Buckling and Post Buckling of Imperfect Cylindrical Shells, G.simitses

Buckling and postbuckling of imperfect cylindrical shells: A rewiewGeorge J Simitses School of Engineering Science and Mechanics, Georgia Institute of Technology, Atlanta GA 30332 Thin-walled cylinders of various constructions are widely used in simple or complex structural configurations. The round cylinder is commonly found in tubing and piping, and in offshore platforms. Depending on their use, these cylinders are subjected (in service) to individual and combined application of external loads. In resisting these loads the system is subject to buckling, a failure mode which is closely associated with the establishment of its load-carrying capacity. Therefore, the system buckling and postbuckling behavior have been the subject of many researchers and investigators both analytical and experimental. The paper is a state-of-the-art survey of the general area of buckling and postbuckling of thin-walled, geometrically imperfect, cylinders of various constructions, when subjected to destabilizing loads. The survey includes discussion of imperfection sensitivity and of the effect of various defects on the critical conditions. I. INTRODUCTION Thin-walled cylinders of various constructions find wide uses as primary structural elements in simple and complex structural configurations. The round cylinder is popular in column design, in tubing and piping, and in offshore platforms. Stiffened and unstiffened metallic and laminated composite thin (large diameter to thickness ratios) shells are used extensively in underwater, surface, air, and space vehicles as well as in the construction of pressure vessels, storage bins, and liquid storage tanks. During their service, thin-walled cylinders are often subjected to individual and combined application of external loads. In resisting these loads, the system is subject to buckling, a physically observed failure mode, which is closely associated with the establishment of its load-carrying capacity. Therefore, the buckling strength of thin shells along with knowledge of its postbuckling behavior have been the subject of many researches and investigations both analytical and experimental. The knowledge derived from these studies is very essential in the safe design of such configurations. It is not surprising then that several hundreds of papers and reports have been written on the subject. Moreover, several review articles have found their way into the open literature, and through these reviews the interested readers and, more importantly, the potential designers are informed of the several specific questions that the reviewed articles address. In particular, attention was paid to the degree of approximation involved in the use of various kinematic relations (which led to several linear and nonlinear shell theories), to the reasons for the discrepancy between classical (linear theory), theoretical predictions for critical loads, and experimentally obtained buckling loads, to the use of stiffening for improving resistance to buckling, and to the effect of cutouts of various shapes, of foreign rigid inclusions and other defects. Moreover, as the complexity of shell-like structures increased and as the computational capability improved, efficient computer codes became necessary for the stability analysis of these configurations. Furthermore, in the more recent years, the constant demand for lightweight efficient structures led the structural engineer to the use of nonconventional materials, such as fiber-reinforced composites. The correct and effective use of these materials requires more complex analyses in order to achieve good understanding of the system response characteristics to external causes. Finally, one should mention that designers of thin cylinders have always strived towards achieving the best possible design (lightest, most reliable, fail safe, most economical, easiest to maintain and repair, etc). An effort has been exerted in the last 15 years or so to accomplish this in a formal manner. Many refer to this effort as structural optimization. The purpose of the present paper is twofold: (1) to review the entire field of buckling and postbuckling, which is primarily accomplished by referring to pertinent reviews, books, and recent technical articles, and (2) to provide a state-of-the-art accounting of the effect of imperfections on the response characteristics of thin cylindrical shells. The imperfections include initial geometric imperfections (out-of-roundness, load eccentricities a n d / o r load misalignments), as well as constructional and material defects, such as small holes, cutouts, rigid inclusions, delaminations, and others. II. BRIEF HISTORICAL REVIEW Because of the tremendous and continuous interest in shell buckling and because of the multitude of the reported theoretical and experimental investigations, a short, historical sketch is presented. For the sake of brevity, this sketch takes rather giant steps in moving through time. The reader, who is interested in smaller steps and more details on every addressed aspect of the problem, is referred to 1517 © Copyright 1986 American Society of Mechanical Engineers Transmitted by A M R Associate Editor Arthur Leissa. Appl Mech Rev vol 39, no 10, Oct 1986 Downloaded 23 Aug 2009 to 129.5.16.227. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms_Use.cfm he may also refer to the proceedings of several symposia12"14 addressing shell buckling problems and to tests 15 " 18 that present a collection of papers on various topics of shell buckling. The single and most important conclusion of all the theoretical investigations of cylindrical shells is that the primary reason for the discrepancy between (linear) theoretical critical loads and buckling loads is that the system is extremely sensitive to initial geometric imperfections. 47-52 fall into this category. Circular tubes and shells. Another approach for imperfection sensitivity studies is to deal directly with the imperfect configuration and employ nonlinear kinematic relations.5. Batdorf62 used simplified Donnell-type 63 of shell equations to predict critical loads. The reader must also consider their cited references. while the books by Bushnell 8 and Kollar and Dulacska 20 include several shell geometries (cylindrical. His theory is limited to the neighborhood of the classical bifurcation load (immediate postbuckling). fall in the range of 40-90% of the classical critical load.82 Reese and Bert.65 Flugge. Note that Ref.31 " 33 (b) effect of in-plane boundary conditions. many experimental studies were performed with the same objective in mind (explain the discrepancy). creep. Moreover.20 Yamaki's book19 deals very thoroughly with the buckling and postbuckling behavior of elastic and isotropic cylindrical shells. The initial theoretical investigations were based on many simplifying assumptions. 