Buckling of Sherical Shells Revisited John W. Hutchinson School of Engineering and Alied Sciences, Harvard University Abstract A study is resented of the ost-buckling behavior and imerfection-sensitivity of comlete sherical shells subject to uniform external ressure. The study builds on and extends the major contribution to sherical shell buckling by W.T. Koiter in the 1960 s. Numerical results are resented for the axisymmetric large deflection behavior of erfect sheres followed by an extensive analysis of the role axisymmetric imerfections lay in reducing the buckling ressure. Several tyes of middle surface imerfections are considered including dimle-shaed undulations and sinusoidal-shaed equatorial undulations. Buckling occurs either as the attainment of a maximum ressure in the axisymmetric state or as a non-axisymmetric bifurcation from the axisymmetric state. Several new findings emerge: the abrut mode localization that occurs immediately after the onset of buckling, the existence of an aarent lower limit to the buckling ressure for realistically large imerfections, and comarable reductions of the buckling ressure for dimle and sinusoidal equatorial imerfections. Keywords: Buckling, bifurcation, sherical shells, geometric imerfections 1. ntroduction The intense study of the nonlinear buckling behavior of shells and, in articular, of sherical shells largely ended almost five decades ago with the ublication of W. T. Koiter s [1] monumental aer on the ost-buckling and imerfection-sensitivity of sherical shells subject to external ressure. Sherical shells under external ressure and cylindrical shells under axial comression dislay extraordinarily nonlinear ost-buckling behavior with a sudden loss of load carrying caacity triggered by buckling. These two shell/loading combinations are the most imerfection-sensitive in the sense that exerimentally measured buckling loads are often as little as 0% of the buckling load redicted for the erfect structure. As a consequence, design codes for elastic buckling of these thin shell structures often stiulate that the design load is knocked down to 0% that of the buckling load of the erfect structure. This emirical rule was
roosed over fifty years ago based on large body of collected exerimental buckling data and is still the governing design rule. No attemt will be made here to survey the rior extensive literature on sherical shell buckling. Nevertheless, related to resent aims, it should be mentioned that the highly nonlinear character of sherical shell buckling was areciated in the first half of the 1900 s and an imortant ste in coming to terms with the nonlinearity and strong imerfection-sensitivity was taken by von Karman and Tsien [] who set in motion a quest for a quantitative criterion governing the low buckling loads of thin sherical and cylindrical shells. A large literature addressing this toic accrued over the next thirty years investigating criteria such as a minimum ost-buckling load or a load at which the energy in the buckled state equals that in the unbuckled state. So far, no accetable criterion of this tye based on the resonse of the erfect shell has emerged. nstead, more rogress has resulted from the consideration of initial imerfections and their role in reducing the buckling load, although rogress along these lines for sherical shells was limited as will be described in this aer. Koiter [3] develoed a general theory of elastic stability which connected imerfection-sensitivity to the initial ost-buckling behavior of the erfect structure. Relevant to the resent study, it must be noted that a limitation of the Koiter theory is that the imerfection-sensitivity redictions are asymtotic and only valid for sufficiently small imerfections. The range of validity is not known from the theory itself and varies from roblem to roblem. Sherical shell buckling is articularly challenging in this regard because the direct alication of Koiter-tye theory to full sheres under external ressure, first resented by Thomson [4] and somewhat later by Koiter [1], turns out to be valid for only extremely small imerfections, too small to be reresentative of those in actual shells. The resent aer brings out clearly the reason underlying the small range of validity of the Koiter theory for full sherical shells and resents accurate buckling ressure results for reresentative imerfection amlitudes and shaes. The lull in research on sherical shell bucking over the ast several decades has been suerseded by a resurgence of activity driven from several quarters. Recent advances with soft elastomeric materials have made it ossible to fabricate sherical shells that are either nearerfect or having recisely controlled imerfections thereby oening the way for systematic exerimental imerfection-sensitivity studies [5,6]. These laboratory develoments align with
efforts underway at NASA and others interested in large shell structures to relace the longstanding emirical buckling knockdown factors emloyed in the design codes with an aroach that comutes buckling loads using commercial finite element codes by incororating realistic imerfections tied to manufacturing rocesses and direct measurement. Sherical shell buckling has also attracted interest for alications as diverse as attern formation and in the life sciences in the study of casules, ollen grains and viruses [7-11]. From a mathematical ersective, sherical and cylindrical shells are interesting because of their comlicated bifurcation behavior and their highly unstable ost-buckling resonse. Recent work in the nonlinear dynamics community has focused on these structures with the aim of gaining both qualitative understanding of the nonlinear system and a quantitative understanding of what sets the aarent lower limit of the buckling loads [1-14].. Three shell theories Three nonlinear shell theories for analyzing buckling of sherical shells will be emloyed in this aer. The rationale for doing so is to establish the range of alicability of the two most commonly used sets of nonlinear buckling equations, small strain-moderate rotation theory and Donnell-Mushtari-Vlasov (DMV) theory, in alication to sherical shell buckling. A theory which emloys exact stretching and bending strain measures for the middle surface of the erfect shell undergoing axisymmetric buckling deformations will be used to benchmark the two commonly used theories. Most of the results in this aer are comuted with small strainmoderate rotation theory, but DMV theory is also used to carry out the classical bifurcation analysis and discussed throughout because of its widesread use in the analysis of sherical shell buckling. All three theories are so-called first order theories intended for alication to thin shells. Such theories rovide an accurate reresentation of the energy in stretching and bending in the constitutive model to first order in t/ R with t as the shell thickness and R as its radius. Moderate rotation theory will be secified first, followed by DMV theory and finally the exact theory..1 Small strain, moderate rotation theory For the all roblems of interest here, the middle surface strains remain small. n addition, for all but an initial set of examles, the rotations remain moderately small. n nonlinear shell
