THE UNIVERSITY OF MICHIGAN INDUSTRY PROGRAM OF THE COLLEGE OF ENGINEERING AN ASYMPTOTIC SOLUTION TO A PROBLEM IN SHELL STABILITY Robert L Armstrong A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the University of Michigan Department of Engineering Mechanics 1964 October, 1964 IP-682

Doctoral Committee: Professor Ernest F. Masur, Chairman Professor Robert Mo Haythornthwaite Professor Albert E. Heins Assistant Professor Ivor K. McIvor

ACKNOWLEDGMENTS The author wishes to express his sincere appreciation to his Chairman, Professor Ernest F. Masur for his patient encouragement and his excellent instruction, and also to the other members of his committee for their interest and time. The author also wishes to thank the National Aeronautics and Space Administration for their financial support and the Industry Program of the College of Engineering for their assistance in the reproduction of this work.

TABLE OF CONTENTS Page ACKNOWLEDGMENTS..........................o.......... i. LIST OF FIGURES............................................ iv NOMENCLATURE............................................. v CHAPTER I. INTRODUCTION................................... 1 CHAPTER II. DERIVATION OF THE SHELL EQUATIONS.............. 8 CHAPTER III. SOLUTION OF THE SHELL EQUATIONS................ 16 3.1 Nondimensional Form of the Shell Equations................................. 16 3.2 Solution Near a Valley.................... 22 3.3 Solution Near a Ridge..................... 35 3.4 Equilibrium Considerations................ 53 CHAPTER IV. RESULTS....................................... 55 CHAPTER V. CONCLUSIONS........................... 69 APPENDICES A. GEOMETRIC RELATIONS FOR THE BUCKLED SHELL........... 71 B. EQUILIBRIUM CONSIDERATIONS........................ 75 C. EXPERIMENTAL RESULTSo.............................. 78 D. SOLUTION OF THE BOUNDARY LAYER EQUATIONS........... 79 REFERENCES............................................ 85 iii

LIST OF FIGURES Figure Page 1.1 Range of Experimental Values for the Buckling Load..oo..... o.......................................o 2 1.2 Effect of Shell Imperfections on the Buckling Behavior......................................... 4 2.1 Geometry of a Prebuckled Panel.................... 8 3.1 Geometry of a Buckled Panel........................ 17 3.2 Orientation of the p,. coordinate system...... 38 3.3 Relationship Between m and k for Equilibrium for Force Prescribed Buckled State................ 54 4.1 Final Shape of the Buckled Panel for s = 0...... 57 4.2 Final Shape of the Buckled Panel for s = 0, x > 0 for Various Values of Axial Load......... 58 4.3 Final Shape of the Buckled Panel for s = 0, x > 0 for Various Values of k,....... o....... 59 4.4 Buckled Shape for a Ridge of the Buckled Shell for p = 0...... o..oo........................ 61 4.5 Circumferential Stress in a Valley................ 62 4.6 Circumferential Stress in a Valley for x > 0 for Various Values of the Axial Force............. 63 4.7 Normal Stress Along a Ridge...................... 64 4.8 Axial Displacement ux/h Along the Line x = 0.. 66 A-i Geometry of a Buckled Panel.....o................ 72 A-2 Geometry of a Buckled Panel..........,.......... 72 B-l Equilibrium of a Valley........................... 75 B-2 Equilibrium of a Ridge............................ 76 iv

NOMENCLATURE A1 parameter (A1 = )'P/PE ) (Hak) A2 parameter (A2 = P/F A5 parameter (A5 k4 A4 parameter (A4 = JZ D flexural rigidity of the shell E Young's modulus F stress function referred to HiF' stress function referred to H2 Ncb additional stress resultants referred to HI N - additional stress resultants referred to H2 N~ prebuckling stress resultants ap N' N', N' additional stress resultants PP 5P, referred to H2 P applied axial stress resultant PR P nondimensional P (P =- ) Eh PE classical buckling load R radius of prebuckled cylinder dS element of surface area ST surface area where tractions are specified T surface traction vector TR force along a ridge v

TRz TRx z and x components of TR Tv force along a valley Tvz z component of T V potential energy change during buckling Vw potential energy change of existing stresses during buckling V potential energy change due to membrane stresses VB potential energy change due to bending stresses W deflection of the prebuckled shell from Hi W' deflection of the prebuckled shell from H2 W Y nondimensional W (Y = -) nh Y' nondimensional W' (yk= WL) a1 parameter (a1 (1 - k2)) a2 parameter (a2> v (1 k2)) 25 (2 k2)) a parameter (a - ( - k a parameter (a3, t2R (2 + k2)) a4 parameter (a4 ) - k) a2R a5 parameter (a i t ) 5 8n bi (i = 1, 2,.... 8) constants of integration c. (i = 1, 2,.... 8) constants of integration di (i = 1, 2,....8) constants of integration f nondimensional F (f = ) vi

F! f' nondimensional F' (f' - Eh2) n3h g parameter (g = R ) h shell thickness k deformation parameter ~~~~k ~parameter (k = 8 2)2 (1+ p2)2 1 representative buckled shell dimension 2, ) axial and circumferential buckled shell wave lengths m parameter indicating the magnitude of the stress in the field m parameter (m = ( s "' ) n number of circumferential buckles (2 4y2k2 p parameter (p = 1 - ) m p 2 472k2 q parameter (q = 2 - 1) s mcircumferential coordinate in 1 s circumferential coordinate in H2 s circumferential coordinA 2sin s nondimensional s (s = -) t parameter (t = e- fkn) 2gy Ua components of the displacement of shell middle surface A u components of the displacement of a general point A u displacement vector dv volume element of the shell w displacement of the shell middle surface normal to I1 vw displacement of the shell middle surface normal to 12 vii

w displacement of a general point Wr radial displacement of the shell middle surface x axial coordinate in H1 x axial coordinate in H2 A n2x x nondimensional (x = -x) y nondimensional w (y = -) y' nondimensional w' (y' = w nh Yi. Yi (i = 0 1 2,...) components of yi and yi Yi (i = 1,2,..., j = 1,2,..) components of y. and y' z coordinate normal to I1 z' coordinate normal to 12 ^, +v, V^ differential operators Ii,' 112 reference planes X angle ai a. (i = 1, 2,... 5) nondimensional ai (i = 1 1 9"- nh 7 parameter (y2 1= 1 56,8:~~~.Kronecker delta parameter (2 = ) n Eca additional strain components nondimensional coordinate ( =- - + 1 x T-s F2 TI*~~~n ~nondimensional coordinate (T = -) Q parameter X parameter (X = $2k/y X parameter (X = P2k/y X. (i = 1, 2,... 8) parameter 1I~~~~~Vi

pi aspect ratio of a buckle (. = x) Rs v Poisson's ratio e; nondimensional coordinate ( = ) p nondimensional coordinate (p = - + +) Ix Is 2 a nondimensional coordinate (a = s) 7 T additional stress components at a au general point T~ prebuckling stress components at a general point nondimensional coordinate (i= - ) Subscripts C, D, and y take on the values x and s. If any of these subscripts are repeated the quantity is summed. ix

CHAPTER I INTRODUCTION The problems of determining theoretically the conditions under which a thin circular shell under axial compression becomes unstable, and of determining the postbuckling behavior of the shell have been of interest to engineers and scientists for nearly sixty years. The first theoretical work on this problem was done by such noted investigators as Lorentz (1), Timoshenko (2) Southwell (3), and Flugge(4). They found what might be called the classical or Euler buckling load. This is the load at which an equilibrium configuration differing from the initial configuration by an infinitesimal displacement can be found. In other words, it is the load at which a bifurcation in the load-axial deflection curve exists. When a cylindrical shell buckles, the change in the potential energy of the shell can be expressed as a sum of second, third, and fourth order terms in the radial displacement, wr. The equilibrium equation in the radial direction can be found by setting the first variation of this additional potential energy equal to zero. If only the second order terms are used, the resulting equilibrium equation is linear. The resulting system is homogeneous, and the lowest value of axial load for which a nontrivial solution exists is the Euler load, PE * In terms of force per unit length of shell circumference this is given by PE - k -. 60^5 7 (.=.30). (1.1) C Ga