71.69 and Donnell. which consider the simultaneous application of two or more destabilizing loads. and touch upon influences of plasticity.61 Several investigations based on linear analyses appear in the literature. III. Oct 1986 reviews and surveys that have appeared in the open literature since the 1950s. The first analysis is attributed to von Mises. 21.cfm .1518 Simitses: Buckling and postbuckling of imperfect cylindrical shells Appl Mech Rev vol 39. Finally. The first theoretical investigations on the subject dealt with axially loaded configurations. constant directional pressure. These are: (a) effect of prebuckling deformations. and Sheinman85 also included multiple loads in their work. When the comparison is extended to stiffened configurations. 27).84 and Simitses. Sheinman.19 Some of these references deal with nonisotropic constructions. 54 " 58 with use of carefully manufactured specimens. Soong64 employed Sanders' shell theory. A similar development was followed for the case of buckling under lateral loading (pressure). (b) effect of boundary conditions. For a fairly complete list. In parallel to the above analytical investigations. especially of the in-plane ones. 34 " 37 and (c) effect of initial geometric imperfections. The theoretical predictions are based on a Galerkin procedure with the emphasis on post-lirnit point equilibrium positions. 63 Sobel70 studied the effect of boundary conditions on the critical pressure.asme.227. no 10. some of these investigators explained that the minimum postbuckling equilibrium load is a measure of the load carrying capacity of the system. Postbuckling and imperfection sensitivity analyses appear in the literature. the discrepancy is still present but less pronounced. and they were performed by Lorenz. 83 Sheinman and Simitses. 22. In addition.68 Sanders. Before closing this section. etc) and various constructions (stiffened.16. Koiter 53 was the first to question the use of the minimal postbuckling load as a measure of the load carrying capacity.65 Simitses and Aswani 66 compared critical loads for the entire range of radius to thickness and length to radius ratios and for various load behaviors during the buckling process (true pressure. and Giri. He also dealt directly with the imperfect configuration. conical. 29 Wilson and Newmark. 38 " 43 In addition. Shaw. including Becker and Gerard. and Mayers44 concluded from their calculations that the minimum postbuckling load tends towards zero with increasing number of terms in the series expansion of the transverse displacement component and with diminishing thickness. 67. 46 Efforts reported in Refs. stiffened. 4.75 and Simitses. and therefore to small initial imperfections. spherical. 24 ' 25 and Southwell. The first attempt is Donnell's. and centrally directed pressure) for a thin cylindrical shell. From the analytical point of view.27 and Donnell (his experimental results are reported in Ref. These studies follow the same pattern and include strictly postbuckling analyses. Redistribution subject to ASME license or copyright.73 Meek. see http://www. the reader is also referred to three recent books. This latter thinking came to an end when Hoff. Initially. 54 obtained also postbuckling curves for both axially and pressure loaded cylinders.26 The first experimental studies are those of Lilly (see Ref.78 " 80 is greater for the case of axial compression than for the case of either pressure or torsion. and Lundquist 78 and Nash79 are two of several that reported experimental results. corrugated. 63 Several studies followed with the emphasis on different considerations. Comparison between theoretical predictions (critical loads) and experimental results (buckling loads) revealed discrepancy of unacceptable magnitude. the imperfection sensitivity of the system was established through strict postbuckling analyses of the perfect configuration.30 Lundquist. etc). and most of their investigations are reported in Ref. Thielemann and Esslinger 59. Madsen. and residual stresses. the initial simplifying assumptions were later reevaluated and removed. employing several linear shell theories: Koiter-Budiansky.72 and Koiter-type of analyses. Hayashi 77 dealt with orthotropic cylinders. 8 ' 19. 76 Buckling analyses for torsion started with the work of Donnell. primarily for the benefit of a large class of practicing engineers.72.23 Timoshenko. 29. Robertson. Hess. of Ref. The interested reader is referred to Yamaki's 19 book for a complete review. and (c) the effect of initial geometric imperfections. 27). In trying to explain this discrepancy. as in the case of axial compression. Many researchers adopted this approach.1"11 Moreover. BOUNDARY CONDITIONS AND PREBUCKLING DEFORMATIONS The discrepancy between theoretical predictions based on classical buckling analyses and experimental results 9. A tremendous effort was made in order to explain the discrepancy both analytically and experimentally.4 Orthotropic. Hayashi and Hirano 80 and Budiansky81 are among those who reported on postbuckling analyses. the new ones. While the old buckling loads fell in the range of 15-50% of the classical critical load. The first Downloaded 23 Aug 2009 to 129.71. and other constructions were considered by several researchers.60 extended their research to include theoretical postbuckled state calculations on the basis of observed experimental results. mention should be made of chapter 13. three sets of simplifying assumptions were identified in connection with the classical approach.org/terms/Terms_Use. Only a few of them are mentioned here in order to discuss certain distinguishing features. This was first established for cylindrical shells of isotropic construction. and they reduced the mathematical model to a linear eigen-boundary-value problem (classical bifurcation approach). mention should be made of at least a few studies. the reader is referred to chapter 5 of Yamaki's book. The theoretical study of these two papers belongs to the group of postbuckling analyses of perfect configurations. This led to studies which attempted to attribute the discrepancy to (a) effect of prebuckling deformations. 