theory, this means that middle surface strains satisfy 1 and rotations about the middle surface tangents and normal satisfy 1. As a rough rule of thumb, the rotations should not exceed about 15 o to 0 o for this theory to remain accurate. Rotations about the middle surface tangents are the largest while the rotation about the normal to the shell middle surface will turn out to be very small in the shere buckling roblem. Nevertheless, the equations accommodate moderate rotations about the normal. Our calculations will also show that there is almost no difference between dead ressure (force er original area acting in the original radial direction) and live ressure (force er current area acting normal to the deformed middle surface) for the behavior of interest in this aer, but both loadings will be modeled to establish this fact. t should be noted that in this aer, dead ressure does not imly that the ressure is rescribed to be fixed, as the terminology dead sometimes imlies. n this aer, rescribed ressure will be the terminology used to characterize a loading condition in which is held constant whether the ressure is dead or live. Equations for a first order shell theory with small strains and moderate rotations were given by Sanders [15], Koiter [16,17] and Budiansky [18]. These are secialized below for initially erfect sherical shells followed by the introduction of small initial geometric imerfections. Euler coordinates (,, r) are emloyed with r as the distance from the origin, as the circumferential angle and as the meridional angle ranging from 0 at the equator to / at the uer ole. The radius of the undeformed middle surface of the shell is R. A material oint at (,, R) on the middle surface of the undeformed shell is located on the deformed shell at r uiui ( R w) i r (.1) where ( i, i, i r ) are unit vectors tangent and normal to the undeformed middle surface associated with the resective coordinates. For general deflections, the dislacements ( u, u, w) are functions of and ; for axisymmetric deflections, u 0 while the other two dislacements are functions only of.
The nonlinear strain-dislacement relations make use of the linearized middle surface strains ( e, e, e ) and the linearized rotations (,, r ) with the rotation comonents about i, i and i r, resectively, which are 1 1 u e tan u w R cos, 1 u e w R, e 1 u 1 u tan u R cos (.) 1 1 w u R cos, 1 w u R, 1 1 u u r tanu R cos (.3) n the small strain-moderate rotation theory, the middle surface strains are nonlinear E 1 1 e, r E 1 1 e, r E 1 e (.4) while the bending strains are linear K 1 tan R, 1 K, R K 1 1 tan R cos (.5) n this aer, imerfections in the form of a small, initial stress-free radial deflection of the middle surface w from the erfect sherical shae is considered with ( u, u) 0. merfections in the form of thickness variations or residual stresses will not be considered. Thickness variations can give rise to both non-uniform re-buckling stresses and initial middle surface undulations but in most structures thickness variations are controlled to a much high tolerance than middle surface undulations. n addition, in this aer attention is limited to axisymmetric imerfections such that w is a function of but not. Assuming that w itself
roduces small middle surface strains and moderate rotations (a condition always met in all our examles), denote the strains in (.4) arising from w by E. Then, evaluate the total strains due to ( u, u, w w), where w is additional to w, and denote the result by arising from dislacements additional to and these are given by w, which roduce the stresses, are U E. The strains U E E E E 1 1 e, r E 1 1 1 dw e, Rd r E 1 1 dw e (.6) Rd where the linearized strains and rotations are evaluated in terms of ( u, u, w). Because the bending strains are linear in the dislacements and their gradients, the same rocess reveals that the relations (.5) still hold for the relation between the bending strains and the additional dislacements with no influence of w. From this oint on, the additional dislacements ( u, u, w) will simly be referred to as the dislacements. An imerfection contribution also arises for live ressure loading which will be introduced shortly. The stress-strain relations for a shell of isotroic material in each of the three first order theories emloyed here are 1 N (1 ) E E (1 ) Et, M D (1 ) K K (.7) with t as the shell thickness, E as the Young s modulus, as Poisson s ratio and 3 D Et / 1(1 ) as the bending stiffness. The resultant membrane stresses are ( N, N, N ) and the bending moments are ( M, M, M ). With S denoting the reference sherical surface secified by r R and the Euler angles (, ), the elastic energy in the shell is 1 SE( u, u, w) MK NE ds (.8) S
For the erfect shell, the otential energy PE of the uniform inward ressure on the shell is the negative of the work done by the ressure. For dead ressure (force er unit original area of the middle surface acting in the original radial direction), PE wds (dead ressure) (.9) S For live ressure (force er area of the deformed middle surface acting normal to the deformed middle surface), the otential energy is the ressure times the change of volume V within the middle surface. The results in [15,16,18] can be used to obtain the following exact exression for V in terms of the middle surface dislacements and their gradients, here in the surface tensor notation of [18]: 1 V w Q u R e w e e ds S 3 (.10) r r with e as the determinant of e, Q e e r and as the surface alternating tensor. We omit listing this exression in terms of hysical comonents as it is rather lengthy. However, in hysical comonents for axisymmetric deformations, (.10) becomes 1 V w 1 e u we e Rwee ds S 3 (.11) These exact results, (.10) and (.11), are alicable to a full sherical shell or any segment of the shell which is constrained such that u and u vanish on the boundary. The integrand in each of the exressions for worth recording that a general exression for to include errors or misrints. V is a cubic function of the dislacements and their gradients. t is V in [17], an alternative to that in (.10), aears Now introduce the effect of an axisymmetric initial imerfection w on the otential energy of the ressure loading of the using the rocess described for the strains where w becomes additional to w. Because it is linear in w, the PE for dead ressure remains as (.9). For live ressure, the resulting exressions derived from (.10) or (.11) involving lengthy and will not be listed. The energy functional of the loaded shell system is w are
