M E C H A N I K A T E O R E T Y C Z N A I S T O S O W A N A 2/3, 21 (1983) S T A B I L I T Y P R O B L E M O F A S H A L L O W C O N I C A L S H E L L U N D E R L A T E R A L P R E S S U R E S T E F A N J O N I А К Politechnika Poznań ska F E R D Y N A N D T W A R D O S Z Politechnika Poznań ska 1. Stability Equations Fig. 1 The set of stability equations for a conical shell under external pressure is o f the form: ( 1 ) V2 V2 F-Eh Jctg/9 ( 2 ) D V2 V2 v v + ctg/5 J 82w x 8x2 I 82 F f jl i—dw YT _ ^l l L * " \ \ [dx\x dcpJi 8 x2 \ x2 8<p\+ x 8x J\ x 8x2 82 w ex2 + 2 Bx
+
-i d2 F x2 Sep F\ 82 Fl 1 11 ex2 \x dw x dx+ •
1 82 w x2 8<p I 1 8w 1 8г у / V / \ x2 dq>i x SxScp^JX 1 (9F i d2 F x2 8<Pt x 8xd<p + pl3 tgP d 2 w 1 dw I 82 w 2 8x2 x 8x+ -
8(pZ)
:)
+
+
where: л ' = т * 9г т ^sin/S (see F i g . 1),shell deflection, force function, 8* _2 8_A __ 1_ J * * _ '2 83 2 _83 _ <9x4 x2 8x2 8(p\ x4 8q>\ x dx3 x3 dxdq>\ + 4 d2 _J_ 82 1 8 x4 ; 8<p2 x2 d x2 x3 dx ' Equations (I), and (2), given here in a transformed form, were derived for the conical shell of an arbitrary shape, c.f. [1]. In equation (2) p l4 c o s4 a should be substituted instead of the underlined term for the stability problem of a shallow conical shell (for a shallow shell tga < 0.2). In this paper the solution of the shallow conical shell stability problem is presented, where the equation (2) in a "full" (with under lined term included) and in a "simplified" form are used. It can be concluded from the analysis which of the equations of (1) and (2) are better in use. The analysis of the influence of shell dimensions on the critical load is also presented. iv — F — V2 V2 = 2 . Solution of the Equations. The strain compability equation (1) was solved by Papkowicz — type procedure. The deflection function was taken as (3) w = ( x2 l )2 / + x4 ( x2 I )2 / , cosncp., w h e r e : / / ! — u n k n o w n parameters, The function (3) satisfies the conditions for clamped shell edge at x = 1, i.e.: (4) w = 0; ~ = 0. 8x • When the deflection function (3) is introduced into righthand side of equation (1), this can be written as follows: (5) V2 V2 F = Eh(A0 + A„cosnq>i + / l2/ . c o s 2 n g 91) ,
where A0, A„, A2„ are the functions of x.
The parameters of deflection function and shell dimensions are also included in these functions. The equations arc of the form given i n ref. [3]. The solution of equation (5) we accept in the form of power series (6) F(x tyQ= У FJx) cos m y , . m-i The coefficients in equation (6) can be determined when the set of four differential
S T A B I L I T Y OF C O N I C A L S H E L L 207
equations, obtained by substituting the function (6) into equation (5) and comparing by identity the corresponding terms of the left — and righthand side, is solved. Thus the force function takes the form of
(7) ffaiC'i) =f F0 + F„cosfi<f, + F2ncos2mp,.
F0, Fn, F2„ are functions of x and of deflection function parameters and they are o f a complex structure. When the force function is known, then we can approximately solve the equilibrium equation (2) assuming a deflection function w. Л BubnovGalerkintype procedure is used for solving the equation (2). The " f u l l " and also the "simplified" equations are solved. Orthogonalization of equation (2) requires 2л i (8) , J ' f K(x,<piix^-l)2 dxdq>smP = 0, о 0 In 1 J J K(x,<p1)x5 (x2 -l)2 cos/ir/>i</.\<j<psin/3 = 0, о 0 where: K ( A \ if,) is left — hand side of equation (2). When the conditions ( 8 ) are expanded we obtain a set of two algebraic equations in the vector of deflection functions parameters. For the " f u l l " equation (2) one obtains A,p4,+A2:,+A3i:\+A^\+A5:,ti+A6?;i = o , ( 9 ) B,p* + B2 + B3;,+B4!;2 +B5i;2 2 = o , and for the "simplified" equation (2) there is A\p* + A2:,+A^2
+ A^\ + As<:,'il + Ab:2
= 0, ( 1 0 )
с
2(в
2+ в
3:,+в
Л1;
2+в
5с
2 2) = о .
The next quantities are introduced i n equations (9) and (10): ™ h ' h ' E 'The coefficients A-, and Bt include shell dimensions and parameter n. Their structure is
very complicated. When parameter Ј2 is eliminated from equations (9) we obtain an expresion form which we calculate the pressure U l ) p - K0 Ci+tfx The same operation made on equations (10) gives (12) p* = н , + н 2!:, + н 31:2 + н л \ . Since the directions of the pressure and the deflection (see F i g . 1) i n equation (11), and (12) are opposite one has to put t, ^ 0.
