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AER-56-12 A New Theory for the Buckling of Thin Cylinders Under Axial Compression and Bending B y L. H. DONNELL,2 AKRON, OHIO The resu lts o f exp erim en ts o n axial lo a d in g o f cylin d rical sh ells (th in en o u g h to bu ck le b elow th e e la stic lim it an d to o short to b u ck le as E uler co lu m n s) are n o t in good agree­ m e n t w ith previous th eories, w h ich have b een based o n th e a ssu m p tio n s o f perfect in itia l sh ap e an d in fin ite sim a l d e­ flections. E xperim en tal fa ilu re stresses ran ge fro m 0.6 to 0.15 o f th e th eo retica l. T h e d iscrep an cy is a p p a ren tly considerably greater for brass an d m ild -s te e l sp ecim en s th a n for d u ra lu m in an d in creases w ith th e r a d iu s-th ic k n ess ratio. T here is a n eq u a lly great d iscrep an cy b etw een observed an d predicted sh ap es o f b u ck lin g d eflection s. In th is paper a n ap proxim ate la rg e-d eflectio n th eo ry is developed, w h ich p erm its in itia l e c c e n tr icitie s or d ev ia ­ tio n s from cylin d rical shap e to be con sid ered . T rue in ­ sta b ility is, o f course, im p o ssib le u n d er su ch c o n d itio n s; th e stress d istrib u tio n is n o lo n g er u n ifo rm , an d it is assu m ed th a t final failure ta k es place w h en th e m a x im u m stress reaches th e y ield p o in t. T h e effect o f in itia l ec c en ­ tricities an d o f large d eflections is m u c h greater th a n for th e case o f sim p le stru ts. M ea su rem en ts o f in itia l e c ­ cen tricities in a ctu a l cylind ers have n o t b een m a d e; h o w ­ ever, it is sh ow n th a t m o st o f th e se d iscrep an cies c a n be explained i f th e in itia l d ev ia tio n s fro m cy lin d rica l fo rm are assu m ed to be resolved in to a d o u b le h a rm o n ic series, an d i f certa in reason ab le a ssu m p tio n s are m a d e as to th e m a g ­ n itu d e s o f th e se co m p o n e n ts o f th e d ev ia tio n s. W ith th e s e a ssu m p tio n s th e fa ilin g stress is fo u n d to b e a fu n c ­ tio n o f th e y ie ld p o in t a s w ell as o f th e m o d u lu s o f e la s tic ­ it y a n d th e r a d iu s-th ick n e ss ra tio . O n th e b asis o f th is a te n ta tiv e d esig n fo rm u la [5] is proposed, w h ic h involves re la tio n s su g g ested b y th e th eo ry b u t is b ased o n experi­ m e n ta l d ata. I t is sh o w n th a t sim ila r d iscrep a n cies b etw een experi­ m e n ts a n d previous th eo ries o n th e b u ck lin g o f th in c y lin ­ ders in p u re b en d in g ca n be rea so n a b ly exp lain ed o n th e sa m e b asis, a n d th a t th e m a x im u m b en d in g stress ca n be ta k e n as a b o u t 1.4 tim e s th e va lu es giv en by E q u a tio n [5]. It is a lso sh o w n th a t p u z z lin g fea tu res in m a n y o th er b u ck lin g p rob lem s c a n prob ab ly be exp lain ed by sim ila r co n sid era tio n s, a n d it is h o p ed th a t th is d iscu ssio n m a y h elp to o p en a n ew field in th e stu d y o f b u ck lin g p rob lem s. T h e la rg e-d eflectio n th eo ry developed in th e paper sh o u ld be u se fu l in exp lorin g th is field, a n d m a y be u sed in o th er a p p lica tio n s a s w ell. T h e paper p resen ts th e resu lts o f a b o u t a h u n d red new te s ts o f thin, cy lin d ers in ax ial c o m p ressio n a n d b en d in g , w h ich , to g e th er w ith n u m e ro u s te s ts b y L u n d q u ist,3 form th e ex p erim en ta l evid en ce for th e c o n c lu sio n s arrived a t. Sym bols U sed radial displacement under load (w = w2— is the movement due to the load) W , W i = numbers proportional to the amplitudes of w and Wi f = a stress function, defined by Equation [15] L r, Ls = wave-lengths of the deformation in the x and s directions €xj €sj €x8) Kxy KSf Kxs = extensional and flexural strains of the middle surface of the cylinder wall T„ Txs, Ox, Gs, Gxs = internal forces and moments per unit length of section as shown in Fig. 11 a, n = constants having to do with the assumed shape of the initial displacement, defined by Equation [4] = the internal strain energy due to the deformation c = \ / 12(1 — m2) ( ~ 3.3 for engineering metals) E, n, <r,, = the modulus of elasticity, Poisson’s ratio, and yieldpoint stress of the material a = average compressive stress in the axial direction, produced by the external load r, t = mean radius and wall thickness of cylinder x, s = axial and circumferential coordinates u, v, w = axial, circumferential, and radial displacements of the middle surface of the wall as shown in Fig. 10. Wi, u>2 = the initial radial displacement considered, and the 1 The experim ental work and m uch of the theoretical work were carried out in the G uggenheim A eronautical Laboratory of the Cali­ fornia In stitu te of Technology. Presented at the Fourth Interna­ tional Congress for Applied M echanics, Cam bridge, England, 1934. 1 G oodyear-Zeppelin Corporation, Akron, Ohio. M em . A .S .M .E . Dr. D onnell w as graduated from the m echanical-engineering course of the U niversity of M ichigan, 1915, and received the degree of P h .D . in m echanics from the sam e un iversity in 1930. H e had autom o­ bile engineering experience from 1915 to 1922. H e was instructor and assistant professor of engineering m echanics at the Univer­ sity of M ichigan from 1923 to 1930. H e w as in charge of the struc­ tures laboratory, Guggenheim Aeronautical Laboratory, California In stitu te of Technology, 1930-1933. Contributed b y the A eronautics D ivision for presentation at the Annual M eeting, N ew York, N . Y ., D ecem ber 3 to 7, 1934, of T h e A m e k ic a n S o c ie t y of M e c h a n ic a l E n g in e e r s . D iscussion of this paper should be addressed to the Secretary, A .S.M .E ., 29 W est 39th Street, N ew York, N . Y ., and will be accepted until January 10,1935, for publication in a later issue of Transactions. N o t e : Statem ents and opinions advanced in papers are to be understood as individual expressions of their authors, and n ot those of the Society. E G e n e r a l D is c u s s io n o f P r o b l e m a n d R e s u l t s HIS paper applies to cylinders thin enough to buckle below the elastic limit, and too short to buckle first as Euler columns. Such cylinders, if carefully made and tested, buckle in small regular waves, as shown in Fig. 1. T s N.A.C.A. R eport No. 473. 795 796 TRANSACTIONS OF TH E AM ERICAN SOCIETY OF MECHANICAL ENGINEERS I t is shown in Appendix 1 that, if length and end conditions are neglected, a theory based on the assumption of infinitesimal deflections and perfect initial shape leads to theoretical values for the buckling stress under axial load, and the wave-length of the buckling deflection given by the following: These results were obtained by Robertson4 by a simplification of equations obtained by Southwell. The same results can be ob­ tained by neglecting items which experiments show to be negli­ gible in a still more complete solution obtained by Timoshenko.5 In Appendix 1 a much shorter derivation is given, based on simpli­ fied equilibrium equations developed by the author.6 The same results can also be obtained by energy considerations. In Fig. 2 this theoretical value of P is compared with the values of P given by some hundred tests made by the author and by Lundquist.3 Numerous striking discrepancies will be noted: (а) There is a great scattering of the experimental points. (б) All the experimental values of P, and therefore of the buckling stress <r, are very much lower than the theoretical value. F ig . 