SE( u, u, w) f ( u, u, w) (.1) where PE f, with f given by (.9) for dead ressure or derived from (.10) or (.11) for live ressure. The resence of. DMV theory w in (.1) is not exlicitly noted. Donnell-Mushtari-Vlasov theory introduces two aroximations to the moderate rotation theory: i) the square of the rotation about the normal, r, is neglected in the in-lane strains (.6), and ii) the deformations are assumed to have a short wavelength relative to R, referred to as shallow deformations, such that the dislacements, u and u, in the rotations and in (.3) are neglected. Thus, in DMV theory for sherical shells with small axisymmetric initial imerfections, the strain dislacement relations are E 1 1 w e R cos, E 1 1 w 1 dw e, R R d E 1 1 w 1 w 1 dw e Rcos R R d (.13) K 1 w w sin R cos, K R 1 w, K 1 R w w tan (.14) where the linearized stretching strains in (.13) are still given by (.). Generally only dead ressure is reresented when DMV theory is used with otential energy given by (.9). The stress-strain relations are given by (.7) and the elastic strain energy by (.8). Thus, for DMV theory, the energy functional for the sherical shell subject to uniform ressure is given by (.1) with f wds. S.3 Exact theory first order theory for axisymmetric deformations
The stretching and bending strain measures of moderate rotation theory and DMV theory are aroximate and their accuracy deteriorates as the shell dislacements and rotations become sufficiently large. The following exressions for the Lagrangian stretching strains, ( E, E ), and changes in curvature, ( K, K ), of the sherical shell middle surface are exact and can be obtained from the aers [15,16,18]. A first order shell theory based on these exact measures will be used to benchmark the other two theories using aroximate strain measures by illustrating the range over which the measures remain accurate. The exact measures are limited to axisymmetric deformations ( u 0, u ( ), w( )). The linearized stretching strains ( e, e ) and rotation,, are defined in (.) and (.3). The Lagrangian stretching strains are 1 E e e, 1 1 E e e (.15) while the changes in curvature are 1 K 1 1 tan tan 1 1 e e e e e e e e (.16) R 1 d de K 1e e ee 1 e 1e 1 R d d (.17) onsistent with a first order theory for a material with a linear stress-strain resonse, the membrane and bending stresses are still given by (.7). n all the roblems investigated in this aer the stretching strains, E and E, remain small. Thus, the distinction between the Lagrangian strains and other measures is insignificant and the linear constitutive relation (.7) is meaningful. The strain energy, otential energy of the live ressure loading, and the energy functional of the system have the same form as those given earlier for the moderate rotation theory in (.1). 3. Bifurcation ressure and modes for the erfect shell based on DMV theory Koiter [1] cites the PhD thesis of van der Neut [19] as the first rigorous demonstration that the eigenvalue roblem for the buckling of erfect sherical shell subject to uniform
ressure has simultaneous axisymmetric and non-axisymmetric eigenmodes with bifurcation from the uniformly comressed state at the critical ressure E t R 3(1 ) (3.1) Koiter [1] rovides his own derivation of van der Neut s results using moderate-rotation theory but in so doing invokes a series of aroximations that follow from the fact that the eigenmodes are shallow, i.e., have short wavelengths relative to R. His derivation is tantamount to invoking DMV theory. t is insightful and useful for resent uroses to rovide a derivation of the classical results for buckling of a erfect sherical shell using DMV theory from the start. This section rovides that derivation. We will demonstrate later in this aer that DMV theory with dead ressure loading is accurate for nearly all asects of the buckling behavior of interest in this aer. The uniform membrane solution for the erfect sherical shell subject to uniform ressure according to either moderate rotation theory or DMV theory is 0 (1 ) R u u 0, w with Et 0 0 N N t and 1 R (3.) t where is the equi-biaxial comressive stress in the shell in the uniform state. Bifurcation from 0 this uniform state in the form ( u, u, w) ( u, u, w w) is sought where the nonlinear equations are linearized about the uniform state with resect to the -quantities. The wellknown formulation using the Airy stress function F to satisfy in-lane equilibrium is emloyed along with the additional comatibility condition. The erturbation rocess leads to a air of couled artial differential equations from DMV theory governing the buckling eigenvalue roblem: 1 0 and R 4 D w F t w 1 1 Et R 4 F w 0 (3.3) where Eliminating is the Lalacian oerator on the sherical reference surface and F from the air of equations in (3.3) gives 4 ( ).
Et R 4 D w w t w 0 (3.4) sin m m m The sherical harmonic Snm(, ) Pn (sin ) cosm, with P n as the associated Legendre function of degree n and order m, satisfies S n( n 1) R S. With n and m (0 m n) restricted to be integers to ensure circumferential continuity and smooth behavior at nm nm the oles, it follows from (3.4) that eigenmodes of the form eigenvalue w S are associated with the nm tr D nn ( 1) q 4 0 1 n( n1) with q R t 4 0 1(1 ) (3.6) Anticiating that for thin shells n will be reasonably large, x n( n 1) can be regarded as a continuous variable, ignoring the fact that n must be integer, to minimize with resect to x. This rovides a lower bound estimate of the lowest eigenvalue, i.e., the buckling stress and ressure for the erfect shell: E 3(1 ) t R, E t, R 3(1 ) nn (3.7) t ( 1) 1(1 ) R This reroduces the result in [1]. For R/ t values for which n is integer, (3.7) is the lowest eigenvalue. However, as Koiter [1] notes, for other values of R/ t the difference between the lower bound in (3.7) and the slightly larger eigenvalue for integer n is of relative order t/ R. This difference is very small for thin shells such that and in (3.7) are universally referred to as the critical buckling stress and ressure of the erfect shell. Moreover, numerical calculations with the moderate rotation theory with either live or dead ressure along the lines of those reorted later reveal that, even for a shell with R/ t as small as 50, the difference between the classical formulas for and in (3.7) and the lowest eigenvalue comuted based on integer n is never more than 1%.