-,2(For a solution of the linear problem, see Timoshenko and Gere(5).) Experimental work, however, indicates that cylindrical shells under axial compression fail at values of axial stress only one tenth to nine tenths the Euler loado Along with this sharp reduction in the actual strength of the shell a wide range of scatter is also observed. This is indicated in Figure 1.1. (Donnell and Wan(14).) I.00 RANGE OF EXPERIMENTAL DATA P.So 0 /o00 2O000 3000. Figure 1.1o Range of Experimental Values for the Buckling Load. Thin cylinders under axial load buckle either generally into a pattern consisting of a large number of circumferential and axial rows of diamond-shaped buckles, or they buckle locally into isolated buckles or into only a few axial rows of circumferential buckles. As buckling progresses the number of circumferential buckles (n) decreases, the value of n being near ten for cylinders which buckle in the manner described. The final buckled shape is observed to consist of regions of small curvature connected by ridges and valleys of very high curvature. This is easily seen in the photographs shown by Fung and Sechler6 and by

-3Lundquist(7). This is the postbuckled shape which is analyzed in this work An explanation of the discrepancy between theory and experiment has been attempted by several investigatorso Donnell(8) in 1934, was the first to use a finite-deflection analysis which included the effect of initial imperfections. Unfortunately his work was not general enough (9) and attracted therefore only limited attention. Von Karman and Tsien( in 1941, extending the idea of Donnell, also considered finite displacements from the prebuckled cylindero They found equilibrium states which could exist at values of axial stress much less than the buckling load. (10) This method was refined and extended among others by Leggett and Jones Michielson(11) Kempner(12), and finally by Almroth(l3) in 1963, Almroth showed that a possible equilibrium state can exist when the external load is only ten per cent of the Euler loado An answer to the question of how the shell reaches its postbuckled state, which seems to account for the wide scatter in experi(14) mental data, was put forth by Donnell and Wilan). in 1950. They postulated an initially imperfect shell, the initial imperfections being of the same form as the buckled shape. They determined that the shell was very sensitive to these imperfections. A series of load-deformation curves were found for various values of the imperfection parameter, (See Figure o12.) Koiter(l5) demonstrated the extreme sensitivity of cylindrical shells to imperfections erby showing that the curve givingthe buckling load as a function of the imperfections amplitude may have infinite slope as the latter approaches zero,

1,0 | X- Perfect Shell Average Stress Closlcol fj3ckllng Stress / /^.^^^ ^ss^"Very Imperfect Shell 0 0 /.0 Averse. Strclin Oassi&cnl Bick\nw Strncw Figure 1.2. Effect of Shell Imperfections on the Buckling Behavior. Several investigators have investigated the dynamics of the postbuckling problem, such as Kadashevich and Pertsov (16), Agamirov and Volmir(17), and Yao(18)0 The results of these studies have shed no significant light on the basic controversy and are therefore not discussed here any further. It is of significance in connection with the present work that none of the previous investigations cited here represent exact solutions to the relevant shell equations, whose nonlinearity has made an exact analysis prohibitively difficult. Instead, the approach which has been

utilized most widely to obtain approximate solutions has been to set up an expression for the potential energy and to minimize that expression within an aggregate of kinematically admissible deflection functionso This is, of course, a permissible scheme, provided that the number of function considered is sufficiently large and the functions themselves represent good approximations. In particular, if the actual deflected surface is sufficiently smooth, an aggregate of trigonometric functions is usually workable. If sharp discontinuities in the functions or their derivatives occur convergence becomes slow or altogether questionable. This phenomenon has been observed in the present case, in which the addition of ever increasing numbers of terms has led to approximate solutions of formidable algebraic complexity without displaying satisfactory convergence as the deflections become large. It may be conjectured that this is a basic shortcoming of the method selected. Indeed, the observed presence of diamond-shaped buckles separated by sharp creases (or internal "boundary layers", as discussed later on) raises the question of the suitability of the representation by an aggregate of simple trigonometric waveso The method employed herein is also approximate, but in,a different sense. Energy techniques are not employed,, Instead, the shell equations are solved approximately through perturbation expansion in terms of a small parameter which is related to the thickness of the shell. Since this parameter (after a number of order-of-magnitude assumptions based on observed behavior) appears as the coefficient of the highest

-6derivatives in the equations, the expansion is singular and gives rise to boundary layers separating "fields" of relatively smooth buckles of vanishing Gaussian curvature. Aside from certain inaccuracies in satisfying some of the kinematic boundary conditions (believed to be of minor significance), the solutions obtained, though not unique, may therefore be considered exact in the limit, that is, as the shell thickness approaches zero. The idea of using a boundary layer approach to problems in shell stability is not new. Friedrichs(l9), in 1941, investigated the problem of the buckling of a spherical cap by using a boundary layer analysis. Other early work using boundary layer analyses in the investigation of the behavior of structures was done by Friedrichs and Stoker(20'21) in connection with the problem of a circular plate under uniform radial compression. More recently the development of a boundary layer in a flat plate with fre edges has been investigated by Fung and Wittrick(22) and (23) by Masur and Chang, There are also examples of boundary layer analyses in the investigation of problems of linear shell theory. See, for example, fph ps) (26) the recent work of Reiss,(2425) and Johnson(6) in the treatment of the linear problem of a cylindrical shell under axial compression. The development of internal boundary layers in cylindrical shell buckling is suggested by the observed buckled shapeo The boundary layers are those regions which include the valleys and ridges which delineate the individual buckles. It is expected that there will be large bending strains in the boundary layers, but that the bending will be almost negligible in the "field" (the region remote from the boundary layers). This behavior has been noted by several investigators. (For example, see Fung and Sechler (6)).

-7The present investigation is in the spirit of von Karman and Tsien in that equilibrium states for the postbuckled shell are sought. An initially perfect cylinder is postulated, although the results can also be shown to be valid for an imperfect cylinder. Another feature of this work is that by considering each buckle as a shallow shell both the local and general buckling problem are investigated simultaneously.

CHAPTER II DERIVATION OF THE SHELL EQUATIONS Consider a shell of constant thickness, h, whose initial middle surface is defined by the relationship W - W (x,s. (2.1) The coordinates x and s are chosen to lie in a reference plane II; the distance of the middle surface from H1 being W. W is measured in the z direction which is normal to II (Figure 2.1). /Prebuckled Cylinder W = W(x,s) \~a~ An/' / Figure 2.1. Axial Section Showing the Prebuckled Panel. It is assumed that for the shell considered W'<< 1 (2.2) -8

-9in which I is a representative shell dimension and Wmax is the maximum rise of the shell from T1. This type of shell is commonly called a shallow shell. Another way of considering the shell is to think of it as a plate with an initial deflection. The Love-Kirchoff assumptions concerning the deformations of the shell are made. The shear deformation is therefore neglected. Also, the volume element of the shell is taken to be dv dX ds x cdZ (2-3) which is consistant with Love's first approximation and the usual shallow shell theory. It follows from the Love-Kirchoff assumptions and the assumption (2.2) that the displacement components of a general point in terms of the middle surface displacement components are U~ (XS, ZL = Lu - Z Wj oa =: x,s (2.4) w (X,S,Z) - \W in which the tangential displacements u. and the normal displacements w of the middle surface are functions of x and s only. (The summation convention is adopted for the subscripts a, - and -; the range of subscripts is indicated in (2.4).)

-10For the deformations being considered the displacement components ua are taken to be of an order-of-magnitude smaller than the displacement component w, that is, (,-O'"h)^^~ ~~(2.5) u<, O(h) j in which n is an integer such that VW -x 0O(nh). (2.6) As a consequence of the order-of-magnitude assumptions which have been made a shallow shell theory with deflections of the order-ofmagnitude of the shell rise is being considered. For a circular cylindrical shell the shallow shell being considered is one of the diamonirhaped buckled panels. For such a shell n is taken to be the number of circumferential buckles. The strains are expressed in terms of the middle surface displacements as follows, iA |,,, t (W +(w W),mlit -2, e] (W i- 2. 7) In the usual theory in which w^O(h) the terms w,, w,p and zw,7 are both o() h2 For the displacements considered here, however, is0 whilr e (2 bh zwxo is 2O( w) while w~ wa, is o(n2 /; thus the term zwof is * The expression wvO(nh) should be interpreted to mean that maximum values of the displacement approximately n times the shell thickness are expected.