74 Hutchinson and Amazigo. but with varied success. 28 Flugge. sandwich. In this limiting case the Yoshimura buckle pattern 45 can be achieved. The results are shown on Fig.3. These buckling equations are subject to radial and in-plane boundary conditions. 19. especially the in-plane type. This yields a linearized set of buckling equations (three partial differential equations) in the additional but small longitudinal. the critical load is reduced by approximately 10% from the classical value. however. and a constant transverse displacement takes place throughout. > R/t = 1000. Moreover. Boundary conditions The other possible source for the discrepancy is the effect of boundary conditions. R = 10. the linearized buckling equations can be obtained by employing the perturbation technique.2 The numbering is different in Ref. For the case of uniform axial compression. and w".Appl Mech Rev vol 39.5 X 106 psi). Redistribution subject to ASME license or copyright. The axial stress may or may not be zero. By employing the standard notation of Nxx. In-plane: N!' (1) where L. and then SS-3 and SS-4. the buckled state can be assumed as close to the primary state as desired. (/' = 1) to the strongest (/' = 4).5 1. 0.).0 = v" = 0. CC: w" = w" = 0.l.68. the order of Eqs.. and skin thickness. the next one to SS-2.4.2 the effect of in-plane boundary conditions for infinitely (very) long cylinders is such that for SS-1 and SS-2 the critical load is approximately half of the classical load. Nrv. by providing complete fixation against radial translation. 86 in investigating the imperfection sensitivity of laminated thin cylindrical shells. According to Hoffs 2 and Ohira's 34 results. which corresponds to zero hoop stresses. while for all other cases it is approximately equal to the classical load. with /' = 1. 1. Mvy. L/R = l. The imperfection shape was taken to be (almost) axisymmetric and the imperfection amplitude parameter £ ( = W^/t. studied numerous effects including the effect of in-plane boundary conditions and the effect of eccentricity in axial load. R. no 10. u". the additional quantities can be made infinitesimal. Note that for small £ values the results support the trend suggested by Hoff2 and Ohira. Therefore. in the case of lateral pressure the ends are free to contract or expand radially. circumferential. (2) corresponds to depicting a move from the weakest configuration FIG.2 For SS-1 the ratio of critical load to classical load is 0.org/terms/Terms_Use. the ends are not completely free to expand or contract. Small additional quantities are added to the primary membrane state in order to take us to the buckled state. 34 ie. where Transverse: SS: w" = AfA°v = 0.Nxy for stress resultants and Mxx. Mxv for moment resultants. Downloaded 23 Aug 2009 to 129. Simitses et al. ! (2) 0. and for SS-3 and SS-4.5.98. Similar results are reported by Yamaki 19 for axial compression and for pressure. four sets of in-plane boundary conditions exist. This effect by itself then cannot totally explain the discrepancy.34 results were obtained by using a linear buckling analysis. In the real and practical case. III. 0.16. the weakest configuration corresponded to SS-1. = °. The nonlinear solution scheme used accounts for prebuckling effects and initial geometric imperfections of specified amplitude and shape. 4 - 7 * " X • X 1 - ' = W". J = 0. the results at £ = 0 (obtained through extrapolation as | -»0) agree well with those of Hoff. the boundary conditions can be written as SS-/' or CC-/'. Results were generated for an isotropic configuration with the following structural geometry: E = 12 A GPa (10. Please note. for each case of radial or transverse boundary conditions. for SS-2.2. This means that in the case of compression the ends of the cylinder are free to expand.227. Moreover. Prebuckling state First. see http://www. respectively. through this approach. and radial displacement components. the prebuckling state has both membrane and bending stresses. According to Hoff. and / denote the length. = : iv" = 0. 1. one of the assumptions in the classical theory is that the prebuckling state is one of a pure membrane. The in-plane boundary conditions have been arranged (1-4) according to Hoff. In the classical approach. Therefore.30. thus resulting in a uniform radial displacement and constant in-plane stresses. (3) III. one seeks the existence of a bifurcation point. depending on whether the ends experience pressure and whether they are free to move axially. Stein31'33 and Fisher 32 independently showed that. v".asme. radius. Oct 1986 Simitses: Buckling and postbuckling of imperfect cylindrical shells 1519 two of these are discussed with some detail in this section. and the last one in the next section.16 cm (4 in. The radical boundary conditions that have received most attention are those corresponding to simply supported ends and to fixed or clamped ends.cfm . that their 2.55. Since. Effect of in-plane boundary conditions on the imperfection sensitivity of isotropic geometry (SS-/).2. Similarly. maximum imperfection amplitude/shell thickness) was varied from 0 to 2. N". 86 approach has been demonstrated 91 for axially loaded imperfect Although these observations are based on results from only cylindrical shells. different solution schemes. The approach that seems to have a better chance for accomplishing both is the one that deals directly with the imperfect configuration. 9 3 and it was obtained IV. for low values of £ (imperfection amplitude). a margin that is well within the accuracy that can be expected for that most of the discrepancy between classical theoretical preimperfection-sensitive buckling load calculations. As already mentioned in the introduction. 1. which Downloaded 23 Aug 2009 to 129. 1).cfm There are two important observations in connection with the results of Fig. The feasibility of this metric laminated geometry. Oct 1986 It has been suggested and efforts have been made towards the establishment of an International Initial Imperfection Data 87. These results cannot fully account for the tremendous scatter in Arbocz. a global outof-roundness. nor can they explain the stiffened specimen. it is the generally accepted conclusion between this and the experimental value is only 7%. In view of the results two questions can be raised: (1) Do we need to apply so complicated a shape for IV. in some cases by a factor of 4. experimentally obtained imperfection distributions into a statisSimilar conclusions are drawn from the results on an asymtical imperfection-sensitivity analysis. and the CC-1. the u" = 0 boundary conditions yield stronger configurations (SS-4 and SS-2) than the N°x = 0 conditions (SS-3 and SS-1). or a small initial curvature in a flat plate or rod. It is reasonable to expect that this approach could and should yield critical loads which compare well with the experimentally obtained buckling loads.2. tally supporting a thin cylinder in a laboratory experiment.3. which may or may not be dependent on axial position. IMPERFECTION SENSITIVITY by using the STAGS code (30-modes model). 