m 0 and There are n 1 modes associated with the lowest eigenvalue: the axisymmetric mode, 0 m w P (sin ) P (sin ), and n non-axisymmetric modes, w cos m P n (sin ) n n m and w sin m P n (sin ) with 1 m n. The shae of the axisymmetric mode is shown in Fig. 1 for shells with 0.3, in one case with R/ t 103.5 corresonding to a mode ( n 18 ) that is symmetric about the equator and in the other case for R/ t 9.6 having an anti-symmetric mode ( n 17 ). Throughout this aer, the inward deflection at the uer ole is defined as wole w( /). n Figs. 1b and 1c, the shae is lotted with w / R 0.1 for visualization uroses. As will be seen, this far exceeds the amlitude for which the bifurcation mode has any relevance. ole Fig. 1 n a) and c), the symmetric axisymmetric bifurcation mode for the erfect shell with R/ t 103.5 and 0.3. n b), the antisymmetric axisymmetric bifurcation mode for R/ t 9.6 and 0.3. The couling of the multile modes in the nonlinear ost-buckling range is one of the reasons for the extremely strong imerfection-sensitivity of the buckling of the sherical shell under external ressure [1,3]. t will be useful at this stage to make contact with Hutchinson s [0] analysis based on behavior of interacting modes in the vicinity of the shell equator. Let m w cos m w( ) be any eigenmode with w( ) P n (sin ), with w satisfying the equation for Legendre s associated functions:
1 cos d dw m cos nn ( 1) w 0 d d cos (3.8) Near the equator where 0, (3.8) is aroximated by dw nn ( 1) mw 0 d (3.9) Thus, in the vicinity of the equator, symmetric modes have w( ) cos q0 m different multilicative factor where, by (3.6) and (3.7), aart from a nn ( 1) q. With and denoting the wavelengths of the mode in the circumferential and meridional directions, the simultaneous symmetric modes in the vicinity of the equator have the form 0 R R w cos cos with R R q 0 (3.10) These are the modes considered in the analysis of mode interaction and imerfection-sensitivity in shallow sections of a shere [0]. The buckle wavelengths are on the order of Rt and small comared to R for thin shells. This reresentation of the modes will be used in the sequel. 4. Axisymmetric ost-buckling of the erfect shell Selected results for the axisymmetric ost-buckling behavior of the erfect shell will be resented emhasizing behavior at both small and large deflections. From a structural standoint, it will be seen that the imortant action occurs at small deflections that are usually not more than several times the shell thickness. The numerical results in this aer make use of highly effective algorithms (c.f., Aendix) for solving nonlinear ode s which were not available in the 1960 s when nearly all the rior studies of sherical shell buckling were carried out. Koiter s [1] study of the imerfection-sensitivity of sherical shell buckling emloyed analytical methods based on erturbation exansions about the bifurcation oint of the erfect shell, although he attemted to extend the range of validity of these exansions by analytical means. We begin by resenting an examle of axisymmetric large deflection behavior based on the formulation in Section.3 that emloys exact middle surface strain measures. The numerical
method for this formulation is described in the Aendix. The shells in Fig. have R/ t 5 and 50 with 0.3. Symmetry with resect to the equator has been imosed. Following bifurcation the ressure falls monotonically to the oint where the oosite oles make contact. Under rescribed ressure, the ost-buckling resonse would be unstable over the entire range of deformation lotted. Even under rescribed volume change V the resonse at bifurcation is unstable until the ressure dros to / 0. ( w / R 0.3) for R/ t 5 or / 0.15 ( wole / R 0.) for R/ t 50 ole. The ost-buckling shae seen in the insert in Fig. a does not resemble the classical axisymmetric mode described in Section 3 for reasons which will be discussed shortly. nstead, the advanced buckling shae in the vicinity of the ole is aroximately an inverted ca with radius of curvature R. For sufficiently small ole deflections, the buckle is shallow and the rotation of the middle surface is small. However, as the ole deflection increases, the magnitude of the maximum rotation becomes larger than 90 o in the limit when wole / R 1, clearly exceeding the range in which moderate rotation theory is exected to hold. Fig. Large axisymmetric deflections of two erfect shells based on the exact formulation. a) Pressure versus ole deflection normalized by the shere radius. b) Pressure versus change in volume normalize by the negative of the volume within the middle surface of the undeformed 3 shere, V0 4 R /3. The deformed middle surface is lotted in the insert in a) at three levels of deformation, including that at which the oosing oles first make contact.
An imortant oint brought out in Fig. 3 is that much of the essential buckling behavior of thin shells lays out in the range of deflections on the order of several shell thicknesses. The solid line curves in Fig. 3 have been comuted using moderate rotation theory of Section. together with live ressure using the exact exression (.11) for the change in volume. For axisymmetric deformations, the small strain, moderate rotation equations reduce to a system of 6 nonlinear first order odes given in the Aendix. The dashed line curves in Fig. 3 have been comuted using the exact formulation for the case R/ t 50. For this case, divergence between the moderate rotation theory and exact theory for the linearized rotation are first evident at 0.5radians ( 15 o ) but are still relatively small at twice that level. n fact, the moderate rotation rediction for the relation of the ressure to the ole deflection for R/ t 50 remains accurate for ole deflections as large as wole / t 10 or wole / R 0.. The same is true for the relation of ressure to change in volume. Equally imortant, is the observation that, in the range of behavior lotted, the relation of / to w / t is essentially indeendent of R/ t and ole becomes increasingly so for even larger R/ t. As evident from Fig. 3c, the relation of / to V / V is not indeendent of R/ t due to the fact that the dimle-like buckle at the ole diminishes in size relative to the shere as R/ t increases. A comlete characterization of this behavior will be given in a subsequent ublication.
