ENGINEERING RESEARCH INSTITUTE UNIVERSITY OF MICHIGAN ANN ARBOR STRESS-STRAIN RELATIONS IN PLASTICITY AND RELATED TOPICS TECHNICAL REPORT NO. 5 DEFORMATION OF ELLIPSOIDAL SHELLS OF REVOLUTION By P. M. NAGHDI C. NEVIN DeSILVA Project 2027 OFFICE OF ORDNANCE RESEARCH U. S. ARMY CONTRACT DA-20-01-ORD-12099 PROJECT NO. TB 20001(234), DA PROJECT 599-01-004 November, 1953

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN DEFORMATION OF ELLIPSOIDAL SHELLS OF REVOLUTION SUMMARY A single complex differential equation is deduced, and its solution by means of asymptotic integration is obtained, for thin elastic shells of revolution of both uniform and nonuniform thickness. Specifically, the stress distribution is determined for ellipsoidal shells of revolution under edge loadings. It is found that the stresses in such shells are essentially the same whether the thickness is uniform or varies in a certain specified manner. 1. Introduction The formulation of the theory of small deflection of shells of revolution and its application,; since H. Reissner's work' on spherical shells of uniform thickness, has been the subject of numerous investigations. A more recent formulation of finite deformation theory of shells of revolution, which also contains the theory of small deflection (linear theory) and where an account of the historical development of the; subject may be found, was given by E. Reissner2, The present paper considers the smaller deformation of thin elastic ellipsoidal shells of revolution, with both uniform and nonuniform thickness, under axisymmetric loading. The solutions obtained are by means of the method of asymptotic integr.ationi ofc a Icomple:x'differe.ntial e&quation involving a large parameter. An interesting feature of the results, which may be of practical interest, is that the stress distribution in an ellipsoidal shell of uniform thickness is found to be essentially the same as that of a she.l.l of nonuniform thickness (see Fig. 4) under the same loading. While attention is devoted mainly to ellipsoidal shells of revolution, both the complex differential equation mentioned and its solution are entirely general and are applicable to all shells of revolution of uniform thickness, as well as to a large class of shells of nonuniform thickness.. —----------. ----------—.1

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN 2. The Basic Equations of Shells of Revolution In this section, we discuss briefly the basic equations of small deformation of elastic shells of revolution, with reference to a more recent formulation of the theory given by Eric Reissner2. Our discussion is general and pertains to shells of variable, as well as uniform, thickness. Using cylindrical coordinates r, @, z, the parametric equation of the middle surface of the shell (Fig. 1) may be represented by r = r(), z = z(). (2.1) Denoting by 0 the slope of the tangent to the meridian of the shell, then r' = a cos 0, z' = a sin 0, (2.2) where 1/2 a = [(r')2 + (z1)2]2 (2.3) and prime denotes differentiation with respect to {. We note for future reference that the principal radii of curvature r1 and r2 are, respectively, the.radius of curvature of the curve generating the middle surface and the length of the normal intercepted between this curve (generating curve) and the axis of rotation. It follows from the geometry of the middle surface that r = r2 sin 0. (2.4) The stress resultants Nf, Ng, and Q, and the stress couples MS and Mg acting on an element of the shell are shown in Fig. 1. Also, as in Ref. 2, it is convenient to introduce; "horizontal" and "vertical" stress resultants H and V, given by Na = r'H + z'V, Q = -z'H + rV. (2.5) ____________ 2

sI Y x Fig. 1 Fig. 1 5~~aq

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN We now record the basic equations of the small deflection theory of elastic shells of revolution with axisymmetric loading. r V -r p d= V a N = (rH)' + r a pH r N = (rH) cos 0 + (rV) sin r Q = -rH sin 0 + rV cos0 (2 v (2.6) M -V 1, r' Mg = D [t + v __ A] s a r M = D [r + v,] a r u = Eh [N-v N" ] [ -.(N - v N0) -r' B] dt, where p is the negative change in 0 due to deformation; u and::w are the components of displacement in the radial and axial directions; pH and pV denote the components of load intensity in the r and z directions; h is the thickness of the shell, and c = D Eh (2.7) 12 (1-v2) E and v being Young's modulus and Poisson's ratio, respectively. The components of stress, due to stress couples (bending) and due to stress resultants N and N. (membrane), as well as the shearing stress T, are defined in tie usual manner by ~eb —' ( (ab)max = b (Fbmax = h2 h2 h (2.8) crYm = a T = --- — m 8m'h 2h where subscripts b and m refer to bending and membrane stresses respectively. _____________________ 4 _______________