30-modes model. In order to remove this restriction. has been used extensively as a benchmark in large discrepancy. because of initial geometric imspline fit are not as close to the experimental results as the 30 perfections. provided that one has complete knowledge of the initial geometric shape imperfections and that this knowledge is accommodated in the mathematical model. There exist two types of initial geometric imperfecmodes analysis. this is the case. for all £ values considered. recalculated by using 30 Fourier coefficients (30-modes There exist several types of imperfections and defects that analysis) does not physically resemble the measured initial affect the response of the configuration. The limit point is a measure of the critical condition.3 to CC-2 to CC-4. the no.8 N/cm. / = 1. STAGS-code. However.227. one (SS-4 and SS-2) (see Fig. 92? clude a local bubble in a thin circular cylindrical shell. for the clamped case. and various computer codes.3 N / c m 9 2 ) was presented. 93. 92 96 89 The above statements suggest that the primary reason for the buckling loads. improve the best numerical prediction. Thus. it appears that all cases (CC-. Moreover. for which carefully measured imperfection data are reported.4) are as these lines Elishakoff90 proposes a method for incorporating sensitive as the SS-4 (and SS-2) configuration.5.asme. and cubic response of the configuration.1520 Simitses: Buckling and postbuckling of imperfect cylindrical shells Appl Mech Rev vol 39. . they suggest that a substantial part of the The effort to establish an Imperfection Data Bank can be discrepancy may be attributed to various ways of experimentraced to the experimental buckling program reported by Singer. as dictions and experimental results can be attributed to the was pointed out in Ref. as etc). First. The shell is subject to limit point instability. and a comparison with the experimentally obtained value discussed in the next section. the initial geometric imperfection presence of small initial geometric imperfections. Initial geometric shape imperfections the imperfection in order to accurately predict critical loads? (2) These imperfections refer to deviations in shape of the Do CC-4 boundary conditions accurately describe the physical structural configuration.96 The best numerical prediction listed in these references is 243.92 In particular. several approaches have been used to establish the imperfection sensitivity of circular cylindrical shells and to predict critical conditions for these configurations.47"52 In this approach nonlinear shell theory is employed. the u" = 0 boundary conditions (CC-1 and CC-2) yield stronger configurations than the N"x = 0 conditions. and he also employed the STAGS code to calculate the group is addressed in this section. especially for commercially manufactured shells. to make these values) weak configurations (SS-1 and SS-3) are not as sensitive to initial geometric imperfections by comparison to the stronger data available to future potential users. 1. and. and the second group in the corresponding critical loads. 132 modes. SS-2.. shape. The difference As already mentioned. The CC-1 and CC-3 results are indistinguishably close. to present all available The second important observation is that the initially (low £ Bank imperfection data in identical form. see http://www.88 with a dual purpose 89 : first.93 The imperfection shapes in Ref. On the other hand. Arbocz96 grouped into two broad categories: (a) initial geometric impertried various fitting methods to recompute the imperfection fections and (b) material or constructional defects. ~ ' In these efforts several solution methodologies were used (Koiter-type Mactor analyses. Redistribution subject to ASME license or copyright. For higher values of the imperfection amplitude (£). Along conditions. the v" = 0 boundary conditions (SS-3 and SS-4) yield stronger configurations than the N"y — 0 boundary conditions (SS-1 and SS-2). and SS-4. between correlating theoretical results with experimentally obtained classical theoretical predictions and experimental results. The order in going from the weakest to the strongest configuration for £ > 0. this is a virtually impossible task. Examples of these imperfections inboundary conditions for shell AS-2 of Ref. 96 are much tions: initial geometric shape imperfections and load eccentricicloser to the true imperfection than the shape used in the ties. no 10. They are based on different kinematic approximations. SS-1.3 results are consistently smaller than the CC-4 results by approximately 12%. They are generally imperfection shape. Unfortunately. All of these calculations employed CC-4 boundary conditions. makes the geometry locally nonsymmetric. For this same range u" = 0 yields a stronger configuration in comparison to N"x = 0 (SS-2 stronger than SS-1 and SS-4 stronger than SS-3). The only difference is that the u" = 0 effect is not the same for the two sets. The first shape. discrepancy must lie elsewhere. AS-2 stringerexperimentally obtained critical loads. the CC-4 results are slightly (2-5%) higher than the SS-4 results.16. and Babcock.org/terms/Terms_Use. Several analytical investigations exist. that makes the geometry globally noncircularly cylindrical. For the clamped boundary free of many currently used simplifying assumptions. His new computations 96 did not next.93 It is indeed surprising The phrase "imperfection sensitivity" means to sensitivity of to see that computations using 74 modes. (226. who may devise improved computer codes based on nonlinear shell analyses. a few geometries. for the clamped case (not shown herein) the order of going from the weakest to the strongest configuration is the same for the entire range of £ values considered (0 < £ < 2) and it is CC-1.7 appears to be SS-3. Indeed. second. while the CC-2 results are slightly