1 T y p ic a l F a il u r e s o f T h in C y l in d e r s in A x ia l C o m p r e s ­ s io n , In all the experiments cited in this paper the ends of the cylin­ ders were clamped or fixed in some way. This stabilized the wall of the cylinder near the ends to such an extent th a t buckling always started a t some distance from the ends. When cylinders are tested free-ended, eccentricity of loading and other local condi­ tions at the ends are likely to obtain. Hence, buckling failure may take place at the ends a t a lower load than would be required to buckle the main part of the cylinder. Such local effects are not im portant in most practical applications (as the ends are usually fastened with some degree of fixity) and will not be con­ sidered in this paper. As is evident from the illustrations, the buckling waves ob­ served in experiments are comparatively small. In most cases there were about ten waves around the circumference, and the wave-length in the axial direction was invariably of about the same size as in the circumferential direction. This immediately suggests that the length of the cylinder should have little effect on its buckling load unless it is very short (with a length less than one or two wave-lengths). This conclusion is entirely borne out by the tests. No correlation could be observed between the length and strength, although many series of cylinders, identical except for length, were tested. In a great many cases buckling occurred over only a comparatively small part of the length, the rest of the cylinder remaining entirely unbuckled. I t is evident from these facts that, except for very short cylinders, the exact degree of end fixity (provided it is sufficient to insure against local weak­ ness as already mentioned) can have little effect on the buckling strength, as different degrees of end fixity are roughly equivalent to different effective lengths. I t will be assumed in this paper th a t the cylinders are long enough so th at length and end condi­ tions can be neglected. (The tests indicate th at they can be ne­ glected even when the length-radius ratio is considerably less than one.) F ig . 2 ders C o m p a r is o n o f E x p e r im e n t a l S t r e n g t h s o f T h in C y l in ­ U n d e r A x ia l C o m p r e s s io n W it h C l a s s ic a l T h e o r y (c) Instead of being constant, the experimental values of P show a very decided tendency to become smaller with increasing r/t. (d) The experiments made by the author, which were made on brass and steel specimens, give consistently lower values of P than those made by Lundquist, which were made with duralumin specimens. As to the shape of the buckling deflections, a first discrepancy is th a t in all experiments the wave-lengths in the axial and circum­ ferential directions are consistently nearly equal, which would require th a t X = S, whereas Equation [2] requires no definite relation between X and S b u t only th a t (X + S )2/ X have a definite value. Thus [2] would be satisfied, for instance, if 4 R . & M . N o. 1185, B ritish A .R .C ., 1929. • S. Tim oshenko, “Theory of E lasticity," 1914 (in R ussian), p. 392. * N .A .C .A . R eport N o. 479. AERONAUTICAL E N G IN EER IN G S = 0, X = 1, in which case [1] and [2] reduce to the “sym­ metrical” theory for the buckling of cylinders under axial com­ pression.7 If we assume that X = S, as indicated by the tests, [2] gives Fig. 3 shows this value compared with the values found from the experiments. I t will be seen that again there is great scattering, but th at the experimental values of X or S are consistently much smaller than the theoretical; that is, the experimental wave­ lengths are much larger than the theoretical. Attempts to explain these discrepancies have not been very satisfactory. The author believes th a t there is nothing incorrect in the classical analysis described, but th a t the assumption it makes th a t there are no initial deviations from true cylindrical shape (or unevennesses in the physical properties of the material or other imperfections—for the purposes of our discussion these can be assumed to be replaced by equivalent deviations from perfect shape) is not permissible in this case, at least for ordinary test specimens or ordinary practical applications. Theories neg­ lecting initial inaccuracies, which are, of course, always present to some extent, give good approximations in other buckling prob­ lems, but these inaccuracies seem to be much more im portant in this case. I t is easy to explain why this is so. When a developable sur­ face, such as a flat or a cylindrical surface, is deformed to a nondevelopable surface, there are produced (besides the flexural and stretching strains due to change of radius, which are considered in usual theories) strains which might be called “large-deflection strains,” th at result from stretching and compressing the develop­ able shape into a non-developable one. These are of more im­ portance than is commonly realized. In the bending of a strut a large-deflection theory is not needed unless the deflections are of the order of magnitude of the length of the strut, and as such deflections are of little practical interest, large-deflection theories have received little attention. B ut the aforementioned “largedeflection strains” in sheets become of importance in general when the deflections are of the order of magnitude of the thick­ ness of the sheet. For the very thin cylindrical shells which are under considera­ tion (especially those rolled up from sheets, as were the speci­ mens in the tests, and as is the case in all common applications), the initial deviations from cylindrical form are already of this order of magnitude. When the compressive load is applied these initial displacements are, of course, increased. The stresses due to the “large-deflection strains” accompanying this movement in­ crease very rapidly, and combined with the direct compressive stress and the other stresses commonly considered, reach the yield point of the material at certain points long before the load has risen to the value given by Equation [1 ]. Beyond this point it is evident th at the resistance of the cylinder will rapidly fall, so th at complete failure must take place soon afterward. I t may be argued that the same effect must take place in the buckling of flat panels, and that, in this case, the ultimate failure is far above th at given by the usual stability theory. This is true, but it does not invalidate the explanation given. There is nothing surprising in the fact that in one case the ultimate load—reached soon after the most highly stressed material passes the yield point—comes well above the classical stability limit, while in the other case it comes well below. In the case of a thin flat panel the stability limit itself is very low; the load at which the combination of common and “large-deflection stresses” reaches the yield point (at the edges of the panel) is much higher, and the panel can go through the stability limit without complete 7 S. Timoshenko, Zeitschrift filr Mathematik und Physik, vol. 58, 1910. AER-56-12 797 failure, because of the artificial support given the edges. In the case of a cylinder the classical stability limit is comparatively very high, and the large-deflection stresses increase very fast ow­ ing to the small size of the waves, so that failure is brought on by yielding a t a lower load than is indicated by the classical theory. B ut the two cases are similar in principle, and a complete solution of the problem of the ultimate strength of flat panels can only be obtained by using large-deflection theory.8 The simplified F ig . 3 C o m p a r is o n o f S iz e o f B u c k l in g W a v e s T e s t s a n d b y C l a s s ic a l T h e o r y as G iv e n by large-deflection theory developed in this paper should be useful in making such a study. The initial displacements are probably not important for this case, so th a t it should be possible to obtain a more definite result than in the problem of the present paper. In other buckling problems, such as the buckling of struts and the buckling of thin cylinders under torsion, initial displacements or other inaccuracies must also be present, and yet in these cases a reasonable check with experiments is obtained without con­ sidering these questions. The explanation is th a t in these cases, as in the case of the flat panel, the classical stability limit is below the load a t which the most highly stressed point would pass the yield point (because the large-deflection stresses are absent in the case of a strut, and they are less important in the case of torsion of a cylinder, as the buckling shape is more nearly a developable surface) and there is no artificial support which can carry some part of the structure through the general stability limit without failure, as is the case with the edges of flat panels. But there may be unexplored ranges of dimensions or materials where these questions are important, even in these cases. The author has noticed th a t in the buckling under torsion of very short cylinders, for which the buckling shape is further from a developable sur­ face than it is for a long cylinder, the yield point is usually reached a t about the theoretical stability limit. In this case the stiffen­ ing effect of the large-deflection stresses (and in general such stresses must always have a stiffening effect, because the increase in internal energy due to them must be supplied by an increase in the external load) just about balances the opposite effect of the yielding of the material, so th a t yielding and deepening of the buckling deflections continue for a long time with little change in 8 A v ery approxim ate solution for this problem has been given by von KArm&n. See “T he Strength of T hin P lates in Com pression,” A .S .M .E . Trans., vol. 54, 1932, paper A PM -54-5. 