3 . Analysis of the Solution The analysis has been performed for shells with ~ = 100, 200, 300 and with angle x varied (tga was from 0.1 trough 0.5 by step of 0.1). p N lO6 h tQ«0.1 0 -1- - 2 - 3 - 4 - 5 - 6 - 7 - 8 - 9 -10 -11 - 1 2 - 1 3 -14 -15 -16 g Fig. 2 F r o m equations (11) and (12) for each pair o f and tga one obtains an infinite num ber of solutions, because they both include the parameter \Щ У T h e only s i Sn i " ficant solution is the solution which gives a minimum p* value. Fig. 2 is a plot of curves obtained from the solutions of equation (11). They refer to a shell for which — = 100 and t g a = 0.1. Each of the solutions brings two extremal values h of the pressure. The lowest from maximum pressures is the upper critical load, signed />*, the lowest taken from minimum pressures is the lower critical load p*. The lowest pressures were obtained at к = 1. These are p* = 6.6489 • 10~6 and pf = = 1.437410~6 . The line for f2 = 0 is also presented. It represents a symmetrical form of buckling and it is of a first approximation of the solution. The minimum value is 2.859 1 0 6 . Change of dimensions and angle a do not influence the quality changes. The critical load is then obtained from the equation at к = 0. The solutions of equation (12) are of the same form. However the buckling critical loads are much higher (for к = 0) here then buckling loads obtained from equation (11).
S T A B I L I T Y OF C O N I C A L S H E L L 209 " o f Fig. 3 Fig. 3 presents the lines of lower critical load p* versus angle a for three different values °f^ • The lower of the two lines presented by the same type of line is referred to equation (11), the upper line is referred to equation (12). It is worthenoting to show that by using the "full" equilibrium equation (2) one obtains in each case, the lower critical load smaller than the critical load of the "simplified" equation. The decrease is as much as 50% of the pressure obtained from "simplified" equation. The critical load increases rapidly with the increase of angle a but the increase is not so rapid when the — ratio is larger. T o evaluate theoretical results the use is made of the experimental data given in ref. [4]. These data are pointed aut by crosses in F i g . 3, and they refer to shells o f -jr** 200, t g a = 0.1 and of ~ = 300 and a = 30°. h The experimental result for a shallow shell is contained within the solutions of equa tions (11) and (12), but the result for a shell of a = 30° differs very much from the theo retical predictions (when the latter are extrapolated for the angle of 30°). Since the other experimental data are not a vailable the range of valid solutions is not resolvable correctly. One may say with cortainty that the accepted deflection, while using a Papkowicz
type procedure and " f u l l " equilibrium equation, makes the results valid for shells of small angle a; it is also to say that the regime of solutions can be enlarged up to tg ft! 0.3. especially when ~ > 200.
References
1. H . M . M U Ś T A R I, К . Z . G A L I M O V , Nelinejnaja teorija uprugich oboloć ek, Tatknigizdat, K a z a ń , 1957.
2. Spravocnik Proć nost', ustojcivost', kolebanija, t. 3, „Masinostroenie"', Moskwa 1968. 3. F. T W A R D O S Z , Rozważ ania nad nieliniową statecznoś cią dynamiczną powłoki stoż kowej,
Zeszyty Naukowe Politechniki G d a ń s k i e j, Mechanika VI, 43, 1963.
4. 1.1. T R A P E Z I N , Eksperimentalnoje opredelenije rielić in kritić esKicli davlem'j dlja konić eskich oboloć ek, Resć oty na proenost' 6, MaSgiz, Moskwa 1960. Р е з ю м е З А Д А Ч А О Б У С Т О Й Ч И В О С Т И П О Л О Г О Й К О Н И Ч Е С К О Й О Б О Л О Ч К И С О В С Е С Т О Р О Н Н И М Г И Д Р А В Л И Ч Е С К И М Д А В Л Е Н И Е М Р а б о т а с о д е р ж и т с р а в н е н и е р е ш е н и й п р о б л е м ы у с т о й ч и в о с т и п о л о г о й к о н и ч е с к о й о б о л о ч к и с п р и м е н е н и е м у п р о щ е н н о г о и н е у п р о й д е н н о г о у р а в н е н и я р а в н о в е с и я . А н а л и з и р у е т с я в л и я н и е р а з м е р о в о б о л о ч к и н а с т о и м о с т ь к р и т и ч е с к и х д а в л е н и й . С р а в н и в о ю т с я т а к ж е т е о р е т и ч е с к и е р е з у л ь т а т ы с в з я т ы м и с л и т е р а т у р ы э к с п е р и м е н т а л ь н ы м и р е з у л ь т а т а м и . « S t r e s z c z e n i e
Z A G A D N I E N I E S T A T E C Z N O Ś CI M A Ł O W Y N I O S Ł E J P O W Ł O K I S T O Ż K O W EJ P O D
D Z I A Ł A N I E M C I Ś N I E N IA
W pracy dokonano p o r ó w n a n i a rozwią zań zagadnienia statecznoś ci p o w ł o k i s t o ż k o w ej o małej wy n i o s ł o ś ci przy zastosowaniu uproszczonego i nieuproszczonego r ó w n a n i a r ó w n o w a g i . Przeanalizowano
w p ł y w w y m i a r ó w i kształtu p o w ł o k i na w a r t o ś ć o b c i ą ż eń krytycznych. Oceniono r ó w n i e ż p r z y d a t n o ś ć
otrzymanych w y n i k ó w na podstawie danych d o ś w i a d c z a l n y ch wzię tych z literatury.