smaller than the CC-4 ones. because of the Poisson effect. In the intermediate range of load eccentricities (±0. Since. The values of Akl and Bkl are part of Table 3 of Ref.16. but only one column is used in the imperfection shape of Eq.97 In so doing. imperfection shapes.52 in which the imperfection can be expressed as a Fourier series in the circumferential direction with " axial coordinate" dependent coefficients. which induces outward prebuckling motion. and 11 are W^/t = 0. Critical load for symmetric imperfection (CC-3). the response seems to be insensitive to the eccentric application of the load.3 0. and this observation supports the contention.10. for axial compression.1 0.227.org/terms/Terms_Use. This implies that the response has a single / representation and the imperfection shape is similar to the response shape (transverse displacement. The critical loads corresponding to calculations based on simplified shapes. no 10.lt. 95. it reaches a maximum expansion at the critical load.cfm . positive eccentricity. While for the virtually axisymmetric imperfection. IV. so that an outward prebuckling deformation is present. 2. the Au and Ba terms are shown for several / values. In spite of these findings.2 0. and the actual imperfection data are only used to establish the maximum amplitude of the /-dependent (circumferential. These findings support the contention of several researchers 21. Both of these are extremely close to the experimental value of 226.2 0 0. (4). finally. For very large eccentricities (+10. It is suspected that one possible reason for this behavior (stabilizing effect for large positive eccentricities) may be associated with the Poisson effect.3 N / c m . more concrete proof is needed to fully assess the effect of initial geometric shape imperfections. Oct 1986 Simitses: Buckling and postbuckling of imperfect cylindrical shells 300 i i 1521 These questions were addressed by Simitses and Shaw. orthotropic. Critical load for axisymmetric imperfection (CC-3).1 _ i _ o 0-3 . Eq. buckling is followed by a predominantly inward lateral deflection. Note that for the symmetric imperfection.4 0. They employed several imperfection shapes from the measured data for the AS-2 shell (converted into double Fourier series). they employed a solution procedure developed earlier. Similarly negative eccentricities have a relatively small destabilizing effect. and laminated.98 that the critical load degradation is most significant when the imperfection shape is similar to the buckling mode shape. In Table 3 of Ref. £ is a measure of the imperfection amplitude. Eqs. Redistribution subject to ASME license or copyright. 0. the imperfection shape is taken as: N k=0 _ ___| _T_ - 260 £ •y 2 <£. Load eccentricities Another important class of initial geometric imperfections consist of load eccentricities. when converted to Fourier series coefficients95 the maximum amplitudes corresponding to / = n = 9. they used two simplified. are 228 and 223 N/cm. (4) implies that. 86 for shells of various construction. This effect was studied and reported in Ref. respectively.079. - _J£C kmx kirx / )cos — h cos —y~\Akl ny cos ny h Bkl sin — (4) J _ 0. The second group employs a virtually axisymmetric imperfection which has the form 2ITX mx ny W" = £t\ cos —— + 0. and 0.5-10 times the skin thickness) an irregularity is observed. and / is computed by requiring both minimum potential energy response and lowest limit point load. From the imperfection data. 86 do not support the above suggestion. from the description of the experiments CC-3 is a better boundary condi- 260 - Z240 200 160 120 0.£ U n^9 _n 1 ^ ^ — 10 | 180 " 140 0 1 1 0. According to Ref. tion than CC-4. Downloaded 23 Aug 2009 to 129. The findings reported in Ref. has a stabilizing effect. 97. (5) and (6). only one circumferential wave shape of measured imperfection is used. it is possible to apply the in-plane destabilizing loads (axial compression and shear) with an eccentricity relative to the neutral surface for metallic configurations. one axisymmetric and one symmetric. The solution procedure 52 employs a single / representation. at and after collapse. Moreover. CC-3 and CC-4. though. In the case of cylindrical shells. 95.082 0. or more times the skin thickness). As the load is applied quasistatically. isotropic. it has been suggested by some to stabilize the shell by applying the load eccentrically.4 0. In the first group. corresponding to wave number /. and then an inward motion takes place. 86 for small eccentricities (minus half to plus half the isotropic shell thickness). 2 and 3) are shown herein to support their findings of Ref.5 W /t FIG.asme. In each one of these.5.2.Appl Mech Rev vol 39. (6) In Eqs. Moreover. see http://www. 3. l = n) imperfection data. the cylindrical shell moves outward.062. this inward motion continues.052. This sequence of events and the corresponding stabiliza- where Akl and Bkl are Fourier series coefficients of measured imperfection shape. and observations. in order to study the possibility of simplification of imperfection shape. Moreover. Two figures (Figs. £ = W^/l.5 Wman/t FIG. agreements. Such imperfections have been used to establish the imperfection sensitivity99 or insensitivity100 of structural elements. (5) and (6). £ = I f ^ / / . only one / value is used. Two kinds of boundary conditions were used in calculating buckling loads.1 sin — cos — (5) The third group employs a symmetric imperfection of the form •nx ny W° = £t sin — cos — . The conclusions is that a simple axisymmetric or symmetric imperfection shape can be employed. The status of experimental buckling investigations of shells. VI. Gulf Publishing. 3. Eds (1982). Because of the importance of the effect of this material type of imperfections. E E (1958). E E (1974). Z Ver deut Ingr 52. 50th Anniversary Issue). Cambridge Univ Press. Ed (1976). Wiley. Eds Prentice-Hall. All of them123"125 use special construction and linear classical buckling analysis. Koiter. Appl Mech Rev 23(12). Offshore structures engineering III. performed by the author and his collaborators. S Y Zamrik and D Dietrich. Citerley. small rigid inclusions. Eds (1980). Computerized buckling analysis of shell. 10. D. in Thin shell structures. The effect of rigid inclusions on the stress field of the medium in the neighborhood of the inclusion has received limited attention in the past 25 years. REFERENCES 1. Many have viewed this as a combined bending-axial compression loading. 