798 TRANSACTIONS OF TH E AMERICAN SOCIETY OF MECHANICAL ENGINEERS This probably explains another puzzling experimental fact, the external load.9 We evidently have here a third type of prob­ lem in which these questions are important. I t is thought th at noted by Lundquist,10 which the author corroborates, th at local these few remarks, and the discussion of the case of the cylinder buckling frequently occurs somewhere in the cylinder without pre­ cipitating complete failure, as the load can continue to increase under axial compression, may help to clarify other points in the study of buckling problems, which hitherto have been puzzling. without any general buckling until complete failure takes place suddenly over a large part of the cylinder at the load which Let us now consider in more detail just what happens when a compressive load is applied to a thin sheet or strut having initial would be expected if no preliminary buckling had taken place. I t now seems evident th a t final failure will, in general, take deviations from straightness in the direction of the compression. Consider first the simple case of a hinged-end strut of a certain place when the stresses due to one of the components of the initial size. If it has an initial displacement in the shape of a half sine displacement (combined with the direct stress) reach the yield wave, the amplitude of the displacement will be increased when point. The component which produces final failure (and which a compressive load is applied. The ratio of increase will be small evidently has the same wave-lengths as the final failure, Fig. 1) for small loads, but will become very great when the load ap­ is thus the one which causes yielding to take place at the lowest proaches the theoretical stability limit. If the initial displace­ value of the external load. We can determine the characteristics of this component from this fact by using minimum principles ment were in the shape of a full sine wave, the amplitude of this displacement would also increase when a compressive load is ap­ or their equivalent. We would naturally expect th at the component having wave­ plied. The ratio of increase would also be small for small loads and would increase as the load increases. I t would become in­ lengths as given by Equation [2] or [3] would cause this yielding, finite if the load could be raised to four times the theoretical sta­ and hence final failure, sooner than any other component, as bility limit. Now imagine that the initial displacement is a com­ its amplitude will certainly increase faster than th a t of any other bination of the two displacements mentioned. If the displace­ component, as the load is increased. But, as noted before, the ment is small, the action which takes place will be a superposition tests indicate th at the component actually causing failure always has a considerably longer wave-length than this. A reasonable of the two actions just described. The rates of increase of the explanation for this is th at the initial amplitudes of the different two components will be enormously different when the load is near components are not equal. It is certainly natural to suppose that the theoretical stability limit, but they will not be so very different the components of the initial radial displacements which have when the load is small. Similar actions will take place in the case of a cylinder under longer wave-lengths will, in general, have larger amplitudes as well. Hence, failure may be precipitated by a component with axial compression. For convenience in discussing this question a longer wave-length than [3], rather than by the component with we may consider the total initial radial displacement from true cylindrical form to be made up of numerous component displace­ this wave-length, because the first has a much greater initial amplitude, in spite of the fact th a t the amplitude of the second ments, each of wave form (similar to the displacements shown in Fig. 1), but having different wave-lengths in the x, or s, or both increases faster, and in spite of the fact, also, th at the stresses directions. Now suppose th a t only one of these components of produced by the deformation decrease as the wave-length in­ the displacement is present. As the compressive load is in­ creases. The calculations given later in this paper show that this is easily possible. In this connection it should be remem­ creased, the amplitude of this component will increase until, at some point of the wave, because of the combination of the direct bered th a t failure takes place at only a fraction of the theoretical stress (the average compressive stress in the axial direction re­ stability limit given by [1 ], at which time the component with sulting from the load) with the other stresses produced by the wave-length [3] does not increase so very much faster than its deformation, the yield point of the material is reached. Such rival components. We have been speaking of “components” of the initial displace­ yielding will occur simultaneously at corresponding points in ment of the cylinder without being very definite as to their shape. each of the waves of the displacement all over the cylinder. Actually this is, of course, as stated in the beginning, only a con­ When the load is increased beyond this point, it is obvious th at venient concept for the purposes of the previous discussion. To the resistance to this form of displacement will be greatly lowered, put the question on a definite basis, actual measurements should and the displacement will increase very rapidly (with probably a be made of the initial displacements in many specimens. The falling resistance to the load, which would explain the sudden, explosive failure which always takes place in tests) until we have author has not had the opportunity of making such measure­ ments and has no data of this sort from other sources. In the such a condition as is shown in Fig. 1. Now let us suppose th a t all the components of the displacement absence of such data the only thing th at can be done is to dis­ are present. When a compressive load is applied to the cylinder cover if there is a possibility of explaining the experimental facts on the basis of reasonable assumptions as to the initial dis­ the amplitudes of all the components will tend to increase. placement. Stresses will be produced due to each one of these components. The initial displacement, whatever it may be, can be analyzed (It is shown later th a t the components can be defined so th a t the question of interaction between them is of minor importance; into a double harmonic series. When an external compressive in any case stress systems having the wave patterns of each of the load is applied, all the terms of the series will, in general, tend to components will be produced—this is all th a t is necessary for the increase in amplitude. But their effects will not be independent, following argument.) Combination of the stresses produced by as with similar harmonic terms in the case of a strut. I t can be several different components, together with the direct stress, may said with confidence that, in most cases, the action of one term cause yielding at certain points in the cylinder before yielding will be nearly independent of the action of another, th at is, the due to any single component, such as described in the last para­ combined effect will be nearly the same as the sum of their effects when present separately, but th at there will probably be certain graph, takes place. But, due to the different wave-lengths of the different components, such yielding because of combinations of groups of terms whose action will decidedly not be independent. components will be local. I t will not occur simultaneously at cor­ These will be groups of terms which, taken together, form a shape which it is “easier” for the cylinder wall to deflect to than a single responding points in each of the waves of one of the components, as described before, and hence it will not produce a great weaken­ pure harmonic shape. We would expect such groups to consist of a primary term, which determines the wave-lengths of the group, ing in the resistance to any one of the components. "IV A .C .A . R eport N o. 479, p. 12. 