-11of an order-of-magnitude smaller than the other terms. However, because of the higher derivatives involved this term becomes significant in a boundary layer whereas other terms which have been neglected in (2.7) and which are of similar order to the bending term involve lower order derivatives and remain small when compared to the membrane terms, even in the boundary layers. For a given load the difference in the potential energy between the buckled and unbuckled state is given by the expression WVf[ ~,+T2d f, v -fToU 8cS (2.8) J~~~~~V ~~~~ST in which T' are the additional stress components which arise during A buckling and Tag are the prebuckling stress components. T is a vector which represents the applied surface tractions and U is a vector which represents the displacements through which the tractions act. The work done by the external tractions is equal to the work of the prebuckling stresses acting through the linear portion of the additional strains. The potential energy change can therefore be written V= VW +Ve +V (2.9) in which VW = 2j N w V iW), dJS. (2.10)

-12x S [(i-v)w^(a ^ t W)/ wRcl dS (2.11) Vm= iN^~^ clS. (2.12) Ne and N Q are defined by h _ -b (2.13) kI - In obtaining (2.11) it has been assumed that the additional strains are related to the additional stresses by Hooke's law for plane stress, that is, E A?d \ 2 ( I-V) Ed + Sd S^] (2.14) The additional stress resultants in terms of the middle surface displacements for a material with a stress strain law as given in (2.14) are

-13i* - ( _ ) [ ( A W+V W+ -+/ t Z ( 2 U,Y +(W W j, g(w W ), y- W Y r) ]. 6 h(2,15) The additional middle surface strains are Ea 2[u.+^uA, U ( +W)4, (W+VW),V -WY VY] (2.16) The equilibrium equations are obtained by equating the first variation of the additional potential energy to zero, that is,,F[~~~~~~~ ~~~(2.17) ~V-o. ) These lead to NMd ~ Q (2.18) Dw,ddiC -N^owo$ -M^(+WU~),> = o.. ~(2.19) Since N.,, rather than uc, have been selected as dependent variables an additional equation "(compatibility) must be added for

completeness. This equation is N40i -Ads,., =^ [(W+W)Aq( W + v) i-(W W)wj(W+ W,^ -V YcVV +i.vWiA/ ] (2.20) The governing set of equations for the shallow shell thus consists of (2.18), (2.19), and (2.20). Consider now a segment of a circular cylindrical shell. The original shell middle surface segment can be approximated by a parabola for segments which can be considered as shallow shellso The initial shape is given by the relation \A/( ) x -a3 + (2.21) Theprebuckling state of stress is taken to be one of uniform axial compression: Nx -P MXs s (2.22) Equations (2.18) are satisfied identically if a stress function F is introduced by means of Nx, iss N - F s Ns - Fx,, (2.23)

-15In view of (2.21), (2.22), and (2.23), the shell equations are DV7w + PwXX-FE, W Z FvxsWXs -x (Wss +F = O (2.24) V4F -Eh ( -wXXWS - W/) =,, Equations (2.24) are the Donnell equations for a cylindrical shell (8'27) These equations are also Marguerre shallow shell equations for the (28) special case of a cylindrical panel. Equations (2.24) are usually consistant only for deflections of the order of the thickness of the shell. However, for modes of deformation which involve boundary layers these equations can be used for larger deflections,

CHAPTER III SOLUTION OF THE SHELL EQUATIONS 3.1. Non-dimensional Form of the Shell Equations The experimentally observed buckled shape for a thin cylindrical shell loaded in axial compression consists of a series of triangular shaped regions of nearly zero curvature separated by ridges and valleys of large curvature. These triangular regions form a set of n circumferential buckles. These buckles can extend over the entire lateral surface of the shell, or there may be only a few axial rows of buckles (6,7) (6, In the following analysis a typical buckle will be considered. A drawing of an idealized buckle is shown in Figure 3.1. The reference plane, IIl is chosen so that the point ( and ( lie in Il when the shell has buckled. The lines (,,, and ( represent ridges, while the line 2 represents a valley. The triangular fields (l) and 1 are assumed to retain some curvature in the buckled state. The unbuckled shell is shown by means of the dashed curve, the buckled shell by means of the solid curves. The parameter k is the ratio of the curvature of a buckled panel to the curvature of the original cylinder. k = 0 means that the "fields" are flat. k = 1 means there has been no deformation. al, a2, a3, and a4 are parameters which aide in the description of the buckled panel. They are functions of R, k, and n. The assumed buckled shape for the shell is almost developable. This fact can be rationalized if it is assumed that the buckled shape will be such as to minimize the potential energy of the shell. The -16

-172_22x x.~A /3 f1_ (33 z- z 7 A \/ R k Section AA Figure 3.1. Geometry of a Buckled Panel.

-18energy associated with membrane strains is proportional to the shell thickness, while the energy of bending is proportional to the cube of the shell thickness. Thus, since shells are of small thickness, the buckled shape will be one which minimizes the membrane strain energy, i.e., a developable surface. The developability of the buckled surface has been noted and discussed by several investigators. (2930) The assumption of developability allows the deformation parameters a2, a3, and a4 to be expressed in terms of k, n, and the undeformed shell parameters R and h by using geometrical considerations alone (Appendix A)o The results of doing this under the assumption that the angle is small are En 03 As (> (3.1.1) F~rrX 1 2n~ The membrane stresses in the regions remote from the boundaries are assumed to be uniform axial tension in the buckled stateo A tensile stress is necessary for equilibrium. (See Appendix B). Equation (2.24) can be put in nondimensional form in the following way. Because of the order of magnitude assumption concerning w(x,s) and W(x,s), let W - nhY (3.1h2) (3.1.2)

-19The initial stress P is taken to be similar in magnitude to the buckling load as obtained from a linear analysis, that is, PO'. O( R (3.1.3) The radius of the undeformed shell is not, in general, representative of the deformed shape; rather, a more realistic choice of a representative of the deformed shape; rather, a more realistic choice of a representative length is made by means of r'- nnRh. (3.1.4) In terms of I rather than R (3.1.3) becomes P-Of( ) * (3.1.5) Dimensionless space variables are introduced by means of X =eL (3.1.6) S-.t With the assumption that differentiation with respect to ~ and a does not significantly change the magnitude of the function being differentiated and also on the basis of (3-.15), let P-. (3.1.7)

-20Finally, define a nondimensional stress function f by means of F nEh3F. (3.1.8) This choice implies that the additional stresses are of the same order as the original stresses. Let ao nh o(2 o3 =- nho(3 " (3.1.9) a4 nho4 \() and assume that the number of circumferential buckles is of such a magnitude that R o ) (3.1.10) This assumption can be justified by experiment. The tests of Lundquist7) Donnell(17) and Tennyson(31) show that, if only the final buckled shape is considered, for all of the cylinders tested n3h 2 - 1l is a better choice than n -h (Appendix C). The assumpR 2 n h tion h 1 can be used to show the consistancy of the Donnell equations for radial deflections of the order of the shell thickness (consistancy in the sense that all terms are of the same magnitude).

-21It is also noteworthy that the number of buckles decreases as buckling proceeds. By the use of (3.1.2) through (3.1o9) Equations (2.24) can be written s2y'v t( -f,,.)y, 2fy,- -, (y,,,- L i) = (3.1.11) "V4 - -.Y. ~y (ys- + ) - ~o. Equations (3.1.11) are the governing shell equations in nondimensional form. If the order-of-magnitude assumptions which have been made are valid then all terms except those with coefficients 52 and 5 7 are near one in magnitude. 2 and 5272 are significantly less than one. The terms in (3,1.11) which are multiplied by 58 and 2 2 5 y are those exhibiting the highest derivatives. This is typical of equations which describe problems for which a boundary layer type solution is expected. (32) To solve Equation (3.1.11) consider only the region and its boundaries. Take the solution to have the form y =yo y, +y *+y3 (3.1.12) f - is t $, + f7 ifs The functions yo and fo describe the deflections and stresses in the field. The functions y1 and fl describe the

-22additional deflection and stresses in the region including the valley O and are negligible in the rest of the region. Similarly Y2, f2, y3, and f are functions which are nonnegligible only near ridges and ) respectivelyo 3o2. Solution Near a Valley Consider first the solution near and including the valley O. In this region the only nonnegligible functions are yo, Y1, fo, and fl ~ The functions yl and fl are expanded in a power series in o Since y1 and fl represent the boundary layer portion of the solution near the independent coordinate normal to the valley is stretched to magnify the effect of the boundary layer. Then y =yO (, ) + y,I( dCr) +S2y (Yat)+,, ( in which? ~?|-. (3.2.2) The functions yo and f should be solutions to (3.1.11) with B equal to zero, that is yo and f should satisfy (3.2-3) Yo^p-Y~iy0^r0 * ->