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN With P and rH as basic variables, proper elimination between equations (2.6), differential equations of equilibrium and compatibility, leads to the following two second-order differential equations:,, + (rD/)' _ [(r)2 - v (r'D/O)'] 3 + Z (rH) = r (rV) (2-9 (rD/c) r (rD/) (rD/oa) (rD/oa) (rH)" + (rC rH)' - [ (r)2 + v (rH) - (r/Ca) r-(r/Ca) (r/Ca) z t r t,l Za = [r + v (z /Cc)'] (rV) + v z (rV)' (2.10) r2 (r/Ca) r. [r/Ca)' + v- () - (rap)' (r/Ca) r ) 3. Normal Form of the Differential Equations Substitution of the quantities C and D from (2.7) into (2.9) and (2.10) and rearrangement of terms result in rh h Eh2 L1 (3) +- [(r'/~) + 3 h ] () co (r/a) h r r2n0 - (35.2) + 2[h + 2v h'r' + h' (r/a) = z m h h r h (ra) Eh2 where Z denotes the right-hand side of (2.10), ho is the value of h at some reference section (say ) = t0), and I —— 5

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN ( ) = ( )" + [ + / ) ) h (r' )2 ) (r/a) h r (3.3) = mr m = [12(1-v2)] Eh2 In (3.1) and (3.2), let C2m =- 22 f(0 ) (3.4) r2h0 where,u is constant and it is to be noted that f(|) is independent of the thickness h(S). Then multiplication throughout (3.1) and (3.2) by {h[ho f(~)]1tI results in L (p) + vXp + 2a12r = F (3-5) L (4) - (vX-6),r - 2,2p = G, (3.6) where L ( ) = [h f()]' L1 (' = [ f()] [3h' r' +. - h (r/a) h r (3.7) 2= ho " fh r'h + h' f r/a)' 6 2[Ef( )] + 2v.-T + F-7r/I F = 22 mrV cot 0 Eh2 G m [ho f(.)]- Eh2 h Introducing the complex function U = 5 + ikr; i =.- (3.8) ---------------------- 6

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN where k is an arbitrary function of | to be determined, the differential equations (3.5) and (3.6) may be combined to read vx a I L (U) = 2p2 (ik - ) 2t + [ik (2 ) - 2p2) -1] l 2i2 vk (ik- )(9) + i [-f(S)](l (k" + [2 L + w h' ).(rc/) + h J + (F + ikG) Taking k in the form k = -i2 {1 - [ ( )21 (5 10) 2~2 2 22 21J with the restriction (the implication of this restriction will be discussed later) that k' = k" = 0, (3 11) (3.9) transforms into L (U) = 2p2 (ik - v.) U + (F + ikG). (3.12) 21p2 By putting the last complex differential equation in the form vX o f(0 L1 (U) - i 212 0o (k + i 22) f() = 0- (F + ikG) h h and observing that the coefficient of U', resulting from the application of the operator L1 defined by (353) is R [(r)' + 3h'] (r/) h -7 —--

ENGINEERING RESEARCH INSTITUTE. UNIVERSITY OF MICHIGAN and that exp [1/2: Rdt] = h3/2 (r/a)1/2 then, with the aid of the transformation W (h )32 (.)2 U (5.13) ho a we obtain W" + [2 i3 +2 (e) + A()] = [.h ]/2 f(|) (F + ikG), (5.14) ho a which is the normal form of (3.11) and where i2 = (k + i 2,,) ( ) f(O) 22 h(5.15) 23. 35) =A -.1/2 ra) +.1/4 [ L2(r/c) 2 (rLL)2 / (cr/rc) (r/o)r - 3/2 (/ h - 3/2 h - /4 ( )2 (r/a) h h h We now return to (3.12) and observe that condition (3.11) is fulfilled only if k is a constant, and this may be achieved by proper choice of X and 8* In particular, we note the following two cases: (a). For shells of variable thickness, and with reference to differential equation (3.12), the condition (3.11) is satisfied, provided (vX - _) is constant. Thus, by (3.7) 2 5 h" + [(r/)' _ ] hr - r' / h = Kf(), (3.16) (r/ac) r (r/c) -----------—.. —----------