10 N.A.C.A. R eport No. 473, pp. 5 and 14. AERONAUTICAL EN G IN E ER IN G AER-56-12 799 are actually present should be sufficient for our purpose—to see and higher harmonics, which modify the shape of the primary if the required magnitudes are reasonable. I t might be empha­ term to an “easier” shape; or combinations of such terms with a secondary term or terms which enable some of the large-deflection sized here th a t it is not expected th a t the assumptions made stresses to be annulled by stresses due to change of radius. (We as to the nature of the initial displacements are anywhere near exact. I t is only necessary, for our contention", th a t they repre­ shall use such a term in the calculations given later.) sent average tendencies—the great observed scattering in the Each of the “components” of the initial displacement in the experimental results explains the wide deviations from the as­ previous discussion can be considered to represent one of these groups of harmonic terms. Since we try to include in each group sumptions which must be expected. In setting up the theory, a combination of the equilibrium and all the terms whose actions greatly affect each other, and since it has been shown previously th a t it is not important to con­ energy principles is used. Expressions for the extensional and flexural strains of the middle surface of the wall are first set up sider combinations of the stresses due to different groups, it seems that we shall make no very great error if we consider only one of by adding terms describing the large-deflection strains to the the groups and neglect the effect of all the others. The one we usual expressions. Using the usual relations between the internal forces and the strains, the equilibrium equations of the elements consider will of course be the one which precipitates failure, of the cylinder wall in the axial and circumferential direction are chosen by the condition th at it gives the lowest final failure load. Any conclusions which we draw from this calculation will be on set up, the same as in small-deflection theory. These enable a “stress function” to be used, and it is then possible to derive, the conservative side, because the chief contention to be made is that the low failure load found in experiments can be explained first, a relation between the stress function and the radial dis­ by reasonable initial displacements, and any parts of the initial placement; and second, an expression for the internal elastic energy in terms of these two variables and the properties of the displacement which we neglect could hardly have had any other cylinder. If, now, we assume an expression for the radial dis­ effect than to reduce the calculated failure load. placement, we can obtain the corresponding expression for the In the calculations, which are given in detail in Appendix 2, it is assumed that the initial displacement and the final displace­ stress function from the first relation, and with the aid of the ment, and hence also the movement, are of the same geometrical second expression and the principle of virtual work we can ob­ shape. The assumed shape of displacement consists of a pri­ tain the external compressive load required to produce the dis­ placement. Using the expression for the displacement already mary term taken as a harmonic function of x and s, with the wave-lengths L x and L a, and the amplitude W \t/c or W t/c (for described, we obtain P (defining the external load) as a function of X , S, W lt and W. the initial displacement and the movement), and one secondary W ith the assumed displacement and the expressions previously term designed to annul as much as possible of the large-deflection found for the internal forces, we now set up the condition for yield­ stresses with stresses due to change of radius. Fig. 12a shows the ing at any point of the displacement wave by the maximum-shearprimary term. It is evident that at p — p the material is on the average circumferentially stretched, while the material at q — q energy theory. With this expression it would be possible to is on the average circumferentially compressed, to have equili­ determine the exact point in the wave a t which yielding first takes place, by maximum-minimum principles. This would brium in the circumferential direction. By superposing the symmetrical deformation shown at Fig. 126 of the proper ampli­ be a very complicated calculation, hpweyer, so instead it is as­ tude, we annul this average circumferential stretching and com­ sumed th a t yielding first takes place at the nodes of the waves, pressing, and although we introduce a certain amount of bending where trial indicates th a t the stress condition is at least approxi­ in the longitudinal direction, the total internal energy is con­ mately as severe as anywhere. W ith this assumption we ob­ ' f _ cay r \ siderably reduced, th at is, the addition of the symmetrical term as a function of X , S, Wi, and W. tain P, makes it a much “easier” form. The use of this term simplifies V \ ~ E ~ t) the calculations, and trial shows it to be very effective in reducing If, now, we knew the actual value of Wi, we could eliminate W the final failure load. Without it, it is impossible to explain at between the two relations and obtain P as a function of Pv, X , and all the low values of failure loads found in experiments. I t is, S. We could then, by trial or by using minimum theory, de­ of course, quite probable that a refinement in the magnitude of termine the X and S which make P a minimum (which means this term, or the addition of other terms, would be found to determining th a t “component” of the deflection which precipi­ be still more effective in reducing the final failure load. No at­ tates failure, as we previously decided to do), and thus find P tempt has been made to make such refinements because of the as a function of P„, which means finding a as a function of the complexity it would involve. properties of the cylinder E, r/t, and E /cay. As to our justification for assuming th a t the initial displacement As we do not know Wi, we shall do the reverse of this and deter­ and the movement will have the shape assumed, there is no doubt mine the magnitude of Wi, which, by the above process, gives th at the movement will take this shape, or a still “easier” one, if values of P as low as shown by the experiments, and then see if possible. And in neglecting possible refinements in the shape, the this value of W i is reasonable. However, although we do not conclusions which we draw from the calculation will be on the know the absolute magnitude of W i we can say something as to conservative side for the same reasons as were given in a previous its probable variation—or the probable “average tendencies” case. The amplitude of the secondary term is proportional to of its variation—with I, Lx, L,, etc. Thus for the flat sheets from W 2, th at is, it is a second-order term, and its absolute magnitude, which cylinders are made we can certainly expect the initial dis­ as given by the calculations, is always very small compared to placement to decrease with increasing thickness and th a t com­ the primary term, so that its presence would not be noticeable in ponents of the displacement of shorter wave-length will have tests. smaller magnitudes. Thus in Fig. 13 we would not expect two Of course we have no right to assume th a t the two terms are sheets rolled by the same method, one thin and one thick, to present