Fig. 3 Axisymmetric ost-buckling behavior of the erfect shell in the range of relatively small deflections for shells with several R/ t, assuming symmetry with resect to the equator. Solid lines denote results based on moderate rotation theory and dashed lines are based on the exact formulation for R/ t 50. n c), V 4 R w is the change of volume of the erfect shell at bifurcation where the associated normal dislacement is w (1 ) t/ 3(1 ). An extremely small initial imerfection is used to trigger bifurcation from the sherical shae. This imerfection is then reduced to zero on the ost-bifurcation branch. For sherical shell buckling it aears that the moderate rotation theory retains a reasonably high level of accuracy for the main quantities of interest to deflections considerably beyond those based on the reviously quoted rule of thumb that the rotations should not exceed 15 to 0 degrees. All of the subsequent results resented in this aer lie within the range w / t 10 and they have been comuted with the moderate rotation theory of Section.. ole Selected results have been recomuted using the less accurate DMV theory. Aart from one case
involving non-axisymmetric bifurcation, we have not observed any areciable difference between the redictions of the two theories. Moreover, excet for the very large deflection results in Fig., there is almost no difference between imosing live or dead ressure for the moderate rotation theory for any of the other results resented in this aer. Finally, and imortantly, the ossibility of non-axisymmetric bifurcation from the axisymmetric state will be analyzed in Section 7 and reorted for all the axisymmetric solutions based on the moderate rotation theory. n articular, non-axisymmetric bifurcation from the axisymmetric ostbuckling solutions in Fig. 3 does not occur. Once initiated these axisymmetric solutions are resistant to non-axisymmetric bifurcation and remain axisymmetric. We end this section on buckling of the erfect sherical shell by dislaying the extraordinarily raid transition from the classical bifurcation mode seen in Fig. 1 to the localized dimle-like mode seen in the insert in Fig. a. For this urose, a erfect shell with R/ t 103.5 and 0.3 is considered, corresonding exactly to n 18 by (3.7) and to the axisymmetric bifurcation mode shown in Fig. 1a,c. The dramatic evolution of the ost-bifurcation buckling mode is shown in Fig. 4. Almost immediately following bifurcation the classical mode gives way to a dimle-like mode localized at the ole. The deflection over most of the shell away from the ole is simly the uniformly comressed state. As discussed in connection with Fig., this dimle becomes an inverted ca near the ole with radius of curvature aroaching R. Evkin et al. [1] have resented an asymtotic analysis of the dimle mode using shallow shell theory alicable to thin sherical shells in which the dimle is confined to the vicinity of the ole. Based on their asymtotic analysis, the authors derive a formula for the deendence of on w / t. Their formula is indeendent of R/ t and reroduces the results in Fig. 3b / ole with an error as large as 5% for moderate values of wole / accurate as wole / t but becoming increasingly t becomes larger. Further details of the asymtotic results in [1] will be discussed in a subsequent ublication.
Fig. 4 Evolution of the buckling mode in the initial ost-bifurcation range for a erfect sherical shell with R/ t 103.5 and 0.3 with indicators ointing to the associated location on the ressure-deflection curve. The classical bifurcation mode with w (sin ) bif P18 is evident only for an extremely small range beyond bifurcation whereuon the mode becomes fully localized at the ole to an inward dimle-like shae. The immediate localization of the ost-buckling mode to the ole of the shere was not well understood to researchers in the 1960 s, although there is already recognition by Karman and Tsien [] that exerimentally observed buckling modes were more dimle-like than shaed like the classical mode. The localization exlains why the initial ost-bifurcation exansions of Thomson [4] and Koiter [1] based on the axisymmetric classical bifurcation mode have such an excetionally small domain of validity. The initial ost-bifurcation aroach tacitly assumes that the classical bifurcation mode rovides the first order, and dominant, aroximation to the ostbuckling mode. As soon as mode localization occurs, this ceases to be a good assumtion. While evidently not aware of the localization henomenon, Koiter was aware that the range of validity of the exansions was severely limited for shere buckling. A substantial ortion of his study [1] was devoted to an attemt to extend the validity of the exansions.