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN where K is any constant. A particular choice of thickness h(t) may be obtained from (3.16) by setting K = O. This corresponds to the vanishing value of (vX - -), or by (3.10) to k = 1. (b). For shells of uniform thickness, 5 vanishes identically and we have x = f —( ) [(i/a)'] = 0 (5.17) (r/a) Clearly, x is a function of t and its form is determined by the geometry of the middle surface. However, for numerous shell configurations X, and by (3.10) k, are either exactly or very nearly constant. It should be mentioned that-whenever the radius of curvature of the generating curve r1 is a constant (r2 may be a function of I), then with proper choice of e ( 0 = 0) and by (2.2), (3.4), and (3.17), A and thus k are in fact constant. The cases of conical shell and toroidal shell treated recently by dClark3 are included in.this cclass. 4. Solution of Differential Equation by Asymptotic Integration Let us first consider the homogeneous differential equation associated with (3.14), namely w" + [2 i3 2 i2(t) + A(S)] W = 0. (4.1) According to Langer4, the solution of (4.1) admits asymptotic representation with respect to p2 (as a large parameter) as dictated by the coefficient of W; i.e., both;2 and L.Aare suitably regular and bounded over a finite interval of the a-axis. Also, there exists a related differential equation (Langer's related differential equation) of the form Y" + [2 i32 E2()) + W(t)] Y = 0 (4.2) whose solution Y = ~[: -dS]_2 fAJ.i( 1i) + BJ.I - (4.3) n+2 n+2......... —--- -- 9 --—, —-—,, —

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN is dominant in the asymptotic representation of W. Thus, (4.3) is an approximate solution of (4.1). In (4.3), A and B are arbitrary constants, n is the order to which C 2 vanishes at to, and w and'l are defined by 1 [(I2)1t _ 3(t )2 + ( )2] 42 n+2 t (4.4) = (2i3) 1/2 to where -d Q = [, ide f = invr [ an. ho For the sake of clarity and completeness, it is expedient to describe briefly the so-called Stokes' phenomenon45 which often arises in the solution of the differential equations of the type (4.2). With n 2 0, i2 is bounded from zero everywhere in the interval Of convergence (S-axis) and (4.3) becomes Y = ['I l ~d,]/2 (AJ-1/2 (n) + BJ+l/ ) (4.6) which, by means of well-known relations, may be written as Y = -1/2 AO exp(-iT) + Bo exp(i)l, (4.7) where Aoand Bo are constants and -= (2i3)1/2 ^ i d (4.8)..10

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN Clearly, (4.7) is an asymptotic solution of (4*1), and with the necessary provision that ~ 2 vanishes nowhere in the interval of convergence, the constants Ao and Bo will be single-valued. If, on the other hand, E2 vanishes at some point o within the interval, then (4.7) tends to infinity at ~o and is multi-valued in the neighborhood of o. This multi-valued character of the solution (or of the constants Ao and B ) in the interval excluding ~o is Stokes' phenomenon. By reversing the procedure that led from (4.3) to (4.6) and to (4.7), we may represent solution (4.7) in terms of Bessel functions and then generalize to obtain (4.3) with single-valued coefficients A and B, when [2 vanishes to the degree n at some point to and only o0, within the interval of convergence. Thus, solution (4.3) is free from the difficulties associated with Stokes' phenomenon. If in (4.1) the restriction on.Ais relaxed so that it is no longer bounded everywhere in the interval, but has a pole of, at most, order 2 at to, then the Stokest phenomenon is present and quantitatively depends on the nature of the pole. In such cases the representation of W in terms of Bessel functions is still possible, and furthermore is valid at to 5 We now return to the inhomogeneous equation (3.14) and note that in the presence of load intensity PH and pV, the right-hand side of (3.14) does not vanish. In addition to solution (4.3), it then becomes necessary to obtain an appropriate particular integral of (3.14). Such particular integrals are relatively easy to obtain and it will suffice to state that they may be determined approximately by the membrane theory of shells or use may be made of a more recent method developed by Clark and Reissner7. 5. Ellipsoidal Shells of Uniform Thickness. Let us consider an ellipsoidal shell of revolution whose middle surface in rectangular cartesian coordinates is specified by 2 + y2 + 2 = 1 (5.1) a2 c2 a and c being the semi-major axes of the ellipsoid. Choosing the independent variable 5 as 0, it follows from the geometry of the middle surface and (2.2) that a = rl. The radii of curvature are r = a c, r a (5.2) a [1+ p2 cos2~]3/2 [1 + p2 cos20]V2.11