in the actual initial displacement with the relative magni­ have components of displacement of like wave-length with the tudes which we have assumed, although it is possible th a t the same amplitudes, as a t a and 6, but would rather expect the ampli­ action of curving flat sheet into cylinders tends to bring this about. tude to be smaller in the thicker sheet, as at c. And in Fig. 14 we But the effective magnitude of the whole group will be some kind would not expect components of different wave-length in the same of average determined by the magnitudes of the terms actu­ sheet to have the same amplitude, as in a and 6, but would expect ally present, and the assumption th a t the calculated proportions a smaller amplitude for a shorter wave-length as at c. 800 F io . TRANSACTIONS OF TH E AMERICAN SOCIETY OF MECHANICAL ENGINEERS 4 D ia g ra m s S h o w in g H o w E x p e r im e n ta l R e s u lt s C a n B e E x p la in e d b y L a r g e - D e f le c tio n T h e o ry i f C e r ta in s u m p tio n s A r e M a d e a s t o I n i t i a l D e v ia tio n s F ro m C y l i n d r i c a l F o rm (T h e th eoretical v a lu e X or S cou ld n o t be d eterm in ed v ery e x a c tly w ith o u t a g reat d ea l of labor. V alu es sh o w n are rough estim a tes.) A s­ It is thus reasonable to expect that for the flat sheets Wi would vary more or less as given by the equation where a and n are non-dimensional quantities more or less con­ stant for sheets rolled by the same process. The process of curving the sheet into cylinders will certainly change this expression for W i considerably, and doubtless intro­ duce the quantity r into it. The nature of the change is some­ thing which could doubtless be analyzed, but a satisfactory analy­ sis is a difficult problem in itself. An attem pt was made to make a rough analysis, but the attem pt was abandoned as it was felt th a t such results were very likely to be misleading. In the ab­ sence of a satisfactory solution, it was felt th a t much could still be learned by using [4] as it stands, as the argument on which it is based, already given, certainly holds for curved sheets as well as flat. Some discussion is given later of possible changes due to the process of curving the sheet. This question has no effect on the main contention of this paper, th a t the very low failure loads found in tests can be explained by reasonably small initial displacements. A principal effect of curving the sheet into cylinders will prob­ ably be to upset the symmetry of expression [4] with respect to L x and L ,. This will greatly affect the ratio between the values of L x and L . given by the theory. W ithout some knowledge of the nature of this effect it is useless to try to see if the theory will explain the fact th a t Lx and L , are nearly equal in tests. Hence, the equality of L x and L , (and so of X and S) was assumed, to see if the other results of tests could be explained. Using the expressions for P and Py, already mentioned with [4], we find P as a function of E/cay, r/t, X (or S), n, and a. As expected, it is found th a t the value of X (or S ) which makes P a minimum depends on the value of n. I t is found th a t we get about the value of X shown by tests if n is 5/4, which is certainly a reasonable value. To bring P down to the level of the test values, a has to be about 1.1 X 10 ~6. Using these values of n, X , and a, we obtain P as a function of E/c<ry and r/t. The value of E/c<ry for the duralumin specimens tested by Lundquist was about 80, while the value of this quantity happened to F ig . 5 V a l u e s o f W t/c a n d W it/c R e q u i r e d b y L a r g e - D e f l e c tio n T h e o ry t o E x p la in T e s t R e s u lts be about 165 for both the steel and the brass specimens tested by the author. For each of these values of E /ca y we obtain P as a function of r/t. In Fig. 4a the corresponding values of P and r/t so obtained have been plotted, and the resulting curves can be compared with the experimental points for the same values of E /ary. I t will be seen th at the accordance is excellent, the theoretical curves showing nearly the same downward slope with increase of r/t, and decrease of P with increasing E /ca y, as shown by the test results. In Fig. 4b the value of X or S corresponding to these results is compared with the test results. Fig. 5 shows the values of Wit/c, and the values of W t/c at which failure starts, as given by the calculations. The values found for Wt/c, giving the magnitude of the movement under load up to the time th at failure starts, are very small, which corresponds to the test experience th a t no general buckling can be noticed by eye up to the instant of the sudden failure. The values found for Wit/c, which indicate the magnitude of the initial displacement which must be assumed to explain the low test failure loads, do not seem particularly excessive. They increase with the radius-thickness ratio, as common observation indicates they should, although this increase is due to the selection of a component with a longer wave­ length, rather than any effect on the initial displacement by the curving of the sheet. Of course, these values represent only a part of the total initial displacement, but probably in practise AERONAUTICAL EN G IN E ER IN G quite a large part; the fact that failure can apparently take place over only a part of the cylinder wall about as easily as over it all, as indicated by the tests, means that it is not necessary for the component most favorable to failure to be large or even to be present all over the cylinder wall, but failure can take place wherever, by accident, it happens to be large over a considerable portion of the wall. If measurements finally show th a t initial displacements are actually not as large as the theory calls for, then the relative crudity of the calculations is probably to blame. I t will be remembered that most of the simplifications were on the conservative side in this respect. Since the crude secondary term used in the displacement had such an enormous effect in reducing the calculated values of P for a given value of W i (or reducing the value of Wi for a given value of P), it is probable th a t further re­ finement would have an important effect in the same direction. There is not much likelihood th a t the result obtained of check­ ing the experimentally indicated decrease of P with increase of r/t and of E/c<ry is accidental and due only to the particular as­ sumptions made. The author has tried many combinations and finds th at any reasonable assumption regarding the initial dis­ placement seems to result in the same general tendency. Thus, varying the value of n in [4] has little effect on the results except to change the value of X or S which gives the minimum P. One effect of curving flat sheets into cylinders is probably to reduce the size of initial unevennesses, so th at we might expect some power of r/t in the numerator of [4], after this effect has been allowed for. Such an addition has also been tried, and it is found that we still get about the same reduction of P with increase of E/cay, and, as might be expected, still greater reduction of P with in­ crease of r/t, a greater reduction than is indicated by experi­ ments. However, the unknown effect of bending the sheets on the relation of Wi to the wave-lengths may easily counterbalance this. It is obvious that our assumptions as to the initial displace­ ments, and to a less extent the calculations themselves, are at best only rough approximations—a fact which may be excused by the newness and difficulty of the problem. Much work, both theoretical and experimental, must be done before the question can be considered as settled. Other factors may enter the prob­ lem besides those which have been discussed. When this paper was presented recently at the Fourth International Congress of Applied Mechanics, Prof. R. V. Southwell made a very interesting suggestion. When a cylinder is compressed, the axial shortening is accompanied by a circumferential expansion. This expansion is largely prevented at the ends, where the cylinders are attached to something else, and this holding-in of the ends relative to the rest of the cylinder produces (since axial elements of the cylin­ der wall are in the condition of a beam