-23The original shape of the shell in terms of dimensionless variables is given by the expression Y -= 3 + a t (3.2.4) The observed buckled shape seems to consist of a sequence of planes joined together at the ridges and valleys. However, equilibrium considerations (Appendix B) imply that some curvature in the circumferential direction is necessary. A null value for k also does not lead to boundary layer equations which have solutions which decay as the coordinate normal to the boundary increases. For these reasons the final shape in the field is assumed to retain some curvature in the a direction and vary linearly with e, that is, Y Y "o+ yo =I4 -2(o2o- 4)yX |^ (3.2.5) or y=Q (03-b0(4)- T (l-k)br -2(((,*+3- )o1.:3.2.6) The initial stress state consists only of a compressive axial stress field equal in magnitude to the applied stressO In the buckled state the load may be expected to be carried also in the ridges and valleys, the ridges carrying compressive loads the valleys tensile loads. If this is the case for equilibrium in the radial direction along a valley, the stress in the field must be tensile. (Appendix B) Therefore let

-24to =(I + r) (3.2.7) in which m is an as yet unknown parameter such that m = 0 means that the total stress in the buckled state in the field vanishes, The functions yo and fo as given in (3.2,6) and (3.2~7) satisfy Equations (3.2.3) except along the line 0 = 0 Substituting (3.2.1) and (3~2.2) in (3.1o11) and collecting terms with like powers of 8 as coefficients leads to the following sets of equations:.y Yi ), f - m Pyl j -k, il Ac = o $^~'hky(3.2.8) )q,,*, v +-kyi ctp = o a yj 2, q -mPYl2, Y|^ - kr ie, k - i,, r^ y-Iiy y -2~fn,,y, iij, -fll,vyW,, i, =~ 0 (3.2.9) f\i,,w.q t kyvz,** -yiy. i ynl^yii/6 r = The remaining sets of equations can be written down in a similar wayo For a first approximation consider only y=yo +Sy,, (3.2.10) f -- o +s,,.

-25The functions Yll, fill must be chosen to satisfy the symmetry and continuity conditions violated by yo and fo, and they should decay as |J increases. Therefore, yll satisfies the following conditions: ( y + Y + yll ) continuous at e =0 (Y + yO y )) 0 at = 0 (3.2.11) (Y + yo+ - Sl )g continuous at e =0 (Y -YO + ytl ),), =0 at 0 In addition symmetry requires that tX (0 a- ) - 0 (3.2.12) i, (, = 0 (3.213) Since only r derivatives appear in Equation (3.2.8) and because of the observed buckled shape assume yl = y,, (N) (3.2.14) i.1 - c1 (() )

-26Then the equations for yll and fll become y2q A -mpd -k fz -o (3.2.15) 2. 4 +k -. Two cases must be considered in solving (3.2.15). Case I 0" _ > O (3.2.16) For this case the solution to (3.2.15) is (Appendix D) AAP I +1 A A,I =be Aos>A4,,, Ybei sin Ari ) y,, -b,e cosPAA,' -~bLeA',n AA,A' -k[(blib ) e Cos AZ + (b +-cb ) e S1n AA,; (3.2,17) + bs] (Y<0) _,i - b3e cos AA0 + bMe -In AA2+

-27in which A, —2 cost Az-^ sin - sin %3- l 2 cos 1e...;' (3.2.19) The solutions to (3.2.15) given in (3.2.17) and (3.2.18) are chosen so that yll and the boundary layer stresses decay as | | increases. The use of (3.2.11) and (3.2,13) gives the following values for the constants in (3.2.17) and (3.2.18): b - ~(3A A )A(do, + -- dA R.- L -b -.A 2 Al3 -c 32A A yZ A (3.2.20) bs -, t - a r 2tP (da r(3t 0(4)n k to a first approximation Thus, to a first approximation,

-28y (..-74)-.... /\- r'( (I...CoS _a I- am As)] l ~~ACO\~~aW A2 21(.a) Pc 43, L\A (3.2.22) F(, Az kPS7 -AXI+ e( ( (a) CO — AI~tc

-29The additional stress are NX= (I+m)P Nxs - 0 (3.2.23) SS'-P., Cos A'^)~-'" I [(++x ) j S k n hi h The axial and circumferential displacements can be calculated by solving Equations (2.15) for ux, us and u + u s Ax IX Eh ( nWxx-,lkss)-yxxx- $ U x s s) us 5 E^( sh -(N u xx)-\X Wjs-VV /- )s (3.2.24) Uxs - sy Sh Nlx s-V s-x-WS - xVs: -w,,W,) Since the compatibility of the strains and displacements has been assured, Equations (3.2.24) can be integrated by using the results of Equations (3.2.23), (3.2.22) and (2.21). The constants of integration can be found by making Nxs, as calculated from Equation (2.15), be zero, and then satisfying condition (3.2.12). It is noted that this condition can be satisfied only in the average; hence the solution is approximate. The expressions for ux and uS are

-30E P - nE k a..LS - CeX oSj X I eA M (AZ- c& K -' a-(i-lo z nXx)^. 2( X I- xR2 + I (a 2. A A2 2Ae [ Co S T A- sin- -x + Men[(A -- 1_ sl)\ 2 X.^As -(At q -.1 ~ A,, (3.2.25) Co X-i (I+$ -L A _ 4' t2A'3. Al~_1A (lk(caL-o..a,: (s- X (X<6o ~ -OLax (x> c > -^ r\ e I + Az C^IX ^A- )sLnXAz a 875& k... ^ i

-31Case II <'c (3.2.26) With the introduction of A 1-' ^P + kC i 7 —=n,Z\~~ ~(3.2.27) A4 = —i l the solution to Equation (3.2.15) corresponding to (3.2.17) and (3.2.18) is AA3 tP - \b=- b7A + beA b (3.2.28) < 0 -AAA A' XA4 W y =- bqe t b,oe fl _z _ IrPA3te^^. b 4eb+ +b, - (3.2.29) S''>o Y

-32The use of the continuity conditions at r = 0 gives the following values for the constants in (3.2.28) and (3.2.29): b9 - b4o = 4 ( IA, - A-), I, (3.2.30) 7 4(dz +d3-d()Q b -- - I, Thus y = ((o-(a (I-k _ 2( =)^iQ l SAm. -i)Ai l gAz -AA, I' el AA(A-A) +AtCA4-4)A ] (3.2.31) -~'~(I +niZ~4)X kSx q.... f A -, A ( A (A-4[xl A4 -+Al At. -AA41'I+ y W3(A3- ^A(A3-nA, J J

-33or in terms of the original variables \w ~-~- C,4) A- /Sl 2( + -| X,A- 1x1A +,A-a - 1x1- (3,2.32) A4 - A A —... Ae )] A,(A~-^A l A-' The additional stresses are NxxClnt )P A Nxs O 0 (3.2.533) Ns, mP(oac. i -t Ar A 11 A"3A" i ] LAk kAs-A4 -A_'~ The displacements in the x and s directions are

-34A,3-'P,,g^(arm 3 a rh(X>Q K a2p R 1 P Z(1,3 a 1c)L A 4As I UV (X O): -xXL X _ A^ l - k"P 3 43AAs _,-, A e-2\,S:'''.-A(, 4 - 4-. a "t _^l j+ -e^ JIs'3es.

=353.30 Solution Near a Ridge Consider the solution to (3.lll.) near the ridge ( Because of the symmetry of the buckled shape, the solution for this ridge suitably modified is valid for all other ridges. To consider the ridge (o, a new reference plane, 12, and a new coordinate system are chosen. The coordinate system is selected to lie in a plane containing a straight line joining the points ( and ( in the buckled state and orthogonal to a radius of the unbuckled cylinder at the midpoint of the line (. The co ordinates used to describe the solution are taken to lie in 112 in the directions along the ridges (l) and (.(Figure 3.1.) This new reference plane, 12, does not change the form of the basic equations(33), but the displacement functions are now taken with respect to 12. This change is possible since the number of circumferential buckles is large and therefore the angle - is such 2 that - < 1 0 The slope of IH2 with respect -to H1 is. *. 2n This is a sufficient condition that the form of the equations be the same. The governing equations are rewritten denoting the new dependent variables by pimes and the new independent varihearables by a bar whenever confusion might arise. They are DV''+ PtW;xx'+ -F (W-Vv)X +2 FVS v(w'VVx),-x(vv v - S (3.3.1)

-36The initial stress, P, is the same in either I1l or 12. The coordinates x and s are the projections of x and s on T2. x is an axial coordinate and s a circumferential one. Again (3.3.1) is put in nondimensional form by using the following transformations: W- nhY w' _ nh y' (3.3.2) P rEP F r nEhS v "x -s r e +X V-i a (3.3.3) s iT(e - ) i The nondimensional equations are