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN and by (2.4) r = a sin (503) [.1 + p2 cos20]1/2 where p2 = C2 - a2 (5.4) a2 From (3.4) and (3.7) by (5.2), we have 212 = ()4* (o ) m, f(a) = (1 + p2 cos20)5/2 (5.5) a ho and & = 0, X = - C2 (1 + p2 cos20)3/2, (5.6) a2 where m is given by (3.5). Since we are concerned here with shells of uniform thickness, in the remainder of this section ho will be replaced by h. With a view toward approximating k to a constant, so that condition (3.11) is fulfilled, we note that restriction of (c/a) to 0(1) is consistent with vX < < 2p2, and by (3.10), [1 - (vX/2L2)2]1/2.1 or k. 1. Thus by this approximation, equation (4.1) is valid and the functions:2 and.\ as well as transformation (3.13) read 22 = (1 + p2 cos20)-/2 _= 53 [_ 2, cot20. + 5 [ p2sin20 2 (5.7a) 4 (1 + p2 Cos20)5 16 (1 + p2 COS20 + 3 p2 [ cps0 ] + 1 (C)2 [sin20 + p2 cos20 ].12 2.(1 + p2 cos20) 2 (1 +.p2 Cos20)2............ —------.12 --------------

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN and W = [sin 0 (1 + p2 cos2)]/2 U. (5.7b) Differential equation (4.1), with i2 andl\.given by (5.7a), has regular singular points at 0 = ni (n = 0, + 1,...). Thus by section 4, Stokes' phenomenon is present in the interval 0 < 0 I < it. However, since geometrically 0 is never negative, we may restrict the solution to the subinterval 0 < 0 < i, and adopt the form of (4.7) as the general solution for ellipsoidal shells of revolution (with uniform thickness) under edge loadings. Hence W = 1Z/2 A exp (-ir) + B exp (ir)' (5.8) where A and B are complex constants, and T) = (2i3)1/2 (5.9a) @ = / +EdZ = 0 [1 - p2 + p+ ] J=S~d' ~ 8 256'2 sin ~ p2 [p.+ (5.9b) 1 sin2. COS2 45 1 sin20 cos2 [ p4 +....] +..., 2 128 The function (0;) is tabulated in Table I for various values of c/a, in 2~ increments of tie angle 0. For c/a = 1, which is the case of the spherical shell, 0 reduces to 0 as may be seen from (5.9b). With + ir = (l+i) X, (5.8) becomes W = ~/2 A [cos k 0 + i sin p ] e 1t~'~~~ 0 -) ~(5.10) + B [cos k - i sin ] 0] e- and by (5.7), (5.10), and (3.8) the quantities I, Ir, I', and IVr', which will 13