15. Fung. MATERIAL OR CONSTRUCTIONAL DEFECTS This group consists of small holes or cutouts. Eds (1982). for evaluating this important interaction. B. Y C Fung and E E Sechler. The interested reader is also referred to a recent review by Tennyson. 9. more work will soon appear. In Theory of shells. E E (1974). elliptic. all of the problems and questions related to buckling of metallic structures are subjected to reexamination in view of the new construction. and triangular) for both cylindrical and spherical thin shells. New York. 17-24. ASME. New York. pp 27-46. 12. W T (1970). and it covers various shapes of holes (circular. J N Goodier and N J HofT. P J. 8. J (1982). Die nicht achsensymmetrische Knickung diinnwandiger Hohlzylinder. New York (1986. Proceedings of the symposium dedicated to the 65th birthday of W T Koiter. With the advent of fiber-reinforced composites. R L (1982).16. Eds.1522 Simitses: Buckling and postbuckling of imperfect cylindrical shells Appl Mech Rev vol 39. For metallic materials. and Mikhailov. the interested reader is referred to Bushnell's8 book. such as nonuniform axial compression. Oct 1986 tion or destabilization is heavily affected by the value of the Poisson ratio or the Al2 term in the extensional stiffness matrix [Ajj]. NASA T N D-1510. 20. the effect of rigid inclusion on buckling characteristics has not been studied. AIAA J 4(9). Fung. Y C. the reader is referred to the work of Hoff116 and Shnerenko. Lorenz. Buckling of laminated composite cylinders: A review. 16. and others. J W (1979). Thompson. J W. Achsensymmetrische Verzerrungen in diinnwandigen Hohlzylinder. and Hutchinson. Yamaki. Downloaded 23 Aug 2009 to 129. DeLuzio. Collected papers on stability of shell structures. Questions of shell theory approximation 126 and imperfection sensitivity86'L27. Redistribution subject to ASME license or copyright. New York. Imperfection sensitivity and postbuckling behavior of shells. The effect of delamination is receiving more attention especially for fiber-reinforced composite construction. and Agelidis. Budiansky. Guide to stability design criteria for metal structures. 21. B. The interested reader is referred to the work of Libai and Durban 103 and their cited references. circular. 7.asme. 13. 4th ed. Instability of thin elastic shells. Washington DC. 1-28. N J (1966). for buckling of laminated configurations. 1353-1366. Cambridge. NJ. Moreover. J W (1966). 110 " 115 which deals with the effect of small and large cutouts on the buckling characteristics of cylindrical shells. 4. Wright-Patterson AF Base OH. Harding. This effect was addressed by Stuhlman. / Appl Mech 50(4. The smaller the Au term (low value of v for isotropic construction or placing the strong axis along the cylinder axis for an orthotropic construction). C D (1983). CLOSURE Very little has been mentioned. 935-940. and Koiter. Thin shells structures. J M T. A survey of some buckling problems. Most of the emphasis in the past has been on establishing stress concentrations and local stress distributions in order to predict material failures. one can find several studies which deal with the effect of material imperfections on the fatigue life of the structural component. Isr J Tech 4(1). Postbuckling Theory. Martinus Nijhoff Publishers. cracks. Y C. For a fairly extensive review. nonmonolithic skin-stiffener connections. Bushnell. Tennyson. Eds (1983). Englewood Cliff's. E. and Almroth 101 in the 1960s. the load eccentricity effect is more complex because it ties boundary effects (moments induced at the boundaries) with the effect of stiffener eccentricity. ACKNOWLEDGMENTS Most of the work.227. delaminations in laminated configurations. Proceedings of the first symposium on naval structural mechanics. Moreover. J F Besseling and A M A van der Heijden. Wiley. E Ramm. 241-260. 1985. in Structural mechanics. Buckling of shells for engineers (translation from Hungarian). 129 Because of the complexity of the problem. theory. pp 115-131. Finally. buckling of shells in offshore structures. D (1985). 3rd ed. Hutchinson. The historical development of shell research and design. Kluwer Academic. Johnston. was supported by the United States Air Force Office of Scientific Research.128 are as important a consideration for composites as for metallic construction. 2. 104 " 108 Savin109 has provided an extensive bibliography on the subject. The perplexing behavior of thin circular cylindrical shells in axial compression. L. Springer-Verlag. HofT. and Hutchinson. New York.86 In stringer-stiffened configurations. so far. and Hunt. E (1984). Furthermore. 3-25. in Pressure vessels and piping: Design technology—1981—a decade of progress. several finite element algorithms have been developed for the nonlinear buckling and postbuckling analysis of shells. Buckling of shells.org/terms/Terms_Use. 23. metallic configurations are continuously being replaced by laminated configurations with or without stiffening. B G. Springer-Verlag. no 10. there exists a small number of publications. Lorenz. Eds. Jun 1981. Kollar. Singer. New York. R (1908). Buckling: Progress and challenge. V. 22. G K. the greater the stabilization effect of the positive eccentricity. Pressure vessels and piping: Design technology—1982—a decade of progress. Collapse: The buckling of structures in theory and practice. Shell stability. Phy Z Leipzig J 13. 18. Ed. to be published). Anonymous (1962). Their102 studies showed that differences in buckling load of up to 50% can be attributed to this effect in some practical configurations. Delft Univ Press. North-Holland. Zamrik. 93-116. Budiansky. J E. G W. Pergamon. 17. N (1984). in Buckling of shells. As expected then. and Sechler. see http://www. and Sechler. in the early 1970s102 parametric studies and many tests were carried out at the Technion. in Trends in solids mechanics 1979. Englewood Cliffs NJ. Elastic stability of circular cylindrical shells. Prentice-Hall. 1978. I should mention another loading effect. and Dietrich. Norwell ME. also AFWAL-TR-81-3049. 1505-1510.117 Another material imperfection is the small rigid inclusion. S Y. The problem is extremely difficult and use of linear buckling theory is highly questionable. Sechler.118"122 As far as this author knows. Proceedings of a symposium. 5. Houston. Their financial support is gratefully acknowledged. cylindrical shells. Eds. Very few efforts are reported in the literature that deal with delamination buckling of cylindrical configurations. R C (1975). 