5. The effect of axisymmetric imerfections in the shae of the classical mode Because imerfections in the shae of the classical bifurcation mode are known to cause the largest load reductions in imerfection-sensitive structures [3], at least for sufficiently small imerfection amlitudes, we begin by considering axisymmetric imerfections of the form w ( ) P (sin ) where is the imerfection amlitude and n is related to R/ t by (3.7). n n the next section, it will be seen that dimle-shaed imerfections are more critical for all but small imerfection amlitudes. Fig. 5. Plots of the ressure versus a) the dislacement at the ole and b) the change in volume of the shell, for shells with R/ t 103.5 and 0.3. Results are shown for five levels of imerfection, w (sin ) P18, where the imerfection is in the shae of the buckling deflection of the erfect shell (c.f, Fig. 1a,c). The results in art b) have been comuted to values of wole / t greater than 10; the solid dot on each curve indicates where wole / t 10. The examle in Fig. 5 shows the effect of different imerfection amlitudes on the ostbuckling resonse of a shell with R/ t 103.5 and 0.3 corresonding to n 18 and w ( ) P (sin ). The imerfection shae lotted in Fig. 1a has its largest magnitude at the 18 oles with w ( /). Shells having imerfections with amlitudes less than / t 0.5 dislay a ressure maximum,, in the early stage of deformation at w / t 1 followed by max ole diminishing ressure with increasing buckling amlitude. Under rescribed ressure the shell would become unstable and undergo sna buckling at max. The shells become unstable just beyond attainment of max even under rescribed volume change for sufficiently small
imerfections, i.e., / t 0.5. An unexected finding is the fact that shells with imerfections equal to or larger than about / t 0.5 have no ressure maximum in the early stage of deformation. For these shells, the ressure increases gradually until a eak is finally attained at much larger deflections, e.g., wole / t 10, as illustrated by the examle in Fig. 5. The normalized maximum ressure, / max, as a function of the imerfection amlitude, / t, for four values of R/ t is lotted in Fig. 6. The imerfection for each R/ t is again taken to be w P(sin ) where the integer n is given exactly in terms of R/ t by n (3.7). Each case corresonds to an even value of n and symmetry about the equator is invoked in the comutations carried out using the moderate rotation theory. As noted in connection with the examle in Fig. 5, max is associated with a clearly defined eak ressure at relative small buckling deflection, e.g., wole / t 1.5, for smaller amlitude imerfections satisfying / t 0.45. For larger / t in Fig. 6, no eak ressure is attained in the range of small buckling deflections. Thus, for / t 0.45, the range of ole deflections has been limited to w / t 5 and max is either the eak ressure, if one occurs in this range, or the value of at ole w / t 5. Two notable features are evident in Fig. 6. First, for / t 0.45, increasing the ole imerfection amlitude increases the buckling ressure. Secondly, the imerfection-sensitivity curves are nearly indeendent of R/ t, increasingly so as R/ t increases beyond 100. The issue of non-axisymmetric buckling from the axisymmetric state will be addressed in Section 7. Excet for the largest imerfections in Fig. 6 in the range / t 0.9, non-axisymmetric bifurcation does not occur at ressures lower than max lotted. n fact, over most of the range for / t 0.9, non-axisymmetric bifurcation does not occur even at deflections well beyond the maximum ressure. n the range / t 0.9, non-axisymmetric bifurcation does occur just rior to attaining max in Fig. 6 imlying that buckling in a non-axisymmetric mode would be initiated. For examle, for / t 1, / 0.39 ( m 8) for R/ t 47. and / 0.41 ( m 8) for R/ t 103.5. bif bif
Fig. 6 merfection-sensitivity for shells having imerfections in the shae of the classical axisymmetric buckling mode, w Pn(sin ). Results for four values of R/ t have been lotted and are almost indistinguishable from one another. For imerfection amlitudes satisfying / t 0.45, max is the ressure maximum attained at ole deflections never larger than wole / t 1.5. For / t 0.45, the value of max lotted is either the ressure maximum, if one occurs, or the value of attained at wole / t 5 if a maximum has not yet been obtained. Two asects of axisymmetric sherical shell buckling described thus far have not been revealed in earlier studies: 1) the nature of the localization transition of the buckling mode at the ole in the ost-buckling resonse almost immediately after the onset of buckling, and ) the unexected increase in load carrying caacity with increasing imerfection amlitude seen in Figs. 5 and 6 for the larger imerfections. Neither asect could be exected to be uncovered from Koiter s [1] analytical aroach for reasons already discussed. Koga and Hoff [] were among the last investigators in the 1960 s to rovide numerical results for the axisymmetric buckling of sherical shells with axisymmetric dimle-like imerfections, but their method was not accurate and their results overestimate the reduction in buckling load due to the imerfection. Moreover, the largest imerfection amlitude these authors considered was / t 0.5. Numerical analysis is a owerful tool for accurately caturing the nonlinear henomena revealed here associated with sherical shell bucking articularly because much interesting
behavior is axisymmetric and therefore governed by nonlinear ordinary differential equations. t should be mentioned that numerical analysis codes did exist beginning in the late 1960 s which were caable of uncovering the behavior bought out in Figs. 5 and 6. Secifically, Bushnell [3] develoed accurate methods for solving nonlinear axisymmetric roblems for shells of revolution which later evolved into the BOSOR code [4]. Non-axisymmetric bifurcation from the axisymmetric state could also be addressed by BOSOR. While this and other numerical codes had the caabilities needed to advance the understanding of sherical shell buckling five decades ago, such studies were not carried out. Uon comletion of the resent aer it came to our attention that D. Bushnell, in his Ph.D thesis [5], carried out an extensive numerical analysis of the axisymmetric buckling of clamed shallow sherical cas that did reveal the localization transition discussed above for cas with sufficient height. Other than aearing in his thesis, this work has not been ublished. 6. merfection-sensitivity for dimle imerfections Motivated by the tendency of the buckling mode to localize at the ole, attention is again focused on axisymmetric behavior of the shell but now for dimle imerfections located at the oles secified by w ( / ) e ( ) with / (at the uer ole) (6.1) Here, sets the width of the imerfection. n all cases the imerfection becomes exonentially small for and is effectively zero at the equator. Attention is limited to deflections which are symmetric about the equator. The localized nature of the buckling decoules behavior above the equator from that below, and there essentially no difference in the imerfectionsensitivity for the symmetric case having identical imerfections at each ole and that for an asymmetric case, for examle, with the imerfection at only one ole. ritical buckling wavelengths are roortional to Rt such that the scaling 1/ B / 1 R/ t (6.) ensures that the imerfection-sensitivity will be essentially indeendent of R/ t, as will be seen.