TABLE I Values of 0 (O;c/a) = (1 + p2 cos20) do for Shells of Uniform Thickness 0 C = 0.7 c = 0.8 = 09 = 1.10 = 1.20 a a a a 0 0 0 0 0 0 2 0.069908 0.056965 0.044965 0.027910 0.025211 4 0.139698 0.115860 0.089899 0.055834 0.050423 6 0.209260 0.170619 0.134777 0.083778 0.075637 8 0.278475 0.227170 0.179570 0.111758 0.100853 10 0.347237 0.283453 0.224248 0.139787 0.126074 12 0.415436 0.339399 0.268787 0.167875 0.151305 14 0.482974 0.394950 0.313162 0.196034 0.176552 16 0.549754 0.450049 0.357345 0.224278 0.2018.18.18 0.615685 0.504640 0.401314 0.252614 0.227.116 20 0.680688 0.558671 0.445049 0.281060 0.252455 22 0.744584 0.612099 0.488524 0.309625 0.277848 24 0.807612 0.664882 0.531728 0.338323 0.503311 26 0.8694.10 0.716982 0.574640 0.367164 0.328861 28 0.930027 0.768368 0.6,17243 0.396160 0.354515 30 0.989427 0.819013 0.659530 0.425325 0.380299 32 1.047574 0.868893 0.701483 0.454669 0.40623.1 34 1.104450 0.917995 0.743097 0.484202 0.432336 36 1.160036 0.966304 0.784364 0.513934 0.458639 38 1.214330 1.013819 0.825279 0.543879 0.485168 40.1.267335 1.060534 0.865840 0.574041 0.511948 42 1.319058 1.106451 0.906042 0.604431 0.5359003 44 1.369521 1.151580 0.945890 0.635057 0.566362 46 1.418749 1.195932 0.985385 0.665923 0.594046 48 1.466768 1.239522 1.024532 0.697036 0.622078 50 1.513620 1.282369 1.063336 0.728399 o.650483 52 1.559345 1.324494 1.101803 0.760017 0.679278 54 1.603987 1.365926 1.139946 0.791889 0.708478 56 1.647598 1.406691 1.177772 0.824015 0.738097 58 1.690232 1.446818 1.215295 0.856395 0.768146 60 1.731943 1.48634.1 1.252526 0.889023 0.798628 62 1.772786 1.525293 1.289478 0.92,1893 0.829547 64 1.812826 1.563708 l3126171 0.955004 o.860900 66 1.852.115 1.601624 1.3626,14 0.988342 0.892681 68 1.890719.1.639080 1.398829 1.021901 0.924877 70 1.928696 1.676.109 1.434830 1.055664 0.957474 72 1.966108 1.712755 1.470635 1.089624 0.990451 74 2.002722 1.749052 1.506265 1.123762 1.023785 76 2.039469 1.785040 1.541736 1.158064 1.057447 78 2.075538 1.820760 1.577066 1.192512 1.091404 80 2.111273 1.856248 1.612278 1.227086 1.125623 82 2.146735 1.891545 1.647391 1.261769 1.160064 84 2.181978 1.926689 1.682421 1.296540 1.194686 86 2.217055 1.961715 1.71739.1 1.331377 1.229445 88 2.252026 1.996667 1.752323 1.366257 1.264297 90 2.286941 2.0531580 1.787231 1.401159 1.299196 14

- ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN be needed in the solution of examples that follow immediately, are f (1 + p2 cos20)l/8 el (Ao cos i t - A1 sin i) 0) ssin 0 + e (Bo cos p $ + B1 sin p. (5.11a) (1 e os 18 (e (A sin 1 ) + Al cos ) \) (1 + 02 COS2,)1/8s i stin f- e (B0 sin pi. - B1 cos i ).1/2 9/8'/' = -LL + pi sin 0 (1 + p2 cos20) M _1 -9/8 (5.llb)' = -L $ + p. sin l/2a^(1 + p2 cos20) N, where L = 1 cot 0 (1 + p2 cos20)-1 [2 (1 + p2) - p2 sin20] 4 M = t[(Ao - Al) e - (Bo - B1) e ] cos u - [(A0 + A1) el + (Bo + B1) e-] sin. } A) ] sin (5..12) N = C[(Ao + Al) ez - (Bo + B1) e'-] cos I + [(Ao - A1) eL + (Bo B1) e-p] sin t 0} and A = Ao + i A1, B = Bo + i B1. (5.13) To illustrate the nature of the solution just obtained, two examples for ellipsoidal shells of revolution (closed at the apex 0 = 0) with uniform thickness, under edge loadings only will be considered: (1) uniform stress couple Mo applied around the edge 0 = i/2, and (2) uniform radial stress resultant Ho applied around the edge 0 = xr/2. In both these examples, the transition conditions at the apex 0 = 0 require that 0 = 0; p,', $, P' remain finite. (5.14) Since 0 = 0 is a regular singular point, (5..11a,b) are not valid there. However, guided by the solution of corresponding examples of spherical shells8, and on account of physical requirements, evidently instead of (5.14) we may 15