on an elastic foundation) a symmetrical displacement, similar to that shown at Fig. 126 except th at the amplitude is a maximum at the ends and damps out as we go toward the center. Calculations show th a t the wave-length of this displacement is the same as required for the secondary term of our assumed displacement when X or S is about 0.13. This value of X or <S is about half th a t given by the classical stability theory, but is still not as small as the aver­ age of the tests calls for. This phenomenon undoubtedly takes some part in the buckling action but how important a part it takes is still to be determined. I t is an experimental fact that when the buckling occurs over only a part of the length of the cylinder it usually occurs very near the end, where this displace­ ment is a maximum. (We could not expect the buckling to occur still closer to the ends, on account of the fixity there.) Some of the photographs given show this. I t is possible th at this phe­ nomenon may eliminate the necessity for some of the assump­ tions which have been proposed. In spite of the roughness of the calculations, it is believed that AER-56-12 801 the most important factors have been taken into consideration and th a t the results, while they do not prove th a t the discrepan­ cies between the experiments and the classical theory are to be explained in the general manner indicated, at least make this strongly probable. I t is particularly believed th a t it has been demonstrated th at the dependence of P on r/t and E/cay, indicated by the tests, is not a mere accident, or explainable by variations in experimental technique. I t therefore seems th a t this relation should be con­ sidered in design formulas. The most im portant practical signifi- F ig . 6 C o m p a r is o n op E q u a t io n [5 ] W it h T est R esu lts cance of this relation is th a t some improvement in buckling strength can be allowed for if a material with a higher yield point is used, and vice versa. The following formula for the average failure stress has been found to describe the relations shown by the tests quite well, and to give reasonable values for the extreme condi­ tions E /o v — 0, E/<rv = °°, and r/t = 0. I t gives a negative re­ sult when r /t is very large; in this case, which is far outside the practical range, <r can be taken as zero without great error. Curves obtained from this formula are compared with the test results in Fig. 6. As stated, the formula is designed to give the average strengths to be expected, and if it is desired to know the minimum strength likely to be encountered under any circum­ stances, some factor must be used with it; the value of the factor to be used under any given conditions can best be estimated from the test results shown in the figure. In the absence of anything better, this formula, with or without some factor, is recommended for design purposes. No theoretical work has been done on the allied problem of the buckling of thin cylinders under pure bending. The smallness of the buckling waves found for axial loading immediately sug­ gested th a t buckling will take place on the compression side of the bending specimen in practically the same way as it does in an axially loaded specimen, and th a t any results found for axial loading would apply also to pure bending, with some factor to allow for the fact th a t the stress varies from zero to a maximum instead of being constant around the circumference. 802 TRANSACTIONS OF TH E AMERICAN SOCIETY OF MECHANICAL ENGINEERS Numerous bending tests on specimens similar to those tested in axial compression completely confirmed this opinion. Photo­ graphs of typical specimens are shown in Fig. 7, while the results of the tests are plotted alongside the results obtained in axial compression, in Fig. 8. Buckling occurred over the compression side of the specimens in the same wave form, with approximately the same wave-lengths, as in the axially loaded specimens. It will be noticed from Fig. 8 th at the results show exactly the same decrease of P with increase of r/t as shown by the axially F ig . 7 T y p ic a l F a il u r e s of T h in C y l in d e r s in P ure B T h e E x p e r im e n t s The results of the experiments have already been described. Detailed data are given in Tables 1 and 2 for the axial compression and pure bending tests respectively. The specimens were made in exactly the same way and tested on the same special testing machine (shown in Fig. 9) as the torsion specimens described in a previous paper by the author,6 and the reader is referred to this paper for a detailed description of the testing machine and the technique of making the specimens. The axially loaded speci­ mens were loaded through a very frictionless universal joint very carefully centered to insure against eccentricity. One fact, not previously mentioned, which was noticed in mak­ ing the tests, is th a t no m atter how quickly the loading was stopped after final failure took place it seemed impossible to reach as high a load on reloading as was reached the first time— and this in spite of the fact th a t the average compressive stress e n d in g loaded specimens. In computing P the value of a w-as taken as the maximum stress in bending, according to elementary theory. The values of P found are about 1.4 times the values found in axial-compression tests for all values of r/t. This is just about what would be expected, as it shows th a t general buckling takes place when the stress at a point in the cylinder wall about 45 deg to the neutral axis rises to the value which produces failure in a uniformly stressed specimen. An ingenious theory for the stability of thin cylinders in bend­ ing has been advanced by Brazier.11 Accordingto this, the elastic curvature produced in the initially straight cylinder produces the well-known phenomenon of the flattening of the cross-sections of curved tubes under bending. The cross-section becomes more and more oval until a point is reached at which the resistance to bending starts to decrease, after which, of course, complete col­ lapse takes place. Serious objection can be made to the theoretical derivation given by Brazier in th a t small-deflection theory is used and is assumed to apply after the deflections become very large; th a t is, the small-order terms neglected in the derivation (which can be neglected when the deflections are very small) are, at the critical point, of the same magnitude as the terms considered. However, it is an undoubted fact that this type of failure does take place in comparatively thick tubes made of a material with a low modulus, such as rubber tubes and thick metal tubes stressed above the yield point. I t would seem that Brazier’s type of failure and the small-wave type are more or less indepen­ dent types of failure, and in an actual tube failure is produced by whichever type happens to occur first; that is, whichever type requires the least load. For thin metal tubes of the type con­ sidered in this paper, failure of the small-wave type, the same as in axial compression, probably always occurs first. For design purposes it seems safe to say th a t the maximum bending stress given by the elementary bending formula can rise to about 1.4 times the value given by [5 ] before buckling will, on the average, occur. F ig . 8 F ig . 9 11 R . & M . N o. 1081, British A .R .C . 1926. t io n o f C o m p a r is o n of A x i a i .- C o m p r e s s i o n T ests and P u r e - B e n d in g S p e c ia l T e s t in g M a c h in e W h ic h A p p l ie s T o r s io n , B e n d in g , a n d C o m p r e s s io n (o r T any C o m b in a ­ L oads e n s io n ) AERONAUTICAL EN G IN EER IN G AER-56-12 803 T A B L E 1 A X I A L C O M P R E S S IO N T E S T S '^ due to the load was in many cases ,-------------------- s —S teel T u b es----------- B rass Tubesonly a small fraction of the yield T h ick ­ N um T h ick N um ­ point. This tends to bear out D ia m e­ ness X E F a ilin g ber of E D ia m e ­ n ess F ailin g ber of ter, L en gth , 103, X10-6 load, w a v es in ter, L en g th , X 103, X io - « load , w a v es in the contention that, even for very (in.) (in.) ( lb /i n .2), (lb) circum . (in.) (in.) ( lb /in . 