-37-,Y P (y,/iy,'ee 2y y,) -,[f'ee(\y'Y -2e,~(y'+Y'),&~tf;(y'+Y')ie| — / (3.3.4) 8 47 f 4 (y^^-YSCe Yo +2Y1<vxt coy-y^yjg^ \ -Y/?Pe Y) ~)Yee -) O in which r (3.3.5) +2(3-22P 3Z)a 3 ) T1le coordinate system p, ~ is oriented as shown in Figure 3,2. It should be noted that the coordinates p and ~ are orthogonal only in the special case p. = 1 The original shape of the cylinder is again approximated by the parabola yI= ('?- o )e- ^ (3-3.6)

-38J -X~o.... f Figure 3.2. Orientation of the p, ~ Coordinate System. in which C5 g a ~ ea —t (Appendix A) (3.3.7) By the use of (3o3-3), Equations (303.4) become S Y VytTV P(y eeB y Yes)-32 F ( fe + a i,2 d+ -41fee ys -2Y-f, s v,' e? (3.3.8):0 =Z

-39As in the solution for the region including the valley the functions y' and f' are taken to be the sum of two terms, the first term being the solution in the field and the other representing the boundary layer. The boundary layer term is expanded in a power series in 5 and the independent variable ~ is stretched to magnify the effect of the boundary layer. Let y'ydYo'(e,~) + y Sy ~ (e, ) S e~~ (e,7)t i (3.3.9) in which (3.3.19) The functions y$ and f~ correspond to yo and fo; they differ in that yO and fl are taken with respect to 112. Y) and f' satisfy (3.3.8) with 6 = 0, in view of the preceding results this implies yO - - d0 + (da + 3 -oq<)( - ) - (o(A+ - 4-o)( j-;) + (ods-4asciCe )' ('7>o) YO~ -01 - - (^ +o(- d4(J>(-ci^ -^C A +4o(95C-)C -) (3.3.11) - ( cd -4z/o(e..: ): (? < o - (1(-r n (e-C-:

-40Substituting (3.3.9) and (3.1.11) in (3.3.8) and collecting terms with like powers of & as coefficients leads to the following sets of equations (only the first two sets of equations are given): 11 yll5 7 -/- k f2,,s- ) (3.3.12) T? 77 I k yaloy = o ) tY82 ) - 6482Pr fj: ^iv^ 8 1 1 (4~{dY)(-~t'V,?9 +')/5; 1 )7 0 (3.3.13) fz.t k"f1 y1::,~"? - 4Y'- (-":" Fj) ( 9 - (S t (,I)tM(~,:;e7 - ~2,; ee~~;r?) = 0 ~ ^( Y2 lr7-Y^^^p Y l r ) / in which (3.3.14)

-41It is easily shown (Appendix A) that 1rr k - k (3,3.15) Again as in the solution for the valley (k, as a first approximation, take only the first two terms in (3.3.9), and let y21 and f21 be functions only of I, that is Y i y/ + 8 yZi (y) (3.3.16) 4' I+-(y. ) The governing equations for y21 and f21 thus become 141C l^d, f t(3.3.17) cl4 + k dY - Because of the similarity between (3.3.17) and (3.2.15), a solution of (3.3.17) is Case I -. (3.3. ^ ^^p^-~~~ -I~ >~)~ ~(3.3.18) rn~n

-42y'=b'i* cosxAA+-b^e. 5l $,A^v^ = k~ [(b-l3-ibJe' o e SA^ - (b,4+,b(3) (3.3.19) yak = bIse'COSAAA + be Sin AA A A 27 = [(bistKb,) e cSs MA^+. (bl-, bt9) (3-3.20) >o A\ ia C-os AP _san- - _CW6s -L ^^^jfrh' f(3.3.21) Cot!lV. _

-43The continuity and symmetry conditions at the ridge are (y'+Y') continuous at = 0 (Y' Y'))X =0 at = 0 (3.3.22) (~Y+Yl)x- continuous at = 0 (\ ( XXX at 0 = 0'-,R 0 at = 0 (3.3.23) The use of (3.3.22) and (3,3.33) to find the unknown constants in (3.3.19) and (33.320) leads to b~~~7 ^-a~~~~~~ —-1 2AJb, - b = (3 A'2 -(3.3~24) b>7 ^b( _ (3A: -A,< ) - XS(3-3.24) bl, =-b18 - 4(d;+d3-594'

-44Thus (d,-idp +d).%4) ^XXAA (3A; AtA (3 ) -SA I I,((-(34)+S(34'-i )sltaJ^A1 - L- Ae (3.3.25) The functions y' and fV are f..., o~ -- ^ ^[.) (3.3.26) CO-sXAu\-Q3A^^ h AA^; (2o >)

-45-'A....- ( (BAa+-A 4I7'wt c ps (1P~(eg2) _ m(d3-d+ eXA1,vj [ (-I - )Cos AAl,3A —A (3.3.28) Atl AAl, A Equations (3.3.26), (3.3.27), and (3.3.28) can be written in terms of the original variables as follows: W = Q2-2 (az +ct3-,4-(a1)i) + ( (a>+C3-2A4- 4CGS) 3r +4<a _as) ^ +G34 _ie[(3AA / cos^A(3 ^ sml(I'a l (3.3.29). (I'A^m,-P,~ S~Nv ~= fa (sA2-Al,,-,. )\ (3Al-A)\ ACos - (3A< A) (3A-AI-) 4\ i

-46v'=.. -4 2 (a,23 -41 +C )-2(,a'a3-4 jS 4(a<S) \ +~X e9"A;M.. Ai F' m (l~.z)J, s Pfa -a4{ err_ r 3.3.30) ( i- ) Co~g ~+((v)?a- A;- -- l v-<0 / The stresses are found again by differentiating the stress function, namely, Nfix F,s (,+m) P-;P(;k. nh ~ p( N.,(.3...1)^ (3.3.31) cos ili+a.-^)sain ]

-47-osi' _m (Pa ^^iVe [( ))] (3.3.31) I ~4 FS VAST Aal (CJA, AaThe components of the additional stress in the p and; directions are also of interest. They are,'-'(I-W')P.(l~~+~)mP(ac+-c~-a)^J^1 -^,1?| NeeA N er = — 77 - (a+(k3 — e -,QA [^(+t)coS/ |+(-,)S d-h'... 3.3.32) /,z'+n)P,/ _ ( +.) P / 7~

-48Np is the normal stress resultant directed in the p direction. N' is a normal stress resultant orthogonal to the ridge. N is a shear stress resultant in the plane of the ridge. The displacement components uo and u could now be calculated. As in the solution for the valley 1, it is not possible to make the displacement u, zero everywhere along the ridge. These quantities do not add significantly to the solution and are not presented. Case II o --- 4 -I < ~ (3.3.33) Let J — 4 - _ (3.3.34) then yal,-b,,e 7+bis( 3 is=k^-^(^ bA, + b, A (3.3.35) _ r-nP

-49L 4 ^M^. -XA4Y7, - A7E +-b- bz2A e'' = —' F[ baj Ae +b3'3 bbtJ: (3.3.36) 1 >c. The use of the symmetry and continuity conditions to evaluate the constants as in Case I gives b h =l _-An (d d3-dq) b9 bZ Az (A7- Az) bzo - ba- -= 3( 7 (3.3.37) b 23 = ba =' A ('"43 -A'). Thus the functions y' and f' are

-50yl -d2 (d2d3- dqA)(et 2)o (dI+ddq+d5( ) + (d4- 4ds)(e _ 2&(d1+o(dA3 —cF4 e7A37 ~ _ y XA4\? 1 A, (A3- A e- ] (3.3.38) A3^ A-A A3-A, y = ^-~+(ddda dO )(C ) -(dafd3 i40- ) t4 -<,Ae-1 X - )[4:F e3A7 ~ (33) 39) _v (AA ()) AVA-A) 1 A

-51f(l+V Yl()jSg mP(d,+d ) A4 41 )eU ) (3.3-39) Equations (3.3.38) and (3.3.39), are written in terms of the original variables; w'= -a C' - 2(ax ga3^J-C<t(l + Q) + 2(a,+c 3-2-4^!.s)30 +4(a|-4a5)Q_ 43LAI A(Qata; -4 A (3.3.40) F' /(-L~)Pv Sni)P((as —a4)[ rO. AZ * - Z ^ knh T A3(A'A4 )?<0Al

-52p -1 (GqLfr S rn (aC+a.-, a) r( agz' " 2 + knh L\ /t'1 g >O. / The additional stress are N''-, - A,~ ^-^+ P') Nn= (A |-) A+ -P(a;3-a() 2 L Ai* AF3-A4' 3.3.42) Nxs =N- g3 (I+?Pr / t[5(t~P

-533.4. Equilibrium Considerations The force in the shell consists of that in the boundary layers and that in the field. The axial force in a ridge is - il r i (34. TIv = 2 ]t htl~xxW F ^ = ^(3.4.2) The total external load must be equilibrated by the internal force in the shell. Therefore -zniP - 2rTlR(a+ lP - 2h7Fi (l' -k^)rP (3.4.3) k or rY = { k Ad (3.4.4) A plot of m versus k is shown in Figure 3.03 Note that if m is to be positive then k must be less than one half, The circumferential and axial equilibrium at a juncture of valleys and ridges is obvious, from the symmetry of the deformed shell. The radial equilibrium equation at a joint is satisfied identically by the stresses found and therefore does not give any new information about the shell o

-543.0 2.5 I-2k 2.51 } | m=k 2.0 1 THE AXIAL STRESS RESULTANT IN A "FIELD" IS mP WHERE 1.5 / P IS THE APPLIED AXIAL STRESS RESULTANT 1.0 / k IS THE RATIO OF THE / |CURVATURE OF THE FIELD TO THE O.50 - / ORIGINAL CURVATURE OF THE SHELL m O0. I " k 0.10.20.30.40.50.60.70.80.90 1.0 -.50 - 1.0 - -i.5/ -2.0 -2. 5 -3.0 Figure 3.3. Relationship Between m and k for Equilibrium for a Force Prescribed Buckled State.