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN require that quantities p, 3', j' be finite in the neighborhood of the pole 0 = O. This is achieved by setting at the outset the constants Bo and B1 equal to zero in (5.11a, b) and (5.12). Example (1): The boundary conditions in this case are 0 = 2; M0 = Mo, Q = 0 (5.15a) or equivalently it2 cfi aMo2 J_ = 0. (5.15b) Applying the above conditions to equations (5.11), the constants Ao and Al are determined as follows: Ao = [2m 1 e cos t Il] Mo Eh2 (c/a)2 (5.16) A, = [2m e-K'l sin O] Mo, 6 Eh2 (c/a)2 where O1 = P(-; c) 2 a Using (2.10), the components of stress were obtained for the case of (c/a) = 0.8 (see Table I), (a/h) = 20, and v = 0.3 (i.e., K = 3.679). Figure 2 shows the ratio of these stresses to that of aM, defined by aM = 6Mo/h2. Example (2): With boundary conditions 0 = Q = -H, M = (517a) or 0= i'a, -- 1, = 0 (5.17b) 2 1. --------------------— 16 ___________

I.0 ~b _0.2 i i 0. - 0.6........... ~I~~ ~ ~ ~ 00 0.2 - _______ 0 10 0 10 30 50 70 90 * IN DEGREES Fig. 2 17

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN and proceeding as in the previous example, the constants Ao and Al are Ao = [ma e-L1 (cos B4 1 + sin [m a1)] Ho (.18) Al = [ma e-11 (cos p O - sin p 01)] Ho Eh2 As in example (1), the ratio of the stresses to that of aH = 3 Ho/2h- are plotted in Fig. 3; again these results are for the case of c/a = 0.8, a/h = 20, and v = 0.3. 6. Ellipsoidal Shells of Nonuniform Thickness It was shown in section 3, that condition (5.11) is satisfied identically if the thickness of the shell is such as to satisfy equation (3.16). Again, we choose the variable S as 0 and recall that relations (5.1) to (5.5), which were obtained from consideration of the middle surface only, are valid here. Thus on substitutions from (5.2) and (5.3), (3.16) reads as follows: h" + c —t 0 [c (l-v) - 3 p2 sin20] h' (1 + p2 cos20) a (6.1) C2_ h -5/2 + v - = K (1 + p2cos20)-5 a2 (1 + p2 COS20) where K is any constant. While any choice of h(o) which satisfies (6.1) falls within the scope of the application of differential equation (3.14), when adapted to ellipsoidal shells of revolution, we shall confine our attention to the case where K = 0, or by (3.10), k = 1. With K = 0, the solution of (6.1) about the analytic point 0 = t/2 in the interval of convergence 0 < 0 < i may be taken in the form oo - = 1 - b cos2", (6.2) ho i 2n n18 ---------------— ^8 ----------------

sr 1 i i i i i 6.....o 4 _- _ a._ Ho ~o -2 -. -........ O 10 30 50 70 90 4 IN DEGREES Fig. 5

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN where 0 = t/2 is the reference section, so that ho = h(2_) and 2 V C2 = v c2 b2 = 2 2 a2 b4 v 2 [4 _2 (2+v)] 8 a2 ~ ~~a2 ~ (6.3) n = n-4 p2 b4 + t2 (n-2) n n a2- [(n-2) + v]bn_2; n > 4. Since, irrespective of the thickness variation, the solution of (4.1) does not hold at 0 = 0, the form of h(0) given by (6.2), which is also invalid at this singular point, should not be disturbing. In fact, as may be seen from Fig. 4, this particular choice of variation in the thickness is of practical interest [In plotting this curve, 12 terms in the solution of (6.2) were used]. 1.0 0.9..-4 0.7 ------- 0.5 _______ _______ 0 10 30 50 70 90 * IN DEGREES Fig. 4 With (6.2), the transformation (3.13) is W = [sin (1 + p2 COS20)]1/2 (h:)3/2 U (6.4) ---..... —-- 0 —.... —-- 20 --

