n c e o f n o n - z e r o v a l u e s o f t h e d . a n d f „ j :
^nl'
4 vO
^ 1 mTOC. mTOC. ^ mToc. n m x ,
j -^ ^ i ("Pf^f )J ^i"-T^ ^i^V" I ^ ^ ^"Pd-^ ^d^i ^ ^ ^ ^ ^ " ^
L [l^^ (n3 Y, . n\) . .
-
^ 5^^ . T{ I^, ( n ^ r^\) ^
2(l-v^)
2 •} mrac, nmx. „ •] mTix, mrac, + A^; (n Pf+nYPj s i r ^ s i n r ^ | + A '^^^ ^^^T^a^J « i " ^ ^ ^ ^
where i , j , I , k = 1, 2, . . . , g,
= 0 (3.1.2)
13
-By (2.2.6) the c o e f f i c i e n t s under the siom signs above are
4 3o n^ ri-vf„ 4 2 / 2 ^NI v2j'v/' 2 ,N /1+V 2N M ^ . n X^ 1 - v r ^ V 2 2 1 nP^ + Yf = — ~ ' ï^ -^ - V n J n \ . . \ = H L r , 2 i ^ 2 ( ^ _ ^ ) ^ l | V ( , _ ^ 2 ) l ^ ^2^(^^2^ 1 ^ 2 ) (3^2_^^)^(^,^2^i] *^Jln "An (3.1.3) where ^i^ is given by (2.2«7)
Equation (3«1.2) is the final expression for the determination of the critical loads. Inspection of the
coefficients (3«1«3)s remembering that A^ = 0{\°) shov/s that —2
the last two are of order X while the first two are of order ^'" . Depending on the magnitude of n, it will normally be
sufficient to retain only a few terms of these series. However, the numerical application of this equation should be the
object of a separate investigation, and it will be satisfactory at present, that the series involved converge. In the
following section, one special case will be studied in greater detail and it will be shov/n that for axially symmetrical
buckling a closed expression can be obtained.
3.2 Cylinder with one ring at the centre. Axially symmetric buckling
In this case (3o1.2) reduces to:
-I %"
+ / 2e''p ra=1
I (ljj(nS-^+n3p^)+A^(np^4Yp|sin^iTO ; Y„. [l^(nV^+n3p^)+A^(nP^HY^)!'| 2 ra=1
. 2 . s m roK
l_ (I^(n-^Yf.+n P^)+A^(n P^+«r^)j sin m . - h% +^„. [l^(n-'Y,+n p j + ra=1 2 ' ^/^ 2\ m=1
2(l-v'')
fAi(n p,+nY-,)j sin rm
= 0
2 !
11 -T'-r
Thls equation is still rather complicated and because of the complexity of the coefficients (3*1«3) no attempt will be made to sum the series here. However, in the case of axially
sjrmmetrical buckling, i.e. when n = 0, a closed expression is easily obtained, as v/ill now be shown; but unfortunately this case is of little practical importance.
For n = 0, (3.2.1) becomes, using (2.2.3)., (2.2.6) and (2.2.7)
4 -.•. n , . 2 DOT
•.J^ r~- n 2, 4 i ü sxn
rr-X \ f , . 2 rnTt a h.% , , „ •::— 2
- - + ^.> .td r s i n " ^ = — " r, + KK ; - — , f r , = 0
Slamming the s e r i e s , using the r e l a t i o n
4- ~ 2= h *^"f^ (3.2.2)
° (2m + 1)2 - K^ ^^ 2
which holds for all complex values of ^ , except for ^ = ±''» ± 3 » etc. one finds finally
J £ ^ 3 ^ / 2 ^ 2 / I 2 2 , %l f 2 2 / / 2 2 , % f f2 2 41 (3.2.3) where 4) L_^2 /k^a h 0 N^a ƒ . _ •
^= /"ir " >/"èr"' V=2P=
ÉT'h^h
- K / + 1 ,+3, ... (3.2.
Since N i s p o s i t i v e for compression and a l l o t h e r
q u a n t i t i e s are p o s i t i v e , a l l t h e r o o t s w i l l be r e a l provided 2 2 f > H , i . e . 9 2 N-a k h .
- V > -f-'-h (3.2.5)
4E I ^ X / 4 . . .15
-CONCLUSI ONS
Using special solutions of the stability equations the problem of general instability of circular cylinders can be solved exactly and, in the general case, when there are only rings or stringers present, the characteristic equation for the critical loads will be a determinant of order three times the number of reinforcing members.
On the basis of a simplified shell theory, requiring assumptions additional to those of the theory of thin shells, the conrplete solution of the problem of a cylinder with
distributed stringers and discrete rings has been obtained. In this case the order of the determinantal equation for the critical loads is twice the number of rings. Special
consideration has been given to the case of a cylinder
with one ring at the centre and a closed expression has been deduced for the case of axially symmetrical buckling.
The method of solution of problems of thin walled structures with stringers and rings, developed in this
report, is equally applicable to problems, other than those of elastic stability. Its advantage lies with the fact that it does not lead to systems of equations with
infinitely many unknowns. Since it is based on types of solutions of the equilibrium or stability equations which appear to be inherent to the case of plates or shells, reinforced by discrete members, it may well be said that these solutions may be conceived as a suggestion towards the use of such reinforcements.
AGKNOYlOiiEDGEMENT
The author is indebted to Professor W.S. Hemp
for a number of lively discussions which greatly assisted the development of the method of solution presented above.
16
-LIST OF REFERENCES
No.
Autho r Title» etc.S. Timoshenko
Theory of e l a s t i c s t a b i l i t y
McGraw H i l l Book Co„ 1936
2.
A. van der Neut The general instability of stiffened cylindrical shells.Nationaal Luchtvaartlaboratoriian Report S.31U 19U6.
W.J. Goodey The stresses in a circular fuselage. R.Ae.S. Journal Nov. 19U6 pp 831-871
u.
S. Butler A theoretical study of cylindrical shell structures 0Thesis - College of Aeronautics, May 1950 (Unpublished).
S. Timoshenko Theory of plates and shells McGraw Hill Book Co. 19^+0.