2) (in.) (in.) (lb) circum . thin cylinders, the cause of final 5 .0 7 6 2 .88 3 1 .3 2 38 11 5 .6 7 6 5 .8 3 1 6 .3 670 10 5 .6 7 6 2 .7 8 3 1 .3 245 11 3 .7 5 6 5 .9 6 1 6 .3 503 9 failure is the fact th at at certain 3 .7 5 6 2 .9 2 3 1 .3 244 10 3 .7 5 6 5 .9 0 1 6 .3 476 8 3 .7 5 6 2 .7 8 3 1 .3 306 10 1 .8 8 5 6 5 .8 3 1 6 .3 682 8 points, strategically located to 1 .8 8 5 2 .8 4 282 6 3 1 .3 7 5 .6 7 6 2 .9 8 1 5 .7 119 10 weaken the cylinder for the type 1 .8 8 5 6 2 .7 2 3 1 .3 382 9 1 .8 8 5 6 141 2 .9 5 1 5 .7 7 3 1 .3 5 .6 7 6 2 .1 7 76 12 1 .8 8 5 6 2 .9 6 1 5 .7 158 7 of failure involved, the local stress 5 .6 7 6 2.11 3 1 .3 82 12 5 .6 7 6 2.11 44 1 5 .7 11 3 .7 5 2 .1 8 3 1 .3 6 133 10 5 .6 7 6 2.01 42 1 5 .7 11 passes the yield point. 3 .7 5 2 .1 3 3 1 .3 6 159 10 3 .7 5 6 2 .1 3 1 5 .7 48 10 Another short series of tests 1 .8 8 5 6 2 .0 5 3 1 .3 134 7 3 .7 5 6 41 2 .0 7 1 5 .7 10 3 1 .3 1 .8 8 5 6 2 .0 3 159 7 5 .6 9 12 5 .9 4 1 6 .3 622 9 which was made, the data for 12 2 .6 4 5 .6 7 3 1 .3 181 10 5 .6 9 12 5 .8 5 1 6 .3 855 9 12 2 .7 4 3 .7 5 3 1 .3 277 9 5 .6 9 12 5 .8 5 1 6 .3 815 9 which are given in Table 3, bears 12 1 .8 8 5 2 .7 4 3 1 .3 223 7 3 .7 5 12 5 .9 8 1 6 .3 547 7 out the same contention. In this 12 5 .6 7 1 .9 9 3 1 .3 81 10 1 .8 8 5 12 5 .8 7 1 6 .3 803 8 12 3 1 .3 3 .7 5 2.01 98 9 5 .6 7 12 1 .9 3 1 5 .7 50 10 eight cylinders were tested in axial 24 1 .8 8 5 2 .8 0 3 1 .3 239 5 3 .7 5 12 10 2 .0 7 1 5 .7 51 1 .8 8 5 30 1 .9 9 3 1 .3 91 7 5 .6 7 30 5 .9 2 6 12 1 6 .3 7 compression. The cylinders were 3 .7 5 30 612 5..91 1 6 .3 7 identical except that instead of 1 .8 8 5 30 5 .9 7 1 6 .3 848 8 being bent to the proper radius be­ TA BLE 2 PU R E B E N D IN G T E S T S » fore making the joint, as was done ■ ■ -------Bra i,S8 T ubes—S teel T u b es-----with all the other specimens, some T h ick ­ N um ­ T h ick ­ N um ­ of them were bent to the proper F a ilin g ber of E ness D ia m e ­ E ness F a ilin g ber of ter, d iam eter, L en gth , X103 X 1 0 -6 m o m en t w a v es in L en g th , X108 X 1 0 -6 m o m en t w a v es in radius, while others were sprung (lb /in . 2) (in .-lb ) circum . (in.) en .) (in.) (in.) ( lb /in . 2) (in .-lb ) circum . (in.) (in.) into shape from the flat sheet or 3 1 .3 458 11 5 .6 7 6 5 .6 7 6 2 .8 6 5 .8 1 1 6 .3 1320 10 3 1 .3 382 11 5 .6 7 6 2 .7 8 6 5 .8 9 1 6 .3 1500 5 .6 7 11 from an entirely different radius. 2 70 11 3 1 .3 3 .7 5 2.88 6 5 .8 9 1 6 .3 1108 3 .7 5 6 9 3 1 .3 276 10 3 .7 5 2 .8 0 6 5 .8 9 1 6 .3 1222 3 .7 5 6 8 The results of these tests are shown 162 12 3 1 .3 1 .8 8 5 2 .1 3 6 5 .9 6 322 6 1 6 .3 8 5 .6 7 in Pig. 15, in which the value of P 12 3 1 .3 196 5 .6 7 2 .0 9 6 3 .0 3 6 1 5 .7 198 10 5 .6 7 150 3 1 .3 10 •5.67 2 .1 3 6 3 .0 9 3 .7 5 6 1 5 .7 220 10 obtained is plotted against the 136 11 3 1 .3 3 .7 5 2.11 6 3 .0 0 1 5 .7 252 10 3 .7 5 6 3 1 .3 428 9 1 .8 8 5 12 2 .7 0 6 2 .9 7 1 5 .7 156 5 5 .6 7 difference between the final and 3 1 .3 224 8 5 .6 7 6 2 .1 3 12 2 .7 4 1 5 .7 122 11 3 .7 5 initial curvatures. I t will be seen 122 10 5 .6 7 6 2 .0 5 114 11 12 1 .9 9 • 3 1 .3 1 5 .7 5 .6 7 3 1 .3 126 10 3 .7 5 6 2 .0 9 12 1 .9 9 1 5 .7 88 10 3 .7 5 th at the specimens with the high­ 72 3 1 .3 6 3 .7 5 6 2 .0 9 24 1 .9 9 1 5 .7 86 10 1 .8 8 5 5 .6 9 12 6 .0 3 1284 1 6 .3 8 est initial stresses, due to the 5 .6 9 12 5 .8 5 1462 1 6 .3 springing, consistently gave the 5 .6 9 12 5 .8 5 1 6 .3 1570 12 3 .7 5 5 .9 6 1 6 .3 1118 *9 lowest values of P. To be sure, th variation of P is within 5 .6 7 12 1 .8 9 1 5 .7 92 11 3 .7 5 12 2.11 64 1 5 .7 9 the limits of the scattering observed l other tests, so th at this 5 .6 7 30 5 .9 3 1 6 .3 692 9 variation could possibly be accidental This seems hardly likely, 1 .8 8 5 24 5 .9 5 1 6 .3 468 8 however, in view of the number of the tests and the fact that T A B L E 3 T E ST S TO D E T E R M IN E E F F E C T OF IN IT IA L S T R E S S E S two specimens of each kind were tested and the same tendency O N F I N A L C O M P R E S S IV E S T R E N G T H 13 was exhibited by both specimens. The fact th a t all specimens O riginal rad iu s of O riginal rad iu s of cu rv a tu re of F a ilin g F a ilin g cu rv a tu re of were cut from the same roll of material probably reduced the sh e et, in. lo a d , lb sh e et, in . lo a d ,lb real scattering in this case. — 0 .5 6 ° 125 1 .5 180 — 0 .5 6 ° 122 1 .5 152 These results have an important practical implication which is — 4 . 3° 157 0 .5 6 119 quite obvious. They also bear out the contention th a t final — 5 . 0“ 142 0 .5 6 123 failure is precipitated by yielding of the material, as obviously A ll cy lin d ers w ere of b rass, w ith d ia m eter 3 .7 5 in ., le n g th 6 .0 in ., th ic k ­ ness 0.00295 in ., a n d E 1 5 ,7 0 0 ,0 0 0 lb per sq in . the initial stresses due to springing the sheet (present on the ° N e g a tiv e sig n in d ica te s original cu rvatu re w as in o p p o site d irectio n outer and inner surfaces all over the cylinder) will combine with from final cu rvatu re. the stress due to other causes to produce yielding before it would b2 d2 otherwise occur. Of course, if the wall is bent to the proper where V2 = — + V4 signifies the application of V 2twice; shape before making the joint, some initial stresses will also be present, but they will be much smaller and more localized than and V8 four times. This equation is satisfied if where the sheet is sprung to shape. Also, the cold-working of the material may raise its yield point slightly, which would also fall in with our contention. Appendix 1. Theory on the Assumption of Perfect Initial Shape and Infinitesimal Displacements In a previous paper12 the author has shown th a t the conditions for equilibrium of an element of the wall of a thin cylinder, under uniform axial compression, when the displacement consists of several waves around the circumference, can be simplified to We neglect edge conditions entirely because, due to the small size of the waves, it is not im portant whether there are an even num­ ber of waves in the circumference, or what the conditions are at the ends of the cylinders. (In the tests, buckling frequently oc­ curred over only a part of the length or circumference.) Substi­ tuting [7] in [6] and using the symbols of the present paper, we obtain Equation [8] gives the values of P required to maintain various states of equilibrium involving different values of X and S. ! N.A.C.A. R eport No. 479, pp. 13-15. 19<jy for th e s teel u sed oy w as arou n d 5 7 ,0 0 0 lb per sq in . in all te s ts . for th e b rass w as b etw een 2 8 ,0 0 0 an d 3 0 ,0 0 0 lb p er sq in . in all te s ts . .804 TRANSACTIONS OF T H E AM ERICAN SOCIETY OF MECHANICAL E NG IN EER S Buckling will take place as soon as P rises to the lowest of these values. B y inspection, or using minimum principles, the lowest -value of P obtainable from [8] is 2, obtained when the quantity >(X + S ) 2/ X = 1. We thus obtain Equations [1] and [2], which have been discussed. The equations of equilibrium of an element in the x and » directions can be taken the same as in the small-deflection theory,12 because the only new internal forces which we are con­ sidering (which are not considered in the small-deflection theory) are the large-deflection stresses, which form a part of Tx, T„ and Tx,. And T x, T», and Tx, are fully considered in these equations: Appendix 2. Theory Considering Initial Dis­ placements and Finite Deflections T h e strains of the middle surface of the cylinder wall are ob-tained in terms of the displacements u, v, wlt w2 from the geomet­ rical relationships between them. We find, for the linear strains in the x and s directions, the shear strain, the changes in curva­ tures in the x and s directions, and the unit twist These equations will be satisfied if we take * The terms involving squares or products of derivatives are largedeflection strains representing the change in length of elements due to their slope. The other terms are the same as in the smalldeflection theory.11 The second expressions for ex, kx, etc. are obtained by substituting the relations where / is the usual stress function, or Airy function, except for the constant factor Et*/c, which is used to sim­ plify the results. Equating the expressions for T*, T,, and T*, in [12] and in [15] and solving for Cxj and ta, we find We next eliminate u and v between the three equations of [9] by applying