CHAPTER IV RESULTS The results of this work consist of mathematical expressions for the deformations and stresses for a buckled panel of a cylindrical shell. The mathematical expressions and typical graphical representations of them are presented here. The final buckled shape for the region of the buckled panel which includes a valley is W.,.w n2k ^2rT r'Ol-k^- rr ) -tA,ixl I sostahi i- ('i n tAx IX (ctkO) (4.1) n~ =.... ZAi-<- A-3 A _-) e_ A (AAej (A-A< (4.2) in which 15 ~A, -3/ )-.-zk AZ - R -A^w =.,' —z' - l/-55-J -55

-56-. A - 2-z -,!, n- ik.~ ___ tx_ _ i__ _. -I (4o3) =zsl zgV-\lPIE n cont'( The coordinates x and s are chosen for convenience in displaying the results graphically. Figure 4.1 shows the buckled shape for an axial section through the panel at s = 0. Figure 4.2 shows how the equilibrium configuration changes as a function of the applied axial load. Figure 4.3 shows how the equilibrium configuration changes as a function of the curvature remaining in the field. It should be noted that for all of the results shown in Figures 4.1, 4.2 and 4.3, the wavyness of the mathematical solution is not evident because of its relatively long wave length. This is significant since no wavyness has been noted experimentally, The final buckled shape for a region of the buckled panel which includes a ridge is W\w/ rT (-k7 (tf)- O e<- k+- - L(-? (3A- A t) 2i (:+ - ( sin (, AZ,< ( >o +'nh za e ( ) (

-5.0 W+ / nh -4.0 PREBUCKLED CYLINDER \ -13.0 / / P FINAL SHAPE — 2.0 nh W -1.0 -.80 -.60 -.40 -.20 0.20.40.60.80 ^ 1.0 Figure 4.1. Final Shape of the Buckled Panel for s = 0 I ~~k=~~~~~~~~.I0Q~ I fl12 -.0 -.80 -. 60 -. 40 -. 20. 0.20.40.60.80 A 1.0

-58-5.0 / - 4.0 W+w nh - 3.0 - nh W +Wo -2.0 - h.h IN k =.10 o 10 n=12 O E-.50 IT 1[R PE='2 —.. = 1728 P -= 80 -* *o,-I'' = I'0 -= 1.0 0.20.40.60.80 1.0 Figure 4.2. Final Shape of the Buckled Panel for s = 0 x > 0

-59-5.0 W+oW nh -40 w 0 k=.05 P/pE=. O k=.05 1 T m n = 12 k=.10 _- 1.0i! R/h=1728 k=.20 _~~~-1.0l~- /J,0r = 1.0 1g k=.30, I, I I I... 0.20.40.60.80 1.0 Figure 4.3. Final Shape of the Buckled Panel for s =, x > 0.

+ (Bta-tI tl, A2pi (P<C (I 5) /+\ e m ('SinA ^AJ(4.5) in which e t 2 —Q + (4.6) Figure 4o4 shows the final buckled shape for a section at p = 0o The circumferential stress in a valley is given by the expressions Ns_ t(l-kP -t*A, IX A A Ps= tI-kP e_ Xe ) cos tALzi 4 A +'( >o) (4.7) PeNs! t(l-k) IP At A -AI AI A Ai - Jt (4A8) - l(l-Zk) 3'A AAe (4-8 ) Similarly the total stress resultant directed along the ridge is __+N_ _) _kP _ _ t (-k)P P- )' ^ I (l+tuV)(1-2 k) P 4(l- k-P)E e' Figures 4.5 and 4.6 show how -p varies with x and how tE the load parameter P/PE affects this stress distributiono Figure 4 7 shows how -pQ varies with Co

1.0.90.80.70.60.50.40.30.20.10 0 -.10 -.20 -.30 -.40 -.50 -.60 -.70 -.80 -.90 -1.0 ",,,,..,''',,,,,'", I, I'' __.. i~/ b eI.^-cPREBUCKLED CYLINDER 2/ ~<.a-BUCKLED SHAPE FOR = 0 BUCKLED SHAPE 4A///<?>5 //~~~~~~ 6 //'~ 7- __ W +We0 8/ / W+ k.10 R/h 1728 I/,, -nh ~ nh P/pE = =.50 J= 1.0 0n =1 Figure 4.4. Buckled Shape for a Ridge of the Buckled Shell for p = O.

-62c_<X CM c ~ II II 0 O O0 C0 o / 2 lI II ^ ale?1 c 0 d / o d ~ r) - 0 bD o 0 H It o ~ o03 0 0 C) b2 o H $law 0i rl0 O d O 0 d

-632.5 NSS 2.0 PE k =.10 ] — 10 1.5 F\ i | R^h~l728 I PE R/h 1728 PeE A 1.0S.50 Fg\r L w6 icmeeta tes naVle o 0 0.20 0.40 0. 1.0 -.50 Figure 4.6. Circumferential Stress'in a Valley for x > 0

-640 -% 0 0 C6 ui,* ag o 5: <\ I I Us 14~ I0 0 i so ~0~ 0 0 CM f 0a) 0 CD? o 0 0 0 L0 0 \ s tl \ I'l *H 0>

-65The axial tensile stress in the field is given by the expression (_Nx - P (tP - (P-2k) e (4.10) The significant result here is that if k is significantly less than 1/2, the tensile stress in the field will be small (e.g., for k =.10, p =.50, the tensile stress in the field is only.0625 PE ) The axial displacement ux can not be made continuous along the valley and the displacement normal to the ridge can not be made continuous at the ridge using the approximations employed herein. A plot of ux/h along a valley indicates the extent to which the continuity condition at the valley is violated, (Figure 4.8). The magnitude of this discrepancy decreases as k decreases. The mathematical expressions other than those listed above which were found in this investigation are as follows: For the region including a valley, x_ = zrr(I(-k)P _ rr_(_- _K)P -otAI X( tA h n (l-21]k) PE /n U —Z -k)P~ -e co tA (X / <% _L \,, A\-I n~l-rk)2 I X (- A - Sir7 i Sn A2 X -- 12 t - t e L (BA'-AI /.o A (3Az-A, A -A| A Cos tA, x ^ s A X +s %A AA-A 2 (l + /f -J- F z- ('3- 21At a Am (4.::) [(Az- 9) A-2 sin t+,, A i- (A A IL COo.s A, 9-^ ^ -k^^]^^^-^ ^(^~~~~~~~~~~

-66Us h I.0 \'.80 /I.80.60.40.20 A n3h = 1.0 -1.0 -.80 -.4 -.26=0.80 1.0 I E k =.05 u Ai k =.10 |1[ k =. 20 Figure 4.8. Axial Displacement _x Along the Line x = 0.