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN and the functions i2 and J-in (4.1) become h -5/2 =2 _ (1 + i( + ) ( ) (1 + p2 cos20) 4L h (6.5). = Ao - +y ^' h +33 (l)2] 2 (r/a) h 2h 4 h where = _-(h ) (.l+p2 cos20)3/2 C2 (I 3 cos20 (2b 2 + 4b cos20 +.) ho a2 o1i - be cos2 - b cos4 + h' =_ [2 + 2si42 [b + 2b4 os0 + b c0 +...] (66 h [1 - b cos20 - b4 cos40 +...] h" = [-sin20 (2b2 + 12b4 cos20 +...) + cos20 (2b2 + 4b4 cos20 +.. h.1 - b2 cos2 - b4 cos4 +... (rl/c,)' = cos 0 [ 1 + p2 (cos20 - 2 sin20)] (r/a) sin 0 (1 + p2cos20) andAo =I -() is given by (5.7a). As in section 5, again i2 is bounded from zero everywhere within the interval of convergence and the solution of (4.1) may be written as W = 1 -/ 2 A e-iT + B eilj (6.7) where A and B are constants and = (2i3)1/2 L - / f( + i v.)1/2 (1 + p2 cos20)- (h)l/ d 0. (6.8) J (.1+ ~ 2p2 h In combination, (6.7) and (6.8) constitute an "exact" asymptotic solution of (4.1) for ellipsoidal shells of revolution, when the thickness is specified by (6.2). 21

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN If now c/a is restricted to 0(1), then (1 + iv.)l/2 -1 and the function $ of (6.8) becomes = (I + p2 cos20)-5/4 (O~)/2 d0 (6 ) h 0 [.1 -. 3 p2 p +. bp4 + +1 b2 p2 + 3 (b4 + b) +.... 8 256 2 - 4 - sin20 [5 p2 135 4.... b2 + 5 b2 p2 (4 + b) +...] 2 8 256 4 64 16 4 in2cos2 p+ 1 sin2- cos20 p 4 + p2 + (b + b2) +....] 2.128 32 and (h,)l/2 (I + p2 cos2)"4) (6.9b) h Expressed in terms of trigonometric functions, W will have the form of (5.10), except that E and M are now given by (6.9). With the aid of (6.4), the quantities, Ir, B', and 1' are [=G (kh)-5/4 sin1/2 (1 + p2 cos20) {cos I [Ao exp (IA) + ho Bo exp (-[A)] + sin lA [B1 exp (-Il) - A1 exp (1)]j) (6.10a) (-)-54 sin-/2 (1 + p2 COS20)1/8 {cos L [A1 exp (4) + B, exp (-4O)] + sin po [A exp ( B ) - B exp (-t)]j1. = 4 (h)-7/4 sin1 (1 + p2 Cos20)-9/8 M - [L + 5 h' ] h, ^~~~~~~~4h A. (6.10b) ho 4 h sin- 2I - 22

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN where L, M, and N are defined as in (5.12), with 0 and i: given by (6.9), and Ao, Al, Bo, and B1 are related to A and B by (5.13)* In order to study the effect of the variable thickness on the stress distribution, let us consider the case which corresponds to example (1) of the previous section, namely: an ellipsoidal shell of revolution of nonuniform thickness [specified by (6.2)] under the action of a uniform stress couple Mo applied around the edge 0 = /(2. Again, the transition conditions at the apex 0 = 0 require that Bo = B1 = 0 (see section 5), and the boundary conditions 0 2 = = M Q = result in Ao - Eh2 (ca) (6.11) A1 = [ 2m e 1, 1 sin ]] M Eho (c/a) MO. where (1 = 0( <;) 2 a Table II provides values of ~(0; c/a) appropriate to (6.2) for the case c/a = 0.8, in 2~ increments of the angle 0. TABLE II Values of 0 (0; c/a) = [ (1 + p2 cos20)-5/ (h,)l/2 d0 for h(0) given by 6.2) and c/a = 0.8 ____________ 0~_D 0~ D 0 0 24 0.704565 48 1.302374 72 1..782562 2 0.060573 26 0.759346 50 1.546302 74 1.8.19017 4 0.121060 28 0.813279 52 1.389395 76 1.855.127 6 0.18.1384 30 0.866338 54 1.431687 78 1.890936 8 0.241463 32 0.918493 56 1.473211 80 1.926487 10 0.301219 34 0.969729 58 1.514007 82 1.961826 12 0.360574 36 1,020055 60 1.554110 84 1.996996 14 0.419460 38 1.069404 62 1.593561 86 2.052036 16 0.477765 40 1.117836 64 1.632406 88 2.066992 18 0.555544 42 1.1653355 66 1.670683 90 2.101900 20 0.592618 44 1.211914 68 1.708440 22 0.648974 46.1.257587 70 1.745776 23