Plots of imerfection-sensitivity for three widths of the dimle imerfection are given in Fig. 7a for R/ t 100. For relatively small amlitudes, e.g., / t 0.5, imerfections with width secified by B 1 give the largest reductions in buckling ressure, while for larger amlitudes the largest reductions are caused by somewhat wider imerfections with B 1. n almost all the cases in Fig. 7, max is associated with a well-defined eak ressure, such as those cases seen in Fig. 5, occurring at relatively small ole deflections. The only excetions are the shells with B and / t 1.6 where no eak occurs at modest deflections. n these cases is defined as either the maximum, if one occurs, or the ressure at w / t 5 if no max maximum has yet been reached. The small jum in the imerfection-sensitivity curve at / t 1.6 in Fig. 7a for this case results from this transition in behavior. n Section 7, it is found that non-axisymmetric bifurcation from the axisymmetric state rior to attainment of does not occur for any of the cases resented in Fig. 7. omarison of the results in Fig. 7a with those in Fig. 6 reveals that an imerfection in the shae of the classical mode does indeed reduce the buckling ressure slightly more than the comarable dimle imerfection at sufficiently small amlitudes, i.e., / t 0.5, but otherwise the dimle roduces larger reductions. Fig. 7b, for shells with dimle imerfection size secified by B 1.5, demonstrates again there is essentially no deendence on R/ t for thin shells as long as the imerfection width scales according to (6.). A remarkable feature of these imerfection-sensitivity trends is the lower limit, or lateau, / 0. max, for buckling in the resence of imerfections with relatively large amlitudes, indeendent of R/ t. This is another feature which might have surfaced decades ago had calculations at these larger imerfection amlitudes of been erformed. The lateau was first noted by Lee, et al [6] in a combined exerimental and theoretical study of the buckling of clamed hemi-sherical shells. The lateau behavior has imortant imlications for identifying the knock-down factor for sherical shell buckling which will be discussed in the concluding remarks. ole max
Fig. 7 merfection-sensitivity based on axisymmetric buckling for dimle imerfections (6.1). a) Shells with R/ t 100, 0.3 and three imerfection widths set by B in (6.). b) Shells with different R/ t, with B 1.5 and dimle width scaling according to (6.). For B and / t 1.6 in a), a maximum ressure at modest deflections does not occur and max is defined as the ressure at wole / t 5. n all other cases, including those in b), max is associated with a ressure maximum attained for buckling deflections satisfying wole / t 5. 7. Non-axisymmetric bifurcation from the nonlinear axisymmetric state The roblem of non-axisymmetric bifurcation from the nonlinear axisymmetric solution is addressed to ascertain whether non-axisymmetric buckling solutions at bif exist rior to attainment of max in the axisymmetric state. Such a non-axisymmetric bifurcation would indicate a lower buckling ressure than max. The bifurcation solution is linearized about the nonlinear axisymmetric solution. Axial symmetry ensures a searated solution for the dislacements in the form u, u, w Rsin( m) u ( ),cos( m) u ( ),cos( m) w( ) (7.1) where m 1 is the unknown integer number of circumferential waves in the bifurcation mode. From (.)-(.5) for the moderate rotation theory, the linearized rotation and strain measures have the form:
,, sin ( ),cos ( ),sin ( ) r E, E, E cos E ( ),cos E ( ),sin E ( ) r (7.) 1 R K, K, K cos K( ),cos K ( ),sin K ( ) where the barred quantities are readily obtained in terms of u ( ), u ( ), w( ), m and the A rotation ( ) in the axisymmetric state, c.f., Aendix. Denote the quadratic functional of the non-axisymmetric dislacements governing bifurcation from the axisymmetric state by P u u w,,. Attention is limited to axisymmetric solutions that are symmetric about the equator. This allows consideration of symmetric and antisymmetric bifurcations with resect to the equator with P defined above the equator. The investigation of non-axisymmetric bifurcation occurs in the range of small rotations where the dead ressure is an excellent aroximation to live ressure. As dead ressure is considerably simler to imlement, it will be used for the bifurcation analysis. When use is made of the searated form of the bifurcation mode listed above and the trivial integrations with resect to are carried out, one can show that P can be reduced to the following dimensionless functional: (1 ),,, P u u w P REt 0 0 1 E E E K K K d ˆ (1 ) (1 ) cos A A A A r r E E E E cosd (7.3) A A where E ( ) and ( ) deendence on or E are the strains in the axisymmetric solution, ˆ 1( R / t) and the w ole arises through the axisymmetric solution. Symmetry at the equator requires u u w w 0 at 0 while anti-symmetry requires u u w w 0. For integers, m 1, conditions at the ole require u u u u w w 0. The mode for m 1 does not require w 0 at the ole; it is an unknown, reresenting a ossible tilt of the
shell at the ole. For all the axisymmetric roblems considered in this aer, w ole is monotonically increasing and it has been used as the rescribed loading arameter. For any non-zero set of admissible bifurcation dislacements u, u, w, P 0 for w ( w ) and there exist non-zero dislacements such that P 0 for w ( w ) ole ole bif ole ole bif. The ole dislacement at bifurcation, ( w ole) bif, is the lowest value of w ole, considering all ossible integers m, for which an admissible non-zero mode u, u, w exists with P 0. The associated ressure at bifurcation is denoted by bif. The numerical solution of the non-axisymmetric bifurcation roblem emloys cubic slines to reresent each of the admissible dislacements u, u, w with nodal dislacements as unknowns. With a, j 1, M j, denoting the vector of unknowns, (7.3) becomes P( a, m, ) A aa (7.4) M M i1 j1 ij i j where the symmetric matrix A is comuted using numerical integration at each value of, or equivalently at each value of ole dislacement w. For w ( w ) ole ole ole bif, A is ositive definite for all integer m ; ( w ) is the lowest w ole over all m such that A 0. ole bif As reorted in the earlier sections, no non-axisymmetric bifurcation was found for the axisymmetric ost-buckling of the erfect shell for ole deflections as large as wole / t 10 in Fig. 3 once the axisymmetric bifurcation mode had been established. Further, no nonaxisymmetric bifurcation was found for the axisymmetric solutions associated with the dimle imerfection rior to attainment of the maximum ressure in the axisymmetric state. n fact, for all cases investigated for the dimle imerfection, non-axisymmetric bifurcation from the axisymmetric state did not occur even well ast the maximum ressure. While it is not ossible to claim that non-axisymmetric bifurcation never occurs for axisymmetric dimle-like imerfections, it aears from the examles investigated here that the axisymmetric deformation becomes locked in and resistant to non-axisymmetric deformation. As noted in Section 5, for imerfections in the shae of the classical mode, the only instance where bifurcation from the