-67x (X<O) =-^ (x> 0 (4.12) uS 2rrTIY(-k P rryt(L-k) P -tAII x"'i I \ h 0- (l-2k)F'E 2 n Kn(I- )1 L A, k cos+AalX + ( - sintAlXl A]^ T _l s3 (k A 3 h _ r'O(k)P X_ ^ _ Fl. (l-k)P -- t ih - e(- la 3 PE 23n/-\I' - 1 A + "A%' X) tA(AzA C Z A ^tA3 2 A3 A -tA ) 2Aq ls tAq(A -A) t ^(+A A3"- A 4 ( t A3AX )A + t(A3 i e A, Al _,-e.- A- / J s _A^ es _s A (, a h - n (I-2t Pe' 4S-kPeL c_ A^^4 et^All ^ S (i-k') ^s

-68\x's = 0 (4.166) For a region including a ridge, Np ( -k p t9 - (I.-2 _ P (4.17) PNe &- Plt)(1-kBP p (4.18)

CHAPTER V CONCLUSIONS The solution, presented in this work, to the problem of finding an equilibrium configuration for a buckled cylindrical shell is not exact, since the axial displacement can not be made continuous within the approximations used in the analysis (Figure 4.8). The effects of the interactions between the ridges and valleys of the buckled shape were also not considered. However, the nature of the results seems to indicate that this is a reasonable approximation to the theoretical determination of the postbuckled state of the shell. The solution was carried out by assuming some residual curvature in the field of the buckled state. There is little experimental evidence to indicate that the buckles are not flato The results however indicate that very little curvature is necessary in the buckled state; the experimental evidence is therefore inconclusive (Figure 4.1). The amount of the discontinuity in the axial displacement decreases as the buckle becomes flatter (Figure 4.8). Comparing the experimental buckled shape with the theoretical shape is further complicated by the fact that when buckling has progressed to the point where the valleys and ridges are sharply defined, the shell material will have yielded and the effects of having a nonelastic material will become apparent. -69

-70An improvement in the solution could conceivably be obtained by considering higher order terms in the expansions of the boundary layer functions. Another possibility would be to consider the field stress to be nonuniform. The curvature necessary for the boundary layer might also be considered to be inherent in the boundary layer rather than the field. These observations are speculation and no proof that they will work is available. One interesting result of this investigation is the fact that the no unique equilibrium load was found, since many buckled configurations are possible for the same external load (i.e., Many values of k are possible (Figure 4.3)). This work sheds no light on the problem of how the shell departs from the prebuckled state. However, it has shown that a buckled equilibrium configuration which resembles quite well the buckled shell can be found analytically and that the axial load necessary for equilibrium is significantly less than the buckling load. This agrees with the experimental work on this problem.

APPENDIX A GEOMETRIC RELATIONS FOR THE BUCKLED SHELL The relations between the shell deformation parameters a2, a"3 a4, a5, n and k and the natural shell parameters R and h are developed in this appendix. The deformation in the field is assumed to be inextensiable, therefore X = k. (A-l) n It follows from the geometry of the deformation that a cos 2 = a (A-2) I n 2' 2 For large n - is negligible when compared with 1, therefore a1 a2 (A-3) Other geometric relationships are a2 + a3 = R( - cos ) (A-4) R/k sinX = (R + al) sin - (A-5) a4 = R/k (1 - cosX). (A-6) a2 is found by substituting (A-2) in (A-5) to give a2 - sin k cot - R cos - (A-7) k n n n' -71

-7203.- ~ / \\~r00' ~ ~' o 7r1 04 a ~R~~~R/ / Figure A-1. Geometry of a Buckled Panel. a -05 Figure A-2. Geometry of a Buckled Panel.

-73a3 is now calculated using (A-7) and (A-4), this gives a3 = R(1 - sin k cot ), (A-8) n n Again as in Equation (A-3) the assumption of - a small angle leads n to 2R a2 n2 (1 - k2) (A-9) 72 a3 R (2 - k2) (A-10) 6n2 a 2 R k (A-ll) 2n It follows from the geometry of the shell that (R + a) cos = R + a - a (A-12) 1 2n R+a1 - In line with the approximations being employed a5 is found to be 2 a5 82 (A-13) The following combinations appear in the text: a3 - a4 6n (1 - k)(2 - k) nh (1 - k)(2-k) (A-14) a2 + a - aR 72nh (1 - k) (A-15) 2 3( 4 2nk6 2g x2 +3 4.. (1 - k) (A-16) x 4pn

-74a4 - 5 4 a 2 (1 - k) = nh (1 - k) (A-17) a2 + a3 - 2a4 + 4a5 (1 - k) =- k) (Aa2 + a3 - 4a5 0 (A-19) Another useful relationship which has been used in the text is my2 m 4i2n R 1 2n3h m( 8C = =8 (n )(t=2Rk - (A-20) n2 2Rk k 8a~ -4 n nRh gxR k

-75APPENDIX B EQUILIBRIUM CONSIDERATIONS If the limiting case of an extremely thin shell is considered so that in the buckled shape the ridges and valleys are lines, it can be shown that the field must have a tensile stress. Consider the equilibrium of avalley in the z direction. Let T, the force in the valley, be tensile, then if the final shape of the valley has some curvature, as has been assumed, a component of this force in the positive z direction will exist. ~J~pVM -'R Figure B-1. Equilibrium of a Valley. The z component of the force in a valley is Tz =Ty W. ~,X=o (B-1) or TvZ - TV (B-2)

-76The only way that this valley could be in equilibrium is for an axial tensile stress to exist in the field. If this stress is assumed to be uniform, then the z component of the force in the field is TFZ -t Yn PS a( + (B-) or TFZ = /P (al 3- (B-4) An equilibrium requirement is that the resulting force in the z direction equal to zero. That is, 2TFz - 2Tz = 0. (B-4) If follows from (B-2), (B-4) and (B-5) that T = tR(l - k)mP (B-6) v nk If the force TV is compressive then the field stress is also compressive. In contrast if the equilibrium of a ridge is considered in the same way the conclusion is that for a tensile axial stress in the field the force in the ridge is compressive. "-~ TR O v1P Figure B-2. Equilibrium of a Ridge.

-77The z component of the force in a ridge is R T - w 2 (B-7) The z component of the force in the field is -r (W+~wo) s _P(ata_-a_ TFz r=n MP ark=p n-s-;P) M(B-8) Again since there are two ends and two sides to the ridge 2mP(Cq+aCz-(.q) TRZ A 2r oP(Q%+63 c it= (B-9) or _- nR T]'/ —(I-k)rmP o- k n (B-10) It should be noted that the Equation (B-6) agrees with Equation (3.4.2) and if the x component of (B-10) were to be found it would be the same as (3.4.1). Since a compressive external load is being considered the forces in the ridges should be compressive; this leads to the conclusion that a tensile field stress must exist. It is important to note, at this point, that the conclusion that a tensile stress in the field must exist can also be drawn on less physical grounds. A negative value of m does not allow for a buckled shape which closely resembles the observed shape since solutions to the boundary layer equations for negative m to not decay as the coordinate normal to the boundary increases.

APPEND IX C EXPERIMENTAL RESULTS To demonstrate the validity of the assumption that in n3h the buckled state R is near one in value, typical experimental n2h results are quoted. For comparison R is also calculated. n3h n2h Investigator n R/h R R Tennyson (31) 6 154 1.40.23 Lundquist (7) 9-10 333-362 2.01-3.33.224-. 33 10 455-460 2.2.22 10-13 625-714 1.4-3.5.14-.27 11 679-757 1.7-2.0.16-.18 11-13 909-920 1.4-2.4.13-.19 12-15 1270-1415 1.2-2.7.10-.18 Donnell (17) 10 483 2.1.21 12 1284 1.3.11 10.5 1383.84.08 8 314 1.6.20 10 897 1.1.11 -78

APPENDIX D SOLUTION OF THE BOUNDARY LAYER EQUATIONS The solution of Equation (3.2.15) is presented in this Appendix. Equation (3.2.15) is be y,, p &Ya - dk 2,, -O dl4 +4 (A - (D-1) ^^^-,+ k o Let yt1 -- co e ^ - i = 1, 2, 3,... 8 (D-2). The substitution of (D-2) in (D-l) leads to ey,, - - mPA - - (D-3) ci k A + d~ A,4 =O (D-4) For a nontrivial solution to (D-3) and (D-4) to exist A (X1 P + )=Q. (D-5) The roots Xi = 0 are not associated with exponentially decaying solutions but with polynomial solutions. These polynomial -79

-80solutions must be rejected for yll since all solutions must decay as |*| increases. A polynomial of first order will not effect the stresses and can be retained. Such a solution will be necessary to satisfy the condition that f,A be zero at t = 0. The four roots to (D-5) which give rise to exponential solutions are A>-" [I - — S- i = 1, 2, 3, 4 (D-6) Case I ip2 >1 |(D-7) Let zP- q X >O (D-8) Then X.A=$ * 2 * X(D-9) To facilitate the solution the following parameters are defined: sine - \ cosQ -1 ~l+2 > (D-10) A 2 cos 2 1A - j2 sin 2 A9 = 2 sin'~ A2r 2

