Buckling of Spherical Shells under Concentrated Load

Transcription

1 TECHNISCHE MECHANIK, Band 18,Hef 2, (1998), Manuskripleingang: 16. September 1997 Buckling of Spherical Shells under Concentrated Load and Internal Pressure S.M.Bauer, P.E.Tovstik Stability of a complete spherical shell under concentrated load was first considered by D.Bushnell (1967) by means of a finite diflerence method. In the present paper the effect of the internal pressure on the critical value of a concentrated load is studied by means of a combination of asymptotic and numerical methods. 1 The Equation of the Axisyrumetric Deformation We start by examining the pre buckling axisymmetric stress-strain state of a spherical shell. Let the shell with radius R be subjected to the concentrained load P applied at two poles 90 = 0, 60 = 7r, where 00 is the angle between the shell normal and the axis of symmetry (see Figure 1), and to the internal pressure Pn. Figure 1. Sperical Shell We use the dimensionless system of equations (Tovstik, 1996). (bv) = bp(1 +,uel)(1 + #82) (3056 u(bu) = 52 + ul/(u ( Vsin 0) bp( )(1 + M52) sinö u(b62) : (1 + p61) C059 cos 60 (1) u(bm1) : b(1 + 1L 1)(Usin9 Vcosd) + umg c056,at : (1 + u51)(m1+ u(1- VH2» where 2 d() h" B sin6 b I 2 4 z : = ' ' - : (l " also 12(1 1/2)R2 b R b e1 : z/5-2 + [1(1 1/2)(Uc050 + Vsind) 6/ M = lt + = 2 M 2 I M) I éi 1+H51 Here V and U are the projections of the stress resultants onto the axial and onto the normal to its directions, correspondingly, T1:Uc056+Vsin6 leusinq Vcosd 135

2 where f) is the angle between the shell normal and the axis of symmetry after deformation, um and usg are the tensile deformations of the meridian and of the parallel, Ml and M2 are the stress couples, h is the shell thickness, 1/ is the Poisson ratio, 11 > 0 is a small parameter, B is the distance between the point in the neutral surface and the axis of symmetry. Dimensionless variables in equations (1) are related to the corresponding dimensional variables (marked by bars) as Ell/ 2 U z UEhu V : VEh/L M7; : MiEhRM p.n : p E Firstly, we study the dependence of axisymmetric deformation of the shell on the applied load P. It is clear that the large deformations occur at the neigbourhood of the points where the load is applied. We study only the neigbourhood of the point 60 : 0. From the first of equations (1) we get in that neigbourhood. C p60 V (93) where P : 27TREh/L2C (2) 0 System (1) contains the small parameter u, at the derivatives. Therefore we can use asymptotic method. We rescale the variables 2 C2117!) or 7rEh3c %=u 9:uw U:' <3 6 By taking into consideration formulas (2) and and by expanding the trigonomical functions, system (1) is transformed with error of the order [12 into two nonlinear equations of the second order. / S Z li/l) (3 (w) 2ää W) The study of a system similar to equations (4) is reported in Shilkrut (1974). 2 Integration of System (4) For an arbitrary load parameter c < O we seek a solution of system (4) conditions which satisfies the following 1) limited at the point where the concentrated load is applied, uzw:0 when 5:0 (5) ii) which turns into the membrane solution when we go away from the domain of large deflections wag u >E+I_ 5 as g aoo (6) g 2 In order to get such a solution, we construct its asymptotic expansion in the neighbourhood of the point where the load is applied f : 0 and in a domain which is far away. Taking into account conditions (5) and (6) we get as f > O wd= ;ba+0&+0@ma> Mv=0a+owm e (n 136

3 andaséaoo, if lpl < 4 then 1W) = 6 + Cam(\/LzeÄ +ic/4) (1+ 0(54)) u<+> 3 E + I; _ 63%(«51A26Ag+16'4) (1+ O( _1)) (8) and if (p) > 4 then w<+> : + (CajzeÄ16+ Cliüexgg) +O( 1)) u é+ 2 (Ca/ms +C4\/z/\äe >(1+O({ )) (+) E 2 - Mg 1 A25 1 (8') where 2 Xifiäi f G l 9?{ \}<0 Both expansions depend on two unknown constants Ci, i : 1, 2, 3, 4. Using the condition that these two solutions (Lil ME) and w(+)(f), u( )( ) and u(+)({) (and their derivatives) should be equal at some intermediate point f : 5*, the constants may be evaluated numerically. After determination of the constants Ci, using expansions (7) and (8) or (8 ) we can get the general solution of system (4). as boundary conditions, The plot of the deflection wo at the pole vs the load parameter c for some values of internal pressure p is shown in Figure 2. Figure 2. The Deflection at the Pole vs the Load Parameter 3 Equations of Axisymmetric State Bifurcation We search the adjacent nonaxisymmetric equilibrium mode with m waves in the circurmferential direction in the form w( w) = 10(6) C03lm90l In order to construct the adjacent equillibrium mode we use the Donnell system of equations for shallow shells. After separation of variables and introduction of the dimensionless variables this system may be written as AAw Atw Aké :0 AAQ+Akw20 In equations (9) the quantities 10(6) and <I>(f) are the additional displacement and the stress function, and the differential operators A, Ak and A1, are given by formulas Aw I {"2 [((fw') )] Akw 2: 6 2 [ (fk211)')' m2k'1'm] (10) Alu) : { 2[{({t1w')' ] 137