11. Dowling. square. Ramm. W T. and Dulacska.cfm . Before closing this section. Amsterdam. 19. Proceedings of the third IUTAM symposium on shell theory. NorthHolland. R (1911). Amsterdam. New York. N. experiment and design. ASME. Composites 6. Ed (1982). 14. 6. as far as the effect of small holes on the response of shells and plates of nonmetallic construction is concerned. including cylindrical configurations. Babcock. there exists a number of investigations that deals with the effect of small cutouts on the stress and deformation (local) response of thin.5. 1766-1793. H. AIAA J 15(12). Lundquist. 463-506. Effects of large deflections and imperfections on the elastic buckling of cylinders under torsion and axial compression. 267-271. Springer-Verlag. Isr J Tech 17. Soong. M E (1967). AIAA J 2. Sheinman. von Mises. Japan. 709-714. hit J Mech Sci 7. Singer. thin cylinders. NACA TR-473. 47. Koiter. N J (1962). 54. G J (1977). An experimental analysis of the buckling of thin initially imperfect cylindrical shells subject to torsion. thesis. 86. A M C. Hess. Postbuckling behavior of circular cylindrical shells under hydrostatic pressure. Local buckling theory of axially compressed cylinders. Z Math Phys 58 337-357 (abstract of paper published in 1907 in Bull Polytech Inst Kiev).org/terms/Terms_Use. Compos Sci Tech 22.5. Notes on nonlinear shell theory. I. N J. and Crouzet-Pascal. Houston TX. R P N (1942). 739-744. Stability of imperfect laminated cylinders: A comparison between theory and experiment. N A C A TR 479. 72. On the postbuckling behavior of thin cylindrical shells. Imperfections.und Innendruck. 61. / Appl Mech 32. Abramovich. W H. Lee. Sheinman. ARC rep and mem no 3366. Sheinman. J C (1967). 63. and Brush. Arbocz. 207-212. G J (1977). Donnell. H R (1965). J Aero Sci 29. 71. J. Univ of Houston. 48. J Aero Sci 21(5). 1962. F I Niordson. Springer-Verlag. The buckling strength of a uniform circular cylinder loaded in axial compression. J Aircraft 11. Buckling of cylindrical shells under pressure by using Sanders' theory. Torsional buckling of orthogonal-anisotropic cylinders. Almroth. 30. A J (1964). 59-66. Initial Imperfection measurements of stiffened shells and buckling predictions. Sobey. The behavior of thin cylindrical shells after buckling under axial compression. Trans ASME Ser B 87. and Sheinman. F I Niordson. Trans ASME 56. Horton. Berlin. L H (1933). J Appl Mech 32. Ohira. Washington DC. Nachbar. 42. 87. N J. pp 37-40. Bel III 5. 795-800. Thielemann. D. I. 36. Timoshenko. Netherlands. Redistribution subject to ASME license or copyright. Simitses. and Hoff. Elishakoff. Budiansky. I (1985). B (1969). Leggett. L H (1964). Buckling of shells with general Downloaded 23 Aug 2009 to 129. Hoff. C D . R V (1914). 91. Arbocz. 39. J L.cfm . 34. Pandalai K A V. B O. and Yaffe. pp 143-152. Meek. Hayashi. Univ of Illinois. 80. 162-171. K Y. 29. 21-36. W F (1962). Buckling of imperfect stiffened cylinders under destabilizing loads. Southwell. Sheinman. E E (1933). D. in Proceedings of 70th anniversary symposium on the theory of shells to honor L H Donnell. 62. pp 345-346. 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A survey of methods of stability analysis of ring stiffened cylinders under hydrostatic pressure. 38. / Aero Sci 24 (Apr). M (1974). 74. AIAA j 5(3). and Durham. 28. and Abramovich. M (1962). Buckling of circular cylindrical shells in axial compression. and Simitses. 393-401. 82. in Collected papers on stability of shell structures. Thielemann. Donnell. 77. D Muster. J (1982). The strength of thin cylindrical shells as columns. Corona Book Co. The effect on the buckling of perfect cylinders of prebuckling deformations and stresses induced by edge support. A (1929). J (1974). Elastic stability of orthotropic shells. 57. Shaw. see http://www. and Hoff. J. in Proceedings of 2nd US congress of applied mechanics. G J. On edge buckling of axially compressed circular cylindrical shells. Buckling of thin cylinders under uniform lateral loading. The behavior of a cylindrical shell under axial compression when the buckling load has been exceeded. G J. Yoshimura. and Soong. 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W T (1967). On the mechanism of buckling of a circular cylindrical shell under axial compression. 827-829. Imperfection sensitivity of eccentrically stiffened cylindrical shells. Buckling analysis of geometrically imperfect stiffened cylinders under axial compression. H F (1948). H (1961). Statik und Dynamik der Schalen. The postbuckling equilibrium of axially compressed circular cylindrical shells. Reese. Postbuckling behavior of thin cylindrical shells. Exp Mech 4(9). Springer-Verlag. no 10. AIAA J 23(7). IngArch. 505-516.16. Ed. Sobel. W R. 73. B (1968). Imperfection sensitivity of fiber-reinforced. pp 203-214. pp 264-293. Nash. J Appl Mech 41(3). Ed. and Esslinger. 55. Patel. Z Ver deut Ingr 58. NACA TN 427. Michielsen. Simitses. Houston TX. W F. 53. Springer-Verlag. 70. Effects of imperfections on buckling of thin cylinders and columns under axial compression. 393-401. 31. and Wan. J (1985). and Answani. ARC rep and mem no 1185. 43. 68. a main contrib- 59. Madsen. Ed. Stability of thin-walled tubes under torsion. Stein. Th. S C (1965). R (1914). 411-416. NASA TN D-1510. G J. in Collected papers on instability of shell structures. 329-334. The Initial Imperfection Data Bank at the Delft University of Technology—part I. N J. The strength of tubular struts. Shell stability analysis: Theory and practice. AIAA J 4. in Proceedings of the eleventh Japan national congress of applied mechanics. NACA TR 874. Berlin. J (1957). Buckling of axially compressed circular cylindrical shells at stresses smaller than the classical critical value. 55-68. T T (1954).asme. Buckling of orthotropic sandwich cylinders under axial compression and bending. 89. 88. 41. 385-390. 26. Effects of boundary conditions on the stability of cylinders subject to lateral and axial pressure. Lansing MI. in Collected papers on instability of shell structures. Y (1964). The effcct of unrein forced circular cutouts on the buckling of circular cylindrical shells undcr axial comprcssion. 1l0. Sheinman. J (1984). pp 173-210. and. M V V (1969). 1970. S. Nonlinear stability analysis of an eccentrically loaded two-bar frame. D.cfm . 