-81The use of (D-10) allows the roots (D-9) to be expressed as 1 = X(A1 + iA2) 2 = -X(A1 + iA2)'-2 1 2) (D-11) 3 X(A1 -iA2) 4 = -X(A1 - iA2) The general solution is y~-c~e +cc. ~c3e +c. (D-12) The substitution of (D-12) in (D-3) leads to dl = (-1+ iq) C1 1 2k = (-1 + iq) C2 2 - (D-12) ^d 2k (-1 - iq) C ^d _ mP (-1 - iq) C4 2k Thus the solution for f11' is fi[- c,(-~lite2\+ c,(-l ^<e^ t ^ (D-14) The real parts of (D-12) and (D-14) are y,= be cos OsAA. b + be sin AA+ (D-15) b3A cos A^A-b4 Sl nAAL

-82-, = —k[ (b,-%bz)e cos AA^ +(b+ib,) Si nA (D-16) +(b3 tb)e cos ^A2 + (bi -ba)e Sn AA,], A linear function of 4 is added to fll to allow condition (3.2.13) to be satisfied. This gives for the solution,AAiq' AA i y~, = ble cos \AA, + b2e Sin,A2' Ir A 4 2iAA (D-174 = jbr, Co s,x& +(b2+ b,)e A ^:-:. -, r~AA~ (D-AA tb, l (<o0) ) y,, ='be CosAAa +tb e Sin AA2' =2k (b3b4)e osAA +(b-tb35 (D-18) e" sin AA +b'. (q>) ) Case II m1 P2 < (D-19) Let - t5 (-(20) ~f1 = I - M^Tp-. (D-20)

-83Then XA I Ac (D-21) Let A - + p, A4 1 - p. (D-22) A 14Y 7, A4 %FYC~ (D-22) The use of (D-22) allows the roots to be expressed as X = X A X = -k A3 ^ (D-23) X3 X A4 4 = -X A4' The general solution for yll is given by (D-12). The solution for fll is found by substituting (D-12) in (D-3) and using the relations for Case II. This leads to, mP 2 ci = mP A2C 2 A4C2 2 =2k 4 2k i_-~ 2 j~~ ~(D-24) d = mP. A2C 3 2k 3 3 mP 2 d42k A3C4 therefore the solution for f1l is,, 2( 4 cAA3 4f A3, AA'z -A )\ ^'"zkv^Cie +Ajze + Ace A3Ce(D-25)

-84The renaming of the constants of integration and the addition of a linear function to fll gives'<o yM, |A' -)\A4 b Y11 = b2e + bo-C:~> ro. P 2^-AA 3 P IL^ \ il i (^ (beAie + booA 3P -t- b2 P) (D-27) >0.

REFERENCES 1. Lorentz, Ro, "Achsensymmetrische Verzerrungen in Dunnwandigen Hohlzylindern," Zeito Ver Deuto Ingr., 52, (1908), 1766. 2. Timoshenko, S., "Einige Stabilitats Probleme der Elastizitatstheorie," Zeit. Math. Physik., 58, (1910), 337. 3. Southwell, R. Vo, "On the General Theory of Elastic Stability," Phil. Trans. Roy. Soc. London, Series A, 213, (1914), 187. t! It 4. Flugge, W., "Die Stabilitat der Kreiszylinderschale," Ingenieur-Archiv, 3, (1932), 463-506. 5. Timoshenko, S., and Gere, J. Mo, The Theory of Elastic Stability, McGraw Hill, Inc., 1961, 6. Fung, Y. C., and Sechler, E. E., "Instability of Thin Elastic Shells," Proceedings of the First Symposium on Naval Structural Mechanics, Stanford University, (August, 1958), 115-168. 7. Lundquist, E. E,, "Strength Tests of Thin Walled Duraluminum Cylinders in Compression," NACA Rep. 473, 1933. 8. Donnel, L. Ho, "A New Theory for the Buckling of Thin Cylinders Under Axial Compression and Bending," Trans. of ASME, 56, (1934), 795. 9o von Karman, T,, and Tsien, Ho SO, "The Buckling of Thin Cylinderical Shells Under Axial Compression," J. Aero. Sci,, 8, Noo 8, (1941), 302o 10. Leggett, Do M. A,, and Jones, Ro P. No, "The Behavior of a Cylindrical Shell Under Compression When the Buckling Load has been Exceeded," Rand M 2190, British Aero, Res. Comm., 1942. 11o Michielson, Ho F., "The Behavior of Thin Cylindrical Shells After Buckling Under Axial Compression," J. Aero. Sci., 15, No. 12, (1948), 738. 12. Kempner, Jo, "Postbuckling Behavior of Axially Compressed Circular Cylindrical Shells," J. Aero Sci., 21, No. 5, (1954), 239. 130 Almroth, B. 0O, "Postbuckling Behavior of Axially Compressed Circular Cylinders," AIAA, 1, No. 3, (1963), 630. 14. Donnell, L. Ho, and Wan, C. C., "Effect of Imperfections on Buckling of Thin Cylinders and Columns Under Axial Compression," J Appl. Mech., 17, No. 1, (1950), 735. -85

-8615. Koiter, W. T., "Elastic Stability and Postbuckling Behaviors" presented at First All-Union Congress of Theoretical and Applied Mechanics, Moscow, 1960. 16. Kadashevich, Iu. I., and Pertsov, A. K., "On the Loss of Stability of Cylindrical Shells Under Dynamic Loading," IZV. AKAD. SSSR. Oclt. Tekhn, Nauk. Mech., Mashinosti, No. 3, (1960), 30-33, Translation in American Rocket Society Journal Supplement, 32, No. 1, (January, 1962), 140-143. 17. Agamirov, V. L., and Volmir, A. S., "Behavior of Cylindrical Shells Under Dynamic Loading by Hydrostatic Pressure or by Axial Compression," Translation in American Rocket Society Journal Supplement, January, 1961. 18. Yao, J. C., "The Dynamics of the Elastic Buckling of Cylindrical Shells," Proc. of the Fourth U. S. Nat. Cong. of Appl. Mech., 1, (1962), 427. 19. Friedrichs, J. C., "On the Minimum Buckling Load for Spherical Shells," Th. von Karman Anniversary Volume, Cal. Inst. Tech., Pasadena, (1941), 258-272. 20. Friedrichs, K. 0., and Stoker, J. J., "The Non-Linear Boundary Value Problem of the Buckled Plate," Proc. of the Nat. Acad. of Science, 25, (1939), 535. 21. Friedrichs, K. 0., and Stoker, J. J., "Buckling of a Circular Plate Beyond the Critical Thrust," J. of Appl. Mech., 9, (1942). 22. Fung, Y. C., and Wittrick, W. H., "A Boundary Layer Phenomenon in the Large Deflection of Thin Plates," Quart. J. of Mecho and Appl. Math., 8, (1955), 338. 23. Masur, E. F., and Chang, Co H., "Development of Boundary Layers in Buckled Plates," J. of the Engineering Mechanics Division, ASCE, 90, No. 2 (1964), 33. 24. Reiss, E. L., "A Theory for the Small Rotationally Symmetric Deformations of Cylindrical Shells," Comm. on Pure and Appl. Math., 13, (1960). 25. Reiss, E. L., "On the Theory of Cylindrical Shells," Quart. J. of Mech. and Appl. Math., 15, (1962). 26. Johnson., M. W., "Boundary Layer Theory for Unsymmetric Deformations of Circular Cylindrical Elastic Shells," J. of Math. and Physics, 42, No. 3 (1963), 167. 27. DonnellL. H., "Stability of Thin Walled Tubes Under Torsion," NACA Rep., 479, 193355

-8728. Marguerre, K., "Zur Theorie der Gekrummten Platte Grosser Formbnderung," Proc. of 5th Inter. Congo of Appl. Mech., (1938). 29. Yoshimura, Y., "On the Mechanism of Buckling of a Circular Cylindrical Shell Under Axial Compression." NACA TM 1390, 1955. 30. Coppa, A. P., "The Buckling of Circular Cylindrical Shells Subject to Axial Impacts" NACA TN D1510, Collected Papers on Shell Structures, (1962), 36i1.31. Tennyson, R. C., "A Note on the Classical Buckling Load of Circular Cylindrical Shells Under Axial Compression," AlAA, 1, No. 2, (1963), 475. 32. Carrier, G. F., "Boundary Layer Problems in Applied Mechanics," Advances in Applied Mechanics, III, Academic Press Inc., (1953), 1-18. 33. Sanders, J. L., "Nonlinear Theories for Thin Shells," Quarterly of Applied Math., 21, 1, (1963), 21-35.

UNIVERSITY OF MICHIGAN 3 9015 02499 5501