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN The stresses which were obtained for the case c/a = 0.8, a/ho = 20, and v = 0.53 are not plotted because for all practical purposes they are the same as those (for ellipsoidal shell of uniform thickness) shown in Fig. 2. To substantiate this, a comparison is made in Table III between the stress distribution of the present case with the corresponding quantities for the shell of uniform thickness [Example (1), section 5]. TABLE II V "0~0 _ 20 40 60 70 80 90 b fh=ho -0.00111 -0.03975 +0.06863 +0.33327 +0.73274 1.00000 Mo lh=h(0) -0.00017 -0.03937 +0.06996 +0.33329 0.72856 1.00000 f( h=ho 0.00145 0.01723 0.01527 0.10406 0.23156 0.30000 aMoh=h(0) 0.00178 -0.01688 +0.01558 0.10404 0.23030 0.30000 Ca (h=ho 0.00125 0.00148 -0.00692 -0.00920 -0.00555 0.000000 M~ [ h=h(0) 0.00099 0.00159 -0.00687 -0.00921 -0.00555 0.00000 r (fh=h 0.00421 -0.00924 -0.09672 -0.10054 +0.06062 0.55075 aMo h=.h(0) 0.00409 -0.00875 -0.09575 -0.09807 +0.06263 0.55075 T hho — 0.00068 -0.00187 0.01797 0.03792 0.04550 0.00000 aMo h=h(0) -0.00054 -0.00200 0.01785 0.05794 0.04548 0.00000 7. Concluding Remarks The very close agreement between the results for shells of uniform and nonuniform thickness is, at least at first glance, somewhat surprising and warrants the following comments. The results for the case of nonuniform thickness (communicated in Table III) should not be considered as affected by the approximation introduced immediately following (6.8), since this very same approximation was also used for the case of uniform thickness. Returning, however, to (6.5), an examination of N\(0)) reveals that the quantity in the bracket, which:'is due to the variation of thickness, is finite and small with respect to gL2 everywhere within the interval of convergence. Hence, this quantity will have but a small- effect on the asymptotic integration of the differential equation (4.1) with respect to Bp2. _____ 24

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN Furthermore, as may be seen from Fig. 4, the variation of h/ho should not lead to much deviation of 0 in (6.9) from that of (5.9b). That this is indeed the case is apparent from proper comparison of Tables I and II. It is reasonable to expect that other forms of variation in the thickness, as solutions of (6.1) with K f O, should result in a more appreciable deviation of the stress distribution from that of the case of uniform thickness. For, in such cases the quantity k will no longer be unity, and this may result in a more pronounced effect on the stresses. In conclusion, we reiterate that although attention has been given specifically to ellipsoidal shells of revolution, the results obtained in sections 2 - 4 are quite general and applicable to all shells of revolution. --------------- 25 -----------------

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN REFERENCES 1. H. Reissner, "Spannungen in Kugelschalen (Kuppeln)", Festschrift Mueller-Breslau, 181-193 (1912). 2. Eric Reissner, "On the Theory of Thin Elastic Shells", H. Reissner Anniv. Vol., 231-247 (1949). 3. R A. Clark, "On the Theory of Thin Elastic Toroidal Shells, J. Math. Phys. 29, 146-178 (1950). 4. R. E. Langer, "On the Aaymptotic Solution of Ordinary Differential Equations", Trans. Am. Math. Soc. 33, 23-64 (1931). 5. R. E. Langer, "On the Asymptotic Solution of Ordinary Differential Equations, with Reference to the Stokes' Phenomenon about a Singular Point", Trans. Am. Math. Soc. 37, 397-416 (1935). 6. F. B. Hildebrand, "On Asymptotic Integration in Shell Theory", Proc. 3rd. Symp. in Appl. Math. 3, 53-66 (1950). 7. R. A. Clark and E. Reissner, "Bending of Curved Tubes", Advances in Applied Mechanics II, Academic Press, pp. 93-122, 1950. 8. 0. Blumenthal, "Uber asymptotische Integration von Differentialgleichungen mit Anwendung auf die Berechnung von Spannungen in Kugelschalen", Proc. Fifth Int. Cong. Math. 2, 319-327 (1912). ---- ~26