97. NASA CR-1l63. A N. N. pp 130-139. Y. Jr (1978). A K (1974). Lekkerkerker. Statics of thin-walled elastic shells (in Russian). in Thin-shell structures. Kiev. J Rhodes and A C Walker. M V V. IUTAM symposium. Libai. G J (1977). Jr (1971). J. Redistribution subject to ASME license or copyright. V P (1983). J. Simitses.227. Roco. Orlando FL. 123. Almroth. Stress distributioll around holes. K P. J. George J Simitses has taught structural mechanics and design in the Schools of Aerospace Engineering and Engineering Science and Mechanics at Georgia Tech. A M C (1972). Eds. C D. Stress concentrations in cylindrically orthotropic compositcs plates with a circular hole. Appl Meeh 38(2). also English translation. J Appl Mech 44(4). Buckling of cylinders with cutouts. Delft Univ of Tech. I (1985). Aug. 104. Stress state around a circular cylindrical shcll with an elliptic hole under torsion. Imperfection sensitivity of laminated cylindrical shells in torsion and axial compression. S I (1971). SES Inc. Lur'e. 120. Proc Roy Soc Ser A. Analysis of design methods for composites shells with holes. Starnes. 119. and Simitses. G J. K J (1981). 105-107. J Appl Mech 36(1). 714-719. 121. J Appf Meeh 48(4). B (1970). Compos Strllet. Scmenyuk. R C (1968). Giri. Kinoshita. He is the author of Elastic stability of structures (Prentice-Hall. and Batterman. Stress field due to a cylindrical inclusion with constant axial cigenstrain in an infinite elastic body. AIAA J 4. Collapse load calculations for axially compressed imperfect stringer stiffened shells. Buckling of shells with cutouts. D. experiment and design.39-46. Berlin. Stress about a circular hole in a cylindrical shell. and Babcock. AEC-TR-3798. Etlect of a slot on the buckling load of a cylindrical shcll with a circular cutout. Simitses. Elsevier. L J (1967). 96. Hellenic Society of Theoretical and Applied Mechanics. J.236-240. Eds. Sallam. Influence of eccentricity of loading on buckling of stringer-stiffened cylindrical shells. R E Krieger. Simitses. I. ATAA J 5(1). R C (1974). 108. pp 306-317. Brogan. 113. and Babcock. N P (1981). and is active in various committees and task groups of these societies. NASA IT F-607. 563-567. Society of Sigma Xi.1733-1742. Delft. Ed. 105. Stuhlman. Effect of longitudinal delamination in a laminar cylindrical shell on the critical external pressure. 872-877.5. 95. lilt J Solids Stmet 10. Prikl Mat Mekh (Ellgl Trallsl) 17(1). efl'ect of general imperfections on the buckling of cylindrical shells. G J. D~pt of Aerospace Eng. I (1985). B 0 (1966). fllt. M.24-30. 103. 102. ATAA J 3. Arbocz. in Buckling of structures. 68-75. 101. Strcss in a cylindrical with a rigid inclusion. also English translation. Prikl Mat Mekh (Ellgl Transl) 17(4). Comput Stmet 20(6). Englewood Cliffs NJ. F.asme. and Shaw. Tennyson. J. see http://www. 128. 124. Simitscs. Part I. J Eng Tnd 90. 1955 and 1956) and Stanford University (PhD in Aeronautics and Astronautics. Buckling of cylindrical shells subjected to nonuniform axial loads. A. 126. and Shaw. pp 283-288. A I (1947). J H. 118. NASA CR-331O. Arbocz. 114. B Budiansky. Moscow. P (1967). AfAA J 10(2). Starnes. imperfect cylindrical shells.706. Kounadis. Prentice-Hall. J G (1964). Ed. structural optimization.16. Englewood Clifl's NJ. Savin. Elastic fields of inclusions in anisotropic media. G J. and Almroth. no 10. 335-360. The elastic field of an inclusion in an anisotropic medium. Stress around an elliptic hole in a cylindrical shell. VanDykc. J Appl Meeh 48(3). Oct 1986 92. Hoff. 1986). ATAA J 9(1). The accuracy of Donnell's equations for axially-Ioadcd imperfect orthotropic cylinders. J. 58-63. Takao. T W (1981). Arbocz. Springer. Phys Staltls Solidi Sa. Arbocz. Chou. and Simitses. 534-536. Jr (1972). He received his degrees from Georgia Tech (BS and MS in Aerospace Engineering. 701. 98. Wellcr. Murthy. pp 251-273. The effect of general imperfections on the buckling of cylindrical shells J Appl Mech 36(1). 1466-1472. J. Englewood Cliffs NJ. Shaw. pp 139-146. and Mura. nonlinear response of structures. In. Downloaded 23 Aug 2009 to 129. 129. F Schroeder. D (1985). 115. 106. in Proceedings of AIAA / ASME/ ASCE/ AHS 26th SDM conference. Van Dyke. 13 0. Collapse of axially-loaded. 541-546. G J (1976). in Stability in the mechanics of continua. He has published extensively in these areas. 125. 1243-1269. 159-166.org/terms/Terms_Use. N J (1981). and Durban. Stress concentrations around circular holes in cylindrical shells. 563-569. Taya. S C (1974). Singer. 759. A. and Babcock. Netherlands. and Sheinman. G N (1968). D (1977). Utilization of STAGS to determine knockdown factors from measured initial imperfections. G J. and has given several lectures on related topics. 99. AfAA J 8. 94. Berlin. Buckling of imperfect stiffened cylindrical shells under axial comprcssion. and other societies. 1976. and Aim roth. C D. New York. Stability of cylindrical shells by various nonlinear shell thcories Z Allgew Math Meeh 65(3). Compos Strtlet 4. On the stress problems of large elliptic cutouts and cracks in circular cylindrical shells. ASME.1524 Simitses: Buckling and postbuckling of imperfect cylindrical shells Appl Mech Rev vol 39. Structural Stability Research Council. and Rocco. 107. Elastic stability of structures. Effect of a circular hole in the buckling of cylindrical shells loaded by axial compression. The effect of shape imperfections and stiffening on the buckling of circular cylinders. to be published. C D (1968). The buckling analysis of imperfection sensitive shell structures. Simitses. in general. His research interests include stability theOly under static and dynamic application of loads. Jr (1980). Jr (1972). A 1AA J 10(11). Influence of stiflener ecccntricity and cnd momcnt on stability of cylinders in comprcssion.28-38. He is a member of AIAA (associate fellow). 127. C E. Arbocz. in Proceedings of AIAA / ASME/ ASCE/ATTS 25th SDM conference. Tennyson. 853-858. 125-137. 93. Arbocz.768. 109. Y C Fung and E E Scchler. American Academy of Mechanics. Melbourne FL. Singcr. Troshin. 122. J H. Shncrcnko. 112. C D. WO. 116. D (1985). Part T. C D (1969). J Compos Mat 17(5). and Sheinman. and Babcock. and Holmes. random imperfections. expcriment and analysis. and Chou. Tennyson. Rep LR-275. Shaw. G J (1986). T (1971). in Proceedings of 11th international congress of applied mechanics. 270-289. and Babcock. T. J. Experimental investigation of the 111. R C (1984). Lond 300.l Solids Stmet 8.227-228. Thin-walled structures in aerospace design. Thcoretical consideration of the impcrfection contact bctwcen layers in bilayered cylindrical shells. Murthy. De Luzio. theOl)'. 1959. Palm Springs CA. Springer-Verlag. 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