4 kl = IP' k2 Z lp/{f t1 Z 11/5 152 = U' (11) where (1:12) are the curvatures of the neutral surface after deformation, t, are the main prebuckling stresses. 4 Integration of System (9) Taking into account relations (11) and expansions (7) we can construct asymptotic expansions of four linearly independent solutions of system (9) in the neighbourhood of the point where the load is applied 6 : O and in the domain which is far away (é : oo). These solutions should be limited at the point where the concentrated load is applied, {wl h <I>( )} > 0 as f > O. wgfl : m+2 + O (3H4) wéfl : O ( m+4) (if) : 0 (ngr4) (by) : Em+2 + O (5m+4) wg?) = Em + O [{m+410g{) (Ii?) : a +210gf+0 ( m+4 log g) p z arm 10g5+0 (5m+4 log g) eff) 2 gm logf + O ( m+410gg) where a, = )/[16(m + 1)]. T0 construct the expansions as E > oo we can take 2/) 2 g, u 2 c/f + pf/z. Then in equations (9) w" m w _ p l c n w. m 2w Au) = Akw : w" + P Alt/w EAw + 2 (w ) f 52 and the solutions of system (9) satisfying the conditions of damping {w(+), <I>(+)} > the form 0 as 5 > 00, have mg) : ÜNZeÄg} (PET) : ÜMÄgZeÄä'} wg : 3{Ze \ } (II?) : J{)\2Z \ } Z_ i +0 wé+l : {mm + O ( 7m72) (I);+l : O ( 7m72) fl win-l) : O (fivaz) (1)5179 : fvm + O (67m4) here /\ is the same as in equations The axisymmetric state bifurcation takes place for load parameter c, if there exist eight nonzero constants Cl ), Cl ), i : 1, 2, 3, 4, such that at some intermediate point f : 5* the functions 4 4 w( ) :ZC;7)w57) and 21,1(+):ZC +)/w +) i=1 i=1 4 4 <I>H and <I><+>}:ZC<+><I><+) i=1 1'21 (and their three derivatives) are equal. The smallest (by the wave number m) load parameter c sponds to the critical value of concentrated load. core 5 Results In Table the values of the load parameter c are given for the different values of the internal pressure p and of the wave numbers m.

5 CSCHHÄGQNS p:0 p:1 p:3 p: Table 1. Load Parameter c as Function of Internal Pressure p and Wave Number m For the case when the internal pressure is equal to zero the results agree well with those obtained by Bushnell (1967). The smallest parameter of load c coresponds to the three waves mode. It is natural that the value of critical concentrated load increases with the internal pressure. When p < 3.56 the smallest load parameter [0 coresponds to the three waves mode, when p = 3.56 the load parameter is the same for the three waves mode and for the four waves mode. if p > 3.56 the smallest load parameter c coresponds to the four waves mode. It is interesting to note that for different values of internal pressure the sizes of the large deformation zones are almost equal. The dent halfangle is approximately equal to 6 h 90* = 6/" Z a (12) Let be 1/ = 0.3. For h/r : 0.02 formula (12) gives : 27, and for h/r = 0.05 it gives 90* : 42". Acknowledgement The research described in this publication was possible in part by Grants N and N from Russian Foundation for Fundamental Research. Literature 1. Bushnell D. Bifurcation Phenomena in Spherical Shells under Concentrated and Ring Loads. AIAA Journ., Vol.55,, (Nov. 1967), No 11, pp Koroteeva P.V., Tovstik P.E., Shuvalkin S.P.: On the Axial Force Value at the Post buckling Axisymmetric Deflections of a Shell of Revolution. Vestnik S.-Peterburg. Univ. Mat.Mekh. Astron, (1995), No. 3, p Shilkrut D.l. The Problems of Qualitative Theory of Nonlinear Shells. Kishinev, (1974), 144 p. 4. Tovstik RE. The Post-buckling Axisymmetrical Deflections of Thin Shells of Revolution under Axial Loading. Technische Mechanik7 16, 2, (1996), Address: Professor Svetlana M. Bauer, Professor Peter E. Tovstik, Department of Theoretical and Applied Mechanics, Faculty of Mathematics and Mechanics. St. Petersburg University, Bibliotechnaya sq. 2, Stary Peterhof. BUS St. Petersburg. Svetlana.Bauer@pobox.spbu.rua Peter.Tovstik@pobox.spbu.ru 139