the operator — to the first equation, — to the da’ H ’ dx3 second equation, and subtracting these two equations from the third equation, to which the operator ----- has been applied. dads This gives us Wi in the first expressions, remembering that K ( = 1 + 2 — by defini­ tion) is a constant with respect to x and s, since w and wi are assumed to have the same geometrical form. The relations between the strains and the internal stresses, given by Hooke’s law, are the same as used in the small-deflection theory.12 The internal forces and moments per unit length of section (see Fie. 11) are Substituting [16 ] in this, we obtain the following relation betweei the stress fu n ctio n /a n d the radial movement w An expression analogous to [18], for the case of a flat plate without initial displacement (r = <o,K = 1) was first obtained by von K&nnfln.14 (However, the author made this derivation indepen­ dently, before learning of von Kdrmdn’s solution.) The internal elastic energy is [substituting Equations [15], [16], [13], and [10] in Equation [19] we obtain an expression for the internal elastic energy in terms of / and w. If / and w are harmonic functions of x and s, this simplifies to 14 Enzyklopadie der Math. Wiss., vol. 4, art. 27. AERONAUTICAL E N G IN E E R IN G AER-56-12 Equations [18] and [20] are general formulas which can be used in solving many other large-deflection problems in which the initial displacement can be taken as geometrically similar to the final displacement, or as zero (in which case K = 1). For flat sheets the first term on the right-hand side of [18] drops out. If an approximate expression for w is assumed, an expression for / can be derived from [18] and the boundary or other conditions, after which [20] can be used to apply the principle of virtual work. We assume for w the shape discussed before " 805 F i g . 11 1 and take Wi — — w, as implied in the definition of K . SubstitutW ing [21] in [18] and using the symbols X and S, we find Fio. 12 From our knowledge of the physics of the problem we know that Tx, T,, and Tx, will be harmonic functions, except for a constant component of T x equal to — t<r. Hence, from [22], f can be taken as The coefficients of the terms in [23] were found by taking them as unknowns, substituting [23] in [22], and solving for the values of the coefficients satisfying [22 ] for any values of x or s. The co­ efficient of the second term in [21] was determined in a similar way, so as to satisfy the condition that the part of T , (found by using [23] in [15]) independent of s shall vanish, as discussed in the first part of the paper. The constant C in [23] is found from the condition that the constant part of Tx shall equal — to- or Using [21], [23], and [24] in [20] and integrating over the circumference and length, we find the internal elastic energy F iq . 15 R e s u l t s o f T e s t s o p C y l in d e r s W i t h I n i t ia l S t r e s s e s D u e t o F o r m in g D if f e r e n t these operations are carried out, after substituting [21 ] in [26] and remembering that K is a function of W, we obtain an expression for P in terms of W, Wi, X , and S where h is the length of the cylinder, or of the part of the cylinder considered. The last term in [25] evidently represents the elas­ tic work due to the ordinary elastic shortening of the cylinder under the load. This has no effect in our problem and the deriva­ tion could have been simplified by omitting the non-harmonic part of [23 ] which produces this term. However, the justification for such an omission might not have been clear. The work done by the external forces during a virtual displace­ ment dW is equal to 2irrta times the average distance the cylin­ der is shortened during such a displacement, or B y the principle of virtual work, this can be equated to the £>E change of E. during the displacement dW which is — dW. dtc When It will be observed that if W i is set equal to zero and terms con­ taining W 2 are neglected as second-order terms, [27] reduces to Equation [8] of the classical theory. In carrying out the integrations of [25] and [26] it is assumed that L x and L , are even multiples of the length and the circum­ ference. This involves little error because of the small size of L x and L„ as discussed in the main part of this paper. We shall now set up the condition for yielding of the material at any point. According to the maximum-shear-energy theory16 which is generally considered to be the most exact expression of 11 See “Plasticity,” by A. Nadai, McGraw-Hill, N. Y., 1931. TRANSACTIONS OF TH E AM ERICAN SOCIETY OF MECHANICAL ENGINEERS 806 the condition for yielding, plastic flow under combined stresses at any point commences when calculations indicate that the material at the surface of the cylinder wall in the nodes of the wave form is in about as unfavor­ able a state as any, in most cases, at least. If we take x = s = 0 and z = t/2 in this expression we obtain where <ri, o-2, and <r3 are the principal stresses at the point and ay is the yield-point stress in simple tension. In our case the stress in the radial direction can be taken as zero and as one of the principal stresses, while the other two principal stresses, in the plane of the cylinder wall, are given by the formula where <Txj and 0*8 are the normal and shear stresses on planes Equation [4], discussed in the first part of the paper, can now be combined with Equations [27] and [32] to eliminate W and Wi, and obtain P as a function of P y, fi, X, S, n, and a. The influence of n, which enters [32], is not important and a value of 0.3 can be taken for it for all engineering metals. We shall also assume that X = S because of the difficulty of checking this ex­ perimentally verified relation, as previously discussed. With these assumptions [27] and [32] are somewhat simplified 0.7 0.9 and Equation [4] becomes Combining [27'], [32'], and [4'], we can obtain P as a function of P y, r/t, X , n, and a. Then we can determine n so as to make the value of X , at which P is a minimum, coincide with test results, and determine a to bring the general magnitude of the values of P down to the level of test results. The complexity of the equations made it impractical to do this directly. Actually, various values of W, Wi, and X were as­ sumed, which enabled the corresponding values of P and P y to be found from [27'] and [32']. This gave sufficient data to plot F ig. 16 T h e o r e t i c a l R e l a t i o n B e t w e e n P , P y, Wi, a n d f o b X = S — 0.07 perpendicular to the x and s directions. for the principal ^tresses in [28], we find W, Using these values The values of ax, <rs, and <rXs at a point in the cylinder wall a dis­ tance z from the middle plane are, assuming a linear distribution of stress, , Substituting [31] in [30] and using the expressions for TX, G X etc. [15], [13], [10], and finally [21], [23], and [24], we obtain an expression for P in terms of P„, W, Wi, X , S, x, s, and z. B y using minimum theory we could determine the value of x, s, &nd z at which P is a minimum—that is, the point at which yield­ ing first occurs, at the lowest value of a. But the expression is too complex to make this very practical. Trials and elementary families of curves giving the relation of P and Pt with W and W 1 for several values of X . Fig. 16 shows such families of curves for X = 0.07, and similar families were drawn for X = 0.04 and X = 0.10. Then, assuming values of a, n, X, and r/t, we find the value Wi from [4']. Taking E / c < t u = 165, as in the tests made by the author, P„ can be calculated, and the corre­ sponding values of W and P found from the curves such as shown in Fig. 16. Then by comparing the different results obtained for the different values of X , and plotting P against X for the same values of r/t, we can roughly determine the value of X at which P is a minimum. This gives sufficient data to plot curves such as shown in Figs. 4 and 5. This process was repeated with differ­ ent values of a and n until the combination used in plotting Figs. 4 and 5 was found. A cknow ledgm ent The author wishes to acknowledge the assistance of Messrs. L. Secretan and K. W. Donnell, who carried out most of the ex­ periments, and he wishes to thank E. L. Shaw and Sylvia K. Donnell for help in preparing this paper.