axisymmetric state was observed was for larger imerfections ( / t 0.9 ) in the range where the maximum ressure increases with increasing imerfection amlitude. Although the imerfection in the shae of the classical mode has otency at small amlitudes, it is not nearly as damaging at larger amlitudes as the dimle imerfection and it is less realistic. The study summarized thus far has established that an axisymmetric dimle imerfection is likely to roduce an axisymmetric buckling resonse when the sherical shell is subject to uniform ressure. merfection amlitudes slightly larger than a shell thickness reduce the buckling ressure to / 0. max. We now show that axisymmetric imerfections at the equator, so-called beltline imerfections, can roduce comarable buckling ressure reductions. For these imerfections, buckling is associated with non-axisymmetric bifurcation from the axisymmetric state. The axisymmetric beltline imerfection is symmetric with resect to the equator and secified by w cos R /, 0 A 0, B / (5) ( ) ( ), A B (7.5) t is lotted in Fig. 8a for the secific choices, 59.41 o and 74.71 o, used in this aer. Here, R / q0 is the critical axisymmetric wavelength given by (3.10) with, A B and (5) ( ) is the fifth order olynomial chosen such w and its first and second derivatives are continuous at A and B. The region in the vicinity of the ole is imerfection-free and the maximum imerfection amlitude is attained in the vicinity of the equator.
Fig. 8 Non-axisymmetric bifurcation from the axisymmetric state for axisymmetric equatorial, beltline, imerfections for a sherical shell with R/ t 100 and 0.3. The imerfection, secified by (7.5), is lotted in a) for / t 1, with 59.41 o A and 74.71 o B. The ressure at bifurcation comuted for the full shere using moderate rotation theory is lotted in b) as the curve which dislays the number of circumferential waves m associated with the lowest bifurcation ressure. ncluded in b) are two results from [0] for the same imerfection based on an analysis of a shallow equatorial section of the shell. The uer dashed curve is the asymtotic imerfection-sensitivity formula (7.6) and the lower solid line curve is a more accurate analytical result not limited to small imerfections. The nonlinear axisymmetric solution for the beltline imerfection (7.5) does not dislay a maximum until becomes almost even for relatively large imerfections amlitudes. Thus, if one restricted consideration to axisymmetric behavior, one would have to conclude that the beltline imerfection does not give rise to buckling imerfection-sensitivity. The conclusion is entirely different, however, when non-axisymmetric bifurcation from the axisymmetric solution is considered. Results of the non-axisymmetric bifurcation study for a shell with R/ t 100 and 0.3 are shown in Fig. 8b where the normalized bifurcation ressure, /, is lotted versus the normalized imerfection amlitude, / t. n this lot the number of circumferential waves, m, associated with the bifurcation mode is shown. The slight rise seen in the bifurcation ressure in Fig. 8b as / t aroaches has been evaluated carefully it is not numerical error. bif
For / t 0, as in Fig. 8b, the lowest bifurcation eigenvalue is associated with a mode u, u, w which is symmetric with resect to the equator, although the analysis assuming an anti-symmetric mode generates only slightly larger eigenvalues. For / t 0, the critical mode is anti-symmetric and the imerfection-sensitivity curve is essentially identical to that in Fig. 8b. The exlanation of the underlying interaction of the non-axisymmetric mode with the membrane stresses generated by the nonlinear axisymmetric solution is similar to that rovided in the study of the effect of axisymmetric imerfections on the non-axisymmetric bifurcation of cylindrical shells under axial comression [6]. Excet for very small imerfections in the shae of the classical mode, the beltline imerfection roduces somewhat larger buckling ressure reductions than the other two shaes considered for imerfection amlitudes with / t 1. For larger imerfection amlitudes, the beltline and dimle imerfections both give rise to buckling ressure reductions that level out at / 0.. ncluded in Fig. 8b are results from two analyses taken from Hutchinson [0] for a sinusoidal imerfection with recisely the same form as (7.5) near the equator. The two analyses emloyed DMV theory but did not consider a full sherical shell. nstead, the analyses exloited the short wavelengths of the modes in the vicinity of the equator and considered eriodic mode interaction in a shallow section of the shell. The dashed curve in Fig. 8 is the relation, bif 9 3(1 ) 1 8 t bif, (7.6) based on a Koiter-tye asymtotic analysis. The second result, lotted as the lowest solid curve in Fig. 8, agrees with (7.6) asymtotically for small / t but is not restricted to small imerfections, although it is aroximate being based on the shallow analysis. This second curve is a slightly more comlicated analytic result taken from the Aendix of [0]. These two results for / do not deend on R/ t, and we strongly exect that the results from the full bif shell analysis in Fig. 8 will be essentially the same for all larger R/ t. This has indeed been verified with R/ t 00 for selected values of / t over the range shown in Fig. 8b. For these